面向非结构化神经算子的可学习样条

# Learnable Splines for Unstructured Neural Operators
Source: [https://arxiv.org/html/2610.22446](https://arxiv.org/html/2610.22446)
Vignesh Gopakumar & Marc Peter Deisenroth††thanks:Correspondence to:v\.gopakumar@ucl\.ac\.ukAffiliation:Centre for Artificial Intelligence, University College LondonAffiliation:London, WC1V 6LJ, UKAnder GrayAffiliation:LIX, École Polytechnique, CNRS, Institut Polytechnique de ParisAffiliation:Paris, 91120, FranceJosh Williams & Nuno NobreAffiliation:Hartree Centre, Science and Technology Facilities CouncilAffiliation:Daresbury, WA4 4AD, UKMatthew FieldJames PanayisAlexander I\. BlairAffiliation:Andrew Davis, Stanislas Pamela & Robert AkersAffiliation:Computing Division, UK Atomic Energy AuthorityAffiliation:Oxford, OX14 3EB, UKZhuoyuan WangValentin Duruisseaux & Anima AnandkumarAffiliation:Computing \+ Mathematical Sciences, CaltechAffiliation:Pasadena, 91125, USA

###### Abstract

Neural operators approximate solution operators of partial differential equations at a fraction of the cost of numerical solvers\. Efficient grid\-based architectures such as the Fourier neural operator \(FNO\) assume uniform rectangular grids, whereas engineering simulations use large, irregular, and locally refined meshes\. We introduce L\-SplineFNO, a memory\-efficient neural operator that couples an FNO backbone with a sparse, learnable tensor\-product B\-spline transfer between unstructured meshes and a latent Cartesian grid\. A local normalised scatter avoids the geometry\-dependent solve of analytic spline transformation while preserving compact support, partition of unity, and spline smoothness by construction\. The neural operator, conditioned on global design and operating parameters, learns the mapping of geometry to the field solution\. Across six three\-dimensional engineering benchmarks, L\-SplineFNO achieves strong predictive accuracy while requiring substantially less memory and compute than baseline operators, scaling to 26\.6 million nodes in training and 46 million in inference on a single A100 GPU\. It also supports zero\-shot super\-resolution and efficient closed\-form spatial derivatives\. Uniquely, the analytic spline decoder yields rigorous prediction bounds over continuous space and uncertain parameters, via Bernstein convex hulls and Taylor models, demonstrated on a 29\.2\-million\-node MAST\-U tokamak model\.

## 1Introduction

Engineering design and qualification rest on repeated numerical simulation, evaluating a system’s governing equations across a range of parameters and conditions\([Winsberg & Alvarado, 2026](https://arxiv.org/html/2610.22446#bib.bib61);[Lavin et al\., 2021](https://arxiv.org/html/2610.22446#bib.bib30)\)\. In steady\-state problems such as thermal management, structural loading and aerodynamic pressures, the quantity of interest is a field defined over an unstructured finite\-element or finite\-volume discretisation of a three\-dimensional domain, determined by a small set of design and operating parameters\([Permann et al\., 2020](https://arxiv.org/html/2610.22446#bib.bib48);[Weller et al\., 1998](https://arxiv.org/html/2610.22446#bib.bib58)\)\. Design sweeps, sensitivity studies and uncertainty quantification each require this map to be evaluated thousands of times, while a single high\-fidelity solve can occupy a compute cluster for hours\([Keyes et al\., 2013](https://arxiv.org/html/2610.22446#bib.bib27)\)\. The cost of the solver therefore limits how much of a design space can be explored\([Gopakumar et al\., 2025](https://arxiv.org/html/2610.22446#bib.bib16)\)\. Safety\-critical settings such as fusion devices also require guaranteed prediction ranges for mesh nodes under uncertain operating conditions\.

Neural operators provide a route past this bound, learning mappings between function spaces with parameters shared across discretisations and approximating solution operators of partial differential equations \(PDEs\) at a fraction of the cost of the solver that generated their training data\([Kovachki et al\., 2023](https://arxiv.org/html/2610.22446#bib.bib29);[Azizzadenesheli et al\., 2024](https://arxiv.org/html/2610.22446#bib.bib2);[Berner et al\., 2026](https://arxiv.org/html/2610.22446#bib.bib3)\)\. The Fourier neural operator \(FNO\)\([Li et al\., 2021](https://arxiv.org/html/2610.22446#bib.bib32)\)has been deployed as a surrogate across fluid dynamics\([Wen et al\., 2022](https://arxiv.org/html/2610.22446#bib.bib59)\), heat transfer\([Yuan et al\., 2025](https://arxiv.org/html/2610.22446#bib.bib63)\)and plasma modelling\([Carey et al\., 2025](https://arxiv.org/html/2610.22446#bib.bib5);[Pamela et al\., 2025](https://arxiv.org/html/2610.22446#bib.bib44)\), reporting speed\-ups of several orders of magnitude while retaining discretisation convergence and zero\-shot super\-resolution\([Gopakumar et al\., 2024](https://arxiv.org/html/2610.22446#bib.bib15)\)\. This efficiency follows from the fast Fourier transform, which assumes the field is sampled on a uniform rectangular grid\. However, engineering meshes are often irregular, feature\-refined and vary across designs, rarely satisfying this assumption\. Existing approaches either process the unstructured mesh directly, which can become memory\-intensive at large node counts, or preserve structured processing by transferring the field onto a latent grid\. Retaining information from the original mesh while keeping the representation memory\-efficient is therefore a key challenge in neural PDEs for large unstructured grids\. Existing learned transfers use global coordinate deformations\([Li et al\., 2023a](https://arxiv.org/html/2610.22446#bib.bib33)\)or kernel integrals over a radius\-limited neighbourhood graph\([Li et al\., 2023b](https://arxiv.org/html/2610.22446#bib.bib34)\), but neither preserves structure by construction \([Section2](https://arxiv.org/html/2610.22446#S2)\)\.

![Refer to caption](https://arxiv.org/html/2610.22446v1/Images/splinefno_diagram.png)Figure 1:End\-to\-end conditional L\-SplineFNO architecture\.The unstructured geometry and global conditioning vector are first mapped to conditioned nodal features through coordinate normalisation, positional encoding, and FiLM modulation\. A learnableCp−1C^\{p\-1\}spline scatter then transforms these features from the unstructured mesh to a structured latent grid, where they are processed by a Fourier neural operator \(FNO\) backbone\. The resulting grid representation is mapped back to the physical mesh using analytic spline reconstruction and a pointwise projection to produce the predicted field\. This construction provides a sparse, structure\-preserving, differentiable mesh\-to\-grid\-to\-mesh mapping with the structured Fourier processing\.Our approach:We build the transfer as a sparse, differentiable and structurally constrained linear map from tensor\-product B\-splines\([Piegl & Tiller, 1996](https://arxiv.org/html/2610.22446#bib.bib49);[Prautzsch et al\., 2002](https://arxiv.org/html/2610.22446#bib.bib51)\)\. Compact support gives each mesh point a local stencil whose size is set by the spline degree, not by a search radius\. Partition of unity preserves constants\. Spline smoothness gives closed\-form evaluation and differentiation at any off\-grid query point\. We call a transfer with all three properties*structure\-preserving*\. The same structure gives a non\-negative Bernstein basis on each decoder cell, so the prediction lies within the convex hull of its coefficients; propagating a Taylor model of the parameters through the network extends this to rigorous bounds over continuous space and uncertain parameters\. The resulting sparse transfer preserves physical locality and gives a memory\-efficient link between unstructured meshes and structured Fourier processing\.

To further address the parametric nature of engineering problems, we incorporate global design and operating parameters into the operator\. With no initial condition to propagate in steady\-state problems, the geometryΩ\\Omegaand a conditioning vector𝝁\\bm\{\\mu\}of design variables, loads, material properties, and operating conditions carry all sample\-specific information\. How𝝁\\bm\{\\mu\}should enter an operator is unsettled, and its effect is confounded by a redundancy that is rarely controlled for\([Koupaï et al\., 2024](https://arxiv.org/html/2610.22446#bib.bib28)\)\. When𝝁\\bm\{\\mu\}restates geometric quantities already visible in the point cloud, a model can ignore the conditioning pathway and still appear conditioned\. Isolating this pathway therefore requires settings in which𝝁\\bm\{\\mu\}carries information that the mesh cannot provide, up to the limiting case of a fixed mesh on which𝝁\\bm\{\\mu\}is the only sample\-specific signal\. Our benchmarks span this range, from the heat sink, where𝝁\\bm\{\\mu\}restates the geometry, to HIVE, whose fixed mesh leaves𝝁\\bm\{\\mu\}as the only sample\-specific signal \([Section4](https://arxiv.org/html/2610.22446#S4)\)\. We combine this conditioning with the learnable spline transfer to form a complete conditional L\-SplineFNO for parametric fields on complex unstructured meshes\. Our contributions are:

- •Learnable spline transfer for unstructured neural operators\.We introduce L\-SplineFNO, which couples an FNO backbone with a learnable, structure\-preseerving spline transformation from unstructured meshes to a structured latent grid\. This replaces the geometry\-dependent linear solve of analytic SplineFNO with alearnable, structure\-preserving B\-spline scatter onto a latent FNO grid, with analytic reconstruction at arbitrary query points\.
- •An end\-to\-end conditional operator framework\.We condition the operator on global design and operating parameters through an affine transformation\([Perez et al\., 2018](https://arxiv.org/html/2610.22446#bib.bib47)\)\. The FiLM\-style conditioning produces a per\-channel scale and shift, applied to the encoded geometry features before the processor backbone\. The conditioning method sits in its own module with interchangeable backbones\.
- •Rigorous spatial and parameter bounds\.Bernstein convex hulls of the spline decoder and second\-order Taylor models of the trained network enclose the prediction over continuous space and uncertain parameter boxes, without reference solutions or retraining\.
- •Comprehensive evaluation on complex engineering problems\.Across six three\-dimensional thermal, structural, and aerodynamic benchmarks, we compare L\-SplineFNO against representative mesh\-native, latent\-attention, and gridded neural operators under a common conditioning pipeline and at matched parameter counts, demonstrating strong predictive accuracy while retaining the efficiency of structured latent processing\. We further demonstrate linear scaling with mesh size, inexpensive analytic spatial derivatives, and deployment on a 29\.2\-million\-node mesh of the MAST\-U tokamak\.

## 2Background and Related Work

##### Conditional operator learning\.

Let𝒰\\mathcal\{U\}denote a space of solution fields on a domainΩ\\Omega, and let𝝁∈𝒫⊂ℝk\\bm\{\\mu\}\\in\\mathcal\{P\}\\subset\\mathbb\{R\}^\{k\}collect design variables, material properties, boundary conditions, or operating conditions\. For the steady problems considered here, with no input field, the*solution operator*maps the geometry and parameters to the solution,

𝒩:\(𝝁,Ω\)⟼y∈𝒰,ℒ𝝁,Ω​\(y\)=f,\\mathcal\{N\}:\(\\bm\{\\mu\},\\Omega\)\\longmapsto y\\in\\mathcal\{U\},\\qquad\\mathcal\{L\}\_\{\\bm\{\\mu\},\\Omega\}\(y\)=f,\(1\)whereℒ𝝁,Ω\\mathcal\{L\}\_\{\\bm\{\\mu\},\\Omega\}is the governing operator andffthe forcing, if any\. All sample\-specific information enters throughΩ\\Omegaand𝝁\\bm\{\\mu\}, so the learning task is both geometric and conditional\.

##### Neural operators\.

Neural operators approximate maps such as[Equation1](https://arxiv.org/html/2610.22446#S2.E1)using architectures whose parameters are shared across discretisations, so that a trained model converges across mesh discretisations rather than to a fixed resolution\([Kovachki et al\., 2023](https://arxiv.org/html/2610.22446#bib.bib29);[Berner et al\., 2026](https://arxiv.org/html/2610.22446#bib.bib3)\)\. The FNO parametrises the kernel integral as a global convolution in Fourier space\([Li et al\., 2021](https://arxiv.org/html/2610.22446#bib.bib32);[Duruisseaux et al\., 2026](https://arxiv.org/html/2610.22446#bib.bib12)\): for a lifted feature fieldvtv\_\{t\}, a layer computes

vt\+1​\(x\)=σ⁡\(W​vt​\(x\)\+ℱ−1​\(Rϕ⋅ℱ⁡\(vt\)\)​\(x\)\),v\_\{t\+1\}\(x\)=\\sigma\\\!\\left\(Wv\_\{t\}\(x\)\+\\mathcal\{F\}^\{\-1\}\\\!\\left\(R\_\{\\phi\}\\cdot\\mathcal\{F\}\(v\_\{t\}\)\\right\)\(x\)\\right\),\(2\)whereℱ\\mathcal\{F\}is the Fourier transform,RϕR\_\{\\phi\}is a learned multiplier on a truncated band of modes, andWWis a pointwise linear map\. Physics\-informed neural operators further constrain the learned solution family by penalising PDE residuals\([Li et al\., 2023c](https://arxiv.org/html/2610.22446#bib.bib35);[Ganeshram et al\., 2026](https://arxiv.org/html/2610.22446#bib.bib14)\)\. Evaluatingℱ\\mathcal\{F\}with the fast Fourier transform offers only𝒪⁡\(N​log⁡N\)\\mathcal\{O\}\(N\\log N\)complexity, making the[Equation2](https://arxiv.org/html/2610.22446#S2.E2)efficient and FNOs a popular choice\([de Hoop et al\., 2022](https://arxiv.org/html/2610.22446#bib.bib11)\), but restricts its operation to uniform rectangular grids\.

##### Operator learning on unstructured geometries\.

Extensions to unstructured grids fall into three groups\.*Mesh\-native*models attend over mesh points or learned groupings of them, as in GNOT\([Hao et al\., 2023](https://arxiv.org/html/2610.22446#bib.bib19)\), Transolver\([Wu et al\., 2024](https://arxiv.org/html/2610.22446#bib.bib62)\)and its parallel successor Transolver\+\+\([Luo et al\., 2025](https://arxiv.org/html/2610.22446#bib.bib37)\), or compress the point set into latent tokens, as in UPT\([Alkin et al\., 2025](https://arxiv.org/html/2610.22446#bib.bib1)\); they avoid gridding, but cost and fidelity depend on how aggressively the mesh is compressed\.*Latent\-grid*models keep a structured backbone by transferring the data onto a regular domain, through learned deformations in Geo\-FNO\([Li et al\., 2023a](https://arxiv.org/html/2610.22446#bib.bib33)\), balanced non\-uniform partitions in NUNO\([Liu et al\., 2023](https://arxiv.org/html/2610.22446#bib.bib36)\), graph\-kernel encoders in GINO\([Li et al\., 2023b](https://arxiv.org/html/2610.22446#bib.bib34)\), multi\-scale attentional encoders with a vision\-transformer processor in GAOT\([Wen et al\., 2026](https://arxiv.org/html/2610.22446#bib.bib60)\), or optimal\-transport maps for surfaces\([Li et al\., 2025](https://arxiv.org/html/2610.22446#bib.bib31)\); here the transfer becomes part of the approximation, since coarse grids lose thin features and interfaces, while dense grids and large neighbourhood graphs cost memory\.*Precomputed*representations such as PGD\-NO build geometry tokens ahead of time, saving memory at very large mesh sizes but removes the representation from end\-to\-end training\([Zhong et al\., 2026](https://arxiv.org/html/2610.22446#bib.bib65)\)\. Industry\-scale comparisons find no representation uniformly best across accuracy, cost, memory and deployment\([Zhong et al\., 2025](https://arxiv.org/html/2610.22446#bib.bib64)\)\.

##### B\-splines in scientific machine learning\.

Tensor\-product B\-splines are compactly supported, smooth, and form a partition of unity\([Piegl & Tiller, 1996](https://arxiv.org/html/2610.22446#bib.bib49)\)\. Their sparse Jacobians have been used as differentiable layers for geometry reconstruction and finite\-element learning\([Cho et al\., 2021](https://arxiv.org/html/2610.22446#bib.bib7)\); physics\-informed B\-spline networks predict control coefficients directly, so that derivatives and boundary conditions can be imposed analytically\([Wang et al\., 2026](https://arxiv.org/html/2610.22446#bib.bib57)\); and Neural Spline Operators extend this coefficient\-learning view from finite\-dimensional PDE parameters to functional inputs\([Wang et al\., 2025](https://arxiv.org/html/2610.22446#bib.bib56)\)\. In each, B\-splines serve as the*output*representation, rather than as a parameter\-efficient, learnable transfer across an unstructured mesh and a latent structured grid\.

##### Conditioning on physical parameters\.

The parameters𝝁\\bm\{\\mu\}either restate information already visible in the geometry or encode quantities the geometry cannot reveal, such as loads, materials and operating conditions\. Broadcasting𝝁\\bm\{\\mu\}as an extra input channel is simple, but asks the backbone to wastefully preserve and propagate a signal that is spatially constant across the domain\([Takamoto et al\., 2023](https://arxiv.org/html/2610.22446#bib.bib55)\)\. Alternatives inject𝝁\\bm\{\\mu\}into the network directly: feature\-wise linear modulation \(FiLM\) applies parameter\-dependent, channel\-wise affine transformations\([Perez et al\., 2018](https://arxiv.org/html/2610.22446#bib.bib47)\); adaptive normalisation and cross\-attention modulate intermediate features in a similar spirit\([Peebles & Xie, 2023](https://arxiv.org/html/2610.22446#bib.bib46);[Rebain et al\., 2022](https://arxiv.org/html/2610.22446#bib.bib53)\); and hypernetworks generate the operator’s weights from𝝁\\bm\{\\mu\}\([Jing et al\., 2026](https://arxiv.org/html/2610.22446#bib.bib26)\)\.

## 3Spline\-Gridded Fourier Neural Operators

[Figure1](https://arxiv.org/html/2610.22446#S1.F1)summarises the method: sparse spline operators carry mesh features to a uniform latent grid for a standard FNO, and reconstruct the output at arbitrary query points\. We first introduce conditional input lifting, then formulate analytic SplineFNO, and finally develop L\-SplineFNO by replacing its geometry\-dependent spline transform with a local learnable scatter\.

##### Notation\.

Let\{ωi\}i=1N⊂Ω\\\{\\omega\_\{i\}\\\}\_\{i=1\}^\{N\}\\subset\\Omegadenote the nodes of an unstructured input mesh in ad−d\-dimensional domainΩ⊂ℝd\\Omega\\subset\\mathbb\{R\}^\{d\}\. We introduce a uniform tensor\-product control grid with multi\-index set𝒢=∏ℓ=1d\{0,…,Sℓ−1\}\\mathcal\{G\}=\\prod\_\{\\ell=1\}^\{d\}\\\{0,\\ldots,S\_\{\\ell\}\-1\\\}; a multi\-index is written𝐠=\(g1,…,gd\)∈𝒢\\mathbf\{g\}=\(g\_\{1\},\\ldots,g\_\{d\}\)\\in\\mathcal\{G\}, wheregℓg\_\{\\ell\}is the grid index along coordinate axisℓ\\ell\. The grid containsG:=\|𝒢\|=∏ℓ=1dSℓG:=\|\\mathcal\{G\}\|=\\prod\_\{\\ell=1\}^\{d\}S\_\{\\ell\}control points, with coordinatesξ𝐠\(ℓ\)=ξ0\(ℓ\)\+gℓ​Δℓ\\xi\_\{\\mathbf\{g\}\}^\{\(\\ell\)\}=\\xi\_\{0\}^\{\(\\ell\)\}\+g\_\{\\ell\}\\Delta\_\{\\ell\},ℓ=1,…,d\\ell=1,\\ldots,d, whereξ0\(ℓ\)\\xi\_\{0\}^\{\(\\ell\)\}andΔℓ\>0\\Delta\_\{\\ell\}\>0are the origin and spacing along axisℓ\\ell\. The spacing may differ between coordinate axes, but is constant along each axis\.

### 3\.1Conditional Input Lifting

Many computational physics problems depend on both geometry and global parameters such as material properties, boundary conditions, source terms, or operating conditions\. We introduce conditioning in our framework at the input lifting stage to incorporate these problem\-specific parameters into the nodal features supplied to the operator\. The model receives an unstructured set of geometry coordinates\{ωi\}i=1N\\\{\\omega\_\{i\}\\\}\_\{i=1\}^\{N\}, a global conditioning vector𝝁∈ℝk\\bm\{\\mu\}\\in\\mathbb\{R\}^\{k\}, and a set of output query points\{qj\}j=1Nq\\\{q\_\{j\}\\\}\_\{j=1\}^\{N\_\{q\}\}\. No input field is supplied, and the sample\-specific signal is carried by the geometry and𝝁\\bm\{\\mu\}\.

In the feature path, each normalised coordinateω¯i∈\[0,1\]d\\bar\{\\omega\}\_\{i\}\\in\[0,1\]^\{d\}is encoded by integrated sinusoidal featuresφ⁡\(ω¯i\)∈ℝ2​d​nf\\varphi\(\\bar\{\\omega\}\_\{i\}\)\\in\\mathbb\{R\}^\{2dn\_\{f\}\}atnfn\_\{f\}frequencies\. The global condition is embedded aseμ=Eμ​\(𝝁\)e\_\{\\mu\}=E\_\{\\mu\}\(\\bm\{\\mu\}\)and applied through FiLM, the most efficient of the conditioning methods we compared \([AppendixG](https://arxiv.org/html/2610.22446#A7)\),

𝐟~i=𝜸μ⊙φ⁡\(ω¯i\)\+𝜷μ,𝐡i=Pin​\[𝐟~i;ω¯i\],\\tilde\{\\mathbf\{f\}\}\_\{i\}=\\bm\{\\gamma\}\_\{\\mu\}\\odot\\varphi\(\\bar\{\\omega\}\_\{i\}\)\+\\bm\{\\beta\}\_\{\\mu\},\\qquad\\mathbf\{h\}\_\{i\}=P\_\{\\mathrm\{in\}\}\\bigl\[\\,\\tilde\{\\mathbf\{f\}\}\_\{i\};\\,\\bar\{\\omega\}\_\{i\}\\,\\bigr\],\(3\)where𝜸μ=Hγ​\(eμ\)\\bm\{\\gamma\}\_\{\\mu\}=H\_\{\\gamma\}\(e\_\{\\mu\}\)and𝜷μ=Hβ​\(eμ\)\\bm\{\\beta\}\_\{\\mu\}=H\_\{\\beta\}\(e\_\{\\mu\}\)are broadcast across all mesh nodes\. The modulation acts only on the encoded featuresφ\\varphi; the normalised coordinates bypass it and are concatenated back before the pointwise lift\. We collect the lifted nodal features as𝒳=\[𝐡1,…,𝐡N\]⊤∈ℝN×c\\mathcal\{X\}=\[\\mathbf\{h\}\_\{1\},\\ldots,\\mathbf\{h\}\_\{N\}\]^\{\\top\}\\in\\mathbb\{R\}^\{N\\times c\}\. In the geometry path, the same normalised coordinatesω¯i\\bar\{\\omega\}\_\{i\}, unmodulated, determine the compact mesh\-to\-grid spline stencils \([Figure1](https://arxiv.org/html/2610.22446#S1.F1)\)\.

![Refer to caption](https://arxiv.org/html/2610.22446v1/heat_sink_error_maps_all.png)Figure 2:Error maps produced by baseline architectures in modelling heat distribution across the heat sink\. L\-SplineFNO produces the most accurate predictions, with minimal errors concentrated at the edges of the heat sink where the temperature gradient is highest\.
### 3\.2SplineFNO

After constructing the conditioned nodal features, we first introduce SplineFNO, which uses fixed analytic spline operators to transform from the unstructured mesh to a latent uniform grid and to reconstruct at the output coordinates, with a standard FNO operating on the latent grid\. Writingti\(ℓ\)=\(ωi\(ℓ\)−ξ0\(ℓ\)\)/Δℓt\_\{i\}^\{\(\\ell\)\}=\(\\omega\_\{i\}^\{\(\\ell\)\}\-\\xi\_\{0\}^\{\(\\ell\)\}\)/\\Delta\_\{\\ell\}for the continuous grid coordinate of mesh nodeωi\\omega\_\{i\}along axisℓ\\ell, and lettingBpB\_\{p\}denote the centred cardinal B\-spline of degreeppintroduced in[SectionB\.1](https://arxiv.org/html/2610.22446#A2.SS1), we usep=3p=3by default and writeB:=BpB:=B\_\{p\}for conciseness\. The unnormalised tensor product basis matrixΦ~∈ℝN×G\\widetilde\{\\Phi\}\\in\\mathbb\{R\}^\{N\\times G\}is defined as

Φ~i,𝐠=∏ℓ=1dB\(ti\(ℓ\)−gℓ\),i=1,…,N,𝐠∈𝒢\.\\widetilde\{\\Phi\}\_\{i,\\mathbf\{g\}\}=\\prod\_\{\\ell=1\}^\{d\}B\\\!\\left\(t\_\{i\}^\{\(\\ell\)\}\-g\_\{\\ell\}\\right\),\\qquad i=1,\\ldots,N,\\quad\\mathbf\{g\}\\in\\mathcal\{G\}\.\(4\)Because the degreeppspline has compact support overp\+1p\+1grid points, each row ofΦ~\\widetilde\{\\Phi\}contains at most\(p\+1\)d\(p\+1\)^\{d\}nonzero entries\. To preserve constants near the boundary, contributions outside𝒢\\mathcal\{G\}are discarded and the remaining basis values are normalised row wise asΦi,𝐠=Φ~i,𝐠/∑𝐤∈𝒢Φ~i,𝐤\\Phi\_\{i,\\mathbf\{g\}\}=\\widetilde\{\\Phi\}\_\{i,\\mathbf\{g\}\}/\\sum\_\{\\mathbf\{k\}\\in\\mathcal\{G\}\}\\widetilde\{\\Phi\}\_\{i,\\mathbf\{k\}\}, so that partition of unity is maintained:∑𝐠∈𝒢Φi,𝐠=1\\sum\_\{\\mathbf\{g\}\\in\\mathcal\{G\}\}\\Phi\_\{i,\\mathbf\{g\}\}=1, whenever the spline support intersects the control grid\.

The forward pass consists of three stages\.\(i\) Spline transformation: unstructured to structured\.We seek grid coefficients𝒳𝒢∈ℝG×c\\mathcal\{X\}\_\{\\mathcal\{G\}\}\\in\\mathbb\{R\}^\{G\\times c\}whose spline reconstruction approximates the nodal features,

𝒳𝒢=\(Φ⊤​Φ\+λ​IG\)−1​Φ⊤​𝒳,\\mathcal\{X\}\_\{\\mathcal\{G\}\}=\\left\(\\Phi^\{\\top\}\\Phi\+\\lambda I\_\{G\}\\right\)^\{\-1\}\\Phi^\{\\top\}\\mathcal\{X\},\(5\)whereλ\>0\\lambda\>0ensures thatΦ⊤​Φ\+λ​IG\\Phi^\{\\top\}\\Phi\+\\lambda I\_\{G\}is positive definite\.\(ii\) Grid\-space neural operator\.The coefficients are reshaped according to the tensor\-product grid and processed by a standard FNO,𝒴𝒢=ℱθ​\(𝒳𝒢\)\\mathcal\{Y\}\_\{\\mathcal\{G\}\}=\\mathcal\{F\}\_\{\\theta\}\(\\mathcal\{X\}\_\{\\mathcal\{G\}\}\)\.\(iii\) Spline reconstruction: structured to unstructured\.The grid features are evaluated using the same analytic basis,

𝒴=Φ​ℱθ​\(\(Φ⊤​Φ\+λ​IG\)−1​Φ⊤​𝒳\)\.\\mathcal\{Y\}=\\Phi\\,\\mathcal\{F\}\_\{\\theta\}\\\!\\left\(\\left\(\\Phi^\{\\top\}\\Phi\+\\lambda I\_\{G\}\\right\)^\{\-1\}\\Phi^\{\\top\}\\mathcal\{X\}\\right\)\.\(6\)The spline matrices are sparse, analytic functions of the sampling coordinates and contain no learned parameters\. Learning occurs inℱθ\\mathcal\{F\}\_\{\\theta\}, while gradients propagate through both spline transfers\.

### 3\.3Learnable SplineFNO \(L\-SplineFNO\)

The spline transformation in[Equation5](https://arxiv.org/html/2610.22446#S3.E5)requires a geometry\-dependent linear solve, whose inverse is generally dense\. L\-SplineFNO replaces this solve with a local, learnable scatter that directly transfers nodal features to the latent grid\. The learned quantities are the transfer kernel shapes, while the grid nodes and analytic spline basis remain fixed\. Along each coordinate axisℓ\\ell, we use

κθ\(ℓ\)​\(t\)=B⁡\(t\)\+∑j=1Jζj\(ℓ\)​B​\(t−τjh\),ζ\(ℓ\)=V0​αθ\(ℓ\),\\kappa\_\{\\theta\}^\{\(\\ell\)\}\(t\)=B\(t\)\+\\sum\_\{j=1\}^\{J\}\\zeta\_\{j\}^\{\(\\ell\)\}B\\\!\\left\(\\frac\{t\-\\tau\_\{j\}\}\{h\}\\right\),\\qquad\\zeta^\{\(\\ell\)\}=V\_\{0\}\\alpha\_\{\\theta\}^\{\(\\ell\)\},\(7\)that is, the analytic splineBBplusJJnarrower B\-spline corrections with fixed centresτj\\tau\_\{j\}and relative widthh∈\(0,1\)h\\in\(0,1\)\. The columns ofV0V\_\{0\}span the null space of the partition\-of\-unity constraints, so only the unconstrained coordinatesαθ\(ℓ\)\\alpha\_\{\\theta\}^\{\(\\ell\)\}are trained\. The constrained correction preserves compact support, partition of unity to numerical precision, andCp−1C^\{p\-1\}regularity\. Details are given in[SectionB\.2](https://arxiv.org/html/2610.22446#A2.SS2)\.

The transfer weights are tensor products of the per\-axis kernels, normalised over the valid stencil of nodeii:𝒮ℓ​\(i\)\\mathcal\{S\}\_\{\\ell\}\(i\)holds the grid indices on axisℓ\\ellwithin the kernel support ofti\(ℓ\)t\_\{i\}^\{\(\\ell\)\}, and𝒮⁡\(i\)=∏ℓ𝒮ℓ​\(i\)\\mathcal\{S\}\(i\)=\\prod\_\{\\ell\}\\mathcal\{S\}\_\{\\ell\}\(i\)the at most\(p\+1\)d\(p\+1\)^\{d\}grid points of its tensor\-product stencil \([Equation29](https://arxiv.org/html/2610.22446#A2.E29)\),

\(Wθ\)i,𝐠=∏ℓ=1dκθ\(ℓ\)​\(ti\(ℓ\)−gℓ\)∑g′∈𝒮ℓ​\(i\)κθ\(ℓ\)​\(ti\(ℓ\)−g′\),𝐠∈𝒮⁡\(i\),\(W\_\{\\theta\}\)\_\{i,\\mathbf\{g\}\}=\\prod\_\{\\ell=1\}^\{d\}\\frac\{\\kappa\_\{\\theta\}^\{\(\\ell\)\}\\\!\\left\(t\_\{i\}^\{\(\\ell\)\}\-g\_\{\\ell\}\\right\)\}\{\\displaystyle\\sum\_\{g^\{\\prime\}\\in\\mathcal\{S\}\_\{\\ell\}\(i\)\}\\kappa\_\{\\theta\}^\{\(\\ell\)\}\\\!\\left\(t\_\{i\}^\{\(\\ell\)\}\-g^\{\\prime\}\\right\)\},\\qquad\\mathbf\{g\}\\in\\mathcal\{S\}\(i\),\(8\)and zero otherwise\. Each grid node then forms a normalised weighted accumulation, withn𝐠=∑i=1N\(Wθ\)i,𝐠​𝒳in\_\{\\mathbf\{g\}\}=\\sum\_\{i=1\}^\{N\}\(W\_\{\\theta\}\)\_\{i,\\mathbf\{g\}\}\\mathcal\{X\}\_\{i\},d𝐠=∑i=1N\(Wθ\)i,𝐠d\_\{\\mathbf\{g\}\}=\\sum\_\{i=1\}^\{N\}\(W\_\{\\theta\}\)\_\{i,\\mathbf\{g\}\}, and𝒳~𝒢​\[𝐠\]=n𝐠/d~𝐠\\widetilde\{\\mathcal\{X\}\}\_\{\\mathcal\{G\}\}\[\\mathbf\{g\}\]=n\_\{\\mathbf\{g\}\}/\\widetilde\{d\}\_\{\\mathbf\{g\}\}, whered~𝐠\\widetilde\{d\}\_\{\\mathbf\{g\}\}is the sign preserving safeguarded version ofd𝐠d\_\{\\mathbf\{g\}\}defined in[SectionB\.3](https://arxiv.org/html/2610.22446#A2.SS3)\. DefiningDθ=diag⁡\(d~𝐠\)D\_\{\\theta\}=\\operatorname\{diag\}\(\\widetilde\{d\}\_\{\\mathbf\{g\}\}\), the learnable spline transformation can then be written compactly as

𝒳~𝒢=Dθ−1​Wθ⊤​𝒳,\\widetilde\{\\mathcal\{X\}\}\_\{\\mathcal\{G\}\}=D\_\{\\theta\}^\{\-1\}W\_\{\\theta\}^\{\\top\}\\mathcal\{X\},\(9\)which provides a local approximation to the spline transformation\(Φ⊤​Φ\+λ​IG\)−1​Φ⊤​𝒳\(\\Phi^\{\\top\}\\Phi\+\\lambda I\_\{G\}\)^\{\-1\}\\Phi^\{\\top\}\\mathcal\{X\}without requiring a geometry\-dependent linear solve\.

To retain mesh coverage information removed by normalisation, we augment the scattered features with an analytic density featureρ\\rhocomputed from the fixed spline basis and form the FNO input

𝒳𝒢=Pρ​\[𝒳~𝒢;log⁡\(1\+ρ\)\],𝒴𝒢=ℱθ​\(𝒳𝒢\),\\mathcal\{X\}\_\{\\mathcal\{G\}\}=P\_\{\\rho\}\\left\[\\widetilde\{\\mathcal\{X\}\}\_\{\\mathcal\{G\}\};\\log\(1\+\\rho\)\\right\],\\qquad\\mathcal\{Y\}\_\{\\mathcal\{G\}\}=\\mathcal\{F\}\_\{\\theta\}\(\\mathcal\{X\}\_\{\\mathcal\{G\}\}\),\(10\)wherePρP\_\{\\rho\}is a learned pointwise projection\. The construction ofρ\\rhois detailed in[SectionB\.3](https://arxiv.org/html/2610.22446#A2.SS3)\. Finally, the output is reconstructed analytically at each query point and projected to the output channels

𝐳j=∑𝐠∈𝒢Φ𝐠​\(qj\)​𝒴𝒢​\[𝐠\],𝐲^j=Pout​𝐳j,\\mathbf\{z\}\_\{j\}=\\sum\_\{\\mathbf\{g\}\\in\\mathcal\{G\}\}\\Phi\_\{\\mathbf\{g\}\}\(q\_\{j\}\)\\mathcal\{Y\}\_\{\\mathcal\{G\}\}\[\\mathbf\{g\}\],\\qquad\\widehat\{\\mathbf\{y\}\}\_\{j\}=P\_\{\\mathrm\{out\}\}\\mathbf\{z\}\_\{j\},\(11\)wherePoutP\_\{\\mathrm\{out\}\}is a learned pointwise output projection\.

##### Region\-specific transfers\.

On meshes with several materials or regions, each node carries a region label and can be routed through its own lift, scatter kernels and output projection, around one shared latent grid and FNO\. The per\-region scatter sums are added before the division in[Equation9](https://arxiv.org/html/2610.22446#S3.E9), so constants are still reproduced\. A per\-region readout lets each region take its own combination of the shared latent features, so the field can remain continuous across an interface while its normal derivative jumps\. Only the lift, projection and kernel weights are duplicated; the FNO is shared\. We use this whenever the mesh is naturally partitioned by regions or materials\.

## 4Experiments

Table 1:RelativeL2L\_\{2\}error \(%, mean±\\pmstd over five seeds; top line of each entry\) and training time \(hh:mm on a single NVIDIA A100; grey line beneath\) for each architecture under FiLM conditioning\. The best mean error for each case is shown in bold\.NNis the number of spatial training nodes\. Heat Sink, Heated Duct and HIVE predict absolute temperature, so their relative errors are small by construction\. L\-SplineFNO outperforms baselines on most experiments, while remaining significantly compute\-efficient\.ModelHeat SinkBracketAircraftBuilding FireHeated DuctHIVE MonoblockN=16,384N=16\{,\}384N=8,192N=8\{,\}192N=131,072N=131\{,\}072N=98,496N=98\{,\}496N=124,800N=124\{,\}800N=194,691N=194\{,\}691BaselinesTransolver0\.040±0\.0020\.040\{\\pm\}0\.0020\.591±0\.0250\.591\{\\pm\}0\.0258\.32±0\.888\.32\{\\pm\}0\.887\.341±0\.2207\.341\{\\pm\}0\.2200\.473±0\.0500\.473\{\\pm\}0\.0500\.301±0\.0610\.301\{\\pm\}0\.06100:4101:4801:2601:1701:1502:28UPT0\.039±0\.0010\.039\{\\pm\}0\.0010\.633±0\.0140\.633\{\\pm\}0\.01413\.87±0\.4313\.87\{\\pm\}0\.437\.162±0\.1587\.162\{\\pm\}0\.1580\.523±0\.0530\.523\{\\pm\}0\.0530\.299±0\.1090\.299\{\\pm\}0\.10900:3301:3000:5801:000:5001:25MSPT0\.038±0\.0020\.038\{\\pm\}0\.0020\.544±0\.0490\.544\{\\pm\}0\.0497\.20±0\.517\.20\{\\pm\}0\.517\.217±0\.2097\.217\{\\pm\}0\.2090\.500±0\.0570\.500\{\\pm\}0\.0570\.315±0\.0900\.315\{\\pm\}0\.09001:2603:2906:1406:1805:2507:40GNOT0\.042±0\.0010\.042\{\\pm\}0\.0010\.534±0\.040\\bm\{0\.534\{\\pm\}0\.040\}6\.86±0\.48\\bm\{6\.86\{\\pm\}0\.48\}7\.456±0\.3947\.456\{\\pm\}0\.3940\.652±0\.0320\.652\{\\pm\}0\.0320\.243±0\.0300\.243\{\\pm\}0\.03001:3304:1003:2503:1602:5208:50NUNO0\.068±0\.0010\.068\{\\pm\}0\.0011\.972±0\.0951\.972\{\\pm\}0\.09511\.72±1\.4111\.72\{\\pm\}1\.416\.805±0\.0876\.805\{\\pm\}0\.0870\.989±0\.1440\.989\{\\pm\}0\.1440\.471±0\.1200\.471\{\\pm\}0\.12000:1400:4000:1400:3400:1000:15GINO0\.052±0\.0020\.052\{\\pm\}0\.0021\.385±0\.0101\.385\{\\pm\}0\.01011\.35±0\.8811\.35\{\\pm\}0\.887\.201±0\.1717\.201\{\\pm\}0\.1711\.095±0\.0671\.095\{\\pm\}0\.0670\.370±0\.0840\.370\{\\pm\}0\.08401:0609:1808:4403:0401:1706:36Spline\-based \(Ours\)SplineFNO0\.035±0\.0010\.035\{\\pm\}0\.0015\.490±0\.0255\.490\{\\pm\}0\.02518\.27±1\.1118\.27\{\\pm\}1\.117\.611±0\.0927\.611\{\\pm\}0\.0920\.542±0\.0630\.542\{\\pm\}0\.0630\.216±0\.0130\.216\{\\pm\}0\.01301:2203:4600:4801:5700:1700:18L\-SplineFNO0\.027±0\.001\\bm\{0\.027\{\\pm\}0\.001\}1\.978±0\.0061\.978\{\\pm\}0\.0068\.55±0\.868\.55\{\\pm\}0\.866\.443±0\.131\\bm\{6\.443\{\\pm\}0\.131\}0\.454±0\.044\\bm\{0\.454\{\\pm\}0\.044\}0\.211±0\.045\\bm\{0\.211\{\\pm\}0\.045\}00:4200:5401:0400:3800:2501:06

##### Data:

We evaluate on six steady\-state, three\-dimensional engineering benchmarks, including a CPU\-boardheat sink\([Zhong et al\., 2026](https://arxiv.org/html/2610.22446#bib.bib65)\), a structurally loadedbracket\([Zhong et al\., 2025](https://arxiv.org/html/2610.22446#bib.bib64)\),aircraftflow fields\([Luo et al\., 2025](https://arxiv.org/html/2610.22446#bib.bib37)\), abuilding firesimulated with FDS\([McGrattan et al\., 2013](https://arxiv.org/html/2610.22446#bib.bib39)\), conjugate heat transfer through a multi\-regionheated duct\([Weller et al\., 1998](https://arxiv.org/html/2610.22446#bib.bib58)\), and theHIVEinduction\-heating benchmark\([Nuno Nobre, Josh Williams and Karthikeyan Chockalingam, 2024](https://arxiv.org/html/2610.22446#bib.bib43);[Horne\-Jones et al\., 2026](https://arxiv.org/html/2610.22446#bib.bib22)\)\. Together they span thermal, structural, and aerodynamic physics, sample\-dependent and shared meshes, and single\- and multi\-region domains\. The suite also varies how much information the conditioning vector𝝁\\bm\{\\mu\}provides beyond the observed geometry\. It is largely redundant for the heat sink; partly non\-redundant for the bracket and for the building fire and essential for Aircraft, heated dict and HIVE, where operating conditions cannot be recovered from the geometry alone\. Further details on the physics, data, and preprocessing are provided in[AppendixD](https://arxiv.org/html/2610.22446#A4)\.

##### Models:

We compare L\-SplineFNO with six representative neural operators for unstructured domains, including mesh\-based attention models Transolver\([Wu et al\., 2024](https://arxiv.org/html/2610.22446#bib.bib62)\)and GNOT\([Hao et al\., 2023](https://arxiv.org/html/2610.22446#bib.bib19)\), latent attention models UPT\([Alkin et al\., 2025](https://arxiv.org/html/2610.22446#bib.bib1)\)and MSPT\([Curvo et al\., 2026](https://arxiv.org/html/2610.22446#bib.bib9)\), and gridded FNO\-based models NUNO\([Liu et al\., 2023](https://arxiv.org/html/2610.22446#bib.bib36)\)and GINO\([Li et al\., 2023b](https://arxiv.org/html/2610.22446#bib.bib34)\)\. We also include the analytic SplineFNO to isolate the effect of learning the spline transfer\. All models use the same outer conditional pipeline in[Figure1](https://arxiv.org/html/2610.22446#S1.F1)and match in parameter count \(2\.32\.3to2\.52\.5million parameters\), so we replace only the backbone and its associated spatial representation\. Full architectural and implementation details are provided in[AppendixF](https://arxiv.org/html/2610.22446#A6)\.

##### Training Objective:

Given a dataset𝒟=\{\(𝝁\(i\),\{ωn\(i\)\}n=1Ni,\{y\(i\)​\(ωn\(i\)\)\}n=1Ni\)\}i=1M\\mathcal\{D\}=\\\{\(\\bm\{\\mu\}^\{\(i\)\},\\\{\\omega\_\{n\}^\{\(i\)\}\\\}\_\{n=1\}^\{N\_\{i\}\},\\allowbreak\\\{y^\{\(i\)\}\(\\omega\_\{n\}^\{\(i\)\}\)\\\}\_\{n=1\}^\{N\_\{i\}\}\)\\\}\_\{i=1\}^\{M\}of conditioning vectors, mesh nodes and target fields, with no input field supplied, we minimise the mean unweighted discrete relativeL2L\_\{2\}error over the sampled mesh nodes \([Equation12](https://arxiv.org/html/2610.22446#S4.E12)\)\.

𝒥⁡\(θ\)=1M​∑i=1M‖𝐲^\(i\)−𝐲\(i\)‖2‖𝐲\(i\)‖2\\mathcal\{J\}\(\\theta\)=\\frac\{1\}\{M\}\\sum\_\{i=1\}^\{M\}\\frac\{\\bigl\\\|\\widehat\{\\mathbf\{y\}\}^\{\(i\)\}\-\\mathbf\{y\}^\{\(i\)\}\\bigr\\\|\_\{2\}\}\{\\bigl\\\|\\mathbf\{y\}^\{\(i\)\}\\bigr\\\|\_\{2\}\}\(12\)Here𝐲\(i\)∈ℝNi\\mathbf\{y\}^\{\(i\)\}\\in\\mathbb\{R\}^\{N\_\{i\}\}is the target field at the nodes\{ωn\(i\)\}n=1Ni\\\{\\omega\_\{n\}^\{\(i\)\}\\\}\_\{n=1\}^\{N\_\{i\}\}of sampleii, and∥⋅∥2\\\|\\cdot\\\|\_\{2\}is the unweighted discrete Euclidean norm over those nodes\. We use the same metric for training and its percentage formulation for the final error evaluation in[Table1](https://arxiv.org/html/2610.22446#S4.T1)\. All models use the same training recipe: Adam with a starting learning rate of10−310^\{\-3\}, halved every 50 epochs, for 250 epochs\.

### 4\.1Benchmarks

![Refer to caption](https://arxiv.org/html/2610.22446v1/hive_error_maps_all.png)Figure 3:Error maps produced by baselines in modelling heat distribution across the HIVE monoblock\. L\-SplineFNO models the temperature across the structure with99\.9%99\.9\\%accuracy \(sample\-specific error shown\)\.As shown in[Table1](https://arxiv.org/html/2610.22446#S4.T1), L\-SplineFNO occupies a favourable accuracy–cost regime across the benchmark suite: it provides the strongest predictive accuracy on the heat sink, building fire, heated duct, and HIVE cases, and remains close to the leading attention\-based models on Aircraft while requiring substantially less training time than MSPT and GNOT\. The bracket case is the principal exception, where direct mesh\-based attention retains a clear accuracy advantage\. The bracket’s peak stresses sit in small, sharp concentrations, which the latent grid’s averaging andC2C^\{2\}decode flatten, whereas attention acts on the mesh nodes directly \([Figure11](https://arxiv.org/html/2610.22446#A4.F11)\)\. Compared with the analytic SplineFNO, however, learning the spline transfer consistently improves accuracy across all six cases with only a modest change in model size\. The faster gridded baselines sacrifice considerably more accuracy, whereas the more accurate attention\-based alternatives incur higher training cost; L\-SplineFNO therefore combines the efficiency of a structured latent grid with predictive performance competitive with, and often better than, several more computationally demanding unstructured operators\.[Figure2](https://arxiv.org/html/2610.22446#S3.F2)demonstrates how L\-SplineFNO achieves near\-perfect accuracy in modelling the heat distribution across a heat sink\.[Figure3](https://arxiv.org/html/2610.22446#S4.F3)shows the error maps produced by each model for the HIVE monoblock case, with L\-SplineFNO modelling the heat distribution with99\.92%99\.92\\%accuracy\. We show further ablations on spline degree and multi\-scale scattering in[AppendixI](https://arxiv.org/html/2610.22446#A9)\.

\(a\)Gradient\-evaluation costs for L\-SplineFNO\.\(b\)Mesh capacity of models across a single GPU\.
Figure 4:Derivative\-evaluation cost and scalability\.\([4\(a\)](https://arxiv.org/html/2610.22446#S4.F4.sf1)\) Mean derivative runtime \(left\) and additional peak GPU allocation \(right\), averaged across the benchmark experiments\. Forward preparation is excluded; memory is measured above the prepared live state, not total GPU use\. Full\-model autograd VJP measures coordinate sensitivity, whereas decoder autograd and analytic spline differentiation evaluate pointwise field gradients\. Closed\-form spline derivatives give exact gradients at a fraction of the cost\. \([4\(b\)](https://arxiv.org/html/2610.22446#S4.F4.sf2)\) Mean largest successful node count across three seeds on an A100 80GB GPU in FP32 with batch size one\. Training includes a complete Adam step\. L\-SplineFNO accomodates 26\.6M nodes in training and 46\.0M in inference, demonstrating superior scalability against the baselines\.
### 4\.2Analytic Gradients

The analytic decoder of L\-SplineFNO yields spatial derivatives without an automatic\-differentiation graph over the query coordinates\. The decoded field of[Equation11](https://arxiv.org/html/2610.22446#S3.E11)is𝐲^​\(q\)=Pout​∑𝐠∈𝒢𝒴𝒢​\[𝐠\]​Φ𝐠​\(q\)\\widehat\{\\mathbf\{y\}\}\(q\)=P\_\{\\mathrm\{out\}\}\\sum\_\{\\mathbf\{g\}\\in\\mathcal\{G\}\}\\mathcal\{Y\}\_\{\\mathcal\{G\}\}\[\\mathbf\{g\}\]\\,\\Phi\_\{\\mathbf\{g\}\}\(q\), where𝒴𝒢​\[𝐠\]\\mathcal\{Y\}\_\{\\mathcal\{G\}\}\[\\mathbf\{g\}\]are the latent\-grid coefficients,Φ𝐠\\Phi\_\{\\mathbf\{g\}\}is the normalised basis \([Equation4](https://arxiv.org/html/2610.22446#S3.E4)\), andPout​𝐳=Wout​𝐳\+𝐛outP\_\{\\mathrm\{out\}\}\\mathbf\{z\}=W\_\{\\mathrm\{out\}\}\\mathbf\{z\}\+\\mathbf\{b\}\_\{\\mathrm\{out\}\}is an affine projection\. The bias𝐛out\\mathbf\{b\}\_\{\\mathrm\{out\}\}vanishes under differentiation, so

∂𝒏𝐲^​\(q\)=Wout​∑𝐠∈𝒢𝒴𝒢​\[𝐠\]​∂𝒏Φ𝐠​\(q\),1≤\|𝒏\|≤2\.\\partial^\{\\bm\{n\}\}\\widehat\{\\mathbf\{y\}\}\(q\)=W\_\{\\mathrm\{out\}\}\\sum\_\{\\mathbf\{g\}\\in\\mathcal\{G\}\}\\mathcal\{Y\}\_\{\\mathcal\{G\}\}\[\\mathbf\{g\}\]\\,\\partial^\{\\bm\{n\}\}\\Phi\_\{\\mathbf\{g\}\}\(q\),\\qquad 1\\leq\|\\bm\{n\}\|\\leq 2\.\(13\)The derivatives∂𝒏Φ𝐠\\partial^\{\\bm\{n\}\}\\Phi\_\{\\mathbf\{g\}\}are evaluated in closed form from the first and second derivatives of the degree\-ppB\-spline, including those of the boundary partition\-of\-unity normalisation\. Each query touches at most\(p\+1\)d\(p\+1\)^\{d\}control points, soNNqueries costO⁡\(N​\(p\+1\)d\)O\(N\(p\+1\)^\{d\}\)with no reverse\-mode coordinate tape, and the result stays differentiable in𝒴𝒢\\mathcal\{Y\}\_\{\\mathcal\{G\}\}, keeping derivative\-based physics losses trainable\.

Full\-modelautogradinstead returns the coordinate sensitivity∂\(∑i𝐲^i\)/∂ωj\\partial\(\\sum\_\{i\}\\widehat\{\\mathbf\{y\}\}\_\{i\}\)/\\partial\\omega\_\{j\}, which for globally coupled operators differs from∇𝐲^​\(ωj\)\\nabla\\widehat\{\\mathbf\{y\}\}\(\\omega\_\{j\}\)because moving nodejjcan alter predictions elsewhere; we therefore compare against autograd through the same decoder with the latent grid fixed\.[Figure4\(a\)](https://arxiv.org/html/2610.22446#S4.F4.sf1)reports runtime and memory across all benchmark experiments\.

### 4\.3Scalability

We benchmark all architectures on synthetic three\-dimensional point clouds with the HIVE settings, on one NVIDIA A100 80GB GPU in FP32 with batch size one \([Figure4\(b\)](https://arxiv.org/html/2610.22446#S4.F4.sf2)\)\. Runtime grows near\-linearly with node count for every model over its feasible range, so memory decides capacity\. L\-SplineFNO trains on up to 26\.6M nodes and runs inference on 46\.0M, well beyond the capacity of existing methods\. Details are in[AppendixH](https://arxiv.org/html/2610.22446#A8)\.

### 4\.4Virtual Tokamak of MAST\-U

Beyond benchmarks, we apply L\-SplineFNO to a whole\-device structural modelling of the Mega\-Ampere Spherical Tokamak Upgrade \(MAST\-U\), a fusion experiment that informs compact spherical\-tokamak power plants\([Milnes et al\., 2015](https://arxiv.org/html/2610.22446#bib.bib40);[Morris et al\., 2018](https://arxiv.org/html/2610.22446#bib.bib41);[Harrison et al\., 2024](https://arxiv.org/html/2610.22446#bib.bib20);[Chapman et al\., 2024](https://arxiv.org/html/2610.22446#bib.bib6)\)\. Such simulations, referred to as*virtual tokamaks*load the vessel, centre column, coils and supports together, so each component sees boundary conditions set by the rest of the assembly\([Ellis et al\., 2025](https://arxiv.org/html/2610.22446#bib.bib13)\)\. A single static solve of the MAST\-U model costs about500\+500\+CPU\-hours on800800cores, challenging the many\-query workloads a structural digital twin needs\([Pamela et al\., 2025](https://arxiv.org/html/2610.22446#bib.bib44)\)\.

![Refer to caption](https://arxiv.org/html/2610.22446v1/mild-python_gravity_thermal_20__eval_02__parameter_study_out_study_app02_full_gt_error.png)Figure 5:MAST\-U virtual tokamak: Displacement magnitude \(mm\) from the MOOSE simulation \(ground truth\) and the L\-SplineFNO \(prediction\) and the absolute error across them \(mm\)\.To alleviate this bottleneck, we learn the static displacement field𝐮=\(ux,uy,uz\)\\mathbf\{u\}=\(u\_\{x\},u\_\{y\},u\_\{z\}\)of the full MAST\-U structural model under thermal expansion, given the thermal expansion coefficient and wall temperature\. The task stresses unstructured operators on several axes at once\. The mesh has29\.229\.2M nodes and134\.6134\.6M tetrahedra; merging3,9383\{,\}938CAD parts leaves element sizes spanning more than four orders of magnitude\. Steel, copper and graphite meet at over a million shared interface nodes\. L\-SplineFNO, with material\-specific spline transfers and a shared FNO backbone, is trained on the full mesh with29\.229\.2M nodes and reaches a relative vectorL2L\_\{2\}error of1\.8%1\.8\\%on a single H100 GPU \([Figure5](https://arxiv.org/html/2610.22446#S4.F5)\)\.[AppendixE](https://arxiv.org/html/2610.22446#A5)details the device, physics, data, and training\.

##### Spatial bounds and parameter uncertainty\.

The analytic spline decoder enables post\-training bounds on the predicted displacement over continuous spatial regions and uncertain operating parameters, without needing additional reference solutions\. These bounds provide a rigorous guarantee that the model prediction lies within a specified range, which is critical for safety\-critical applications such as nuclear fusion devices\.

*Spatial bounds\.*At fixed parameters, each displacement component on a spline cell can be expressed in Bernstein form\([Romani & Sabin, 2004](https://arxiv.org/html/2610.22446#bib.bib54)\)as

u^j​\(x\)=∑kYj,k​bk​\(x\)∑kDk​bk​\(x\),mink⁡Yj,kDk≤u^j​\(x\)≤maxk⁡Yj,kDk\.\\widehat\{u\}\_\{j\}\(x\)=\\frac\{\\sum\_\{k\}Y\_\{j,k\}b\_\{k\}\(x\)\}\{\\sum\_\{k\}D\_\{k\}b\_\{k\}\(x\)\},\\qquad\\min\_\{k\}\\frac\{Y\_\{j,k\}\}\{D\_\{k\}\}\\leq\\widehat\{u\}\_\{j\}\(x\)\\leq\\max\_\{k\}\\frac\{Y\_\{j,k\}\}\{D\_\{k\}\}\.\(14\)
![Refer to caption](https://arxiv.org/html/2610.22446v1/mild-python_sample0_native_parameter_uncertainty.png)Figure 6:Rigorous enclosure of L\-SplineFNO over the uncertain parameter boxℬ\\mathcal\{B\}, obtained combining a Taylor\-remainder bound with the convex\-hull property of the spline\.HereYj,kY\_\{j,k\}andDkD\_\{k\}are the Bernstein coefficients of the decoder numerator and normalisation denominator, respectively, andbkb\_\{k\}are the spatial Bernstein basis functions\. Sincebk​\(x\)≥0b\_\{k\}\(x\)\\geq 0andDk\>0D\_\{k\}\>0, the prediction is a convex combination of the coefficient ratiosYj,k/DkY\_\{j,k\}/D\_\{k\}\([Piegl & Tiller, 1996](https://arxiv.org/html/2610.22446#bib.bib49)\)\. Their extrema therefore enclose every point within the cell, including points between mesh nodes\. The bound is loose when a cell is large compared with the variation of the field, since the Bernstein coefficients then lie well away from the function values\. We tighten it by spatial subdivision\. The de Casteljau algorithm splits a cell exactly in half along its longest side and gives each half its own Bernstein coefficients, describing the same spline on a smaller box\([Boehm & Müller, 1999](https://arxiv.org/html/2610.22446#bib.bib4)\)\. Afternnsuch splits, each cell is covered by2n2^\{n\}sub\-boxes\.[Equation14](https://arxiv.org/html/2610.22446#S4.E14)is applied to each sub\-box, and as the sub\-boxes shrink their coefficients converge to the function values \(see[SectionB\.4](https://arxiv.org/html/2610.22446#A2.SS4)for more details\)\.

*Parameter uncertainty\.*The same enclosure extends to an uncertain parameter boxℬ\\mathcal\{B\}, where𝝁=\(α,Tw\)∈ℬ\\bm\{\\mu\}=\(\\alpha,T\_\{w\}\)\\in\\mathcal\{B\}\. Only the spline coefficients change: at fixed parameters each is a number, whereas overℬ\\mathcal\{B\}each is a function of𝝁\\bm\{\\mu\}\. We obtain these functions by propagating a second\-order Taylor model through the fixed trained network\([Makino & Berz, 1999](https://arxiv.org/html/2610.22446#bib.bib38)\), so that each physical spline coefficient takes the form

ci​\(𝝁\)=ci0\+𝐚i⊤​𝜹\+12​𝜹⊤​Hi​𝜹\+εi,\\displaystyle c\_\{i\}\(\\bm\{\\mu\}\)=c\_\{i\}^\{0\}\+\\mathbf\{a\}\_\{i\}^\{\\top\}\\bm\{\\delta\}\+\\tfrac\{1\}\{2\}\\bm\{\\delta\}^\{\\top\}H\_\{i\}\\bm\{\\delta\}\+\\varepsilon\_\{i\},\(15\)\|εi\|≤ri,𝜹=𝝁−𝝁0\.\\displaystyle\|\\varepsilon\_\{i\}\|\\leq r\_\{i\},\\quad\\bm\{\\delta\}=\\bm\{\\mu\}\-\\bm\{\\mu\}\_\{0\}\.
Hereci0c\_\{i\}^\{0\},𝐚i\\mathbf\{a\}\_\{i\}, andHiH\_\{i\}are the coefficient value, parameter gradient, and parameter Hessian at the expansion centre𝝁0\\bm\{\\mu\}\_\{0\}, whilerir\_\{i\}bounds the remainder uniformly overℬ\\mathcal\{B\}\. The parameters enter only through the input modulation of[Equation3](https://arxiv.org/html/2610.22446#S3.E3); the affine operations that follow \(lifting, scatter, Fourier layers and readout\) carry the signed polynomial terms exactly, and only the nonlinear activations add torir\_\{i\}\.

The Bernstein transformation in[Equation14](https://arxiv.org/html/2610.22446#S4.E14)is linear in the spline coefficients, so applying it to these polynomials gives each numerator coefficient as a quadratic in𝜹\\bm\{\\delta\}plus a bounded remainder,

Yj,k​\(𝝁\)=Yj,k0\+πj,k​\(𝜹\)\+ηj,k,\|ηj,k\|≤rj,k\.Y\_\{j,k\}\(\\bm\{\\mu\}\)=Y\_\{j,k\}^\{0\}\+\\pi\_\{j,k\}\(\\bm\{\\delta\}\)\+\\eta\_\{j,k\},\\qquad\|\\eta\_\{j,k\}\|\\leq r\_\{j,k\}\.\(16\)The transformation acts on the polynomial coefficients*before*any extremum is taken, so all spline coefficients share the same𝜹\\bm\{\\delta\}and their parameter contributions can cancel\. Minimising and maximisingπj,k\\pi\_\{j,k\}overℬ\\mathcal\{B\}and subtracting or addingrj,kr\_\{j,k\}givesYj,k−Y\_\{j,k\}^\{\-\}andYj,k\+Y\_\{j,k\}^\{\+\}\. The denominator coefficientsDkD\_\{k\}depend only on geometry, so the convex\-hull argument of[Equation14](https://arxiv.org/html/2610.22446#S4.E14)applies unchanged:

mink⁡Yj,k−Dk≤u^j​\(x,𝝁\)≤maxk⁡Yj,k\+Dkfor all​x​in the cell and all​𝝁∈ℬ\.\\begin\{gathered\}\\min\_\{k\}\\frac\{Y\_\{j,k\}^\{\-\}\}\{D\_\{k\}\}\\leq\\widehat\{u\}\_\{j\}\(x,\\bm\{\\mu\}\)\\leq\\max\_\{k\}\\frac\{Y\_\{j,k\}^\{\+\}\}\{D\_\{k\}\}\\qquad\\text\{for all \}x\\text\{ in the cell and all \}\\bm\{\\mu\}\\in\\mathcal\{B\}\.\\end\{gathered\}\(17\)
Spatial subdivision tightens this enclosure exactly as before\. Parameter subdivision is the complementary refinement\. It partitionsℬ\\mathcal\{B\}and re\-centres the Taylor model on each part, which shrinks the nonlinear remainders, and the final enclosure is the union over all parts\. In[Figure6](https://arxiv.org/html/2610.22446#S4.F6)we show the displacement bounds over a parameter boxℬ\\mathcal\{B\}such thatα∈\[1\.4228,1\.4428\]×10−5​K−1\\alpha\\in\[1\.4228,\\,1\.4428\]\\times 10^\{\-5\}\\,\\mathrm\{K\}^\{\-1\}andTw∈\[859,869\]​KT\_\{w\}\\in\[859,869\]\\,\\mathrm\{K\}\.

These intervals describe the range of the surrogate prediction, including spatial variation, parameter variation and enclosure conservatism\. They do not quantify prediction error against a reference numerical solution\. Here we present the validated binary64 \(FP64\) results, which carry an end\-to\-end rounding certificate: outward rounding accounts for arithmetic error, so the enclosure contains the real\-arithmetic surrogate defined by the stored weights\. The same enclosure formulas can also be evaluated in ordinary FP32 or FP64, which is cheaper but gives no rounding certificate\. All three are compared in[Table5](https://arxiv.org/html/2610.22446#A2.T5), and[SectionB\.4](https://arxiv.org/html/2610.22446#A2.SS4)gives the construction and arithmetic assumptions\.

## 5Discussion

##### Strengths\.

L\-SplineFNO constructs a structure\-preserving learnable transfer that maintains compact support, partition of unity andCp−1C^\{p\-1\}smoothness\. This allows the model to scale linearly in both memory and compute for mesh size\. L\-SplineFNO achieves better performance on most benchmarks while remaining training\-compute efficient and maintaining accuracy on super\-resolution tasks\. Because the transfer is parameter\-efficient, it can be specialised per material or region at negligible cost\. The explicit spline reconstruction provides inexpensive analytic gradients, useful for sensitivity analysis and physics\-based objectives\. The same reconstruction also yields rigorous bounds on the prediction over whole cells, not just at mesh nodes, and over boxes of uncertain parameters\. Mesh\-native and attention\-based operators have no continuous reconstruction, so they provide no comparable guarantee\.

##### Limitations\.

The spline transfer can smooth features in the target, such as kinks at material interfaces and stress concentrations around small geometric features, where mesh\-native attention remains more accurate\. Per\-region output projections recover interface kinks, but the scatter still mixes regions on a shared latent grid; interface\-aware or deformed transfers and local graph corrections near thin features could address this\([Li et al\., 2023a](https://arxiv.org/html/2610.22446#bib.bib33);[Li et al\., 2025](https://arxiv.org/html/2610.22446#bib.bib31);[Li et al\., 2023b](https://arxiv.org/html/2610.22446#bib.bib34)\)\. The fixed linear decoder also limits each output to the span of the spline basis, which sets an accuracy floor that no encoder or backbone can overcome, most visibly at sharp interfaces and on surfaces embedded in a volumetric grid\. Finally, stencil cost grows rapidly with spline degree in three dimensions, so higher degrees exhaust memory on large meshes without relieving the smoothness bias \(see[appendixI](https://arxiv.org/html/2610.22446#A9)\)\. Adaptive refinement of latent cells only where geometry or error demands it, as in NUNO\([Liu et al\., 2023](https://arxiv.org/html/2610.22446#bib.bib36)\), could ease both the accuracy floor and the memory cost while keeping cubic stencils\. The enclosures bound the surrogate, not its error, and are conservative on account of the rigour\. They loosen with cell size, parameter\-box width and network depth; subdivision tightens them at proportional cost for a fixed geometry\.

##### Future work\.

Several extensions follow directly from the spline structure\. First, interface kinks could be represented exactly\. Our decoder is a cubic B\-spline basis, which isCp−1C^\{p\-1\}at every knot; repeating a knot lowers the continuity at that location alone, so three coincident knots permit a jump in gradient and four a jump in value\([de Boor, 1978](https://arxiv.org/html/2610.22446#bib.bib10);[Piegl & Tiller, 1996](https://arxiv.org/html/2610.22446#bib.bib49)\)\. Isogeometric analysis, which solves PDEs directly in a spline basis, uses exactly this to resolve material interfaces\([Hughes et al\., 2005](https://arxiv.org/html/2610.22446#bib.bib23)\)\. Second, the closed\-form derivatives make physics\-informed training inexpensive\([Raissi et al\., 2019](https://arxiv.org/html/2610.22446#bib.bib52);[Li et al\., 2023c](https://arxiv.org/html/2610.22446#bib.bib35)\); on multi\-region problems, PDE\-residual losses could be paired with penalties on flux continuity across interfaces, as in domain\-decomposed physics\-informed networks\([Jagtap et al\., 2020](https://arxiv.org/html/2610.22446#bib.bib25)\)\. Third, the parameter enclosures supply a natural acquisition function for active learning, placing new simulations where the enclosure is widest\([Musekamp et al\., 2025](https://arxiv.org/html/2610.22446#bib.bib42);[Polanska et al\., 2026](https://arxiv.org/html/2610.22446#bib.bib50)\)\. Fourth, because FiLM conditions on a global vector, time can enter as one more component of𝝁\\bm\{\\mu\}, as in time\-conditioned PDE foundation models\([Herde et al\., 2024](https://arxiv.org/html/2610.22446#bib.bib21);[Takamoto et al\., 2023](https://arxiv.org/html/2610.22446#bib.bib55)\); this would extend the framework from steady states to transient fields, complementing an operator\-split formulation\([Gopakumar et al\., 2026](https://arxiv.org/html/2610.22446#bib.bib17)\)\. Finally, the enclosures currently bound the surrogate rather than the numerical simulation it approximates\. Widening them by a calibrated bound on the surrogate\-to\-solver error, for instance through conformal prediction on held\-out simulations\([Gopakumar et al\., 2025](https://arxiv.org/html/2610.22446#bib.bib16)\), would give intervals that contain the simulated field over the whole parameter box with a stated probability\.

#### Acknowledgments

The Fusion Futures Programme partially funds Vignesh Gopakumar and Ander Gray\. As announced by the UK Government in October 2023, Fusion Futures aims to provide holistic support for the development of the fusion sector\. Ander Gray is partially funded by the Academic and Research Chaire “Architecture des Systèmes Complexes” Dassault Aviation, Naval Group, Dassault Systèmes, KNDS France, Agence de l’Innovation de Défense, Institut Polytechnique de Paris\. Anima Anandkumar is supported in part by Bren endowed chair, ONR \(MURI grant N00014\-23\-1\-2654\), and the AI2050 senior fellow program at Schmidt Sciences\. The authors acknowledge funding from the Fusion Computing Lab collaboration between the Hartree Centre and UKAEA\. This work was performed using resources provided by the Cambridge Service for Data Driven Discovery \(CSD3\) operated by the University of Cambridge Research Computing Service \(www\.csd3\.cam\.ac\.uk\), provided by Dell EMC and Intel using Tier\-2 funding from the Engineering and Physical Sciences Research Council \(capital grant EP/T022159/1\), and DiRAC funding from the Science and Technology Facilities Council \(www\.dirac\.ac\.uk\)\.

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## Appendix ANotation

[Table2](https://arxiv.org/html/2610.22446#A1.T2)lists the symbols used across the paper, grouped by topic, with a pointer to where each is defined\. Bold lowercase denotes a vector at one node, query point or sample \(𝐡i\\mathbf\{h\}\_\{i\},𝝁\\bm\{\\mu\}\); calligraphic letters denote collections over nodes or grid points, operators and sets \(𝒳\\mathcal\{X\},𝒢\\mathcal\{G\},𝒩\\mathcal\{N\}\); a subscript𝒢\\mathcal\{G\}marks a latent\-grid quantity, and a hat marks a prediction \(y^\\widehat\{y\}\)\. A parenthesised superscript indexes a coordinate axis on coordinates \(ωi\(ℓ\)\\omega\_\{i\}^\{\(\\ell\)\}\) and a sample on dataset quantities \(𝐲\(i\)\\mathbf\{y\}^\{\(i\)\}\); it never denotes a power\. Symbols local to one derivation, such as the intermediate Taylor\-model terms in[SectionB\.4](https://arxiv.org/html/2610.22446#A2.SS4)and the internals of individual baselines in[AppendixF](https://arxiv.org/html/2610.22446#A6), are defined where they appear\. Symbols with more than one meaning are collected in[Table3](https://arxiv.org/html/2610.22446#A1.T3)\.

Table 2:Notation\.SymbolMeaningProblem and dataΩ⊂ℝd\\Omega\\subset\\mathbb\{R\}^\{d\},ddSpatial domain and its dimension \(d=3d=3throughout\)\.𝐱∈Ω\\mathbf\{x\}\\in\\OmegaGeneric continuous point; writtenxxin the enclosure formulas\.𝝁∈𝒫⊂ℝk\\bm\{\\mu\}\\in\\mathcal\{P\}\\subset\\mathbb\{R\}^\{k\}Conditioning vector of design, load, material and operating parameters\.𝒰\\mathcal\{U\},u0u\_\{0\}Solution space \([Equation1](https://arxiv.org/html/2610.22446#S2.E1)\); optional per\-node input field, absent in every benchmark \(cin=0c\_\{\\mathrm\{in\}\}=0\)\.ℒ𝝁,Ω\\mathcal\{L\}\_\{\\bm\{\\mu\},\\Omega\}Governing PDE operator\.𝒩\\mathcal\{N\},𝒩θ\\mathcal\{N\}\_\{\\theta\}Exact solution operator and learned surrogate,\(𝝁,Ω\)↦y\(\\bm\{\\mu\},\\Omega\)\\mapsto yandy^\\widehat\{y\}\.yy,y^\\widehat\{y\}Reference and predicted steady\-state field\.𝐲\(i\)\\mathbf\{y\}^\{\(i\)\},𝐲^\(i\)∈ℝNi\\widehat\{\\mathbf\{y\}\}^\{\(i\)\}\\in\\mathbb\{R\}^\{N\_\{i\}\}The same fields at theNiN\_\{i\}nodes of sampleii\.𝒟\\mathcal\{D\},MM,MtestM\_\{\\mathrm\{test\}\}Dataset ofMMsamples and the size of its test split\.𝒥⁡\(θ\)\\mathcal\{J\}\(\\theta\),ℰ\\mathcal\{E\}Training objective \([Equation12](https://arxiv.org/html/2610.22446#S4.E12)\) and reported test error in percent, in physical units \([Equation39](https://arxiv.org/html/2610.22446#A4.E39)\)\.∥⋅∥2\\\|\\cdot\\\|\_\{2\},∥⋅∥∞\\\|\\cdot\\\|\_\{\\infty\}Unweighted Euclidean norm over one sample’s nodes; largest absolute entry\.Mesh and latent gridNN,ωi\\omega\_\{i\},ω¯i\\bar\{\\omega\}\_\{i\}Number of mesh nodes, coordinate of nodeii, and its normalisation to\[0,1\]d\[0,1\]^\{d\}\.ii,ℓ\\ellMesh\-node indexi∈\{1,…,N\}i\\in\\\{1,\\ldots,N\\\}; coordinate\-axis indexℓ∈\{1,…,d\}\\ell\\in\\\{1,\\ldots,d\\\}\.qjq\_\{j\},NqN\_\{q\}Output query points and their number; by default the input nodes\.𝒢\\mathcal\{G\},𝐠\\mathbf\{g\}Latent grid∏ℓ\{0,…,Sℓ−1\}\\prod\_\{\\ell\}\\\{0,\\ldots,S\_\{\\ell\}\-1\\\}and a multi\-index𝐠=\(g1,…,gd\)\\mathbf\{g\}=\(g\_\{1\},\\ldots,g\_\{d\}\)in it\.SℓS\_\{\\ell\},GGControl points along axisℓ\\ell, and in total,G=\|𝒢\|=∏ℓSℓG=\|\\mathcal\{G\}\|=\\prod\_\{\\ell\}S\_\{\\ell\}\.ξ0\(ℓ\)\\xi\_\{0\}^\{\(\\ell\)\},Δℓ\\Delta\_\{\\ell\}Grid origin and spacing along axisℓ\\ell;ξ𝐠\(ℓ\)=ξ0\(ℓ\)\+gℓ​Δℓ\\xi\_\{\\mathbf\{g\}\}^\{\(\\ell\)\}=\\xi\_\{0\}^\{\(\\ell\)\}\+g\_\{\\ell\}\\Delta\_\{\\ell\}\.ti\(ℓ\)t\_\{i\}^\{\(\\ell\)\}Continuous grid coordinate of nodeii,\(ωi\(ℓ\)−ξ0\(ℓ\)\)/Δℓ\(\\omega\_\{i\}^\{\(\\ell\)\}\-\\xi\_\{0\}^\{\(\\ell\)\}\)/\\Delta\_\{\\ell\}\.Features and conditioning\([Sections3\.1](https://arxiv.org/html/2610.22446#S3.SS1)and[G](https://arxiv.org/html/2610.22446#A7)\)cc,cinc\_\{\\mathrm\{in\}\},coutc\_\{\\mathrm\{out\}\}Model width; input\-field and output channel counts\.LL,BbatchB\_\{\\mathrm\{batch\}\}Number of backbone layers; batch size\.φ\\varphi,nfn\_\{f\},cfc\_\{f\}Integrated sinusoidal encoding ofω¯\\bar\{\\omega\}atnf=16n\_\{f\}=16frequencies, of widthcf=2​d​nf=96c\_\{f\}=2dn\_\{f\}=96; withω¯\\bar\{\\omega\}appended, the lift sees 99 channels\.𝐟i\\mathbf\{f\}\_\{i\},𝐟~i\\tilde\{\\mathbf\{f\}\}\_\{i\}Encoded featuresφ⁡\(ω¯i\)\\varphi\(\\bar\{\\omega\}\_\{i\}\)and their FiLM\-modulated form \([Equation3](https://arxiv.org/html/2610.22446#S3.E3)\)\.EμE\_\{\\mu\},eμe\_\{\\mu\}Conditioning embedding network and its outputEμ​\(𝝁\)E\_\{\\mu\}\(\\bm\{\\mu\}\)\.HγH\_\{\\gamma\},HβH\_\{\\beta\},HμH\_\{\\mu\}FiLM scale and shift heads;HμH\_\{\\mu\}denotes the full map𝝁↦\(𝜸μ,𝜷μ\)\\bm\{\\mu\}\\mapsto\(\\bm\{\\gamma\}\_\{\\mu\},\\bm\{\\beta\}\_\{\\mu\}\)\.𝜸μ\\bm\{\\gamma\}\_\{\\mu\},𝜷μ∈ℝcf\\bm\{\\beta\}\_\{\\mu\}\\in\\mathbb\{R\}^\{c\_\{f\}\}FiLM scale and shift, broadcast over nodes; exactly𝟏\\mathbf\{1\}and𝟎\\mathbf\{0\}at initialisation\.⊙\\odot,𝟏N\\mathbf\{1\}\_\{N\}Channel\-wise product broadcast over nodes; length\-NNvector of ones\.PinP\_\{\\mathrm\{in\}\},PρP\_\{\\rho\},PoutP\_\{\\mathrm\{out\}\}Pointwise lift, density\-augmented grid projection \([Equation10](https://arxiv.org/html/2610.22446#S3.E10)\) and output projection\.𝐡i\\mathbf\{h\}\_\{i\},𝒳∈ℝN×c\\mathcal\{X\}\\in\\mathbb\{R\}^\{N\\times c\}Lifted, conditioned features at nodeii, stacked as the rows of𝒳\\mathcal\{X\}\.𝒳𝒢\\mathcal\{X\}\_\{\\mathcal\{G\}\},𝒴𝒢\\mathcal\{Y\}\_\{\\mathcal\{G\}\}FNO input and output on the latent grid \([Equations5](https://arxiv.org/html/2610.22446#S3.E5)and[10](https://arxiv.org/html/2610.22446#S3.E10)\)\.𝒴\\mathcal\{Y\},𝐳j\\mathbf\{z\}\_\{j\},𝐲^j\\widehat\{\\mathbf\{y\}\}\_\{j\}Decoded features at the mesh nodes; decoded latent atqjq\_\{j\}and predictionPout​𝐳jP\_\{\\mathrm\{out\}\}\\mathbf\{z\}\_\{j\}\([Equation11](https://arxiv.org/html/2610.22446#S3.E11)\)\.ℱ\\mathcal\{F\},ℱθ\\mathcal\{F\}\_\{\\theta\}Fourier transform \([Equation2](https://arxiv.org/html/2610.22446#S2.E2)\); grid\-space FNO\.θ\\thetaAll trainable parameters;κθ\(ℓ\)\\kappa\_\{\\theta\}^\{\(\\ell\)\}andℱθ\\mathcal\{F\}\_\{\\theta\}use the scatter and FNO parts\.Spline basis and learnable scatter\([Sections3\.2](https://arxiv.org/html/2610.22446#S3.SS2),[3\.3](https://arxiv.org/html/2610.22446#S3.SS3)and[B](https://arxiv.org/html/2610.22446#A2)\)BpB\_\{p\},BB,ppCentred cardinal B\-spline of degreepp\([Equation18](https://arxiv.org/html/2610.22446#A2.E18)\);B:=BpB:=B\_\{p\}at fixed degree,p=3p=3by default\.RpR\_\{p\},ttSupport radius\(p\+1\)/2\(p\+1\)/2; signed distance along one axis in grid cells\.Φ\\Phi,Φ~∈ℝN×G\\widetilde\{\\Phi\}\\in\\mathbb\{R\}^\{N\\times G\}Row\-normalised and raw spline basis matrices \([Equation4](https://arxiv.org/html/2610.22446#S3.E4)\);Φ𝐠​\(q\)\\Phi\_\{\\mathbf\{g\}\}\(q\)is basis function𝐠\\mathbf\{g\}at a query point\.φgℓ\(ℓ\)\\varphi^\{\(\\ell\)\}\_\{g\_\{\\ell\}\}One\-dimensional factor ofΦ𝐠\\Phi\_\{\\mathbf\{g\}\}\([Equation19](https://arxiv.org/html/2610.22446#A2.E19)\)\.λ\\lambdaTikhonov coefficient of the SplineFNO solve \([Equation5](https://arxiv.org/html/2610.22446#S3.E5)\)\.κθ\(ℓ\)\\kappa\_\{\\theta\}^\{\(\\ell\)\}Learned one\-dimensional scatter kernel on axisℓ\\ell\([Equation7](https://arxiv.org/html/2610.22446#S3.E7)\)\.JJ,τj\\tau\_\{j\},hh,ζj\(ℓ\)\\zeta^\{\(\\ell\)\}\_\{j\}Number, fixed centres, relative width and coefficients of the compact corrections inκθ\(ℓ\)\\kappa\_\{\\theta\}^\{\(\\ell\)\}\.AA,QQ,uqu\_\{q\}Partition\-of\-unity constraint matrixA∈ℝQ×JA\\in\\mathbb\{R\}^\{Q\\times J\}at collocation pointsuq∈\[0,1\)u\_\{q\}\\in\[0,1\)\([Equation27](https://arxiv.org/html/2610.22446#A2.E27)\)\.V0V\_\{0\},αθ\(ℓ\)\\alpha\_\{\\theta\}^\{\(\\ell\)\},rrNull\-space basisV0∈ℝJ×rV\_\{0\}\\in\\mathbb\{R\}^\{J\\times r\}ofAAand free coordinatesαθ\(ℓ\)∈ℝr\\alpha\_\{\\theta\}^\{\(\\ell\)\}\\in\\mathbb\{R\}^\{r\},ζ\(ℓ\)=V0​αθ\(ℓ\)\\zeta^\{\(\\ell\)\}=V\_\{0\}\\alpha\_\{\\theta\}^\{\(\\ell\)\}\([Equation28](https://arxiv.org/html/2610.22446#A2.E28)\)\.ssNumber of scatter scales in the multiscale encoder\.𝒪p​\(u\)\\mathcal\{O\}\_\{p\}\(u\)Active stencil offsets at fractional cell positionuu\([Equation25](https://arxiv.org/html/2610.22446#A2.E25)\);𝒪3=\{−1,0,1,2\}\\mathcal\{O\}\_\{3\}=\\\{\-1,0,1,2\\\}\.𝒮ℓ​\(i\)\\mathcal\{S\}\_\{\\ell\}\(i\),𝒮⁡\(i\)\\mathcal\{S\}\(i\)Valid grid indices of nodeiion axisℓ\\ell\([Equation29](https://arxiv.org/html/2610.22446#A2.E29)\) and their product, at most\(p\+1\)d\(p\+1\)^\{d\}points\.Wθ∈ℝN×GW\_\{\\theta\}\\in\\mathbb\{R\}^\{N\\times G\}Sparse learned transfer \([Equation8](https://arxiv.org/html/2610.22446#S3.E8)\); applied by indexed scatter\-add, never materialised\.n𝐠n\_\{\\mathbf\{g\}\},d𝐠d\_\{\\mathbf\{g\}\},a𝐠a\_\{\\mathbf\{g\}\}Weighted feature sum, signed weight sum and absolute weight sum at grid point𝐠\\mathbf\{g\}\.d~𝐠\\widetilde\{d\}\_\{\\mathbf\{g\}\},DθD\_\{\\theta\}Safeguarded denominator \([Equation32](https://arxiv.org/html/2610.22446#A2.E32)\) andDθ=diag⁡\(d~𝐠\)D\_\{\\theta\}=\\operatorname\{diag\}\(\\widetilde\{d\}\_\{\\mathbf\{g\}\}\)\.sgn\+\\operatorname\{sgn\}\_\{\+\},η\\eta,ε\\varepsilonSign function withsgn\+⁡\(0\)=\+1\\operatorname\{sgn\}\_\{\+\}\(0\)=\+1; relative and absolute denominator floors\.𝒳~𝒢\\widetilde\{\\mathcal\{X\}\}\_\{\\mathcal\{G\}\}Normalised scatterDθ−1​Wθ⊤​𝒳D\_\{\\theta\}^\{\-1\}W\_\{\\theta\}^\{\\top\}\\mathcal\{X\}\([Equation9](https://arxiv.org/html/2610.22446#S3.E9)\)\.ρ𝐠\\rho\_\{\\mathbf\{g\}\}Analytic sampling density∑iΦi,𝐠\\sum\_\{i\}\\Phi\_\{i,\\mathbf\{g\}\}\([Equation34](https://arxiv.org/html/2610.22446#A2.E34)\)\.Spatial and parameter enclosures\([Sections4\.4](https://arxiv.org/html/2610.22446#S4.SS4)and[B\.4](https://arxiv.org/html/2610.22446#A2.SS4)\)𝐮^​\(x,𝝁\)\\widehat\{\\mathbf\{u\}\}\(x,\\bm\{\\mu\}\),u^j\\widehat\{u\}\_\{j\}Predicted displacement and its componentj∈\{1,2,3\}j\\in\\\{1,2,3\\\};𝝁\\bm\{\\mu\}is dropped at fixed parameters\.Yj​\(x\)Y\_\{j\}\(x\),D⁡\(x\)D\(x\)Numerator and geometry\-only denominator of the rational decoder,u^j=Yj/D\\widehat\{u\}\_\{j\}=Y\_\{j\}/D;D=1D=1wherever spline support is untruncated\.kk,bk​\(x\)b\_\{k\}\(x\)Bernstein multi\-index in\{0,…,p\}d\\\{0,\\ldots,p\\\}^\{d\}and tensor\-product Bernstein polynomial on the current box;bk≥0b\_\{k\}\\geq 0,∑kbk=1\\sum\_\{k\}b\_\{k\}=1\.Yj,kY\_\{j,k\},DkD\_\{k\}Bernstein coefficients ofYjY\_\{j\}andDD\([Equation14](https://arxiv.org/html/2610.22446#S4.E14)\); the enclosure requiresDk\>0D\_\{k\}\>0\.wk​\(x\)w\_\{k\}\(x\)Convex weightDk​bk​\(x\)/D⁡\(x\)D\_\{k\}b\_\{k\}\(x\)/D\(x\), sou^j=∑kwk​Yj,k/Dk\\widehat\{u\}\_\{j\}=\\sum\_\{k\}w\_\{k\}Y\_\{j,k\}/D\_\{k\}\.\[Lj,Uj\]\[L\_\{j\},U\_\{j\}\]Component enclosure; at fixed𝝁\\bm\{\\mu\}, themink\\min\_\{k\}andmaxk\\max\_\{k\}ofYj,k/DkY\_\{j,k\}/D\_\{k\}\.nnde Casteljau subdivision depth, giving2n2^\{n\}sub\-boxes per cell\.m−m^\{\-\},m\+m^\{\+\}Lower and upper bounds on‖𝐮^‖2\\\|\\widehat\{\\mathbf\{u\}\}\\\|\_\{2\}derived from the component intervals\.ℬ⊂𝒫\\mathcal\{B\}\\subset\\mathcal\{P\}Axis\-aligned parameter box, partitioned into leaves with their own expansion centres\.α\\alpha,TwT\_\{w\}MAST\-U parameters: thermal expansion coefficient \(K−1\\mathrm\{K\}^\{\-1\}\) and wall temperature \(K\\mathrm\{K\}\)\.𝝁0\\bm\{\\mu\}\_\{0\},𝜹\\bm\{\\delta\}Taylor expansion centre and offset𝝁−𝝁0\\bm\{\\mu\}\-\\bm\{\\mu\}\_\{0\}\.ci​\(𝝁\)c\_\{i\}\(\\bm\{\\mu\}\);ci0c\_\{i\}^\{0\},𝐚i\\mathbf\{a\}\_\{i\},HiH\_\{i\}Scalar physical spline coefficient; its value, gradient and Hessian at𝝁0\\bm\{\\mu\}\_\{0\}\.εi\\varepsilon\_\{i\},rir\_\{i\}Taylor remainder ofcic\_\{i\}and its uniform bound over the box\.Yj,k0Y\_\{j,k\}^\{0\},πj,k​\(𝜹\)\\pi\_\{j,k\}\(\\bm\{\\delta\}\)Nominal Bernstein numerator coefficient and its quadratic in𝜹\\bm\{\\delta\}\.ηj,k\\eta\_\{j,k\},rj,kr\_\{j,k\}Remainder after the Bernstein transform and its uniform bound\.πj,k±\\pi\_\{j,k\}^\{\\pm\},Yj,k±Y\_\{j,k\}^\{\\pm\}Extrema ofπj,k\\pi\_\{j,k\}over the box, and numerator boundsYj,k0\+πj,k±±rj,kY\_\{j,k\}^\{0\}\+\\pi\_\{j,k\}^\{\\pm\}\\pm r\_\{j,k\}\.Baselines\([AppendixF](https://arxiv.org/html/2610.22446#A6)\)QQ,KK,VV,chc\_\{h\}Query, key and value matrices of an attention block; head dimension\.ς\\varsigmaFeature\-wise softmax in GNOT’s linear attention\.nsn\_\{s\}Transolver slices or UPT latent tokens,ns≪Nn\_\{s\}\\ll N\.nbn\_\{b\},nen\_\{e\}GNOT input branches \(nb=1n\_\{b\}=1here\) and mixture experts\.npn\_\{p\},nqn\_\{q\}MSPT patches and supernodes pooled per patch\.ncn\_\{c\}NUNO K\-D tree subdomains\.rr,𝒩r\\mathcal\{N\}\_\{r\}GINO graph radius in normalised coordinates and the neighbourhood it induces\.κG\\kappa\_\{\\mathrm\{G\}\},γG\\gamma\_\{\\mathrm\{G\}\},βG\\beta\_\{\\mathrm\{G\}\}GINO graph\-kernel MLP and adaptive\-instance\-norm scale and shift\.FFNumber of sinusoidal frequencies in the UPT and GINO embeddings\.nτn\_\{\\tau\},cτc\_\{\\tau\}Cross\-attention parameter tokens and their width\.Table 3:Symbols with more than one meaning\. The context, given in the last column, always fixes which is meant\.SymbolMain meaningOther meaning \(context\)iimesh nodedataset sample \(training objective,[AppendixD](https://arxiv.org/html/2610.22446#A4)\); scalar coefficientcic\_\{i\}\(enclosures\)jjquery pointqjq\_\{j\}correction index inκθ\(ℓ\)\\kappa\_\{\\theta\}^\{\(\\ell\)\}; displacement component \(enclosures\)kkdim𝝁\\dim\\bm\{\\mu\}Bernstein multi\-index \(enclosures\)qqquery pointqjq\_\{j\}collocation indexuqu\_\{q\}\([SectionB\.2](https://arxiv.org/html/2610.22446#A2.SS2)\)ppspline degreestatic pressure \(aircraft\)ccmodel widthspline coefficientci​\(𝝁\)c\_\{i\}\(\\bm\{\\mu\}\)\(enclosures\)BBcardinal B\-splineparameter boxℬ\\mathcal\{B\}\(enclosures\)Φ\\Phispline basisnormal CDF in GELU \(enclosures\)φ\\varphi,ϕ\\phipositional encoding1D spline factorφgℓ\(ℓ\)\\varphi^\{\(\\ell\)\}\_\{g\_\{\\ell\}\}; normal densityϕ\\phi\(enclosures\)DDscatter normaliserDθD\_\{\\theta\}decoder denominatorD⁡\(x\)D\(x\),DkD\_\{k\}\(enclosures\)HHFiLM headsHγH\_\{\\gamma\},HβH\_\{\\beta\}HessianHiH\_\{i\}\(enclosures\)α\\alphanull\-space coordinatesαθ\(ℓ\)\\alpha\_\{\\theta\}^\{\(\\ell\)\}thermal expansion coefficient \(MAST\-U\)LLbackbone depthlower endpointLjL\_\{j\}; linear Taylor term \(enclosures\)QQcollocation countattention query \(baselines\); quadratic Taylor term \(enclosures\)rrnull\-space dimensionGINO radius; remainder boundsrir\_\{i\},rj,kr\_\{j,k\}ss,ttscatter scales; grid distanceTaylor\-term bounds \(enclosures\)η\\eta,ε\\varepsilonscatter floorsremaindersηj,k\\eta\_\{j,k\},εi\\varepsilon\_\{i\}\(enclosures\)gggrid indexgℓg\_\{\\ell\}scalar activation \(enclosures\)σ\\sigmaactivation \([Equation2](https://arxiv.org/html/2610.22446#S2.E2)\)von Mises stress \(bracket\)ρ\\rhosampling densityfluid density \(aircraft\)κ\\kappascatter kernelκθ\(ℓ\)\\kappa\_\{\\theta\}^\{\(\\ell\)\}thermal conductivity \([AppendixD](https://arxiv.org/html/2610.22446#A4)\); GINO kernelκG\\kappa\_\{\\mathrm\{G\}\}𝒩\\mathcal\{N\}solution operatorGINO neighbourhood𝒩r\\mathcal\{N\}\_\{r\}𝒪\\mathcal\{O\}stencil offsets𝒪p\\mathcal\{O\}\_\{p\}complexityO⁡\(⋅\)O\(\\cdot\)
## Appendix BLearnable\-Spline Construction

The spline transfers used in this work are built from one\-dimensional cardinal B\-spline kernels and lifted todddimensions by tensor products\. SplineFNO uses the fixed analytic splineBpB\_\{p\}of degreepp, while L\-SplineFNO augments this kernel with a constrained learnable correction\. For clarity, the main text presents the default cubic casep=3p=3, while the construction below is stated for a general spline degreepp\. The higher\-degree variants considered in the spline\-degree ablation follow the same construction with the corresponding support and smoothness\.

This appendix proceeds in three parts\.[SectionB\.1](https://arxiv.org/html/2610.22446#A2.SS1)establishes the tensor\-product properties of the analytic degree\-ppbasis\.[SectionB\.2](https://arxiv.org/html/2610.22446#A2.SS2)introduces the constrained learned kernel and shows how compact support, partition of unity, and spline smoothness are retained\. Non\-negativity is not imposed on the learned correction, and[SectionB\.3](https://arxiv.org/html/2610.22446#A2.SS3)introduces the safeguarded normalised scatter used when signed weights cancel\.

### B\.1From 1D todd\-D: Tensor\-Product Construction

LetBp:ℝ→ℝB\_\{p\}:\\mathbb\{R\}\\to\\mathbb\{R\}denote the centred cardinal B\-spline of degreepp\. We writeB:=BpB:=B\_\{p\}when the degree is fixed\. The one\-dimensional basis satisfies

1. \(P1\)compact support,supp​B⊆\[−Rp,Rp\]\\mathrm\{supp\}\\,B\\subseteq\[\-R\_\{p\},R\_\{p\}\]withRp=\(p\+1\)/2R\_\{p\}=\(p\+1\)/2;
2. \(P2\)partition of unity,∑n∈ℤB⁡\(t−n\)=1\\sum\_\{n\\in\\mathbb\{Z\}\}B\(t\-n\)=1for alltt;
3. \(P3\)smoothness,B∈Cp−1​\(ℝ\)B\\in C^\{p\-1\}\(\\mathbb\{R\}\);
4. \(P4\)non\-negativity,B⁡\(t\)≥0B\(t\)\\geq 0\.

For the default cubic casep=3p=3,Rp=2R\_\{p\}=2andB∈C2B\\in C^\{2\}, with

B3​\(t\)=\{16​\(4−6​t2\+3​\|t\|3\),0≤\|t\|<1,16​\(2−\|t\|\)3,1≤\|t\|<2,0,\|t\|≥2\.B\_\{3\}\(t\)=\\begin\{cases\}\\tfrac\{1\}\{6\}\\left\(4\-6t^\{2\}\+3\|t\|^\{3\}\\right\),&0\\leq\|t\|<1,\\\\\[2\.84526pt\] \\tfrac\{1\}\{6\}\(2\-\|t\|\)^\{3\},&1\\leq\|t\|<2,\\\\\[2\.84526pt\] 0,&\|t\|\\geq 2\.\\end\{cases\}\(18\)
Writing\{ξgℓ\(ℓ\)\}gℓ=0Sℓ−1\\\{\\xi^\{\(\\ell\)\}\_\{g\_\{\\ell\}\}\\\}\_\{g\_\{\\ell\}=0\}^\{S\_\{\\ell\}\-1\}for the control points along axisℓ\\ell, define

φgℓ\(ℓ\)​\(x\(ℓ\)\)=B⁡\(x\(ℓ\)−ξgℓ\(ℓ\)Δℓ\),Φ𝐠​\(𝐱\)=∏ℓ=1dφgℓ\(ℓ\)​\(x\(ℓ\)\)\.\\varphi^\{\(\\ell\)\}\_\{g\_\{\\ell\}\}\(x^\{\(\\ell\)\}\)=B\\\!\\left\(\\frac\{x^\{\(\\ell\)\}\-\\xi^\{\(\\ell\)\}\_\{g\_\{\\ell\}\}\}\{\\Delta\_\{\\ell\}\}\\right\),\\qquad\\Phi\_\{\\mathbf\{g\}\}\(\\mathbf\{x\}\)=\\prod\_\{\\ell=1\}^\{d\}\\varphi^\{\(\\ell\)\}\_\{g\_\{\\ell\}\}\(x^\{\(\\ell\)\}\)\.\(19\)RowiiofΦ\\PhistoresΦ𝐠​\(ωi\)\\Phi\_\{\\mathbf\{g\}\}\(\\omega\_\{i\}\)over the active multi\-indices\.

##### Compact support and sparsity\.

A product vanishes whenever any factor does, soΦ𝐠​\(𝐱\)=0\\Phi\_\{\\mathbf\{g\}\}\(\\mathbf\{x\}\)=0unless

\|x\(ℓ\)−ξgℓ\(ℓ\)Δℓ\|<Rp\\left\|\\frac\{x^\{\(\\ell\)\}\-\\xi^\{\(\\ell\)\}\_\{g\_\{\\ell\}\}\}\{\\Delta\_\{\\ell\}\}\\right\|<R\_\{p\}along every coordinate axis\. Since a degree\-ppcardinal spline has at mostp\+1p\+1active shifts along each axis,

\#⁡\{𝐠:Φ𝐠​\(𝐱\)≠0\}≤\(p\+1\)d\.\\\#\\\{\\mathbf\{g\}:\\Phi\_\{\\mathbf\{g\}\}\(\\mathbf\{x\}\)\\neq 0\\\}\\leq\(p\+1\)^\{d\}\.\(20\)The default cubic case therefore gives a4d4^\{d\}stencil\.

##### Partition of unity\.

Because each axis sum is independent of the remaining indices,

∑𝐠Φ𝐠​\(𝐱\)=∏ℓ=1d\(∑gℓφgℓ\(ℓ\)​\(x\(ℓ\)\)\)=1\.\\sum\_\{\\mathbf\{g\}\}\\Phi\_\{\\mathbf\{g\}\}\(\\mathbf\{x\}\)=\\prod\_\{\\ell=1\}^\{d\}\\left\(\\sum\_\{g\_\{\\ell\}\}\\varphi^\{\(\\ell\)\}\_\{g\_\{\\ell\}\}\(x^\{\(\\ell\)\}\)\\right\)=1\.\(21\)The reconstruction therefore reproduces constant control\-grid fields exactly in the interior\. Near a domain boundary, part of the infinite\-grid stencil falls outside𝒢\\mathcal\{G\}\. Renormalising the surviving basis values restores partition of unity over the valid stencil\.

##### Smoothness\.

Since the tensor\-product factors depend on disjoint coordinates,

∂𝒏Φ𝐠​\(𝐱\)=∏ℓ=1ddnℓ​φgℓ\(ℓ\)d​\(x\(ℓ\)\)nℓ​\(x\(ℓ\)\)\.\\partial^\{\\bm\{n\}\}\\Phi\_\{\\mathbf\{g\}\}\(\\mathbf\{x\}\)=\\prod\_\{\\ell=1\}^\{d\}\\frac\{d^\{n\_\{\\ell\}\}\\varphi^\{\(\\ell\)\}\_\{g\_\{\\ell\}\}\}\{d\(x^\{\(\\ell\)\}\)^\{n\_\{\\ell\}\}\}\(x^\{\(\\ell\)\}\)\.\(22\)Each factor is continuous whenevernℓ≤p−1n\_\{\\ell\}\\leq p\-1, so the tensor\-product basis inherits theCp−1C^\{p\-1\}smoothness of the one\-dimensional spline\.

##### Non\-negativity\.

A product of non\-negative factors is non\-negative\. Together with[Equation21](https://arxiv.org/html/2610.22446#A2.E21), the active basis values form a convex combination, so analytic spline reconstruction cannot overshoot the range of the active control\-point values\.

##### Implication for the transformation\.

Partition of unity makes a constant field exactly representable,Φ⁡\(c​𝟏G\)=c​𝟏N\\Phi\(c\\mathbf\{1\}\_\{G\}\)=c\\mathbf\{1\}\_\{N\}\. ProvidedΦ\\Phihas full column rank, the regularised transformation in[Equation5](https://arxiv.org/html/2610.22446#S3.E5)recovers this constant in the limitλ→0\+\\lambda\\to 0^\{\+\}\. The compact support in[Equation20](https://arxiv.org/html/2610.22446#A2.E20)also bounds the number of basis evaluations per mesh point by\(p\+1\)d\(p\+1\)^\{d\}, independently ofGG\.

### B\.2From Analytic to Learned Kernels

L\-SplineFNO augments the fixed degree\-ppspline with compact spline corrections\. Along coordinate axisℓ\\ell,

κθ\(ℓ\)​\(t\)=B⁡\(t\)\+∑j=1Jζj\(ℓ\)​B​\(t−τjh\),ζ\(ℓ\)=V0​αθ\(ℓ\),\\kappa\_\{\\theta\}^\{\(\\ell\)\}\(t\)=B\(t\)\+\\sum\_\{j=1\}^\{J\}\\zeta\_\{j\}^\{\(\\ell\)\}B\\\!\\left\(\\frac\{t\-\\tau\_\{j\}\}\{h\}\\right\),\\qquad\\zeta^\{\(\\ell\)\}=V\_\{0\}\\alpha\_\{\\theta\}^\{\(\\ell\)\},\(23\)where the correction centresτj\\tau\_\{j\}and relative widthhhare fixed and onlyαθ\(ℓ\)\\alpha\_\{\\theta\}^\{\(\\ell\)\}is trained\. The cubic construction shown in the main text corresponds top=3p=3\.

##### Compact support and smoothness \(P1, P3\)\.

LetRp=\(p\+1\)/2R\_\{p\}=\(p\+1\)/2\. We choose the correction centres and width to satisfy

\|τj\|≤Rp​\(1−h\),0<h<1\.\|\\tau\_\{j\}\|\\leq R\_\{p\}\(1\-h\),\\qquad 0<h<1\.\(24\)The support of each correction is then contained within\[−Rp,Rp\]\[\-R\_\{p\},R\_\{p\}\], soκθ\(ℓ\)\\kappa\_\{\\theta\}^\{\(\\ell\)\}retains the same bounded stencil as the analytic spline\. Since both the base spline and every correction are degree\-ppsplines, the learned kernel remainsCp−1C^\{p\-1\}\. These two properties hold independently of the learned coefficients\.

##### Partition of unity in the interior \(P2\)\.

For a fractional cell positionu∈\[0,1\)u\\in\[0,1\), let

𝒪p​\(u\)=\{o∈ℤ:B⁡\(u−o\)≠0\}\\mathcal\{O\}\_\{p\}\(u\)=\\left\\\{o\\in\\mathbb\{Z\}:B\(u\-o\)\\neq 0\\right\\\}\(25\)denote the active analytic stencil\. Its cardinality is at mostp\+1p\+1, and

∑o∈𝒪p​\(u\)B⁡\(u−o\)=1\.\\sum\_\{o\\in\\mathcal\{O\}\_\{p\}\(u\)\}B\(u\-o\)=1\.Forp=3p=3, this reduces to the familiar four\-point stencil𝒪3=\{−1,0,1,2\}\\mathcal\{O\}\_\{3\}=\\\{\-1,0,1,2\\\}foru∈\[0,1\)u\\in\[0,1\)\.

The learned kernel retains partition of unity when its correction has zero stencil sum,

∑j=1Jζj\(ℓ\)​∑o∈𝒪p​\(u\)B⁡\(u−o−τjh\)=0\.\\sum\_\{j=1\}^\{J\}\\zeta\_\{j\}^\{\(\\ell\)\}\\sum\_\{o\\in\\mathcal\{O\}\_\{p\}\(u\)\}B\\\!\\left\(\\frac\{u\-o\-\\tau\_\{j\}\}\{h\}\\right\)=0\.\(26\)Given collocation points\{uq\}q=1Q⊂\[0,1\)\\\{u\_\{q\}\\\}\_\{q=1\}^\{Q\}\\subset\[0,1\), define

Aq​j=∑o∈𝒪p​\(uq\)B⁡\(uq−o−τjh\),A∈ℝQ×J\.A\_\{qj\}=\\sum\_\{o\\in\\mathcal\{O\}\_\{p\}\(u\_\{q\}\)\}B\\\!\\left\(\\frac\{u\_\{q\}\-o\-\\tau\_\{j\}\}\{h\}\\right\),\\qquad A\\in\\mathbb\{R\}^\{Q\\times J\}\.\(27\)LetV0∈ℝJ×rV\_\{0\}\\in\\mathbb\{R\}^\{J\\times r\}contain the right singular vectors spanning the numerical null space ofAA\. We parameterise

ζ\(ℓ\)=V0​αθ\(ℓ\),αθ\(ℓ\)∈ℝr,\\zeta^\{\(\\ell\)\}=V\_\{0\}\\alpha\_\{\\theta\}^\{\(\\ell\)\},\\qquad\\alpha\_\{\\theta\}^\{\(\\ell\)\}\\in\\mathbb\{R\}^\{r\},\(28\)which givesA​ζ\(ℓ\)≃0A\\zeta^\{\(\\ell\)\}\\simeq 0to numerical precision for every value of the learned parameters\. The matrixV0V\_\{0\}is computed once and remains fixed throughout training\. We additionally verify[Equation26](https://arxiv.org/html/2610.22446#A2.E26)on a dense set of fractional coordinates\.

##### Partition of unity at the boundary \(P2\)\.

For mesh pointωi\\omega\_\{i\}, define the valid one\-dimensional stencil

𝒮ℓ​\(i\)=\{gℓ∈\{0,…,Sℓ−1\}:\|ti\(ℓ\)−gℓ\|<Rp\}\.\\mathcal\{S\}\_\{\\ell\}\(i\)=\\left\\\{g\_\{\\ell\}\\in\\\{0,\\ldots,S\_\{\\ell\}\-1\\\}:\\left\|t\_\{i\}^\{\(\\ell\)\}\-g\_\{\\ell\}\\right\|<R\_\{p\}\\right\\\}\.\(29\)The tensor\-product transfer weights are

\(Wθ\)i,𝐠=∏ℓ=1dκθ\(ℓ\)​\(ti\(ℓ\)−gℓ\)∑q∈𝒮ℓ​\(i\)κθ\(ℓ\)​\(ti\(ℓ\)−q\),𝐠∈𝒮⁡\(i\),\(W\_\{\\theta\}\)\_\{i,\\mathbf\{g\}\}=\\prod\_\{\\ell=1\}^\{d\}\\frac\{\\kappa\_\{\\theta\}^\{\(\\ell\)\}\\\!\\left\(t\_\{i\}^\{\(\\ell\)\}\-g\_\{\\ell\}\\right\)\}\{\\displaystyle\\sum\_\{q\\in\\mathcal\{S\}\_\{\\ell\}\(i\)\}\\kappa\_\{\\theta\}^\{\(\\ell\)\}\\\!\\left\(t\_\{i\}^\{\(\\ell\)\}\-q\\right\)\},\\qquad\\mathbf\{g\}\\in\\mathcal\{S\}\(i\),\(30\)and zero otherwise, where𝒮⁡\(i\)=∏ℓ=1d𝒮ℓ​\(i\)\\mathcal\{S\}\(i\)=\\prod\_\{\\ell=1\}^\{d\}\\mathcal\{S\}\_\{\\ell\}\(i\)\. The explicit normalisation restores partition of unity over the surviving entries whenever the denominator is nonzero\. Each mesh point therefore interacts with at most\(p\+1\)d\(p\+1\)^\{d\}latent\-grid points\.

##### Non\-negativity is surrendered \(P4\)\.

The null\-space parameterisation does not constrain the sign of the learned correction\. Consequently,κθ\(ℓ\)\\kappa\_\{\\theta\}^\{\(\\ell\)\}may contain negative lobes\. The learned transfer retains compact support,Cp−1C^\{p\-1\}smoothness, and partition of unity to numerical precision, but no longer has the convex\-combination property of the analytic spline\. Signed contributions may therefore cancel when they are accumulated at a latent\-grid point\.

### B\.3Stable Normalised Scatter and Density Feature

The main text defines

n𝐠=∑i=1N\(Wθ\)i,𝐠​𝒳i,d𝐠=∑i=1N\(Wθ\)i,𝐠\.n\_\{\\mathbf\{g\}\}=\\sum\_\{i=1\}^\{N\}\(W\_\{\\theta\}\)\_\{i,\\mathbf\{g\}\}\\mathcal\{X\}\_\{i\},\\qquad d\_\{\\mathbf\{g\}\}=\\sum\_\{i=1\}^\{N\}\(W\_\{\\theta\}\)\_\{i,\\mathbf\{g\}\}\.Because the learned weights may be signed,d𝐠d\_\{\\mathbf\{g\}\}can approach zero through cancellation even when the corresponding grid point is well covered\. We therefore also accumulate the total unsigned weight

a𝐠=∑i=1N\|\(Wθ\)i,𝐠\|a\_\{\\mathbf\{g\}\}=\\sum\_\{i=1\}^\{N\}\\left\|\(W\_\{\\theta\}\)\_\{i,\\mathbf\{g\}\}\\right\|\(31\)and use the sign preserving safeguarded denominator

d~𝐠=sgn\+⁡\(d𝐠\)​max​\{\|d𝐠\|,η​a𝐠,ε\},\\widetilde\{d\}\_\{\\mathbf\{g\}\}=\\operatorname\{sgn\}\_\{\+\}\(d\_\{\\mathbf\{g\}\}\)\\max\\left\\\{\|d\_\{\\mathbf\{g\}\}\|,\\eta a\_\{\\mathbf\{g\}\},\\varepsilon\\right\\\},\(32\)wheresgn\+⁡\(z\)=−1\\operatorname\{sgn\}\_\{\+\}\(z\)=\-1ifz<0z<0and\+1\+1otherwise,η∈\(0,1\]\\eta\\in\(0,1\], andε\>0\\varepsilon\>0\. The normalised feature is then

𝒳~𝒢​\[𝐠\]=n𝐠d~𝐠\.\\widetilde\{\\mathcal\{X\}\}\_\{\\mathcal\{G\}\}\[\\mathbf\{g\}\]=\\frac\{n\_\{\\mathbf\{g\}\}\}\{\\widetilde\{d\}\_\{\\mathbf\{g\}\}\}\.Equivalently, definingDθ=diag⁡\(d~𝐠\)D\_\{\\theta\}=\\operatorname\{diag\}\(\\widetilde\{d\}\_\{\\mathbf\{g\}\}\)gives the compact form in[Equation9](https://arxiv.org/html/2610.22446#S3.E9)\.

The relative floor bounds the amplification caused by cancellation,

‖𝒳~𝒢\[𝐠\]‖∞≤1ηmaxi:\(Wθ\)i,𝐠≠0∥𝒳i∥∞\\left\\\|\\widetilde\{\\mathcal\{X\}\}\_\{\\mathcal\{G\}\}\[\\mathbf\{g\}\]\\right\\\|\_\{\\infty\}\\leq\\frac\{1\}\{\\eta\}\\max\_\{i:\(W\_\{\\theta\}\)\_\{i,\\mathbf\{g\}\}\\neq 0\}\\\|\\mathcal\{X\}\_\{i\}\\\|\_\{\\infty\}\(33\)whenevera𝐠\>0a\_\{\\mathbf\{g\}\}\>0\. The absolute floorε\\varepsilonhandles uncovered or numerically degenerate grid points\. For non\-negative weights,d𝐠=a𝐠d\_\{\\mathbf\{g\}\}=a\_\{\\mathbf\{g\}\}, so the relative safeguard is inactive forη≤1\\eta\\leq 1\.

##### Sampling\-density feature\.

Normalised scatter reduces variations caused solely by local point density, but it also removes information about mesh coverage\. We retain this information through the fixed analytic spline basis,

ρ𝐠=∑i=1NΦi,𝐠,𝒳𝒢=Pρ​\[𝒳~𝒢;log⁡\(1\+ρ\)\]\.\\rho\_\{\\mathbf\{g\}\}=\\sum\_\{i=1\}^\{N\}\\Phi\_\{i,\\mathbf\{g\}\},\\qquad\\mathcal\{X\}\_\{\\mathcal\{G\}\}=P\_\{\\rho\}\\left\[\\widetilde\{\\mathcal\{X\}\}\_\{\\mathcal\{G\}\};\\log\(1\+\\rho\)\\right\]\.\(34\)BecauseΦ\\Phiis non\-negative,ρ\\rhoprovides a non\-negative analytic measure of local mesh coverage that is unaffected by cancellation in the learned scatter\. The pointwise projectionPρP\_\{\\rho\}is initialised to preserve the scattered feature channels and assign zero weight to the density feature, so addingρ\\rhodoes not change the initial model output\.

##### Implementation\.

All transfers remain local\. For degreepp, each mesh point contributes to at most\(p\+1\)d\(p\+1\)^\{d\}latent\-grid points, and the products in[Equation9](https://arxiv.org/html/2610.22446#S3.E9)are implemented as indexed scatter\-add operations over these active mesh\-to\-grid edges, as detailed in[AppendixC](https://arxiv.org/html/2610.22446#A3)\.

### B\.4Continuous spatial bounds and parameter uncertainty

##### Setup\.

All bounds are computed post\-training, using the checking across the 29\.2M nodal points\. Geometry, model weights, preprocessing, and material assignments remain fixed\. We consider three local material regions around held\-out sample, one each in steel, copper, and graphite\. No reference displacement values enter the enclosure construction\.

##### Continuous spatial bounds\.

For fixed parameters, each physical displacement component on a decoder cell is a rational tensor\-product cubic spline,

u^j​\(x\)=Yj​\(x\)D⁡\(x\)\.\\widehat\{u\}\_\{j\}\(x\)=\\frac\{Y\_\{j\}\(x\)\}\{D\(x\)\}\.The numerator contains the physical output coefficients, including the affine readout, while the denominator accounts for the normalisation of spline support truncated at the decoder boundary\. In the interior, this denominator is one\.

We cover the requested material region with restricted decoder\-cell boxes\. On each box, a linear basis transformation expresses the numerator and denominator in tensor\-product Bernstein form\([Piegl & Tiller, 1996](https://arxiv.org/html/2610.22446#bib.bib49)\):

Yj​\(x\)=∑kYj,k​bk​\(x\),D⁡\(x\)=∑kDk​bk​\(x\)\.Y\_\{j\}\(x\)=\\sum\_\{k\}Y\_\{j,k\}b\_\{k\}\(x\),\\qquad D\(x\)=\\sum\_\{k\}D\_\{k\}b\_\{k\}\(x\)\.Herekkis a multi\-index over the spatial Bernstein coefficients\. Since the basis functions are nonnegative and sum to one, positive denominator coefficients give

u^j​\(x\)=∑kwk​\(x\)​Yj,kDk,wk​\(x\)=Dk​bk​\(x\)∑ℓDℓ​Bℓ​\(x\),wk​\(x\)≥0,∑kwk​\(x\)=1\.\\widehat\{u\}\_\{j\}\(x\)=\\sum\_\{k\}w\_\{k\}\(x\)\\frac\{Y\_\{j,k\}\}\{D\_\{k\}\},\\qquad w\_\{k\}\(x\)=\\frac\{D\_\{k\}b\_\{k\}\(x\)\}\{\\sum\_\{\\ell\}D\_\{\\ell\}B\_\{\\ell\}\(x\)\},\\qquad w\_\{k\}\(x\)\\geq 0,\\quad\\sum\_\{k\}w\_\{k\}\(x\)=1\.Thus the prediction is a convex combination of the coefficient ratios, giving the continuous enclosure

Lj=mink⁡Yj,kDk,Uj=maxk⁡Yj,kDk\.L\_\{j\}=\\min\_\{k\}\\frac\{Y\_\{j,k\}\}\{D\_\{k\}\},\\qquad U\_\{j\}=\\max\_\{k\}\\frac\{Y\_\{j,k\}\}\{D\_\{k\}\}\.These are the spatial bounds in[eq\.14](https://arxiv.org/html/2610.22446#S4.E14)\. Their validity follows from the basis representation, rather than evaluations at sampled points\.

We tighten the enclosure using de Casteljau subdivision, which restricts the numerator and denominator together to smaller spatial boxes\. Each step splits a box at the midpoint of its longest side, so depthnncovers every original decoder cell with2n2^\{n\}sub\-boxes\. The split is an exact change of basis: each sub\-box carries Bernstein coefficients for the same rational spline, and these approach the function values as the box shrinks\. Taking the smallest lower endpoint and largest upper endpoint across all covering boxes gives a regional enclosure\. Each box uses the decoder head associated with its material\.[Table4](https://arxiv.org/html/2610.22446#A2.T4)reports the resulting trade\-off: the median width halves between depths zero and three, while the cost roughly doubles with every further level\.

DepthnnPartitions2n2^\{n\}Median width \(mm\)Max width \(mm\)Time \(s\)018\.70148\.125\.71127\.65142\.3210\.64246\.62130\.7120\.34384\.2493\.2639\.614163\.8979\.8881\.996642\.7157\.98322\.43Table 4:MAST\-U spatial subdivision ablation at nominal parameters\. Depth zero uses no subdivision within each original spline cell; each subsequent depth doubles the number of sub\-boxes per cell\. Widths are displacement\-magnitude enclosure widths over the same4,077,3124\{,\}077\{,\}312displayed triangles, with equal weight per triangle\. Times are medians of three warm, synchronised bound calculations on one NVIDIA A100 GPU, excluding model loading, source encoding and reporting\. Depth three is the current subdivision setting\.
##### Taylor propagation through the network\.

For the prescribed parameter box, we propagate a second\-order polynomial plus a bounded interval remainder through the unchanged network\([Makino & Berz, 1999](https://arxiv.org/html/2610.22446#bib.bib38)\)\. Writing𝜹=𝝁−𝝁0\\bm\{\\delta\}=\\bm\{\\mu\}\-\\bm\{\\mu\}\_\{0\}, an intermediate scalar quantity is represented as

z⁡\(𝝁\)=z0\+𝐚⊤​𝜹\+12​𝜹⊤​H​𝜹\+e,\|e\|≤R\.z\(\\bm\{\\mu\}\)=z\_\{0\}\+\\mathbf\{a\}^\{\\top\}\\bm\{\\delta\}\+\\tfrac\{1\}\{2\}\\bm\{\\delta\}^\{\\top\}H\\bm\{\\delta\}\+e,\\qquad\|e\|\\leq R\.The parameter endpoints are converted using the model’s fixed normalisation before propagation\. With two parameters, the retained polynomial consists of six coefficient fields: a constant, two linear terms, and three quadratic terms\.

FiLM conditioning introduces the parameter dependence\. Affine lifting, fixed\-geometry scatter, Fourier processing, and physical readout propagate the signed polynomial coefficient fields\. Remainders are enclosed using absolute\-weight or norm bounds; Fourier remainder propagation uses Parseval\-based norm envelopes\. The same parameter symbols are shared across source points, channels, and spline coefficients, rather than treating their perturbations as independent intervals\.

Nonlinear activations require an additional remainder\. For a scalar activation inputz0\+L\+Q\+ez\_\{0\}\+L\+Q\+e, whereLLandQQare the retained linear and quadratic terms, let\|L\|≤s\|L\|\\leq s,\|Q\|≤t\|Q\|\\leq t, and\|e\|≤R\|e\|\\leq R\. The retained second\-order activation polynomial is

g⁡\(z0\)\+g′​\(z0\)​\(L\+Q\)\+12​g′′​\(z0\)​L2\.g\(z\_\{0\}\)\+g^\{\\prime\}\(z\_\{0\}\)\(L\+Q\)\+\\tfrac\{1\}\{2\}g^\{\\prime\\prime\}\(z\_\{0\}\)L^\{2\}\.A valid bound on the discarded terms is

Rnew≤Lg​R\+\|g′′​\(z0\)\|​\(s​t\+12​t2\)\+16​M3​\(z0,s\+t\)​\(s\+t\)3,R\_\{\\mathrm\{new\}\}\\leq L\_\{g\}R\+\|g^\{\\prime\\prime\}\(z\_\{0\}\)\|\\left\(st\+\\tfrac\{1\}\{2\}t^\{2\}\\right\)\+\\tfrac\{1\}\{6\}M\_\{3\}\(z\_\{0\},s\+t\)\(s\+t\)^\{3\},whereLgL\_\{g\}bounds\|g′\|\|g^\{\\prime\}\|andM3​\(z0,v\)M\_\{3\}\(z\_\{0\},v\)bounds\|g′′′\|\|g^\{\\prime\\prime\\prime\}\|on\[z0−v,z0\+v\]\[z\_\{0\}\-v,z\_\{0\}\+v\]\. This accounts both for the incoming remainder and for higher\-order products discarded during composition\. For exact GELU,g⁡\(z\)=z​Φ​\(z\)g\(z\)=z\\Phi\(z\),

g′′′​\(z\)=\(z3−4​z\)​ϕ​\(z\),g^\{\\prime\\prime\\prime\}\(z\)=\(z^\{3\}\-4z\)\\phi\(z\),so the third\-derivative envelope is obtained analytically from interval endpoints and contained stationary points\. The implementation also evaluates a direct composition remainder bound and takes the smaller of the two valid envelopes\.

##### Combining parameter and spatial bounds\.

The joint bound reuses the spatial construction above unchanged; only the coefficients it is applied to differ\. After physical readout, every spline coefficient has a quadratic parameter polynomial and a bounded remainder\. We transform the signed polynomial coefficient fields into spatial Bernstein form before computing their parameter extrema\. This preserves cancellations that would be lost by first replacing each spline coefficient with an independent interval\.

On each spatial patch, write

Yj,k​\(𝝁\)=Yj,k0\+πj,k​\(𝜹\)\+ηj,k,\|ηj,k\|≤rj,k,Y\_\{j,k\}\(\\bm\{\\mu\}\)=Y\_\{j,k\}^\{0\}\+\\pi\_\{j,k\}\(\\bm\{\\delta\}\)\+\\eta\_\{j,k\},\\qquad\|\\eta\_\{j,k\}\|\\leq r\_\{j,k\},whereπj,k\\pi\_\{j,k\}contains the linear and quadratic parameter terms\. We obtain its extremaπj,k±\\pi\_\{j,k\}^\{\\pm\}over the parameter\-offset box by checking corners, admissible stationary points on its edges, and any admissible interior stationary point\. For a singular quadratic Hessian, boundary candidates suffice; degenerate parameter boxes are handled in their remaining dimensions\.

The lower and upper numerator\-coefficient bounds are

Yj,k−=Yj,k0\+πj,k−−rj,k,Yj,k\+=Yj,k0\+πj,k\+\+rj,k\.Y\_\{j,k\}^\{\-\}=Y\_\{j,k\}^\{0\}\+\\pi\_\{j,k\}^\{\-\}\-r\_\{j,k\},\\qquad Y\_\{j,k\}^\{\+\}=Y\_\{j,k\}^\{0\}\+\\pi\_\{j,k\}^\{\+\}\+r\_\{j,k\}\.Since the denominator is parameter\-independent and its Bernstein coefficients are positive, the spatial convex\-hull argument gives

mink⁡Yj,k−Dk≤u^j​\(x,𝝁\)≤maxk⁡Yj,k\+Dk\\min\_\{k\}\\frac\{Y\_\{j,k\}^\{\-\}\}\{D\_\{k\}\}\\leq\\widehat\{u\}\_\{j\}\(x,\\bm\{\\mu\}\)\\leq\\max\_\{k\}\\frac\{Y\_\{j,k\}^\{\+\}\}\{D\_\{k\}\}for every position and parameter setting in the covered product domain\.

##### Refinement and magnitude ranges\.

Spatial subdivision tightens the Bernstein enclosure\. Parameter subdivision reduces the local nonlinear remainders: we partition the complete original parameter box, recenter the Taylor model on every leaf, and take the union of all leaf enclosures\. Every leaf must be included to retain coverage of the original uncertainty range\.

For component intervals\[Lj,Uj\]\[L\_\{j\},U\_\{j\}\], a displacement\-magnitude enclosure follows from the nearest and furthest points of the component rectangle from the origin:

m−=\(∑jdist⁡\(0,\[Lj,Uj\]\)2\)1/2,m\+=\(∑jmax⁡\{Lj2,Uj2\}\)1/2\.m^\{\-\}=\\left\(\\sum\_\{j\}\\operatorname\{dist\}\(0,\[L\_\{j\},U\_\{j\}\]\)^\{2\}\\right\)^\{1/2\},\\qquad m^\{\+\}=\\left\(\\sum\_\{j\}\\max\\\{L\_\{j\}^\{2\},U\_\{j\}^\{2\}\\\}\\right\)^\{1/2\}\.Magnitude enclosures are computed before combining parameter leaves, avoiding the unnecessary combination of component extrema from different leaves\.

At spatial subdivision depth three, with physical parameter half\-widths10−7​K−110^\{\-7\}~\\mathrm\{K\}^\{\-1\}inα\\alphaand55K inTwT\_\{w\}, the reported second\-order results are

Material regionNominal spatial range \(mm\)Joint range,​8×8​partition \(mm\)Steel\[41\.47,42\.37\]\[41\.06,42\.80\]Copper\[39\.86,41\.79\]\[39\.47,42\.22\]Graphite\[35\.62,38\.80\]\[35\.26,39\.21\]\\begin\{array\}\[\]\{lcc\}\\hline\\cr\\text\{Material region\}&\\text\{Nominal spatial range \(mm\)\}&\\text\{Joint range, \}8\\times 8\\text\{ partition \(mm\)\}\\\\ \\hline\\cr\\text\{Steel\}&\[41\.47,42\.37\]&\[41\.06,42\.80\]\\\\ \\text\{Copper\}&\[39\.86,41\.79\]&\[39\.47,42\.22\]\\\\ \\text\{Graphite\}&\[35\.62,38\.80\]&\[35\.26,39\.21\]\\\\ \\hline\\cr\\end\{array\}All parameter leaves cover the same original uncertainty box; these results do not arise from shrinking that box\.

##### Interval arithmetic and rounding certificates\.

We compare arithmetic implementations on the held\-out case as shown in[Figures5](https://arxiv.org/html/2610.22446#S4.F5)and[6](https://arxiv.org/html/2610.22446#S4.F6), using the same frozen checkpoint and the1×1\\timesparameter box with physical half\-widths10−7​K−110^\{\-7\}~\\mathrm\{K\}^\{\-1\}inα\\alphaand55K inTwT\_\{w\}\. Every evaluation encodes all29,183,75429\{,\}183\{,\}754native source nodes and computes displacement\-magnitude intervals at the same4,077,3124\{,\}077\{,\}312display face centres\. These are parameter bounds at fixed positions; they differ from the continuous spatial\-subdivision experiment in[Table4](https://arxiv.org/html/2610.22446#A2.T4)and the partitioned local\-region ranges above\. No parameter subdivision is used here\.

![Refer to caption](https://arxiv.org/html/2610.22446v1/mastu_parameter_box_arithmetic_surfaces.png)Figure 7:Displacement\-magnitude enclosure widths at fixed face centres foreval\_02over the1×1\\timesparameter box, using ordinary FP32, ordinary FP64 and validated binary64\. The FP32/FP64 calculations use verified norms and quadratic extrema with ordinary propagation; validated binary64 encloses the real stored\-weight surrogate\. All panels use the manuscript camera, the same geometry and a shared linear colour scale without clipping\. The distributions correspond to[Table5](https://arxiv.org/html/2610.22446#A2.T5); panel annotations give the median and maximum widths over all4,077,3124\{,\}077\{,\}312face centres\.The original formulation implements the real\-arithmetic Taylor and spline inequalities using ordinary FP32 scatter, FFT, nonlinear and SVD calculations\. The updated ordinary FP32/FP64 formulation retains the same second\-order construction but replaces numerical norm estimates by verified upper bounds: SVD proposals are checked using outward\-enclosed Hermitian LDL pivots, with conservative norm fallbacks\. Quadratic parameter extrema are evaluated with outward binary64 intervals, including stationary values and conservative fallbacks when the Hessian determinant cannot be resolved safely\. These checks validate those individual calculations; the remaining ordinary propagation is not an end\-to\-end rounding certificate\.

Validated binary64 instead propagates outward\-enclosed arithmetic through parameter normalisation, coordinate/positional preprocessing, FiLM, encoder accumulation, Fourier/Taylor propagation, nonlinear remainders and the surface decoder\. Arithmetic uncertainty is included alongside the Taylor remainder\. This encloses the real surrogate defined by the exact stored weight values under the checked arithmetic assumptions\. It does not certify every output of the existing vendor FP32/TF32 inference program\.

The validated implementation specializes interval products by exact stored weights and fuses spline gathers and transcendental series recurrences into CUDA kernels\. These kernels retain explicit directed binary64 rounding, the original accumulation order and analytic remainder bounds\. For the measured box, the saved magnitude endpoints are identical to those from the earlier validated implementation at every face centre\.

[Table5](https://arxiv.org/html/2610.22446#A2.T5)reports medians of three CUDA\-synchronised repetitions in one NVIDIA A100\-SXM4\-80GB allocation\. The original and updated implementations ran in separate allocations of the same GPU model, with matched geometry, parameter boxes, chunk sizes and thread counts\. Timed calls include source/query preprocessing and transfers, FiLM conditioning, encoding, FNO propagation and surface queries\. They exclude model restoration, mesh loading, constant preparation, warmup, statistics, archive writes and plotting\. Width statistics use the first measured repetition, with equal weight per face centre\. Every call rebuilds its Taylor state and output bounds; only loaded fixed geometry and weight/grid\-dependent constants are reused\. Source features and spline stencils are recomputed\.

Interval implementationTime \(s\)Median width \(mm\)Max width \(mm\)Original formulation, FP3274\.632\.436946\.34944Verified norms/extrema, FP32106\.682\.436946\.34944Verified norms/extrema, FP64133\.832\.436946\.34944Validated binary64356\.402\.474116\.45884Table 5:Parameter\-box arithmetic comparison for the evaluation case at1×1\\times, using the optimized validated binary64 implementation\. Times are median per\-box costs over three repetitions in the same A100 allocation with cached constants, excluding setup and warmup\. Widths are displacement\-magnitude enclosure widths at fixed face centres\. The ordinary FP32/FP64 rows have no end\-to\-end rounding certificate; validated binary64 encloses the real surrogate defined by the stored weights\.The original and updated FP32 widths agree at the precision of[Table5](https://arxiv.org/html/2610.22446#A2.T5): their largest pointwise width difference is7\.97×10−67\.97\\times 10^\{\-6\}mm and their largest endpoint difference is3\.41×10−53\.41\\times 10^\{\-5\}mm\. Thus this case shows negligible change in enclosure width from verifying norms and extrema\. The updated FP32 surface\-query stage costs about40\.0740\.07s versus8\.458\.45s for the original, consistent with its more expensive enclosed extremum calculation\. Validated binary64 increases the median width by approximately1\.53%1\.53\\%and the maximum by1\.72%1\.72\\%\. It also encloses real preprocessing, whereas ordinary FP64 promotes the captured FP32 processed features, so this increase is not solely a final rounding pad\. Agreement of computed widths and sampled containment checks are empirical diagnostics; they do not establish the enclosure guarantee of ordinary arithmetic\. Caching reduces repeated constant\-construction cost in both formulations and is independent of rounding validation\.

##### Effect of parameter\-box size in FP32\.

![Refer to caption](https://arxiv.org/html/2610.22446v1/mastu_parameter_box_size_surfaces.png)Figure 8:FP32 displacement\-magnitude enclosure widths foreval\_02over the0\.5×0\.5\\times,1×1\\timesand2×2\\timesparameter boxes in[Table6](https://arxiv.org/html/2610.22446#A2.T6)\. Both parameter half\-widths scale together about the same nominal parameters, with no parameter subdivision\. The geometry and camera are identical to[Figure7](https://arxiv.org/html/2610.22446#A2.F7)\. A shared linear colour scale spanning the largest width in these three panels shows the growth of field variation and enclosure conservatism without clipping or independent panel rescaling\. Widths are evaluated at fixed face centres using the updated ordinary FP32 formulation\.[Table6](https://arxiv.org/html/2610.22446#A2.T6)varies both physical parameter half\-widths together for the same evaluation case, using the updated formulation with verified norms/extrema and ordinary FP32 propagation\. The checkpoint, native source nodes and face centres remain fixed\. Width statistics again use the first measured repetition, with no parameter subdivision\. Larger boxes increase both the allowed field variation and enclosure conservatism\.

Box factorα\\alphahalf\-width\(K−1\)\(\\mathrm\{K\}^\{\-1\}\)TwT\_\{w\}half\-width\(K\)Median width\(mm\)Max width\(mm\)0\.5×0\.5\\times5×10−85\\times 10^\{\-8\}2\.50\.473101\.026621×1\\times10−710^\{\-7\}52\.436946\.349442×2\\times2×10−72\\times 10^\{\-7\}1020\.5516151\.67701Table 6:FP32 displacement\-magnitude enclosure widths foreval\_02across three parameter\-box sizes\. Both half\-widths scale by the box factor about the same nominal parameters\. Median and maximum widths are computed over all4,077,3124\{,\}077\{,\}312fixed face centres\. These are ordinary\-arithmetic enclosure calculations, without an end\-to\-end rounding certificate\.
##### Setup, warmup and constant reuse\.

Model restoration and geometry loading cost10\.6910\.69s for the original run and12\.2512\.25s shared across the three updated methods\.[Table7](https://arxiv.org/html/2610.22446#A2.T7)gives the additional constant preparation/check and warmup costs\. The original formulation builds ordinary spectral constants in2\.642\.64s\. The updated run first builds verified Fourier and skip norm constants in FP32 mode \(336\.39336\.39s\); FP64 and validated binary64 reuse the same exact promoted weight/grid keys, with cache checks of0\.460\.46s and0\.410\.41s, respectively\. The aggregate verified\-constant preparation/check cost is337\.25337\.25s\. An independent cold start of an updated method must build the verified constants; the later modes’ lookup times do not represent cold\-start setup costs\. Each method receives one full\-workload warmup oneval\_02at0\.5×0\.5\\timesbefore any timed repetitions\.

Interval implementationConstant setup/check \(s\)Warmup \(s\)Original formulation, FP322\.6488\.75Verified norms/extrema, FP32336\.39108\.64Verified norms/extrema, FP640\.46134\.67Validated binary640\.41522\.53Table 7:One\-time constant preparation/checks and one full warmup per method, excluding the separately reported restore/geometry cost\. The updated methods ran in order FP32, FP64, validated binary64 in one process: FP32 builds the shared verified constants, while the last two rows report cache checks\. Warmup uses the0\.5×0\.5\\timesbox; the measured comparison in[Table5](https://arxiv.org/html/2610.22446#A2.T5)uses1×1\\times\.These bounds enclose the surrogate prediction\. They do not quantify prediction error relative to a reference solution or bound PDE discretisation, solver convergence, or physical modelling error\.

## Appendix CL\-SplineFNO Implementation

L\-SplineFNO scatters point features onto a regular grid with learned spline weights, processes the grid with an FNO, and gathers the result at query points with the fixed analytic spline\. The listings describe these cubic transfers in PyTorch\-style pseudocode using ordinary autograd\. We describe the single\-scale, single\-region path with an affine readout, followed by the extensions and memory optimisations\.

##### Notation\.

We useBbatchB\_\{\\mathrm\{batch\}\}samples per batch,CCfeature channels \(the widthccof the main text\),NNinput points andNqN\_\{q\}output query points inddspatial dimensions\. The regular grid hasSℓS\_\{\\ell\}nodes along axisℓ\\ellandG=∏ℓ=1dSℓG=\\prod\_\{\\ell=1\}^\{d\}S\_\{\\ell\}nodes in total\. Each point interacts with at mostK=4dK=4^\{d\}grid nodes\. The listings abbreviateBbatchB\_\{\\mathrm\{batch\}\}asB; all tensors and kernels must have compatible devices and floating\-point dtypes\.

##### Step 1: neighbouring grid nodes and their weights\.

The*stencil*of a point is the set of grid nodes it contributes to\. Along axisℓ\\ell, with grid originξ0\(ℓ\)\\xi\_\{0\}^\{\(\\ell\)\}and spacingΔℓ\\Delta\_\{\\ell\}, the coordinateti\(ℓ\)=\(ωi\(ℓ\)−ξ0\(ℓ\)\)/Δℓt\_\{i\}^\{\(\\ell\)\}=\(\\omega\_\{i\}^\{\(\\ell\)\}\-\\xi\_\{0\}^\{\(\\ell\)\}\)/\\Delta\_\{\\ell\}gives the position of pointiiin grid cells\. Its four candidate neighbours aregℓ=⌊ti\(ℓ\)⌋\+og\_\{\\ell\}=\\lfloor t\_\{i\}^\{\(\\ell\)\}\\rfloor\+owitho∈\{−1,0,1,2\}o\\in\\\{\-1,0,1,2\\\}, at distancesti\(ℓ\)−gℓ∈\[−2,2\)t\_\{i\}^\{\(\\ell\)\}\-g\_\{\\ell\}\\in\[\-2,2\)\. Each neighbour is weighted by the kernel evaluated at that distance\. Neighbours that fall outside the grid receive zero weight, and the remaining weights are normalised along each axis before being multiplied across axes:

vi,gℓ\(ℓ\)=𝟏0≤gℓ<Sℓ​κθ\(ℓ\)​\(ti\(ℓ\)−gℓ\)sgn\+⁡\(si​ℓ\)​max⁡\(\|si​ℓ\|,δ\),\(Wθ\)i,𝐠=∏ℓ=1dvi,gℓ\(ℓ\),v\_\{i,g\_\{\\ell\}\}^\{\(\\ell\)\}=\\frac\{\\mathbf\{1\}\_\{0\\leq g\_\{\\ell\}<S\_\{\\ell\}\}\\,\\kappa\_\{\\theta\}^\{\(\\ell\)\}\(t\_\{i\}^\{\(\\ell\)\}\-g\_\{\\ell\}\)\}\{\\operatorname\{sgn\}\_\{\+\}\(s\_\{i\\ell\}\)\\max\(\|s\_\{i\\ell\}\|,\\delta\)\},\\qquad\(W\_\{\\theta\}\)\_\{i,\\mathbf\{g\}\}=\\prod\_\{\\ell=1\}^\{d\}v\_\{i,g\_\{\\ell\}\}^\{\(\\ell\)\},\(35\)wheresi​ℓs\_\{i\\ell\}is the sum of the valid kernel weights of pointiialong axisℓ\\ell,sgn\+⁡\(s\)=−1\\operatorname\{sgn\}\_\{\+\}\(s\)=\-1fors<0s<0and\+1\+1otherwise, andδ\\deltaprevents division by a near\-zero sum\. Wheneverδ\\deltais inactive, each axis’s weights sum to one, and hence so does each row ofWθW\_\{\\theta\}\. We useδ=10−3\\delta=10^\{\-3\}for the learned kernel\. The analytic kernel usesδ=10−12\\delta=10^\{\-12\}; this floor is inactive for points inside the grid bounds\.

The learned kernel follows[Equation7](https://arxiv.org/html/2610.22446#S3.E7), withJ=9J=9correction centres uniformly spaced on\[−1,1\]\[\-1,1\]and widthh=1/2h=1/2by default\. The coefficientsc=V0​αc=V\_\{0\}\\alphapreserve the interior partition of unity to numerical precision\. A float64 SVD of the257×9257\\times 9collocation matrix in[SectionB\.2](https://arxiv.org/html/2610.22446#A2.SS2)gives six free parameters per axis;V0V\_\{0\}is cached per kernel configuration\. Initialisingα=0\\alpha=0recovers the analytic cubic kernel, not the least\-squares encoder of SplineFNO\. The implementation evaluates the correction using a piecewise\-cubic table and Horner’s rule\. Listingstores the four neighbours per axis separately; their tensor product is formed only for the point chunk being processed\.

importtorch

fromtorchimportnn

d=3

kernels=nn\.ModuleList\(\[C2SplineKernel1D\(\)for\_inrange\(d\)\]\)

defstencil\(coords,axes,kernels,floor=1e\-3\):

pairs=\[\_scatter\_axis\(coords\[:,ax\],axes\[ax\],kernels\[ax\],

pou\_floor=floor\)foraxinrange\(len\(axes\)\)\]

indices,weights=zip\(\*pairs\)

return\_FactoredStencil\(indices,weights,\[len\(a\)forainaxes\]\)

Listing 1:Build a stencil: the four neighbouring grid nodes and their normalised weights along each axis, stored per axis\.
##### Step 2: scatter to the grid and gather back\.

*Scatter*adds each input point’s features to its stencil nodes, weighted byWθW\_\{\\theta\}\. For featuresH∈ℝBbatch×N×CH\\in\\mathbb\{R\}^\{B\_\{\\mathrm\{batch\}\}\\times N\\times C\}, each grid node𝐠\\mathbf\{g\}accumulates three sums \(batch index omitted\):

n𝐠=∑i\(Wθ\)i,𝐠​Hi,d𝐠=∑i\(Wθ\)i,𝐠,a𝐠=∑i\|\(Wθ\)i,𝐠\|,n\_\{\\mathbf\{g\}\}=\\sum\_\{i\}\(W\_\{\\theta\}\)\_\{i,\\mathbf\{g\}\}H\_\{i\},\\qquad d\_\{\\mathbf\{g\}\}=\\sum\_\{i\}\(W\_\{\\theta\}\)\_\{i,\\mathbf\{g\}\},\\qquad a\_\{\\mathbf\{g\}\}=\\sum\_\{i\}\|\(W\_\{\\theta\}\)\_\{i,\\mathbf\{g\}\}\|,\(36\)the weighted feature sum, the signed weight sum and the absolute weight sum\. Without safeguards, the normalised valuen𝐠/d𝐠n\_\{\\mathbf\{g\}\}/d\_\{\\mathbf\{g\}\}reproduces constant features whereverd𝐠≠0d\_\{\\mathbf\{g\}\}\\neq 0\. Signed weights can, however, cancel\. We therefore use[Equation32](https://arxiv.org/html/2610.22446#A2.E32)withη=0\.1\\eta=0\.1andε=10−6\\varepsilon=10^\{\-6\}\. This bounds each output channel by1/η1/\\etatimes the largest absolute input value contributing to that node\. For non\-negative weights,a𝐠=d𝐠a\_\{\\mathbf\{g\}\}=d\_\{\\mathbf\{g\}\}, so only the absolute floor can be active\. Uncovered nodes remain zero; constant features are reproduced wherever neither floor changes the denominator\.

After the FNO produces grid featuresZ∈ℝBbatch×G×CZ\\in\\mathbb\{R\}^\{B\_\{\\mathrm\{batch\}\}\\times G\\times C\},*gather*evaluates them at each query pointqb,jq\_\{b,j\}as a weighted sum of that point’s stencil nodes, using the analytic weightsΦ\\Phi:

H^b,j=∑𝐠Φ​\(qb,j\)𝐠​Zb,𝐠\.\\widehat\{H\}\_\{b,j\}=\\sum\_\{\\mathbf\{g\}\}\\Phi\(q\_\{b,j\}\)\_\{\\mathbf\{g\}\}\\,Z\_\{b,\\mathbf\{g\}\}\.\(37\)Scatter and gather build separate stencils, so the source and query clouds may differ in both location and size\. The output projection then mapsH^\\widehat\{H\}to the physical output variables\.

Listingprocesseschunkpoints at a time\. The methodtapsreturns flattened grid indices and weights of shape\[Bbatch,hi−lo,K\]\[B\_\{\\mathrm\{batch\}\},\\texttt\{hi\}\-\\texttt\{lo\},K\], or batch dimension one for a shared mesh\. The first grid axis varies slowest\. Addingb​GbGto samplebb’s indices lets indexed additions and gathers process the batch together\. Gradients propagate to features, grid values and learned kernel coefficients\.

defscatter\(H,src,G,chunk,eta=0\.1,eps=1e\-6\):

B,N,C=H\.shape

offset=\(torch\.arange\(B,device=H\.device\)\*G\)\[:,None,None\]

num=H\.new\_zeros\(B\*G,C\)

mass=H\.new\_zeros\(B\*G,2\)

forloinrange\(0,N,chunk\):

hi=min\(lo\+chunk,N\)

col,w=src\.taps\(node\_start=lo,node\_stop=hi\)

w=w\.expand\(B,\-1,\-1\)

idx=\(col\+offset\)\.reshape\(\-1\)

values=H\[:,lo:hi,None,:\]\*w\[\.\.\.,None\]

num\.index\_add\_\(0,idx,values\.reshape\(\-1,C\)\)

mass\.index\_add\_\(0,idx,

torch\.stack\(\(w,w\.abs\(\)\),\-1\)\.reshape\(\-1,2\)\)

den,absolute=mass\.unbind\(\-1\)

sign=torch\.where\(den<0,\-1\.,1\.\)

safe=sign\*torch\.maximum\(den\.abs\(\),\(eta\*absolute\)\.clamp\_min\(eps\)\)

return\(num/safe\[:,None\]\)\.view\(B,G,C\)

defgather\(Z,qry,chunk\):

B,G,C=Z\.shape

offset=\(torch\.arange\(B,device=Z\.device\)\*G\)\[:,None,None\]

flat=Z\.reshape\(B\*G,C\)

out=Z\.new\_empty\(B,qry\.n\_points,C\)

forloinrange\(0,qry\.n\_points,chunk\):

hi=min\(lo\+chunk,qry\.n\_points\)

col,w=qry\.taps\(node\_start=lo,node\_stop=hi\)

idx=\(col\+offset\)\.reshape\(\-1\)

values=flat\.index\_select\(0,idx\)\.view\(B,hi\-lo,\-1,C\)

out\[:,lo:hi\]=\(values\*w\[\.\.\.,None\]\)\.sum\(2\)

returnout

Listing 2:Scatter features from points to the grid with guarded normalisation, and gather grid features at query points\.chunkis the number of points processed at once\.
##### Step 3: the full model\.

The pointwise lift formsHi=Pin​\[xi;ω¯i\]H\_\{i\}=P\_\{\\mathrm\{in\}\}\[x\_\{i\};\\bar\{\\omega\}\_\{i\}\], as in[Equation3](https://arxiv.org/html/2610.22446#S3.E3)\. For the FiLM\-conditioned benchmarks,xi=𝜸μ⊙φ⁡\(ω¯i\)\+𝜷μx\_\{i\}=\\bm\{\\gamma\}\_\{\\mu\}\\odot\\varphi\(\\bar\{\\omega\}\_\{i\}\)\+\\bm\{\\beta\}\_\{\\mu\}contains 96 sinusoidal features \(d=3d=3,nf=16n\_\{f\}=16\), modulated in the outer wrapper\. The three normalised coordinates bypass modulation, soPin:ℝ99→ℝCP\_\{\\mathrm\{in\}\}:\\mathbb\{R\}^\{99\}\\rightarrow\\mathbb\{R\}^\{C\}\.

Normalisation removes the overall weight magnitude at each node\. We retain coverage information throughρ𝐠=∑iΦ​\(ω¯i\)𝐠\\rho\_\{\\mathbf\{g\}\}=\\sum\_\{i\}\\Phi\(\\bar\{\\omega\}\_\{i\}\)\_\{\\mathbf\{g\}\}, computed on the fine grid from the source coordinates\. This non\-negative feature depends on the sampled geometry but not on the learned kernels\. The channellog⁡\(1\+ρ\)\\log\(1\+\\rho\)is appended to the fused features and mixed back toCCchannels by a1×11\\times 1convolutionPρP\_\{\\rho\}, initialised as the identity on the features and zero on the density, so that the density has no effect at initialisation\. The full forward pass is

Xs\\displaystyle X\_\{s\}=Dθ,s−1​Wθ,s⊤​H,\\displaystyle=D\_\{\\theta,s\}^\{\-1\}W\_\{\\theta,s\}^\{\\top\}H,Xfine\\displaystyle X\_\{\\mathrm\{fine\}\}=Fuse⁡\(\{Ups⁡Xs\}s∈𝒮\),\\displaystyle=\\operatorname\{Fuse\}\\\!\\big\(\\\{\\operatorname\{Up\}\_\{s\}X\_\{s\}\\\}\_\{s\\in\\mathcal\{S\}\}\\big\),\(38\)Z\\displaystyle Z=ℱθ​\(Pρ​\[Xfine;log⁡\(1\+ρ\)\]\),\\displaystyle=\\mathcal\{F\}\_\{\\theta\}\\\!\\left\(P\_\{\\rho\}\[X\_\{\\mathrm\{fine\}\};\\log\(1\+\\rho\)\]\\right\),𝒴^​\(q\)\\displaystyle\\widehat\{\\mathcal\{Y\}\}\(q\)=Pout​\(Φ​\(q\)​Z\),\\displaystyle=P\_\{\\mathrm\{out\}\}\\\!\\left\(\\Phi\(q\)Z\\right\),whereDθ,sD\_\{\\theta,s\}contains the guarded denominators of Step 2,𝒮\\mathcal\{S\}is the set of scales,Ups\\operatorname\{Up\}\_\{s\}interpolates scalessto the fine grid \(Up1\\operatorname\{Up\}\_\{1\}is the identity\) andFuse\\operatorname\{Fuse\}is the1×11\\times 1fusion; both are omitted with a single scale\. When spectral modulation is selected, a condition embedding is also passed to every FNO layer\. Region\-specific variants may use separate lifts, output projections or kernel banks\. With separate kernels, the three scatter sums are added across regions before division; constant reproduction has the same coverage and safeguard conditions as in Step 2\. Region routing uses node labels supplied alongside the coordinates\.

The two halves are exposed separately\.encoderuns the lift, scatter and FNO and returns the grid stateZZ, anddecodegathersZZat a query cloud and appliesPoutP\_\{\\mathrm\{out\}\}\. Evaluating at new query points therefore needs only a newdecodein the path described here\. Region\-specific readouts additionally require query\-region labels, and the optional node skip connection requires lifted features at the query nodes\.

##### Cost and memory\.

At fixed width and stencil size, both transfers are linear in the number of points: writingB=BbatchB=B\_\{\\mathrm\{batch\}\}, scatter costsO⁡\(B​N​K​C\)O\(BNKC\)and gatherO⁡\(B​Nq​K​C\)O\(BN\_\{q\}KC\), plusO⁡\(B​G​C\)O\(BGC\)to initialise the grid buffers\. The encoder needs noG×GG\\times GGram matrix and no factorisation\. Each FNO layer costsO⁡\(B​C​G​log⁡G\)O\(BCG\\log G\)for its real FFTs, and dense spectral channel mixing costsO⁡\(B​C2​Mf\)O\(BC^\{2\}M\_\{f\}\)forMfM\_\{f\}retained modes, with a furtherO⁡\(B​C2​G\)O\(BC^\{2\}G\)for pointwise channel mixing\. The main transfer optimisations are:

- •Factored stencils\.Storing four index–weight pairs per axis takes4​d​B​N4dBNpairs, against4d​B​N4^\{d\}BNfor the full tensor product \(5\.3×5\.3\\timesfewer ford=3d=3\)\. The product is formed only for the chunk being processed\.
- •Fused accumulation\.On both CPU and GPU, each point chunk uses one indexed addition for features and one for the signed and absolute weight sums together:2​⌈N/M⌉2\\lceil N/M\\rceiladditions for chunks ofMMpoints per scale and region\.
- •Recomputed backward passes\.Custom autograd functions save the per\-axis stencil factors and the source features or grid state\. They reconstruct tensor\-product indices and weights in chunks during backward, avoiding retention of every chunk’s expanded intermediates as ordinary autograd would do for the reference listings\.
- •Shared geometry\.When every sample in a batch has the same coordinates and these do not require gradients, one stencil is built and broadcast across the batch, and the weight sumsd𝐠d\_\{\\mathbf\{g\}\}anda𝐠a\_\{\\mathbf\{g\}\}, which depend only on the geometry, are accumulated once\. Samples with different meshes keep their own stencils\.
- •Matrix\-free decoding\.The gather computesΦ​Z\\Phi Zdirectly, so the sparse reconstruction matrix is never assembled in the forward path\.

The parameterscatter\_max\_temp\_mb\(default 128 MiB\) sets an approximate temporary\-buffer budget for both transfers on every device\. It controls chunk size, not total model or training memory\. Gather reduces each query’sKKcontributions within a single chunk; scatter reductions may differ in floating\-point rounding with chunk size or device\. For source clouds too large to lift in one pass, streamed single\-sample inference reads and lifts the inputs in chunks, accumulates the scatter and density sums over all chunks before normalisation, and runs the FNO once\. This matches the complete\-cloud computation up to floating\-point rounding\.

## Appendix DPhysics and Data

All six benchmarks are steady\-state, conditioned operator\-learning problems posed on unstructured meshes\. They were chosen to span two axes that matter for a conditional neural operator\. The first is*geometric*: shared\-mesh problems, where a single node ordering is valid for every case \(Aircraft, HIVE, Heated duct\), against per\-sample\-mesh problems, where node indexiirefers to unrelated physical locations in different cases \(Heat sink, Bracket, Building fire\)\. The second is the*informativeness of𝛍\\bm\{\\mu\}*: how much of what𝝁\\bm\{\\mu\}says is already recoverable from the per\-node input\.

- •Redundant\.Heat sink — every parameter describes a shape the model already sees, so any gain from conditioning measures how hard that shape information is to extract from a sampled point cloud rather than how much new information𝝁\\bm\{\\mu\}adds\.
- •Partly non\-redundant\.Bracket, Building fire — on the bracket the three geometry parameters restate the mesh, but the applied pressure load is invisible in it, so two designs can share a shape and differ only in𝝁\\bm\{\\mu\}\. The fire splits the same way: the refinement box that follows the burner places the two fire coordinates in the point cloud exactly, while the five ventilation parameters move no node at all\. These are the only cases here where𝝁\\bm\{\\mu\}splits into an inferable half and a hidden one, and both loaders expose each half separately so the two can be ablated apart\.
- •Non\-redundant\.Aircraft, HIVE, Heated duct — the five flight conditions of one airframe have identical coordinates, so𝝁\\bm\{\\mu\}is the only signal separating Mach 2 from Mach 7\. HIVE and the heated duct are the same in kind: one shared mesh across every run, so nothing in the geometry distinguishes them\.

A third property is carried by two cases\. The heated duct is*multi\-region*: its domain is three separate meshes that share no cells and couple only across two conjugate interfaces, so temperature is continuous there while its gradient is not\. All other benchmarks here are single structures, and the gridded operators reduce to a single latent grid without loss; on the heated duct and building fire the architecture deploys three different branches for mesh encoding and decoding with a shared latent backbone\.

DatasetPhysics / targetCasestrain/testNativenodesUsednodesdim𝝁\\dim\\bm\{\\mu\}Mesh structureHeat sinkConduction /TT625625\(438/187438/187\)4,6714\{,\}671–32,64532\{,\}64516,38416\{,\}38444Sample\-dependentBracketElasto\-plasticity /σ\\sigma3,0003\{,\}000\(2,100/9002\{,\}100/900\)5,0545\{,\}054–15,21015\{,\}2108,1928\{,\}19244Sample\-dependentAircraftCompressible flow /\(ρ,u,v,w,p\)\(\\rho,u,v,w,p\)150150\(105/45105/45\)331,971331\{,\}971131,072131\{,\}07233Shared correspondenceBuilding fireFire LES \(time\-avg\.\) /\(Δ​T,ρair,ρprod\)\(\\Delta T,\\rho\_\{\\rm air\},\\rho\_\{\\rm prod\}\)547547\(383/164383/164\)116,928116\{,\}92898,49698\{,\}49677Sample\-dependent, three\-storeyHeated ductConjugate heat transfer /TT122122\(103/19103/19\)124,800124\{,\}800124,800124\{,\}80055Shared, three\-regionHIVEInduction heating /TT186186\(149/37149/37\)194,691194\{,\}691194,691194\{,\}69166Shared

Table 8:Overview of the six benchmarks\. “Native nodes” denotes the original mesh size, whereas “used nodes” is the spatial budget presented to the model\. Variable\-size meshes are sampled or duplicate\-padded to this budget\.### D\.1Conventions common to all datasets

Each data loader returns a single tensor𝐅∈ℝN×\(cin\+cout\)×N\\mathbf\{F\}\\in\\mathbb\{R\}^\{N\\times\(c\_\{\\mathrm\{in\}\}\+c\_\{\\mathrm\{out\}\}\)\\times N\}together with𝝁∈ℝN×k\\bm\{\\mu\}\\in\\mathbb\{R\}^\{N\\times k\}andΔ​t=0\\Delta t=0, the latter flagging the steady\-state path\. The firstcinc\_\{\\mathrm\{in\}\}channels of𝐅\\mathbf\{F\}are inputs and the remainder are targets\.

Where the node count varies case to case \(Heat sink, Bracket\), every case is drawn down to one fixed node budgetNNso the cases stack into a dense tensor\. The budget is a free parameter of the sweep rather than a property of the corpus; we set it to the power of two nearest the median mesh, so a majority of cases are drawn without replacement and the remainder are duplicate\-padded with their own real nodes\. Padding with repeats rather than with zeros or a sentinel matters for the gridded backbones: a repeated node adds weight to a knot that a real node already occupies, which the scatter and its density normalisation absorb exactly, whereas an artificial coordinate would invent geometry\. The cost is a reduced effective sample — at the budgets used here, 432 of the 625 heat\-sink cases and 878 of the 3,000 bracket designs are padded, leaving on average 76% and 97% of their slots occupied by distinct nodes \([Table8](https://arxiv.org/html/2610.22446#A4.T8)\)\. The per\-sample geometry path then reads each case’s own coordinates out of its input channels rather than broadcasting a cross\-case mean mesh, which for independently drawn meshes would collapse towards the domain centroid\. The three shared\-mesh cases need none of this: Aircraft draws all 150 cases at the same node indices, and HIVE and the heated duct use every node of their one mesh\.

All six benchmarks are*geometry\-only*: no case carries a field input channel, so the per\-node input is the processed coordinates and nothing else\. Where node positions differ between cases, the three leading channels of𝐅\\mathbf\{F\}hold that case’s own\(x,y,z\)\(x,y,z\)andcin=3c\_\{\\mathrm\{in\}\}=3\. This happens in two ways: either each case has its own mesh \(Heat sink, Bracket, Building fire\), or every case is a deformation of one shared template, so nodekkis the same template node in every case, but its position changes with the airframe \(Aircraft\)\. The per\-sample geometry path consumes these channels as geometry rather than as features, so they are not passed to the backbone as channels\. Where the mesh is genuinely fixed \(HIVE, Heated duct\), the coordinates live in the mesh metadata instead andcin=0c\_\{\\mathrm\{in\}\}=0\. Either way, the coordinates reach the backbone only through the coordinate processor, where an integrated sinusoidal positional encoding at 16 frequencies gives them bandwidth, mappingℝ3→ℝ3\+2⋅3⋅16=ℝ99\\mathbb\{R\}^\{3\}\\to\\mathbb\{R\}^\{3\+2\\cdot 3\\cdot 16\}=\\mathbb\{R\}^\{99\}\. The network therefore consumes𝒳∈ℝBbatch×N×99\\mathcal\{X\}\\in\\mathbb\{R\}^\{B\_\{\\mathrm\{batch\}\}\\times N\\times 99\}and emitsℝBbatch×N×cout\\mathbb\{R\}^\{B\_\{\\mathrm\{batch\}\}\\times N\\times c\_\{\\mathrm\{out\}\}\}throughout\. The first 3 of these 99 channels are the normalised coordinates and the other 96 are the sinusoidal featuresφ⁡\(ω¯\)\\varphi\(\\bar\{\\omega\}\)\. Under the field\-space conditioning methods \(FiLM, AdaLN\-Zero, cross\-attention\),𝝁\\bm\{\\mu\}modulates only the 96 sinusoidal channels, and the 3 coordinates pass through unmodulated \([Equation3](https://arxiv.org/html/2610.22446#S3.E3)\)\.

Figure 9:Distribution of the target field over all nodes of all cases, one panel per benchmark \(pressure shown for aircraft and temperature change for building fire\)\. Counts are normalised to a fraction of nodes and plotted on a log ordinate; the horizontal axis is trimmed to\[p0\.05,p99\.95\]\[p\_\{0\.05\},p\_\{99\.95\}\]and the lighter shading marks the tails outside\[p1,p99\]\[p\_\{1\},p\_\{99\}\]\. The dashed line is the median\. Sample skewness\(γ1\)\(\\gamma\_\{1\}\)and kurtosis\(β2\)\(\\beta\_\{2\}\)are annotated \(a Gaussian hasSkew=0\\mathrm\{Skew\}=0,Kurt=3\\mathrm\{Kurt\}=3\)\.Field normalisation is per\-variable and standardised throughout; conditioning parameters are min–max normalised per variable\. Train/test splits are random permutations under a fixed seed of 42, except on the heated duct, whose designed split is honoured instead \([SectionD\.6](https://arxiv.org/html/2610.22446#A4.SS6)\)\. The field normaliser, the parameter normaliser and the coordinate processor are all fitted on the pooled corpus before the train–test split\. The Building fire departs from these conventions in one respect: a cell\-size\-aware positional encoding, described in[SectionD\.5](https://arxiv.org/html/2610.22446#A4.SS5)\.

Reported errors are the percentage relativeL2L\_\{2\}error, computed per sample in physical \(denormalised\) units and averaged over the test set,

ℰ=100Mtest​∑i=1Mtest∥𝐲^\(i\)−𝐲\(i\)∥2∥𝐲\(i\)∥2,\\mathcal\{E\}=\\frac\{100\}\{M\_\{\\mathrm\{test\}\}\}\\sum\_\{i=1\}^\{M\_\{\\mathrm\{test\}\}\}\\frac\{\\lVert\\widehat\{\\mathbf\{y\}\}^\{\(i\)\}\-\\mathbf\{y\}^\{\(i\)\}\\rVert\_\{2\}\}\{\\lVert\\mathbf\{y\}^\{\(i\)\}\\rVert\_\{2\}\},\(39\)which equals100×100\\timesNRMSE under the definition used in[Section4](https://arxiv.org/html/2610.22446#S4)\. Because[Equation39](https://arxiv.org/html/2610.22446#A4.E39)normalises by the magnitude of the target rather than by its variation, it is generous to fields carrying a large constant offset; all three thermal cases sit on a pedestal —∼300\\sim\\\!300K for the heat sink and HIVE,280280K for the heated duct — and the per\-dataset notes below give the scale of that effect\.

### D\.2Heat sink

##### Governing equation\.

Steady conduction through a finned aluminium heat sink cooling a CPU,

−∇⋅\(κ∇T\)=Q,−κ∂T∂n=h\(T−T∞\)onΓlat∪Γtop,\-\\nabla\\\!\\cdot\\\!\\left\(\\kappa\\nabla T\\right\)=Q,\\qquad\-\\kappa\\frac\{\\partial T\}\{\\partial n\}=h\\,\(T\-T\_\{\\infty\}\)\\ \\ \\text\{on\}\\ \\Gamma\_\{\\mathrm\{lat\}\}\\cup\\Gamma\_\{\\mathrm\{top\}\},\(40\)with forced convection on the lateral surfaces, free convection on the top surface and the chip heat load applied on the baseplate underside\. The material conductivityκ\\kappa, the loadQQ, the film coefficienthhand the ambientT∞T\_\{\\infty\}are held*fixed*across the whole dataset\. Only the geometry varies, so𝝁\\bm\{\\mu\}fully determines the solution and the benchmark is a direct test of whether a conditional operator can read a 3D shape\.

##### Parameter scan\.

A complete54=6255^\{4\}=625factorial over four design variables at five levels each \([Table9](https://arxiv.org/html/2610.22446#A4.T9)\)\. The baseplate is a fixed 26 mm alongxx\(widened only whenW\>26W\>26mm\), so the fin pitch is26​mm/nfin∈\[3\.25,6\.5\]26\\,\\mathrm\{mm\}/n\_\{\\rm fin\}\\in\[3\.25,6\.5\]mm and is not an independent variable\. This is the source of the benchmark’s difficulty for grid\-based methods: fins 1\.5–2\.5 mm thick at a 3\.25–6\.5 mm pitch alias badly on any uniform latent grid, and at least 16 cells alongxxare needed for eight fins to separate at all\.

ParameterSymbolLevelsMeaningSink widthWW20,22\.5,25,27\.5,3020,22\.5,25,27\.5,30mmyyextent; fins span it fullyFin heightHH15,16\.25,17\.5,18\.75,2015,16\.25,17\.5,18\.75,20mmzzextent above the baseplateThicknesstt1\.5,1\.75,2,2\.25,2\.51\.5,1\.75,2,2\.25,2\.5mmfins and baseplate;zzspans\[−t,H\]\[\-t,H\]Fin countnfinn\_\{\\rm fin\}4,5,6,7,84,5,6,7,8plates stacked along the 26 mm baseTable 9:Heat sink: the545^\{4\}factorial design\. All 625 combinations are present exactly once\.
##### Mesh and provenance\.

The dataset is the heat\-sink benchmark of[Zhong et al\. \(2026\)](https://arxiv.org/html/2610.22446#bib.bib65), distributed as a MATLAB struct of nodal coordinates and temperatures with no connectivity\. Every case carries its own tetrahedral mesh, from 4,671 to 32,645 nodes \(median 12,673, with 613 distinct node counts among 625 cases\), so node indexiiis an unrelated physical location in different cases\. At the node budget used here,N=16384N=16384, the 193 cases with larger meshes are drawn without replacement and the remaining 432 are duplicate\-padded; the padding factor reaches3\.5×3\.5\\timeson the smallest mesh, and averaged over the corpus 76% of the budget is distinct nodes\. The pooled bounding box isx∈\[0,30\]x\\in\[0,30\],y∈\[−30,0\]y\\in\[\-30,0\],z∈\[−2\.5,18\.5\]z\\in\[\-2\.5,18\.5\]mm; each design fills its own sub\-box of that domain, and that occupancy*is*the geometry signal, which is why coordinates are min–max scaled per axis over the pooled cloud rather than per sample\.

##### Learning setup\.

𝐅∈ℝ625×4×16384\\mathbf\{F\}\\in\\mathbb\{R\}^\{625\\times 4\\times 16384\}with channels\(x,y,z,T\)\(x,y,z,T\)andcin=3c\_\{\\mathrm\{in\}\}=3, so inputs areℝN×3×16384\\mathbb\{R\}^\{N\\times 3\\times 16384\}and targetsℝN×1×16384\\mathbb\{R\}^\{N\\times 1\\times 16384\};𝝁∈ℝ625×4\\bm\{\\mu\}\\in\\mathbb\{R\}^\{625\\times 4\}\. Coordinates are dropped as features and re\-enter only through the 99\-channel integrated encoding, so the network sees geometry and𝝁\\bm\{\\mu\}and nothing else\. The split is 70/30 \(438 train, 187 test\) at batch size 8, matching the test share used by the source paper\([Zhong et al\., 2026](https://arxiv.org/html/2610.22446#bib.bib65)\)\.

##### Data statistics\.

T∈\[296\.46,343\.76\]T\\in\[296\.46,343\.76\]K with mean 307\.25 K and standard deviation 5\.79 K \(Skew=1\.25\\mathrm\{Skew\}=1\.25,Kurt=5\.60\\mathrm\{Kurt\}=5\.60\)\. Case\-mean temperatures span\[300\.07,322\.86\]\[300\.07,322\.86\]K\. The measurements are taken on the temperature rise above ambient,T−TminT\-T\_\{\\min\}withTmin=296\.46T\_\{\\min\}=296\.46K\.

![Refer to caption](https://arxiv.org/html/2610.22446v1/heat_sink_test_case_95_error_maps_all.png)Figure 10:Error maps produced by various architectures in modelling heat distribution across the heat sink\.

### D\.3Bracket

##### Governing equations\.

Steady elasto\-plastic deformation of a lug, i\.e\. a hollow beam with a bolt hole which is fixed at one end and loaded by a pressure over a marked area:

∇⋅𝝈=𝟎,𝝈=ℂ⁡\(𝜺−𝜺p\),𝜺=12​\(∇𝐮\+∇𝐮⊤\),\\nabla\\\!\\cdot\\\!\\bm\{\\sigma\}=\\mathbf\{0\},\\qquad\\bm\{\\sigma\}=\\mathbb\{C\}\\left\(\\bm\{\\varepsilon\}\-\\bm\{\\varepsilon\}^\{\\mathrm\{p\}\}\\right\),\\qquad\\bm\{\\varepsilon\}=\\tfrac\{1\}\{2\}\\\!\\left\(\\nabla\\mathbf\{u\}\+\\nabla\\mathbf\{u\}^\{\\\!\\top\}\\right\),\(41\)with an elastic–plastic constitutive law under isotropic hardening and the small\-deformation assumption, solved with quadratic hexahedral elements\. The prediction target is the nodal stress field\. This is the only non\-thermal, non\-fluid problem in the set\.

##### Why this dataset earns its place\.

𝝁\\bm\{\\mu\}splits cleanly into a redundant half and a non\-redundant one: the three geometry parameters restate what the node coordinates already show, while the applied pressure is*invisible*in the mesh — two designs can share a geometry and differ only in load\. It is the only benchmark here where half of𝝁\\bm\{\\mu\}is inferable from the input and half is not, and the loader exposes each half separately so the two contributions can be ablated independently\.

##### Parameter scan and normalisation\.

N=3000N=3000designs over four continuous parameters: length, thickness and hole radius \(geometry\) plus the applied pressure \(loading\), each realised at 3000 distinct values\. The published release\([Zhong et al\., 2025](https://arxiv.org/html/2610.22446#bib.bib64)\)is*already*min–max normalised by its authors: coordinates, design parameters and stress all arrive on\[0,1\]\[0,1\], and the scaling factors for coordinates and parameters are not distributed, so physical units cannot be recovered for those channels\. The normalisation is global rather than per sample, so each design occupies its own sub\-box of the unit cube — mean per\-design spans and relative scale between designs is preserved\. The latent grids used for this dataset are consequently isotropic: the pooled cloud fills the unit cube exactly, and the true physical aspect ratio is unknowable\.

##### Mesh and learning setup\.

Every design has its own hexahedral mesh, 5,054 to 15,210 nodes \(median 9,405\), with no node correspondence between designs\. At the node budget used here,N=8192N=8192, the 2,122 designs with larger meshes are drawn without replacement and the remaining 878 are duplicate\-padded, at worst1\.6×1\.6\\times; averaged over the corpus 97% of the budget is distinct nodes, so padding is a marginal effect on this dataset in a way it is not on the heat sink\.𝐅∈ℝ3000×4×8192\\mathbf\{F\}\\in\\mathbb\{R\}^\{3000\\times 4\\times 8192\}with channels\(x,y,z,σ^\)\(x,y,z,\\hat\{\\sigma\}\)andcin=3c\_\{\\mathrm\{in\}\}=3;𝝁∈ℝ3000×4\\bm\{\\mu\}\\in\\mathbb\{R\}^\{3000\\times 4\}\. Split 70/30 \(2100 train, 900 test\) at batch size 8\. The random split is sound here: 3000 independent designs with no grouping structure to leak across, unlike the aircraft case\.

##### Data statistics\.

σ^\\hat\{\\sigma\}fills\[0,1\]\[0,1\]by construction of the authors’ normalisation, with mean 0\.197 and standard deviation 0\.121 \(Skew=1\.19\\mathrm\{Skew\}=1\.19,Kurt=3\.92\\mathrm\{Kurt\}=3\.92\); the 99th percentile is 0\.534, so the upper half of the normalised range is populated only by stress concentrations at the hole and the fixed end\. Case\-mean stresses span\[0\.086,0\.383\]\[0\.086,0\.383\]\. The metric is not degenerate here, so unlike the two temperature datasets, the stress field carries spatial structure that no constant can fake\. See[Figure9](https://arxiv.org/html/2610.22446#A4.F9)for the distribution\.

![Refer to caption](https://arxiv.org/html/2610.22446v1/bracket_error_maps_all.png)Figure 11:Error maps produced by various architectures in modelling stress across the bracket\.

### D\.4Aircraft

##### Governing equations and flight envelope\.

Steady compressible external flow over a parameterised airframe, with the solution sampled on the surface triangulation\. The released fields are the freestream\-normalised density, the three velocity components and the pressure\. The envelope is supersonic to hypersonic, with shocks present on the surface: 30 airframes are each solved at five flight conditions,

\(Ma,α,β\)∈\{\(2,0,0\),\(7,0,0\),\(7,0,2\),\(7,7,0\),\(7,7,2\)\},\(\\mathrm\{Ma\},\\alpha,\\beta\)\\in\\\{\(2,0,0\),\\ \(7,0,0\),\\ \(7,0,2\),\\ \(7,7,0\),\\ \(7,7,2\)\\\},withα\\alphathe angle of attack andβ\\betathe sideslip angle, both in degrees, givingN=150N=150cases\.

##### Why this dataset earns its place\.

All five flight conditions of one airframe share bit\-identical coordinates, so the point cloud carries*no*information about the flight condition: only𝝁\\bm\{\\mu\}distinguishes them\. This is the opposite extreme from the heat sink, where𝝁\\bm\{\\mu\}merely restates the shape\. Unlike HIVE and the heated duct, where the mesh is fixed outright, the geometry here still varies across airframes as we subsample them, so the model must use both: the shape to tell airframes apart, and𝝁\\bm\{\\mu\}to tell flight conditions apart\.

##### Mesh and provenance\.

The data ships with Transolver\+\+\([Luo et al\., 2025](https://arxiv.org/html/2610.22446#bib.bib37)\)as Tecplot ASCIIFEPOINTblocks and is used as one of the six industrial benchmarks of[Zhong et al\. \(2026\)](https://arxiv.org/html/2610.22446#bib.bib65)\. Every case has the same 331,971 surface nodes and 663,938 triangles; the airframes are deformations of a shared template \(mean node displacement between the two extreme geometries is 1\.3% of the bounding\-box diagonal, maximum 13%\), so node indexiiis the same semantic vertex everywhere\. We subsample all 150 cases at the*same*131,072 node indices — 39% of the surface\. The pooled bounding box over the sampled nodes isx∈\[0,5000\]x\\in\[0,5000\],y∈\[−298,774\]y\\in\[\-298,774\],z∈\[−2320,2320\]z\\in\[\-2320,2320\]: a thin sheet inyy, which is why coordinates are min–max scaled per axis; a global scaling would leave theyyaxis of a latent grid nearly empty\. Per\-case force coefficients\(CA,CN,CZ,Cl,Cn,Cm\)\(C\_\{A\},C\_\{N\},C\_\{Z\},C\_\{l\},C\_\{n\},C\_\{m\}\)are carried through, with the lift coefficientCL=CN​cos⁡α−CA​sin⁡αC\_\{L\}=C\_\{N\}\\cos\\alpha\-C\_\{A\}\\sin\\alphaprecomputed, so the aggregate quantity of interest needs no surface integration\.

##### Learning setup\.

𝐅∈ℝ150×8×131072\\mathbf\{F\}\\in\\mathbb\{R\}^\{150\\times 8\\times 131072\}with channels\(x,y,z,ρ,u,v,w,p\)\(x,y,z,\\rho,u,v,w,p\)andcin=3c\_\{\\mathrm\{in\}\}=3, giving inputsℝN×3×131072\\mathbb\{R\}^\{N\\times 3\\times 131072\}and targetsℝN×5×131072\\mathbb\{R\}^\{N\\times 5\\times 131072\};𝝁∈ℝ150×3\\bm\{\\mu\}\\in\\mathbb\{R\}^\{150\\times 3\}\. Split 70/30 \(105 train, 45 test\) at batch size 4\.

##### Split caveat\.

The split is a random permutation over the 150 cases, so the same airframe appears in train and test under different flight conditions\. With only 30 airframes, almost every test geometry has been seen\. The reported numbers therefore measure flight\-condition interpolation on known shapes, not geometry generalisation\.

##### Data statistics\.

Alongside the histogram in[Figure9](https://arxiv.org/html/2610.22446#A4.F9), per\-channel statistics are given in[Table10](https://arxiv.org/html/2610.22446#A4.T10)\. The channels are strikingly unequal\. The stream\-wise velocityuuis nearly determined by the flight condition alone whereas the transverse componentsvvandwware largely geometry responses \(67\.8% and 61\.3%\)\. The pressure is the most heavy\-tailed target in the whole benchmark set \(Kurt=108\\mathrm\{Kurt\}=108, reaching46​p∞46\\,p\_\{\\infty\}at the shock\), and it is the channel that drives the joint score\.

ChannelminmaxmeanstdSkew\\mathrm\{Skew\}Kurt\\mathrm\{Kurt\}per\-conditionmean\(%\)ρ\\rho−0\.017\-0\.01718\.241\.3150\.8672\.3316\.035\.95uu−0\.324\-0\.3247\.6705\.6651\.957−1\.42\-1\.423\.205\.25vv−3\.044\-3\.0443\.5000\.2450\.522−0\.13\-0\.134\.3067\.77ww−3\.900\-3\.9004\.0700\.0630\.714−0\.05\-0\.055\.0061\.33pp−0\.035\-0\.03546\.461\.6502\.1318\.17107\.846\.73joint−3\.900\-3\.90046\.461\.7882\.4712\.0814\.323\.30Table 10:Aircraft: per\-channel statistics over all 150 cases at the 131,072 sampled nodes\. All quantities are freestream\-normalised and dimensionless\. The last column is the relativeL2L\_\{2\}error of predicting the mean field of the matching flight condition, i\.e\. the score of a predictor that uses𝝁\\bm\{\\mu\}perfectly and ignores the geometry; a model must beat 23\.3% on the joint field before it has learned anything from the airframe shape\.![Refer to caption](https://arxiv.org/html/2610.22446v1/aircraft_error_maps_all.png)Figure 12:Error maps produced by various architectures in modelling the pressure distribution across the aircraft\.

### D\.5Building fire: smoke and heat transport in a building

##### Governing equations\.

A 500 kW methane fire in a ventilated three\-storey building, computed with Fire Dynamics Simulator \(FDS 6\)\([McGrattan et al\., 2013](https://arxiv.org/html/2610.22446#bib.bib39)\)in large\-eddy simulation mode\. FDS solves the low\-Mach\-number form of the reacting Navier–Stokes equations for thermally driven flow,

∂tρ\+∇⋅\(ρ𝐮\)=0,∂t\(ρZα\)\+∇⋅\(ρZα𝐮\)=∇⋅\(ρDα∇Zα\)\+m˙′′′α,∂t\(ρ​𝐮\)\+∇⋅\(ρ​𝐮𝐮\)\+∇p~=\(ρ−ρ0\)​𝐠\+∇⋅𝝉,∂t\(ρhs\)\+∇⋅\(ρhs𝐮\)=D​p¯D​t\+q˙′′′−∇⋅𝐪˙′′,p¯=ρRT∑αZα/Wα\.\\begin\{gathered\}\\partial\_\{t\}\\rho\+\\nabla\\\!\\cdot\\\!\(\\rho\\mathbf\{u\}\)=0,\\qquad\\partial\_\{t\}\(\\rho Z\_\{\\alpha\}\)\+\\nabla\\\!\\cdot\\\!\(\\rho Z\_\{\\alpha\}\\mathbf\{u\}\)=\\nabla\\\!\\cdot\\\!\(\\rho D\_\{\\alpha\}\\nabla Z\_\{\\alpha\}\)\+\\dot\{m\}^\{\\prime\\prime\\prime\}\_\{\\alpha\},\\\\ \\partial\_\{t\}\(\\rho\\mathbf\{u\}\)\+\\nabla\\\!\\cdot\\\!\(\\rho\\mathbf\{u\}\\mathbf\{u\}\)\+\\nabla\\tilde\{p\}=\(\\rho\-\\rho\_\{0\}\)\\,\\mathbf\{g\}\+\\nabla\\\!\\cdot\\\!\\bm\{\\tau\},\\\\ \\partial\_\{t\}\(\\rho h\_\{s\}\)\+\\nabla\\\!\\cdot\\\!\(\\rho h\_\{s\}\\mathbf\{u\}\)=\\frac\{\\mathrm\{D\}\\bar\{p\}\}\{\\mathrm\{D\}t\}\+\\dot\{q\}^\{\\prime\\prime\\prime\}\-\\nabla\\\!\\cdot\\\!\\dot\{\\mathbf\{q\}\}^\{\\prime\\prime\},\\qquad\\bar\{p\}=\\rho RT\\textstyle\\sum\_\{\\alpha\}Z\_\{\\alpha\}/W\_\{\\alpha\}\.\\end\{gathered\}\(42\)The three lumped species areα∈\{air,fuel,products\}\\alpha\\in\\\{\\text\{air\},\\text\{fuel\},\\text\{products\}\\\}, with mass fractionsZαZ\_\{\\alpha\}\. Here,hsh\_\{s\}is sensible enthalpy,q˙′′′\\dot\{q\}^\{\\prime\\prime\\prime\}is the volumetric heat\-release rate, and𝐪˙′′\\dot\{\\mathbf\{q\}\}^\{\\prime\\prime\}contains conductive, diffusive and radiative heat fluxes\. Combustion is a single\-step, mixing\-controlled reaction \(soot yield 0\.01, CO yield 0\.005\) and radiation is a grey\-gas solve\. FDS uses the sensible\-enthalpy balance to constrain∇⋅𝐮\\nabla\\\!\\cdot\\\!\\mathbf\{u\}rather than advancing it explicitly, and recovers temperature from the equation of state\.

The flow is turbulent and unsteady, so we predict a late\-time average rather than an instantaneous field,

ϕ¯​\(𝐱\)=1120​s​∫480​s600​sϕ⁡\(𝐱,t\)​𝑑t,\\bar\{\\phi\}\(\\mathbf\{x\}\)=\\frac\{1\}\{120\\,\\mathrm\{s\}\}\\int\_\{480\\,\\mathrm\{s\}\}^\{600\\,\\mathrm\{s\}\}\\phi\(\\mathbf\{x\},t\)\\,\\mathrm\{d\}t,\(43\)taken over a window beginning 450 s after the burner reaches full output\. The averaged field is treated as a static learning target \(Δ​t=0\\Delta t=0\)\.

##### Target fields\.

We predict three cell\-centred averages: the temperature rise above ambient,Δ​T=T−293\.15\\Delta T=T\-293\.15K, and the partial densities\(ρair,ρprod\)\(\\rho\_\{\\rm air\},\\rho\_\{\\rm prod\}\)of the lumped air and combustion\-products species\. These partial densities measure the mass of each mixture per unit volume;ρprod\\rho\_\{\\rm prod\}tracks smoke transport through the building\.

##### Scenario\.

The building measures12×12×1012\\times 12\\times 10m and comprises three levels: a car park at00–44m, a middle level at4\.254\.25–77m, and an upper level at7\.257\.25–1010m\. Concrete slabs separate the levels, with four openings connecting the car park to the middle level and a6×66\\times 6m atrium void connecting the middle and upper levels\. The upper floor is partitioned into adjoining rooms surrounding the atrium void\. A floor\-mounted methane burner has a0\.5×0\.50\.5\\times 0\.5m burning surface atz=0\.25z=0\.25m above the car\-park floor\. Its prescribed heat\-release rate increases linearly over the first 30 s to20002000kW m\-2\(500​k​W500~kW\)\. Two1\.5×1\.01\.5\\times 1\.0m wall vents on opposite faces provide push–pull ventilation: forQ\>0Q\>0, air enters through the vent atx=0x=0and exits through the vent atx=12x=12m;Q<0Q<0reverses the direction\. The enclosure is otherwise closed to gas flow\. Ambient conditions are20∘20\\,^\{\\circ\}C with no stratification\.𝝁\\bm\{\\mu\}contains all prescribed case\-to\-case variation\.

##### Parameter scan\.

Seven randomly sampled parameters control the burner position, the signed ventilation rate, and the position of each wall vent \([Table11](https://arxiv.org/html/2610.22446#A4.T11)\)\. The six positional parameters are restricted to a0\.250\.25m lattice to maintain alignment with the computational mesh, whereas the signed ventilation rate is sampled continuously\. The corpus contains 547 cases\.

ParameterSymbolRangeDiscretisationGroupVisible in the meshBurner centre,xxxfx\_\{\\rm f\}\[2\.25,9\.75\]\[2\.25,9\.75\]m0\.250\.25mfireyes, exactlyBurner centre,yyyfy\_\{\\rm f\}\[2\.25,9\.75\]\[2\.25,9\.75\]m0\.250\.25mfireyes, exactlyVentilation rateQQ\[−4\.0,4\.0\]\[\-4\.0,4\.0\]m3s\-1continuousventilationnoLeft vent centre,yyyLy\_\{\\rm L\}\[1\.0,11\.0\]\[1\.0,11\.0\]m0\.250\.25mventilationnoLeft vent centre,zzzLz\_\{\\rm L\}\[4\.75,6\.5\]\[4\.75,6\.5\]m0\.250\.25mventilationnoRight vent centre,yyyRy\_\{\\rm R\}\[1\.0,11\.0\]\[1\.0,11\.0\]m0\.250\.25mventilationnoRight vent centre,zzzRz\_\{\\rm R\}\[4\.75,6\.5\]\[4\.75,6\.5\]m0\.250\.25mventilationnoTable 11:Building fire: the seven conditioning parameters, their sampling ranges, and positional discretisation for the 547\-case corpus\. Left and right denote the vents atx=0x=0andx=12x=12m, respectively\. The last column indicates whether each parameter is recoverable from the node coordinates; see the redundancy discussion below\.
##### Mesh\.

FDS discretises the domain into 11 rectilinear blocks containing 116,928 Cartesian cells in three cell classes \([Table12](https://arxiv.org/html/2610.22446#A4.T12)\)\. Of these, 111,237 are gas cells and 5,691 lie inside solid obstructions\. Solid cells are excluded from the learning point clouds, targets, field statistics and error metrics\. The counts and index ranges below describe the full Cartesian arrays, before either solid\-cell masking or refinement\-box removal\. A24×24×3224\\times 24\\times 32refinement block, centred horizontally on the burner and extending fromz=0z=0to44m, uses cubic cells of side0\.1250\.125m\. Outside this block, four vertically refined blocks with flattened0\.25×0\.25×0\.1250\.25\\times 0\.25\\times 0\.125m cells form a layer atz=3z=3–44m, immediately below the intermediate floor\. The remaining six blocks use cubic cells of side0\.250\.25m\. We use cell centres as nodes; their pooled bounding box isx,y∈\[0\.125,11\.875\]x,y\\in\[0\.125,11\.875\]m andz∈\[0\.0625,9\.875\]z\\in\[0\.0625,9\.875\]m\. The refinement box follows the burner, resizing the surrounding car\-park blocks and producing 415 distinct point clouds, all with the same node count\. These cases therefore require per\-sample coordinates, as with the heat sink and bracket cases\. The middle and upper meshes are bit\-identical across cases, and the concatenated ordering preserves fixed storey ranges: car park\[0,61632\)\[0,61632\), middle\[61632,89280\)\[61632,89280\)and upper\[89280,116928\)\[89280,116928\)\.

Cell dimensions \[m\]CellsVolume \[m3\]Where0\.12530\.125^\{3\}18,43218\{,\}432\(15\.8%\)3636\(2\.5%\)refinement box around the burner0\.25×0\.25×0\.1250\.25\\times 0\.25\\times 0\.12517,28017\{,\}280\(14\.8%\)135135\(9\.4%\)z=3z=3–44m excluding refinement box0\.2530\.25^\{3\}81,21681\{,\}216\(69\.5%\)12691269\(88\.1%\)everywhere elseTable 12:Building fire: the three cell classes of the 116,928\-cell mesh \(1440 m3in total, including solid obstructions\), before solid\-cell masking and refinement\-box removal\. On this full Cartesian mesh, an unweighted node average over\-weights the refinement box by∼6\.3×\\sim\\\!6\.3\\timesrelative to its volume, so integral quantities of interest over the gas domain must exclude solid cells and be volume\-weighted; the loader carries the cell volume per node for this purpose\.
##### Why this dataset earns its place\.

Its conditioning vector splits in a way no other benchmark here does\. The refinement box is centred on the burner, so the centre of the finest cells reproduces\(xf,yf\)\(x\_\{\\rm f\},y\_\{\\rm f\}\)exactly\. The five ventilation parameters, by contrast, move no cell centre at all\. Two dimensions of𝝁\\bm\{\\mu\}are therefore recoverable from the geometry, and five are invisible in it\. The unconditioned arm is consequently not a geometry\-only model; it knows exactly where the fire is and nothing about the ventilation\. As on the bracket, the loader exposes the fire and ventilation parameter groups of𝝁\\bm\{\\mu\}separately so that they can be ablated apart\.

The case is also*multi\-region*, in a different sense from the heated duct\. The three storeys are separated by slabs that the flow crosses only through the openings, so the fields change near\-discontinuously in value across a slab rather than kinking in gradient across a material interface — but the difficulty for a singleC2C^\{2\}basis on a single latent grid is the same\. The storeys serve as the region set for the per\-region encoding and decoding branches described for the heated duct\. The 11 FDS blocks cannot be used for this, because four of their boundaries move with the fire and block membership is therefore a per\-sample property\.

##### Prediction domain: the smoke, not the flame\.

A fire\-safety engineer does not run this simulation to learn what happens inside the flame\. The design fire is prescribed: the burner’s position and output are inputs to the study, not results of it\. The question is whether the rest of the building stays tenable, meaning how hot the gas gets, how far the smoke spreads and how much combustion product reaches the car park and the upper storeys, where occupants are and where the escape routes run\. The error metric pulls the other way\. The18,40018\{,\}400gas nodes of the refinement box around the burner are 16\.5% of the gas\-cell cloud but carry roughly two thirds of the error energy, because they contain the flame itself \(Δ​T\\Delta Tup to 1002 K there, against 189 K everywhere else\)\. With them left in, every joint metric on this case would score the flame, which is the one region the engineer does not need predicted\.

We therefore remove the box’s gas nodes from the prediction domain, leaving92,83792\{,\}837nodes:43,20043\{,\}200in the car park and49,63749\{,\}637in the middle and upper storeys\. The task becomes predicting heat and combustion\-product transport into the occupied building, away from the source\. Removing the box does not hide where the fire is, because the surrounding mesh blocks still abut it and the retained cloud still moves with the burner\. It does make that information far less accessible\. The two fire\-position dimensions of𝝁\\bm\{\\mu\}can still be recovered from the geometry, but they are no longer cheap to extract\([Panayis et al\., 2025](https://arxiv.org/html/2610.22446#bib.bib45)\)\.

##### Learning setup\.

𝐅∈ℝ547×6×98496\\mathbf\{F\}\\in\\mathbb\{R\}^\{547\\times 6\\times 98496\}with channels\(x,y,z,Δ​T,ρair,ρprod\)\(x,y,z,\\Delta T,\\rho\_\{\\rm air\},\\rho\_\{\\rm prod\}\)andcin=3c\_\{\\mathrm\{in\}\}=3, giving inputsℝN×3×98496\\mathbb\{R\}^\{N\\times 3\\times 98496\}and targetsℝN×3×98496\\mathbb\{R\}^\{N\\times 3\\times 98496\};𝝁∈ℝ547×7\\bm\{\\mu\}\\in\\mathbb\{R\}^\{547\\times 7\}\. The coordinates are consumed as geometry by the per\-sample path, so the network again sees the 99\-channel encoding and𝝁\\bm\{\\mu\}only\. One departure from the common convention: because this mesh has two cell sizes, the integrated positional encoding is given a per\-node footprint, attenuating frequency bandffbyexp⁡\(−12​f2​σ2\)\\exp\(\-\\tfrac\{1\}\{2\}f^\{2\}\\sigma^\{2\}\)withσ=h/12\\sigma=h/\\sqrt\{12\}the standard deviation of a uniform distribution over the node’s cell of sidehh\(in normalised units\); on every other benchmarkσ=0\\sigma=0and the encoding reduces to the plain sinusoidal one\. The storey index is carried alongside the mesh for the region branches and is not an input\. The split is a random 70/30 permutation \(383 train, 164 test\) at batch size 4\. It is sound here, unlike on the aircraft: the cases were submitted in batches, but the within\-batch parameter spread matches the corpus spread on all seven dimensions, so the batches are not design clusters\.

##### Data statistics\.

Per\-channel statistics over the retained gas nodes are given in[Table13](https://arxiv.org/html/2610.22446#A4.T13)\. The three targets sit at very different scales, so a joint relativeL2L\_\{2\}over the raw channels \([Equation39](https://arxiv.org/html/2610.22446#A4.E39)\) comes down to the error inΔ​T\\Delta Talone\. We therefore report the per\-variable normalised error\. Unlike the other thermal cases, the temperature target is a rise over ambient and carries no pedestal\. The cases differ substantially from one another: the case\-mean temperature rise and the case\-mean product density each vary by roughly a factor of two across the corpus\. Air density varies least, only a few percent about its mean, and that small spread makes it the hardest of the three targets to predict in normalised terms\.

ChannelminmaxmeanstdΔ​T\\Delta T\[K\]00189\.4189\.439\.3239\.3216\.6216\.62ρair\\rho\_\{\\rm air\}\[kg m\-3\]0\.7630\.7631\.6301\.6301\.2471\.2470\.08450\.0845ρprod\\rho\_\{\\rm prod\}\[kg m\-3\]000\.10070\.10070\.04620\.04620\.01640\.0164Table 13:Building fire: per\-channel statistics over all 547 cases at the92,83792\{,\}837retained gas nodes \(solid cells and refinement box removed\)\.![Refer to caption](https://arxiv.org/html/2610.22446v1/fds_temperature_rise_floor_slices_film_best.png)Figure 13:Error maps produced by various architectures in modelling temperature change across the building due to a fire in the lower floor\.

### D\.6Heated duct: conjugate heat transfer

##### Governing equations\.

Steady conjugate heat transfer over three materials — a laminar water duct, the metal shell around it, and a volumetrically heated block beneath — coupled only through two interfaces:

fluid:−∇⋅\(kf∇T\)\+ρcp𝐔⋅∇T\\displaystyle\-\\nabla\\\!\\cdot\\\!\\left\(k\_\{f\}\\nabla T\\right\)\+\\rho c\_\{p\}\\,\\mathbf\{U\}\\\!\\cdot\\\!\\nabla T=0,\\displaystyle=0,\(44\)metal:−∇⋅\(km∇T\)\\displaystyle\-\\nabla\\\!\\cdot\\\!\\left\(k\_\{m\}\\nabla T\\right\)=0,\\displaystyle=0,\(45\)heater:−∇⋅\(kh∇T\)\\displaystyle\-\\nabla\\\!\\cdot\\\!\\left\(k\_\{h\}\\nabla T\\right\)=qh,\\displaystyle=q\_\{h\},\(46\)subject at each interface to continuity of temperature and of normal heat flux,

Ta=Tb,ka​∂Ta∂n=kb​∂Tb∂n\.T\_\{a\}=T\_\{b\},\\qquad k\_\{a\}\\frac\{\\partial T\_\{a\}\}\{\\partial n\}=k\_\{b\}\\frac\{\\partial T\_\{b\}\}\{\\partial n\}\.\(47\)The flow is laminar \(Re=21\\mathrm\{Re\}=21–8383\) and, with constant properties and gravity switched off, the momentum equation does not depend on temperature — a fact with sharp consequences for the parameter scan, taken up below\. Data were generated with the OpenFOAM Foundation v13multiRegion/CHT/heatedDucttutorial geometry\([Weller et al\., 1998](https://arxiv.org/html/2610.22446#bib.bib58)\), run to steady state withfoamMultiRun\.

##### Why this dataset earns its place\.

It is the only multi\-material case in the set\. The three regions are three separate meshes that share no cells: 40,560 fluid, 67,440 metal and 16,800 heater cell centres, which the loader concatenates into one 124,800\-node cloud so a single operator spans all of them\. They occupy disjoint sub\-volumes of one bounding box — the heater sits below the duct aty<0y<0, the fluid inside it, the metal in the shell between — so the concatenated cloud is a partition of the domain rather than an overlay, and region membership is a function of position\. That is why the region label can be withheld: the default configuration feeds coordinates alone and the model must read the material off the geometry, while a one\-hot variant hands it over explicitly and measures what naming the material is worth\.

The interfaces are conformal to machine precision — 15,360 matched metal–fluid face pairs and 3,360 heater–metal pairs, with a maximum pair separation of8×10−178\\times 10^\{\-17\}m — so[Equation47](https://arxiv.org/html/2610.22446#A4.E47)is imposed on coincident points and the kink in∇T\\nabla Tis a property of the solution rather than of the discretisation\.

##### Geometry\.

The pooled bounding box isx∈\[0\.5,49\.5\]x\\in\[0\.5,49\.5\],y∈\[−9\.5,49\.5\]y\\in\[\-9\.5,49\.5\],z∈\[0\.5,149\.5\]z\\in\[0\.5,149\.5\]mm\. The structure is cross\-sectional and the geometry is prismatic inzz: the fluid occupiesx,y∈\[5\.6,44\.4\]x,y\\in\[5\.6,44\.4\]mm, leaving a 5\.0 mm metal shell, and the heater spansx∈\[8\.0,42\.0\]x\\in\[8\.0,42\.0\],y∈\[−9\.5,−0\.6\]y\\in\[\-9\.5,\-0\.6\]mm\. Under the per\-axis min–max scaling this puts the fluid–metal wall at0\.1030\.103and0\.8970\.897alongxx, and at0\.2550\.255and0\.9150\.915alongyy, so a latent grid must resolve roughly a tenth of its own extent to see the wall at all\. This is why the latent grids for this case are sized to feature scale inxxandyyrather than to the49:59:14949:59:149bounding\-box aspect, which would put the resolution along the one axis that carries no structure\.

##### Parameter scan\.

Five parameters, drawn on a Sobol sequence and listed in[Table14](https://arxiv.org/html/2610.22446#A4.T14)\. Of 544 proposed designs, 146 converged and were accepted; the rest were rejected for an estimated peak temperature above 900 K or for failing the convergence gate\. Acceptance is tight: heater power spans 131–1050 W, matched by the advected power to an energy imbalance of at most9\.6×10−49\.6\\times 10^\{\-4\}, with a maximum solver residual of2\.3×10−52\.3\\times 10^\{\-5\}\.

ParameterSymbolPhysical rangeTransformShapeHeater power densityqhq\_\{h\}2\.5×1062\.5\\times 10^\{6\}–2\.0×1072\.0\\times 10^\{7\}W m\-3log⁡\(qh/107\)\\log\(q\_\{h\}/10^\{7\}\)noInlet velocityUinU\_\{\\rm in\}5×10−45\\times 10^\{\-4\}–2×10−32\\times 10^\{\-3\}m s\-1Uin/10−3U\_\{\\rm in\}/10^\{\-3\}yesInlet temperatureTinT\_\{\\rm in\}280280–320320K\(Tin−300\)/78\.5\(T\_\{\\rm in\}\-300\)/78\.5noHeater conductivitykhk\_\{h\}4040–120120W m\-1K\-1kh/80k\_\{h\}/80yesMetal conductivitykmk\_\{m\}100100–300300W m\-1K\-1km/200k\_\{m\}/200yesTable 14:Heated duct: the five conditioning parameters\.𝝁\\bm\{\\mu\}is the dimensionless transform, not the raw value — only thelog\\logonqhq\_\{h\}, which spans a factor of eight, is non\-trivial\. The last column records whether the parameter changes the*shape*of the temperature field or only rescales it; see the degeneracy noted below\.
##### A degeneracy in two of the five dimensions\.

Because the momentum equation decouples from temperature, the coupled energy problem is linear inTTand admits the exact separation

T⁡\(𝐱,qh,Uin,Tin,kh,km\)=Tin\+qh​Θ​\(𝐱,Uin,kh,km\)\.T\(\\mathbf\{x\};\\,q\_\{h\},U\_\{\\rm in\},T\_\{\\rm in\},k\_\{h\},k\_\{m\}\)=T\_\{\\rm in\}\+q\_\{h\}\\,\\Theta\(\\mathbf\{x\};\\,U\_\{\\rm in\},k\_\{h\},k\_\{m\}\)\.\(48\)Two of the five parameters therefore carry no shape information at all: they are an exact affine rescaling of the field, and onlyUinU\_\{\\rm in\},khk\_\{h\}andkmk\_\{m\}changeΘ\\Theta\. This has to be read into any conditioning ablation on this case, because a method that captures theqhq\_\{h\}/TinT\_\{\\rm in\}affinity exactly and getsΘ\\Thetawrong will still post a respectable relativeL2L\_\{2\}\.

##### The interface is where the gridded family is stressed\.

Temperature is continuous across a conjugate interface but its gradient is not, and the jump is set by the conductivity ratio in[Equation47](https://arxiv.org/html/2610.22446#A4.E47)— herekh∈\[40,120\]k\_\{h\}\\in\[40,120\]againstkm∈\[100,300\]k\_\{m\}\\in\[100,300\]W m\-1K\-1\. Every gridded backbone in the comparison scatters all three regions onto*one*latent grid through*one*C2C^\{2\}basis, and no number of retained modes lets aC2C^\{2\}function reproduce a kink\. The effect is measurable rather than notional: on the\[48,48,32\]\[48,48,32\]latent grid, 24\.0% of covered knots receive mass from more than one region, and on a coarser\[32,32,24\]\[32,32,24\]grid that rises to 37\.8%\. Each such knot holds a single number serving two materials, at exactly the location where the kink should be\. We treat this as a property of the benchmark to be measured, not a defect to be hidden\.

##### Split\.

The corpus ships a designed split — 85 train, 18 validation, 19 interpolation test and 24 extrapolation corners — and the loader honours it by permuting the samples so the pipeline’s own seeded split lands exactly on the design\. The default mode trains on train plus validation \(103 cases\) and tests on the 19 interpolation cases, dropping the corners; that is theN=122N=122recorded in[Table8](https://arxiv.org/html/2610.22446#A4.T8)\. An extrapolation mode instead trains on all 122 and tests on the 24 corners, answering the harder question of performance outside the sampled parameter box\. A plain random split over all 146 is available but leaks the corner design into training and is not used\.

##### Learning setup\.

𝐅∈ℝ122×1×124800\\mathbf\{F\}\\in\\mathbb\{R\}^\{122\\times 1\\times 124800\}carryingTTalone, withcin=0c\_\{\\mathrm\{in\}\}=0: there are no field channels whatsoever, so inputs areℝN×0×124800\\mathbb\{R\}^\{N\\times 0\\times 124800\}and the network sees only the 99\-channel encoded coordinates,ℝBbatch×124800×99\\mathbb\{R\}^\{B\_\{\\mathrm\{batch\}\}\\times 124800\\times 99\}, with targetsℝN×1×124800\\mathbb\{R\}^\{N\\times 1\\times 124800\}\.𝝁∈ℝ122×5\\bm\{\\mu\}\\in\\mathbb\{R\}^\{122\\times 5\}; batch size 4\. The per\-node region index is carried alongside the mesh for masked losses and per\-region metrics, but is not an input under the default configuration\.

##### Data statistics\.

T∈\[280\.00,880\.63\]T\\in\[280\.00,880\.63\]K with mean 461\.0 K and standard deviation 118\.1 K\. WithSkew=0\.87\\mathrm\{Skew\}=0\.87andKurt=3\.01\\mathrm\{Kurt\}=3\.01this is much the most nearly Gaussian target in the set \([Figure9](https://arxiv.org/html/2610.22446#A4.F9)\), which is a direct consequence of[Equation48](https://arxiv.org/html/2610.22446#A4.E48): the field is a smoothΘ\\Thetascaled and shifted by two broadly sampled parameters, with none of the shock or stress\-concentration tails that make the aircraft and bracket targets heavy\. Case\-mean temperatures span\[319\.5,732\.2\]\[319\.5,732\.2\]K, a factor of 2\.3, so the case\-to\-case variation is far larger than on any other benchmark here\.

![Refer to caption](https://arxiv.org/html/2610.22446v1/heated_duct_error_maps_all.png)Figure 14:Error maps produced by various architectures in modelling the temperature distribution from the conjugate heat transfer across the heated duct\.

### D\.7HIVE monoblock

##### Physics\.

HIVE \(Heating by Induction to Verify Extremes\) is a UKAEA facility for testing fusion divertor components under reactor\-relevant heat loads\([Hancock, 2018](https://arxiv.org/html/2610.22446#bib.bib18)\)\. The simulation is a coupled multiphysics MOOSE model\([Permann et al\., 2020](https://arxiv.org/html/2610.22446#bib.bib48);[Nuno Nobre, Josh Williams and Karthikeyan Chockalingam, 2024](https://arxiv.org/html/2610.22446#bib.bib43)\): an induction coil drives eddy currents in a divertor monoblock, and the resulting Joule heating is the source term of a conduction problem\([Horne\-Jones et al\., 2026](https://arxiv.org/html/2610.22446#bib.bib22)\),

ρcp∂T∂t=∇⋅\(κ∇T\)\+qEM\(𝐱\),−κ∂T∂n=h\(T−Tcool\)onΓpipe,\\rho c\_\{p\}\\frac\{\\partial T\}\{\\partial t\}=\\nabla\\\!\\cdot\\\!\\left\(\\kappa\\nabla T\\right\)\+q\_\{\\mathrm\{EM\}\}\(\\mathbf\{x\}\),\\qquad\-\\kappa\\frac\{\\partial T\}\{\\partial n\}=h\\,\(T\-T\_\{\\mathrm\{cool\}\}\)\\ \\ \\text\{on\}\\ \\Gamma\_\{\\mathrm\{pipe\}\},\(49\)run forward to convergence\. The coolant\-side film coefficient is set by a Nusselt correlation of the standard form

Nu=C​Rea​Prb,h=Nu​κf/Dh,\\mathrm\{Nu\}=C\\,\\mathrm\{Re\}^\{a\}\\,\\mathrm\{Pr\}^\{b\},\\qquad h=\\mathrm\{Nu\}\\,\\kappa\_\{\\mathrm\{f\}\}/D\_\{\\mathrm\{h\}\},\(50\)whose three coefficients\(C,a,b\)\(C,a,b\)are the epistemically uncertain quantities of the study, alongside the three components of the coil\-to\-target offset\.

##### Parameter scan\.

Latin hypercube sampling over the six independent uniform priors of[Table15](https://arxiv.org/html/2610.22446#A4.T15)\. The launched ensemble contains some failed runs; we use the 186 that completed, listed in the release’s completeness manifest\. The two groups act on the physics very differently — the coil offsets move where the induced heating is deposited, while the Nusselt coefficients rescale the coolant\-side film coefficient\.

ParameterGroupPriorRealised rangeRolenusselt\_scaleHTC𝒰⁡\(0\.001,0\.031\)\\mathcal\{U\}\(0\.001,0\.031\)\[0\.00112,0\.03090\]\[0\.00112,0\.03090\]CCin[Equation50](https://arxiv.org/html/2610.22446#A4.E50)nusselt\_re\_powerHTC𝒰⁡\(0\.8,0\.9\)\\mathcal\{U\}\(0\.8,0\.9\)\[0\.80043,0\.89950\]\[0\.80043,0\.89950\]aain[Equation50](https://arxiv.org/html/2610.22446#A4.E50)nusselt\_pr\_powerHTC𝒰⁡\(0\.35,0\.45\)\\mathcal\{U\}\(0\.35,0\.45\)\[0\.35048,0\.44997\]\[0\.35048,0\.44997\]bbin[Equation50](https://arxiv.org/html/2610.22446#A4.E50)coil\_target\_offset\_xgeometry𝒰⁡\(−0\.01,0\.01\)\\mathcal\{U\}\(\-0\.01,0\.01\)m\[−0\.00996,0\.00995\]\[\-0\.00996,0\.00995\]coil positioncoil\_target\_offset\_ygeometry𝒰⁡\(−0\.01,0\.01\)\\mathcal\{U\}\(\-0\.01,0\.01\)m\[−0\.00994,0\.00996\]\[\-0\.00994,0\.00996\]coil positioncoil\_target\_offset\_zgeometry𝒰⁡\(0\.019,0\.039\)\\mathcal\{U\}\(0\.019,0\.039\)m\[0\.01901,0\.03892\]\[0\.01901,0\.03892\]coil standoffTable 15:HIVE: the six uncertain parameters, their sampling priors and the ranges realised over the 186 completed runs\.
##### Mesh and a node\-ordering subtlety\.

The solid meshes \(coil and target\) are identical across runs, with the coil displaced relative to a stationary monoblock\. A separate tool then builds a vacuum mesh around the two bodies, and that step permutes the point ordering on the monoblock\. The release resolves this by taking the first sample’s mesh as a template and storing, per run, the permutation that matches it; we apply those maps, after which all 186 runs share a single 194,691\-node ordering\. The preprocessed archive retains only that template’s coordinates — and the coil is in any case not part of the stored mesh — so the per\-run coil displacement is*not*visible in the geometry presented to the operator\. It reaches the model through𝝁\\bm\{\\mu\}alone, as does everything else that distinguishes one run from another\. The bounding box isx∈\[−11\.5,11\.5\]x\\in\[\-11\.5,11\.5\],y∈\[−75,75\]y\\in\[\-75,75\],z∈\[−11\.5,19\.5\]z\\in\[\-11\.5,19\.5\]mm\. Unlike the other 3D datasets, no spatial subsampling is applied: all 194,691 nodes are used\.

##### Learning setup\.

𝐅∈ℝ186×1×194691\\mathbf\{F\}\\in\\mathbb\{R\}^\{186\\times 1\\times 194691\}carrying the convergedTfinalT\_\{\\mathrm\{final\}\}alone, withcin=0c\_\{\\mathrm\{in\}\}=0: inputs areℝN×0×194691\\mathbb\{R\}^\{N\\times 0\\times 194691\}, the network sees only the 99\-channel encoded coordinates,ℝBbatch×194691×99\\mathbb\{R\}^\{B\_\{\\mathrm\{batch\}\}\\times 194691\\times 99\}, and targets areℝN×1×194691\\mathbb\{R\}^\{N\\times 1\\times 194691\}\.𝝁∈ℝ186×6\\bm\{\\mu\}\\in\\mathbb\{R\}^\{186\\times 6\}; split 80/20 \(149 train, 37 test\) at batch size 5\. No spatial subsampling is applied, so this is the largest per\-sample node count in the set\.

![Refer to caption](https://arxiv.org/html/2610.22446v1/hive_test_case_19_error_maps_all.png)Figure 15:Error maps produced by various architectures in modelling the temperature distribution from inductive heating in a monoblock in the HIVE case\.With one shared mesh and no field input, the model is fitting a6→ℝ1946916\\to\\mathbb\{R\}^\{194691\}map from roughly 148 training points in a six\-dimensional design space\. The conditioning module, not the backbone, is expected to be the bottleneck — which is precisely what makes this case a measurement of the conditioning pathway rather than of the operator\. The corpus also ships the first stored timestepTinitialT\_\{\\mathrm\{initial\}\}; a companion configuration feeds it in as a field channel at the same seed and split, so the pair measures directly what an initial field is worth against the parameter path\.

##### Data statistics\.

Tfinal∈\[300\.00,557\.23\]T\_\{\\mathrm\{final\}\}\\in\[300\.00,557\.23\]K with mean 341\.43 K and standard deviation 36\.67 K \(Skew=1\.30\\mathrm\{Skew\}=1\.30,Kurt=4\.78\\mathrm\{Kurt\}=4\.78\)\. Case\-mean temperatures span\[310\.98,414\.67\]\[310\.98,414\.67\]K\. As on the heat sink, the∼300\\sim\\\!300K pedestal flatters[Equation39](https://arxiv.org/html/2610.22446#A4.E39): a single global constant scores 9\.3% and a per\-case constant 6\.5%\. A linear fit of𝝁\\bm\{\\mu\}to the case mean explainsR2=0\.89R^\{2\}=0\.89of the case\-to\-case variation and scores 6\.9%, so the six parameters set the bulk thermal level well and what the operator must actually supply is the spatial structure\. For scale, the stored initial fieldTinitialT\_\{\\mathrm\{initial\}\}sits at\[294\.06,405\.23\]\[294\.06,405\.23\]K and the rise to convergence spans\[0\.03,162\.58\]\[0\.03,162\.58\]K with mean 39\.34 K — the field the operator predicts is far from the state the simulation started in\.

## Appendix EMAST\-U: full\-device structural reconstruction

##### The device\.

MAST\-U is the upgraded successor to MAST, the Mega\-Ampere Spherical Tokamak at UKAEA’s Culham site\([Milnes et al\., 2015](https://arxiv.org/html/2610.22446#bib.bib40)\)\. A spherical tokamak has a low aspect ratio: the plasma wraps closely around a slender centre column, so magnets, vessel, and in\-vessel structures are packed into a compact volume\. MAST\-U places many of its coils inside the main vacuum vessel\([Cunningham, 2013](https://arxiv.org/html/2610.22446#bib.bib8)\), and adds closed upper and lower divertors that can run configurations from conventional to the long\-leg Super\-X\([Morris et al\., 2018](https://arxiv.org/html/2610.22446#bib.bib41)\)\. Its experimental programme targets the integration of core, pedestal, and exhaust physics\([Harrison et al\., 2024](https://arxiv.org/html/2610.22446#bib.bib20)\), and it informs the design of the Spherical Tokamak for Energy Production \(STEP\)\([Chapman et al\., 2024](https://arxiv.org/html/2610.22446#bib.bib6)\)\. Structurally, the machine is a6\.86\.8m tall,6\.26\.2m wide assembly of roughly110110tonnes\([Ellis et al\., 2025](https://arxiv.org/html/2610.22446#bib.bib13)\), held on four support legs\.

##### Why a whole\-device model\.

Fusion components are usually analysed one at a time, with loads from the rest of the machine prescribed as boundary conditions\. For a tightly coupled assembly those boundary conditions are not known in advance: temperature, stress, and magnetic loads vary with position and are related through the operating state of the whole device, so a component\-scale view makes the mechanical equilibrium problem difficult even to pose\.[Ellis et al\. \(2025\)](https://arxiv.org/html/2610.22446#bib.bib13)demonstrates a single mechanical model of the complete MAST\-U structure from3,9383\{,\}938CAD volumes, using a conformal tetrahedral mesh and the solid\-mechanics module of MOOSE\([Permann et al\., 2020](https://arxiv.org/html/2610.22446#bib.bib48)\), as a first step towards a structural digital twin\. The paper describes approximately127127million elements, whereas the archived mesh used here contains134,609,743134\{,\}609\{,\}743four\-node tetrahedra and29,183,75429\{,\}183\{,\}754unique nodes\. Each static solve takes about500500CPU\-hours on600600–800800cores\([Ellis et al\., 2025](https://arxiv.org/html/2610.22446#bib.bib13)\)\. A digital twin must be queried far more often than that cost allows, across load cases, operating scenarios, material uncertainty, and design changes, which is the role a trained surrogate would fill\([Pamela et al\., 2025](https://arxiv.org/html/2610.22446#bib.bib44)\)\. We take the first step of that programme: a surrogate conditioned on the two parameters of a thermal\-load parameter study\.

##### Governing equations\.

For small\-strain linear elasticity, displacement𝐮:Ω→ℝ3\\mathbf\{u\}:\\Omega\\to\\mathbb\{R\}^\{3\}satisfies

∇⋅𝝈\+ρ​𝐠=𝟎,𝝈=ℂ⁡\(ω\):\(𝜺⁡\(𝐮\)−𝜺th\),𝜺th=α⁡\(T⁡\(𝐱\)−Tsf\)​𝐈,\\nabla\\\!\\cdot\\bm\{\\sigma\}\+\\rho\\mathbf\{g\}=\\mathbf\{0\},\\qquad\\bm\{\\sigma\}=\\mathbb\{C\}\(\\omega\):\\bigl\(\\bm\{\\varepsilon\}\(\\mathbf\{u\}\)\-\\bm\{\\varepsilon\}^\{\\mathrm\{th\}\}\\bigr\),\\qquad\\bm\{\\varepsilon\}^\{\\mathrm\{th\}\}=\\alpha\\,\\bigl\(T\(\\mathbf\{x\}\)\-T\_\{\\mathrm\{sf\}\}\\bigr\)\\mathbf\{I\},\(51\)with𝜺⁡\(𝐮\)=12​\(∇𝐮\+\(∇𝐮\)⊤\)\\bm\{\\varepsilon\}\(\\mathbf\{u\}\)=\\tfrac\{1\}\{2\}\\bigl\(\\nabla\\mathbf\{u\}\+\(\\nabla\\mathbf\{u\}\)^\{\\top\}\\bigr\),ℂ\\mathbb\{C\}the piecewise isotropic stiffness tensor,α\\alphathe thermal expansion coefficient andTsfT\_\{\\mathrm\{sf\}\}the stress\-free temperature\. Displacement is fixed at the four support\-leg footprints\. Stainless steel, copper and graphite have Young’s moduli\(210,125,11\)\(210,125,11\)GPa, densities\(7800,8960,2260\)\(7800,8960,2260\)kg m\-3and a common Poisson ratio of0\.30\.3\([Ellis et al\., 2025](https://arxiv.org/html/2610.22446#bib.bib13)\)\. Merged material interfaces share nodes, enforcing displacement continuity while strain jumps across the stiffness discontinuity\. We predict nodal\(ux,uy,uz\)\(u\_\{x\},u\_\{y\},u\_\{z\}\)in metres; element\-wise stresses are not targets\.

##### Parameter scan\.

The corpus contains a stochastic parameter sampling study giving7070simulations on the same device\. Each solve uses1,1201\{,\}120MPI ranks and samples the two parameters of[Table16](https://arxiv.org/html/2610.22446#A5.T16)over a rectangle\. Each sub\-application writes its own partitioned Nemesis/Exodus output\. The sampled inputs are read from the corresponding study’s parameter table, with the study identifier and sub\-application index cross\-checked against the row; the output files themselves carry no inputs\. The productα​Tw\\alpha T\_\{w\}, which sets the thermal\-loading scale, spans a factor of3\.363\.36\(5\.32×10−35\.32\\times 10^\{\-3\}to1\.79×10−21\.79\\times 10^\{\-2\}\)\.

ParameterSymbolUnitRealised rangeMeanRoleThermal expansion coefficientα\\alpha10−510^\{\-5\}K\-11\.0151\.015–1\.9821\.9821\.501\.50eigenstrain scaleWall temperatureTwT\_\{w\}K508\.6508\.6–996\.2996\.2750\.1750\.1thermal boundary conditionTable 16:MAST\-U: the two conditioning parameters over the pooled7070solves\. Both are min–max normalised per variable using the pooled corpus ranges before entering the conditioning module\.
##### Mesh\.

All7070solves share one mesh with29,183,75429\{,\}183\{,\}754nodes and134,609,743134\{,\}609\{,\}743four\-node tetrahedra; cache construction checks each new solve’s coordinates against the reference mesh node for node\. Comparison with the single archived solve of[Ellis et al\. \(2025\)](https://arxiv.org/html/2610.22446#bib.bib13)confirms exactly matching coordinates, per\-node material membership and lumped control volumes\. The1,1201\{,\}120\-rank decomposition does, however, renumber nodes relative to that archived solve, so the two must be matched by coordinate rather than global ID\. The mesh is strongly multiscale: merging thousands of parts leaves sliver elements, with edge lengths from1\.4×10−61\.4\\times 10^\{\-6\}to4\.7×10−24\.7\\times 10^\{\-2\}m around a33mm median\. Node count is therefore a poor proxy for volume: the lumped nodal volume spans eleven orders of magnitude\. Steel, copper and graphite occupy64\.4%64\.4\\%,22\.3%22\.3\\%and13\.3%13\.3\\%of the13\.1113\.11m3of solid and meet at1,290,1611\{,\}290\{,\}161shared interface nodes \(1,164,4571\{,\}164\{,\}457steel/copper,125,704125\{,\}704steel/graphite, none copper/graphite\)\.

##### Split\.

The split contains6060training solves and1010named test solves\. All ten lie inside the training parameter ranges, so the task measures interpolation in𝝁\\bm\{\\mu\}\.

##### Learning setup\.

The full corpus is𝐅∈ℝ70×3×29,183,754\\mathbf\{F\}\\in\\mathbb\{R\}^\{70\\times 3\\times 29\{,\}183\{,\}754\}, carrying\(ux,uy,uz\)\(u\_\{x\},u\_\{y\},u\_\{z\}\)in metres, withcin=0c\_\{\\mathrm\{in\}\}=0and𝝁∈ℝ70×2\\bm\{\\mu\}\\in\\mathbb\{R\}^\{70\\times 2\}\. Training and the ten\-case evaluation uses the entire 29\.2M nodes with material specific routing for the mesh transfer\. The shared mesh allows positional encodings and spline stencils to be computed once and reused\. Native\-mesh inference for[Figure5](https://arxiv.org/html/2610.22446#S4.F5)uses all29,183,75429\{,\}183\{,\}754nodes as both sources and queries\.

##### Data statistics\.

Over all7070solves and all native nodes,ux∈\[−3\.70,3\.77\]u\_\{x\}\\in\[\-3\.70,3\.77\]cm anduy∈\[−3\.71,3\.73\]u\_\{y\}\\in\[\-3\.71,3\.73\]cm are approximately centred \(means−0\.004\-0\.004and0\.0030\.003cm, standard deviations0\.920\.92and0\.980\.98cm\), whileuz∈\[−0\.18,5\.43\]u\_\{z\}\\in\[\-0\.18,5\.43\]cm has mean2\.102\.10cm and standard deviation0\.920\.92cm: heating expands the machine upward and outward from its supports\. With equal weight per node,uzu\_\{z\}carries74\.3%74\.3\\%of‖𝐮‖22\\\|\\mathbf\{u\}\\\|\_\{2\}^\{2\}, compared with12\.1%12\.1\\%and13\.7%13\.7\\%foruxu\_\{x\}anduyu\_\{y\}, so the joint relativeL2L\_\{2\}is weighted towards the vertical\. Displacement magnitude peaks at5\.895\.89cm and the case\-mean magnitude spans\[1\.52,3\.54\]\[1\.52,3\.54\]cm, a factor of2\.32\.3\. Displacement is pinned to zero at the supports; predicting zero scores exactly100%100\\%under[Equation39](https://arxiv.org/html/2610.22446#A4.E39)\.

##### Model and optimisation\.

The model used for[Figure5](https://arxiv.org/html/2610.22446#S4.F5)is FiLM\-conditioned L\-SplineFNO with a120×120×173120\\times 120\\times 173latent grid, four dense FNO layers of width3232,1212Fourier modes, spline scatter scales\(1,2,4\)\(1,2,4\)and material\-specific lifting, projection and spline kernels around a shared spectral backbone;28,359,62728\{,\}359\{,\}627trainable parameters\. A local bypass carries the conditioned lifted node features directly to the decoder, where they are concatenated with the decoded grid features and passed through a material\-specific GELU readout with hidden width6464\. Coordinates are min–max normalised per axis and encoded with integrated positional encoding at1616frequencies; displacement components are standardised separately\. We train for500500epochs with Adam, an initial learning rate of10−310^\{\-3\}, cosine decay to zero, gradient clipping at1\.01\.0, the relativeL2L\_\{2\}loss on standardised targets and deterministic kernels\. The training takes18\.718\.7h on one Nvidia H100 with9696GB\.

##### Native\-mesh reconstruction\.

[Figure5](https://arxiv.org/html/2610.22446#S4.F5)shows displacement magnitude and absolute vector error for the held\-out solve with\(α,Tw\)=\(1\.43283×10−5​K−1,863\.979​K\)\(\\alpha,T\_\{w\}\)=\(1\.43283\\times 10^\{\-5\}~\\mathrm\{K\}^\{\-1\},863\.979~\\mathrm\{K\}\), on the undeformed device with the front quarter removed\. All native source nodes contribute to a single global encoding and FNO evaluation, and all29,183,75429\{,\}183\{,\}754query nodes are decoded in chunks\. This full\-mesh evaluation reaches a relative vectorL2L\_\{2\}error of3\.40%3\.40\\%on this case, compared with1\.60%1\.60\\%for its saved prediction on the sampled nodes\. The ten\-case mean above therefore describes the sampled\-node evaluation; the native\-mesh result additionally changes the source set used to construct the latent field\. The figure uses millimetres and caps the error colour scale at the9999th percentile,2\.872\.87mm; the maximum nodal vector error is13\.0213\.02mm\.

## Appendix FModels

This appendix describes the six baselines against which the spline\-gridded operators are compared\. They fall into three families:*point\-set transformers*that never leave the mesh \(Transolver, GNOT\),*latent\-token transformers*that compress the mesh into a fixed\-size bottleneck \(UPT, MSPT\), and*gridded neural operators*that transfer the mesh onto a uniform latent grid so that an FNO can be applied \(NUNO, GINO\)\. The last family is the one our method belongs to, and the transfer operators used by NUNO and GINO — nearest\-cell scatter and a radius\-graph kernel integral, respectively — are exactly the components the spline encode/decode pair replaces\.

All six consume the same per\-node tensor produced by the pipeline,

𝐱i=\[ωi∥u0\(ωi\)\]∈ℝd\+cin,i=1,…,N,\\mathbf\{x\}\_\{i\}\\;=\\;\\big\[\\,\\omega\_\{i\}\\;\\big\\\|\\;u\_\{0\}\(\\omega\_\{i\}\)\\,\\big\]\\in\\mathbb\{R\}^\{d\+c\_\{\\mathrm\{in\}\}\},\\qquad i=1,\\dots,N,\(52\)whereωi∈Ω\\omega\_\{i\}\\in\\Omegais the coordinate of mesh nodeiiandu0​\(ωi\)u\_\{0\}\(\\omega\_\{i\}\)the input field features at that node, and all six emit𝒴∈ℝN×cout\\mathcal\{Y\}\\in\\mathbb\{R\}^\{N\\times c\_\{\\mathrm\{out\}\}\}at the same nodes\. In all six benchmarks there is no input field, and the tensor is the positional encoding of[Equation3](https://arxiv.org/html/2610.22446#S3.E3),𝐱i=\[ω¯i∥φ\(ω¯i\)\]∈ℝ3\+96\\mathbf\{x\}\_\{i\}=\[\\bar\{\\omega\}\_\{i\}\\,\\\|\\,\\varphi\(\\bar\{\\omega\}\_\{i\}\)\]\\in\\mathbb\{R\}^\{3\+96\}\. Conditioning on𝝁∈𝒫\\bm\{\\mu\}\\in\\mathcal\{P\}enters through the shared conditioning module \(FiLM, concatenation, or cross\-attention\) applied to the encoded featuresφ⁡\(ω¯i\)\\varphi\(\\bar\{\\omega\}\_\{i\}\)before each model’s input embedding, with the leadingddcoordinate channels passed through unmodulated, except where a model has its own native mechanism — GINO’s adaptive instance normalisation, noted below\. Throughout,ccdenotes the model width,LN\\mathrm\{LN\}layer normalisation, andMHA⁡\(Q,K,V\)=Softmax⁡\(Q​K⊤/ch\)​V\\mathrm\{MHA\}\(Q,K,V\)=\\mathrm\{Softmax\}\\\!\\big\(QK^\{\\top\}/\\sqrt\{c\_\{h\}\}\\big\)Vstandard softmax attention with head dimensionchc\_\{h\};MHSA⁡\(X\)\\mathrm\{MHSA\}\(X\)is the self\-attention special caseMHA⁡\(X,X,X\)\\mathrm\{MHA\}\(X,X,X\)\.

ModelEncoderProcessorDecoderAttention CostTransolverfeature slicingsoftmax MHSAdeslicingO⁡\(N​ns​c\+ns2​c\)O\(Nn\_\{s\}c\+n\_\{s\}^\{2\}c\)GNOTtrunk/branch MLPslinear cross \+ self\-attn\.MLPO⁡\(N​c2\)O\(Nc^\{2\}\)UPTPerceiver poolingsoftmax MHSAPerceiver queryO⁡\(N​ns​c\+ns2​c\)O\(Nn\_\{s\}c\+n\_\{s\}^\{2\}c\)MSPTpatch poolingchunked MHSA \+ supernodesMLPO⁡\(N2/np\+N​np​nq\)​cO\(N^\{2\}/n\_\{p\}\+Nn\_\{p\}n\_\{q\}\)cNUNOK\-D tree \+ scatterFNO on subdomain gridtrig\. interpolation—GINOradius\-graph GNOFNO on latent gridradius\-graph GNO—Table 17:The six baselines by encode–process–decode role\.NNis the number of mesh nodes,nsn\_\{s\}the number of slices or latent tokens,npn\_\{p\}the number of patches andnqn\_\{q\}the supernodes per patch,ccthe model width\. The two gridded operators have no attention term; their cost is dominated by the FFT on the latent grid,O⁡\(G​log⁡G\)O\(G\\log G\)forGGgrid points\.### F\.1Transolver

Transolver\([Wu et al\., 2024](https://arxiv.org/html/2610.22446#bib.bib62)\)is a transformer solver for PDEs on general geometries\. Applying a standard transformer to a discretised 3D mesh is quadratic in the node count and therefore infeasible at the10410^\{4\}–10610^\{6\}nodes these benchmarks carry\. Transolver’s answer is to stop attending between nodes altogether: it learns a soft partition of the nodes into a small numberns≪Nn\_\{s\}\\ll Nof*slices*, attends only among thensn\_\{s\}slice tokens, and broadcasts the result back\. Crucially the partition is a function of the node*features*, not of spatial proximity, so nodes in physically similar states — a boundary layer distributed across a wing, say — are grouped into the same token regardless of where they sit inΩ\\Omega\. This is the*Physics\-Attention*mechanism, and it is what lets the model be geometry\-general without a graph\. Transolver\+\+\([Luo et al\., 2025](https://arxiv.org/html/2610.22446#bib.bib37)\)scales the same idea to million\-node meshes\.

The mesh is treated as an unordered point set\. There is no adjacency structure, no grid, and no requirement that nodes be in correspondence across samples, so the model is permutation\-equivariant in the node index and consumes per\-sample geometries without modification\.

##### Encoding\.

Node features are lifted, then assigned to slices by a softmax with a per\-node adaptive temperaturesis\_\{i\}, which controls how sharply a node commits to a single slice:

hi\(0\)=MLPlift​\(xi\)∈ℝc,si=Softplus⁡\(MLPs​\(hi\)\)\+ε,h\_\{i\}^\{\(0\)\}=\\mathrm\{MLP\}\_\{\\mathrm\{lift\}\}\(x\_\{i\}\)\\in\\mathbb\{R\}^\{c\},\\qquad s\_\{i\}=\\mathrm\{Softplus\}\\big\(\\mathrm\{MLP\}\_\{s\}\(h\_\{i\}\)\\big\)\+\\varepsilon,\(53\)wi​j=exp⁡\(\(Ws​hi\)j/si\)∑j′=1nsexp⁡\(\(Ws​hi\)j′/si\),Ws∈ℝM×c,∑j=1nswi​j=1\.w\_\{ij\}=\\frac\{\\exp\\\!\\big\(\(W\_\{s\}h\_\{i\}\)\_\{j\}/s\_\{i\}\\big\)\}\{\\sum\_\{j^\{\\prime\}=1\}^\{n\_\{s\}\}\\exp\\\!\\big\(\(W\_\{s\}h\_\{i\}\)\_\{j^\{\\prime\}\}/s\_\{i\}\\big\)\},\\qquad W\_\{s\}\\in\\mathbb\{R\}^\{M\\times c\},\\qquad\\sum\_\{j=1\}^\{n\_\{s\}\}w\_\{ij\}=1\.\(54\)Thensn\_\{s\}slice tokens are the weighted means of their member nodes,

zj=∑i=1Nwi​j​hi∑i=1Nwi​j\+ε∈ℝc\.z\_\{j\}=\\frac\{\\sum\_\{i=1\}^\{N\}w\_\{ij\}\\,h\_\{i\}\}\{\\sum\_\{i=1\}^\{N\}w\_\{ij\}\+\\varepsilon\}\\in\\mathbb\{R\}^\{c\}\.\(55\)

##### Processing\.

Full softmax self\-attention is applied among thensn\_\{s\}tokens only, which is where theN2N^\{2\}cost is avoided:

Z′=Z\+MHSA⁡\(LN⁡\(Z\)\),Z=\[z1,…,zM\]⊤∈ℝM×c\.Z^\{\\prime\}=Z\+\\mathrm\{MHSA\}\\big\(\\mathrm\{LN\}\(Z\)\\big\),\\qquad Z=\[z\_\{1\},\\dots,z\_\{M\}\]^\{\\top\}\\in\\mathbb\{R\}^\{M\\times c\}\.\(56\)

##### Decoding\.

Tokens are scattered back to nodes with the same assignment weights \(“deslicing”\), followed by a pointwise feed\-forward block;LLsuch blocks are stacked and a final projection produces the output field:

hi←hi\+∑j=1nswi​j​zj′,hi←hi\+FFN⁡\(LN⁡\(hi\)\),𝒴i=MLPproj​\(hi\(L\)\)\.h\_\{i\}\\leftarrow h\_\{i\}\+\\sum\_\{j=1\}^\{n\_\{s\}\}w\_\{ij\}\\,z^\{\\prime\}\_\{j\},\\qquad h\_\{i\}\\leftarrow h\_\{i\}\+\\mathrm\{FFN\}\\big\(\\mathrm\{LN\}\(h\_\{i\}\)\\big\),\\qquad\\mathcal\{Y\}\_\{i\}=\\mathrm\{MLP\}\_\{\\mathrm\{proj\}\}\\big\(h\_\{i\}^\{\(L\)\}\\big\)\.\(57\)Each block costsO⁡\(N​ns​c\+ns2​c\)O\(Nn\_\{s\}c\+n\_\{s\}^\{2\}c\), linear inNN\. The LinearNO variant used in some of our sweeps replaces the token\-level softmax attention with separate key and query slicing weights and a spectrally\-normalised value projection,zj=∑iwi​jk​\(Wv​hi\)/∑iwi​jkz\_\{j\}=\\sum\_\{i\}w^\{k\}\_\{ij\}\(W\_\{v\}h\_\{i\}\)/\\sum\_\{i\}w^\{k\}\_\{ij\}andh^i=∑jwi​jq​LN​\(zj\)\\hat\{h\}\_\{i\}=\\sum\_\{j\}w^\{q\}\_\{ij\}\\mathrm\{LN\}\(z\_\{j\}\), removing thens2n\_\{s\}^\{2\}term\.

### F\.2General Neural Operator Transformer \(GNOT\)

GNOT\([Hao et al\., 2023](https://arxiv.org/html/2610.22446#bib.bib19)\)is a transformer built for the case where the operator consumes*heterogeneous*inputs — a geometry, one or more input functions, and a set of scalar parameters — which is precisely the conditional setting of this paper\. It keeps a*trunk*stream of query points and one*branch*stream per input function, with separate key/value projections for each branch, so streams of different physical meaning are never forced into a shared embedding\. Two design choices make it usable at mesh scale: all attention is normalised*linear*attention,O⁡\(N​c2\)O\(Nc^\{2\}\)rather thanO⁡\(N2​c\)O\(N^\{2\}c\), and the feed\-forward blocks are mixtures of experts gated by the point’s coordinates, which lets different regions of the domain \(boundary layer against far field\) be handled by different experts — a soft, learned analogue of domain decomposition\.

Inputs are point clouds\. Query points are arbitrary and need not coincide with the points at which the input functions are sampled, so the model imposes no mesh correspondence across samples\.

##### Encoding\.

The trunk and each of thenbn\_\{b\}branches are embedded by residual MLPs,

hi\(0\)=MLPtrunk\(xi\)∈ℝc,yt\(s\)=MLPbranch\(s\)\(bt\(s\)\)∈ℝc,s=1,…,S\.h\_\{i\}^\{\(0\)\}=\\mathrm\{MLP\}\_\{\\mathrm\{trunk\}\}\(x\_\{i\}\)\\in\\mathbb\{R\}^\{c\},\\qquad y^\{\(s\)\}\_\{t\}=\\mathrm\{MLP\}^\{\(s\)\}\_\{\\mathrm\{branch\}\}\\big\(b^\{\(s\)\}\_\{t\}\\big\)\\in\\mathbb\{R\}^\{c\},\\quad s=1,\\dots,S\.\(58\)In our single\-input\-function configurationnb=1n\_\{b\}=1and the branch is fed the same node tensor as the trunk\.

##### Processing\.

Writeς\\varsigmafor the softmax taken over the*feature*dimension\. The normalised linear attention kernel is

LinAttn​\(Q,K,V\)i=ς​\(qi\)⊤​\(∑tς⁡\(kt\)​vt⊤\)ς​\(qi\)⊤​∑tς⁡\(kt\),\\mathrm\{LinAttn\}\(Q,K,V\)\_\{i\}=\\frac\{\\varsigma\(q\_\{i\}\)^\{\\top\}\\Big\(\\sum\_\{t\}\\varsigma\(k\_\{t\}\)\\,v\_\{t\}^\{\\top\}\\Big\)\}\{\\varsigma\(q\_\{i\}\)^\{\\top\}\\sum\_\{t\}\\varsigma\(k\_\{t\}\)\},\(59\)which never forms theN×NN\\times Nattention matrix: the context∑tς⁡\(kt\)​vt⊤∈ℝch×ch\\sum\_\{t\}\\varsigma\(k\_\{t\}\)v\_\{t\}^\{\\top\}\\in\\mathbb\{R\}^\{c\_\{h\}\\times c\_\{h\}\}is accumulated first, givingO⁡\(N​c2\)O\(Nc^\{2\}\)cost\. Cross\-attention sums the per\-branch contributions, and the geometry gate selects amongnen\_\{e\}experts,

CrossAttn​\(h,\{y\(s\)\}\)i=∑s=1nbLinAttn​\(Wq​h,Wk\(s\)​y\(s\),Wv\(s\)​y\(s\)\)i,\\mathrm\{CrossAttn\}\(h,\\\{y^\{\(s\)\}\\\}\)\_\{i\}=\\sum\_\{s=1\}^\{n\_\{b\}\}\\mathrm\{LinAttn\}\\big\(W\_\{q\}h,\\,W^\{\(s\)\}\_\{k\}y^\{\(s\)\},\\,W^\{\(s\)\}\_\{v\}y^\{\(s\)\}\\big\)\_\{i\},\(60\)gi=Softmax⁡\(MLPgate​\(ωi\)\)∈Δne−1,MoE⁡\(hi\)=∑e=1negi​e​FFNe​\(hi\)\.g\_\{i\}=\\mathrm\{Softmax\}\\big\(\\mathrm\{MLP\}\_\{\\mathrm\{gate\}\}\(\\omega\_\{i\}\)\\big\)\\in\\Delta^\{n\_\{e\}\-1\},\\qquad\\mathrm\{MoE\}\(h\_\{i\}\)=\\sum\_\{e=1\}^\{n\_\{e\}\}g\_\{ie\}\\,\\mathrm\{FFN\}\_\{e\}\(h\_\{i\}\)\.\(61\)The gate sees the coordinates alone, so its specialisation is spatial while the attention remains feature\-driven\. One GNOT layer applies both attention types, each followed by a gated mixture:

h\\displaystyle h←h\+CrossAttn⁡\(LN1​\(h\),\{LN\(s\)​\(y\(s\)\)\}\),\\displaystyle\\leftarrow h\+\\mathrm\{CrossAttn\}\\big\(\\mathrm\{LN\}\_\{1\}\(h\),\\\{\\mathrm\{LN\}^\{\(s\)\}\(y^\{\(s\)\}\)\\\}\\big\),h\\displaystyle\\qquad h←h\+LN3​\(MoE1​\(h\)\),\\displaystyle\\leftarrow h\+\\mathrm\{LN\}\_\{3\}\\big\(\\mathrm\{MoE\}\_\{1\}\(h\)\\big\),\(62\)h\\displaystyle h←h\+LinSelfAttn⁡\(LN4​\(h\)\),\\displaystyle\\leftarrow h\+\\mathrm\{LinSelfAttn\}\\big\(\\mathrm\{LN\}\_\{4\}\(h\)\\big\),h\\displaystyle\\qquad h←h\+LN5​\(MoE2​\(h\)\)\.\\displaystyle\\leftarrow h\+\\mathrm\{LN\}\_\{5\}\\big\(\\mathrm\{MoE\}\_\{2\}\(h\)\\big\)\.

##### Decoding\.

A residual MLP maps the final trunk tokens to the output channels,𝒴i=MLPout​\(hi\(L\)\)\\mathcal\{Y\}\_\{i\}=\\mathrm\{MLP\}\_\{\\mathrm\{out\}\}\\big\(h\_\{i\}^\{\(L\)\}\\big\)\. Because the trunk is evaluated pointwise, the operator can be queried at coordinates that appear in no training mesh\.

### F\.3Universal Physics Transformer \(UPT\)

The Universal Physics Transformer\([Alkin et al\., 2025](https://arxiv.org/html/2610.22446#bib.bib1)\)is an encode–process–decode architecture whose defining property is that the processor never sees the discretisation\. A Perceiver\-style pooling stage\([Jaegle et al\., 2022](https://arxiv.org/html/2610.22446#bib.bib24)\)compresses a variable\-length point cloud into a*fixed*numbernsn\_\{s\}of latent tokens; the physics is modelled entirely among those tokens; and a decoder cross\-attends from arbitrary query positions back into the latent set\. Cost in the processor is thereforeO⁡\(ns2​c\)O\(n\_\{s\}^\{2\}c\)and independent ofNN— the model scales with the complexity of the solution, not the density of the mesh — and the encoder and decoder discretisations are decoupled, so the field can be read out on a mesh other than the one it was encoded from\.

Meshes are handled as raw point clouds: no padding, no graph construction, no correspondence between samples\. Our steady\-state configuration drops the latent time\-propagation stage of the original, which advances the latent tokens between time steps; the encoder here pools directly over all nodes rather than over the message\-passed supernodes of the reference implementation\.

##### Encoding\.

Coordinates are embedded with geometrically\-spaced sinusoids,PE⁡\(ω\)=\[sin⁡\(ω\(ℓ\)​fr\),cos⁡\(ω\(ℓ\)​fr\)\]ℓ,r\\mathrm\{PE\}\(\\omega\)=\\big\[\\sin\(\\omega^\{\(\\ell\)\}\\\!f\_\{r\}\),\\,\\cos\(\\omega^\{\(\\ell\)\}\\\!f\_\{r\}\)\\big\]\_\{\\ell,r\}withfr=10000−r/Ff\_\{r\}=10000^\{\-r/F\}, and added to the lifted features:

hi=MLPlift​\(xi\)\+Wp​PE​\(ωi\)∈ℝc\.h\_\{i\}=\\mathrm\{MLP\}\_\{\\mathrm\{lift\}\}\(x\_\{i\}\)\+W\_\{p\}\\,\\mathrm\{PE\}\(\\omega\_\{i\}\)\\in\\mathbb\{R\}^\{c\}\.\(63\)nsn\_\{s\}learnable query tokensQ∈ℝns×cQ\\in\\mathbb\{R\}^\{n\_\{s\}\\times c\}pool the mesh:

Z\(0\)=Q\+MHA⁡\(LN⁡\(Q\),LN⁡\(H\),LN⁡\(H\)\)∈ℝM×c,H=\[h1,…,hN\]⊤\.Z^\{\(0\)\}=Q\+\\mathrm\{MHA\}\\big\(\\mathrm\{LN\}\(Q\),\\,\\mathrm\{LN\}\(H\),\\,\\mathrm\{LN\}\(H\)\\big\)\\in\\mathbb\{R\}^\{M\\times c\},\\qquad H=\[h\_\{1\},\\dots,h\_\{N\}\]^\{\\top\}\.\(64\)

##### Processing\.

LLpre\-norm transformer blocks act on the latent set alone:

Z←Z\+MHSA⁡\(LN⁡\(Z\)\),Z←Z\+FFN⁡\(LN⁡\(Z\)\)\.Z\\leftarrow Z\+\\mathrm\{MHSA\}\\big\(\\mathrm\{LN\}\(Z\)\\big\),\\qquad Z\\leftarrow Z\+\\mathrm\{FFN\}\\big\(\\mathrm\{LN\}\(Z\)\\big\)\.\(65\)

##### Decoding\.

Any query positionωq\\omega\_\{q\}forms a query token from the same lifting and positional embedding, and cross\-attends into the processed latents:

𝒴⁡\(ωq\)=MLPproj​\(hq\+MHA⁡\(LN⁡\(hq\),LN⁡\(Z\(L\)\),LN⁡\(Z\(L\)\)\)\)\.\\mathcal\{Y\}\(\\omega\_\{q\}\)=\\mathrm\{MLP\}\_\{\\mathrm\{proj\}\}\\Big\(h\_\{q\}\+\\mathrm\{MHA\}\\big\(\\mathrm\{LN\}\(h\_\{q\}\),\\,\\mathrm\{LN\}\(Z^\{\(L\)\}\),\\,\\mathrm\{LN\}\(Z^\{\(L\)\}\)\\big\)\\Big\)\.\(66\)

### F\.4Multi\-Scale Patch Transformer \(MSPT\)

MSPT\([Curvo et al\., 2026](https://arxiv.org/html/2610.22446#bib.bib9)\)attacks the same quadratic\-attention problem from the opposite direction to UPT: instead of compressing the whole mesh into a global bottleneck, it partitions the nodes intonpn\_\{p\}patches, runs full attention*inside*each patch, and couples the patches through a small set of*supernodes*—nqn\_\{q\}pooled summaries per patch — that every patch also attends to\. Local detail is therefore resolved at full resolution while global communication is carried bynp​nqn\_\{p\}n\_\{q\}tokens, giving the two scales the name refers to\. Supernodes persist across blocks by residual accumulation, so the global state is refined depth\-wise rather than recomputed\.

Partitioning is either sequential over the node ordering or, for spatially coherent patches, a ball\-tree partition of the coordinates; the latter is what makes patches correspond to compact regions ofΩ\\Omegaon an unstructured mesh\. Rotary positional embeddings may be applied within a patch sequence\. Node counts that are not a multiple ofnpn\_\{p\}are zero\-padded and the padding is stripped on output\.

##### Encoding\.

Tokens are formed from coordinates \(optionally replaced by a*unified position*embedding: distances to a fixedrefd\\mathrm\{ref\}^\{d\}reference lattice, which gives every sample a common frame\) concatenated with field features, and permuted into patches\{Cv\}v=1np\\\{C\_\{v\}\\\}\_\{v=1\}^\{n\_\{p\}\}by the partitionπ\\pi:

hi=MLPpre\(\[ϕ\(ωi\)∥u0\(ωi\)\]\)\+p,\{Cv\}v=1np=π\(\{ωi\}\),\|Cv\|=N/V\.h\_\{i\}=\\mathrm\{MLP\}\_\{\\mathrm\{pre\}\}\\big\(\[\\,\\phi\(\\omega\_\{i\}\)\\,\\\|\\,u\_\{0\}\(\\omega\_\{i\}\)\\,\]\\big\)\+p,\\qquad\\\{C\_\{v\}\\\}\_\{v=1\}^\{n\_\{p\}\}=\\pi\\big\(\\\{\\omega\_\{i\}\\\}\\big\),\\qquad\|C\_\{v\}\|=N/V\.\(67\)

##### Processing\.

Each block poolsnqn\_\{q\}supernodes per patch and adds the previous block’s,

sv,q\(ℓ\)=poolq​\(\{hi\(ℓ\)\}i∈Cv\)\+sv,q\(ℓ−1\),S\(ℓ\)=\[sv,q\(ℓ\)\]∈ℝV​Q×c,s^\{\(\\ell\)\}\_\{v,q\}=\\mathrm\{pool\}\_\{q\}\\big\(\\\{h^\{\(\\ell\)\}\_\{i\}\\\}\_\{i\\in C\_\{v\}\}\\big\)\+s^\{\(\\ell\-1\)\}\_\{v,q\},\\qquad S^\{\(\\ell\)\}=\\big\[s^\{\(\\ell\)\}\_\{v,q\}\\big\]\\in\\mathbb\{R\}^\{VQ\\times c\},\(68\)withpool\\mathrm\{pool\}a mean, top\-nqn\_\{q\}max, or learned linear pooling\. Attention then runs on each patch*concatenated with the full supernode set*, so every node sees a summary of every other patch:

H~v=\[HCv∥S\(ℓ\)\]∈ℝ\(N/V\+V​Q\)×c,H~v←H~v\+MHSA\(LN\(H~v\)\),\\tilde\{H\}\_\{v\}=\\big\[\\,H\_\{C\_\{v\}\}\\;\\big\\\|\\;S^\{\(\\ell\)\}\\,\\big\]\\in\\mathbb\{R\}^\{\(N/V\+VQ\)\\times c\},\\qquad\\tilde\{H\}\_\{v\}\\leftarrow\\tilde\{H\}\_\{v\}\+\\mathrm\{MHSA\}\\big\(\\mathrm\{LN\}\(\\tilde\{H\}\_\{v\}\)\\big\),\(69\)H~v←H~v\+FFN\(LN\(H~v\)\),HCv=H~v\[1:N/V\],S\(ℓ\)←1V∑v=1VH~v\[−VQ:\]\.\\tilde\{H\}\_\{v\}\\leftarrow\\tilde\{H\}\_\{v\}\+\\mathrm\{FFN\}\\big\(\\mathrm\{LN\}\(\\tilde\{H\}\_\{v\}\)\\big\),\\qquad H\_\{C\_\{v\}\}=\\tilde\{H\}\_\{v\}\[1\{:\}N/V\],\\qquad S^\{\(\\ell\)\}\\leftarrow\\frac\{1\}\{V\}\\sum\_\{v=1\}^\{V\}\\tilde\{H\}\_\{v\}\[\-VQ\{:\}\]\.\(70\)The cost isO⁡\(np​\(N/np\+np​nq\)2​c\)=O⁡\(\(N2/np\+N​np​nq\)​c\)O\\big\(n\_\{p\}\(N/n\_\{p\}\+n\_\{p\}n\_\{q\}\)^\{2\}c\\big\)=O\\big\(\(N^\{2\}/n\_\{p\}\+Nn\_\{p\}n\_\{q\}\)c\\big\), minimised atnp∼N/nqn\_\{p\}\\sim\\sqrt\{N/n\_\{q\}\}\.

##### Decoding\.

The final block projects to the output channels and the patch permutation is inverted,𝒴=π−1​\(Wout​LN​\(H\(L\)\)\)\\mathcal\{Y\}=\\pi^\{\-1\}\\big\(W\_\{\\mathrm\{out\}\}\\,\\mathrm\{LN\}\(H^\{\(L\)\}\)\\big\)\.

### F\.5Non\-Uniform Neural Operator \(NUNO\)

NUNO\([Liu et al\., 2023](https://arxiv.org/html/2610.22446#bib.bib36)\)extends grid\-based operator learning to non\-uniform data\. Fast neural operators such as the FNO\([Li et al\., 2021](https://arxiv.org/html/2610.22446#bib.bib32)\)rest on the FFT, which requires a uniform grid; applying one to a tetrahedral mesh means interpolating globally onto a single grid, and the resulting error is set by the worst\-resolved region of the domain\. NUNO’s remedy is decomposition: a K\-D tree recursively splitsΩ\\Omegaintoncn\_\{c\}subdomains of roughly equal node count, so that dense regions receive small cells and sparse regions large ones, and the local interpolation error is bounded within each subdomain rather than globally\. The subdomain grids are then processed by an FNO, retaining FFT speed\.

##### Encoding\.

The tree splits each node setPPat the median along its widest axis,

ℓ∗\\displaystyle\\ell^\{\*\}=arg​maxℓ∈\{1,…,d\}⁡\(maxi∈P⁡ωi\(ℓ\)−mini∈P⁡ωi\(ℓ\)\),\\displaystyle=\\operatorname\*\{arg\\,max\}\_\{\\ell\\in\\\{1,\\dots,d\\\}\}\\Big\(\\max\_\{i\\in P\}\\omega^\{\(\\ell\)\}\_\{i\}\-\\min\_\{i\\in P\}\\omega^\{\(\\ell\)\}\_\{i\}\\Big\),\(71\)P\\displaystyle P↦\{i∈P:ωi\(ℓ∗\)≤mℓ∗\}∪\{i∈P:ωi\(ℓ∗\)\>mℓ∗\},\\displaystyle\\mapsto\\big\\\{i\\in P:\\omega^\{\(\\ell^\{\*\}\)\}\_\{i\}\\leq m\_\{\\ell^\{\*\}\}\\big\\\}\\;\\cup\\;\\big\\\{i\\in P:\\omega^\{\(\\ell^\{\*\}\)\}\_\{i\}\>m\_\{\\ell^\{\*\}\}\\big\\\},recursed untilncn\_\{c\}leaves remain\. Lifted features are transferred to the uniform grid𝒢\\mathcal\{G\}of shapeS1×⋯×SdS\_\{1\}\\times\\cdots\\times S\_\{d\}by nearest\-cell scatter,

𝒳𝒢\[𝐠\]=∑i:γ⁡\(ωi\)=g𝒳i,γ\(ωi\)\(ℓ\)=⌊ωi\(ℓ\)−minj⁡ωj\(ℓ\)maxj⁡ωj\(ℓ\)−minj⁡ωj\(ℓ\)\(Sℓ−1\)⌉,\\mathcal\{X\}\_\{\\mathcal\{G\}\}\[\\mathbf\{g\}\]=\\sum\_\{i\\,:\\,\\gamma\(\\omega\_\{i\}\)=g\}\\mathcal\{X\}\_\{i\},\\qquad\\gamma\(\\omega\_\{i\}\)^\{\(\\ell\)\}=\\Big\\lfloor\\frac\{\\omega^\{\(\\ell\)\}\_\{i\}\-\\min\_\{j\}\\omega^\{\(\\ell\)\}\_\{j\}\}\{\\max\_\{j\}\\omega^\{\(\\ell\)\}\_\{j\}\-\\min\_\{j\}\\omega^\{\(\\ell\)\}\_\{j\}\}\\,\(S\_\{\\ell\}\-1\)\\Big\\rceil,\(72\)after which the normalised grid coordinates are concatenated as channels and lifted\.[Equation72](https://arxiv.org/html/2610.22446#A6.E72)is the piecewise\-constant limit of the transfer operator our spline encoder generalises: it is order\-zero, so it assigns a node’s whole contribution to one cell and provides no gradient with respect toωi\\omega\_\{i\}\.

##### Processing\.

Standard Fourier layers act on the grid, truncated to\|kℓ\|≤mℓ\|k\_\{\\ell\}\|\\leq m\_\{\\ell\}modes per axis:

v\(ℓ\+1\)=σ⁡\(W​v\(ℓ\)\+ℱ−1​\(R\(ℓ\)⋅ℱ⁡\(v\(ℓ\)\)\)\)\.v^\{\(\\ell\+1\)\}=\\sigma\\Big\(Wv^\{\(\\ell\)\}\+\\mathcal\{F\}^\{\-1\}\\big\(R^\{\(\\ell\)\}\\cdot\\mathcal\{F\}\(v^\{\(\\ell\)\}\)\\big\)\\Big\)\.\(73\)

##### Decoding\.

A pointwise MLP expands the grid features intoncn\_\{c\}per\-subdomain copies, and each subdomain’s nodes are read out by evaluating the truncated Fourier series directly at their true coordinates, rather than by interpolating off the grid a second time:

𝒴i=MLPout​\(∑‖k‖∞≤mv^k​e2​π​i​⟨k,ωi⟩\),v^k=∑𝐠∈𝒢v⁡\[𝐠\]​e−2​π​i​⟨k,ξ𝐠⟩\.\\mathcal\{Y\}\_\{i\}=\\mathrm\{MLP\}\_\{\\mathrm\{out\}\}\\\!\\left\(\\sum\_\{\\\|k\\\|\_\{\\infty\}\\leq m\}\\hat\{v\}\_\{k\}\\,e^\{\\,2\\pi\\mathrm\{i\}\\langle k,\\,\\omega\_\{i\}\\rangle\}\\right\),\\qquad\\hat\{v\}\_\{k\}=\\sum\_\{\\mathbf\{g\}\\in\\mathcal\{G\}\}v\[\\mathbf\{g\}\]\\,e^\{\-2\\pi\\mathrm\{i\}\\langle k,\\,\\xi\_\{\\mathbf\{g\}\}\\rangle\}\.\(74\)This is the same non\-uniform forward/inverse transform pair used by Geo\-FNO\([Li et al\., 2023a](https://arxiv.org/html/2610.22446#bib.bib33)\)\. In three dimensions our implementation falls back to trilineargrid\_samplefor the read\-out, since the explicitO⁡\(N​∏ℓ2​mℓ\)O\(N\\prod\_\{\\ell\}2m\_\{\\ell\}\)basis becomes the dominant memory cost\.

### F\.6Geometry\-Informed Neural Operator \(GINO\)

GINO\([Li et al\., 2023b](https://arxiv.org/html/2610.22446#bib.bib34)\)is the other gridded baseline, and the closest architectural comparison to our method: it is an FNO on a uniform latent grid, bracketed by learned transfer operators that move features between the mesh and that grid\. Where we use a tensor\-product spline scatter and its analytic adjoint, GINO uses a Graph Neural Operator on both sides — a radius\-graph kernel integral that aggregates, for each latent grid point, the mesh nodes within a ball of radiusrr, and then the reverse\. The encoder and decoder are therefore Monte\-Carlo/Riemann approximations of a continuous kernel integral, learned as a neural network of the relative displacement\.

The radius graph is built between the mesh and the latent grid, so the mesh needs no fixed connectivity and per\-sample geometries are admitted\. Two practical points follow from the construction\. First,rris measured in the normalised\[0,1\]d\[0,1\]^\{d\}coordinates of the latent grid and must be at least the largest grid spacing, or latent points collect no neighbours and their features are identically zero\. Second, because the neighbourhood structure differs per sample, the reference implementation batches only when the geometry is shared, and our wrapper therefore loops over samples\.

##### Encoding\.

For each latent grid pointξ𝐠\\xi\_\{\\mathbf\{g\}\},𝐠∈𝒢\\mathbf\{g\}\\in\\mathcal\{G\}, with neighbourhood𝒩r​\(ξ𝐠\)=\{i:‖ωi−ξ𝐠‖2≤r\}\\mathcal\{N\}\_\{r\}\(\\xi\_\{\\mathbf\{g\}\}\)=\\\{i:\\\|\\omega\_\{i\}\-\\xi\_\{\\mathbf\{g\}\}\\\|\_\{2\}\\leq r\\\},

𝒳𝒢​\[𝐠\]=1\|𝒩r​\(ξ𝐠\)\|​∑i∈𝒩r​\(ξ𝐠\)κG​\(ξ𝐠,ωi,𝒳i\)​𝒳i≈∫Br​\(ξ𝐠\)κG​\(ξ𝐠,ω\)​u​\(ω\)​𝑑ω,\\mathcal\{X\}\_\{\\mathcal\{G\}\}\[\\mathbf\{g\}\]=\\frac\{1\}\{\|\\mathcal\{N\}\_\{r\}\(\\xi\_\{\\mathbf\{g\}\}\)\|\}\\sum\_\{i\\in\\mathcal\{N\}\_\{r\}\(\\xi\_\{\\mathbf\{g\}\}\)\}\\kappa\_\{\\mathrm\{G\}\}\\big\(\\xi\_\{\\mathbf\{g\}\},\\,\\omega\_\{i\},\\,\\mathcal\{X\}\_\{i\}\\big\)\\,\\mathcal\{X\}\_\{i\}\\;\\approx\\;\\int\_\{B\_\{r\}\(\\xi\_\{\\mathbf\{g\}\}\)\}\\kappa\_\{\\mathrm\{G\}\}\(\\xi\_\{\\mathbf\{g\}\},\\omega\)\\,u\(\\omega\)\\,\\mathrm\{d\}\\omega,\(75\)withκG\\kappa\_\{\\mathrm\{G\}\}an MLP of the sinusoidally\-embedded relative displacementξ𝐠−ωi\\xi\_\{\\mathbf\{g\}\}\-\\omega\_\{i\}\.

##### Processing\.

An FNO acts on the latent grid as in[Equation73](https://arxiv.org/html/2610.22446#A6.E73)\. Conditioning is native here: whenk\>0k\>0the normalisation layers are adaptive instance norms driven by𝝁\\bm\{\\mu\},

AdaIN⁡\(v,𝝁\)=γG​\(e⁡\(𝝁\)\)​v−𝔼⁡\[v\]Var⁡\[v\]\+ε\+βG​\(e⁡\(𝝁\)\),e⁡\(𝝁\)=\[sin⁡\(fr​𝝁\),cos⁡\(fr​𝝁\)\]r=1F,\\mathrm\{AdaIN\}\(v;\\bm\{\\mu\}\)=\\gamma\_\{\\mathrm\{G\}\}\\big\(e\(\\bm\{\\mu\}\)\\big\)\\,\\frac\{v\-\\mathbb\{E\}\[v\]\}\{\\sqrt\{\\mathrm\{Var\}\[v\]\+\\varepsilon\}\}\+\\beta\_\{\\mathrm\{G\}\}\\big\(e\(\\bm\{\\mu\}\)\\big\),\\qquad e\(\\bm\{\\mu\}\)=\\big\[\\sin\(f\_\{r\}\\bm\{\\mu\}\),\\,\\cos\(f\_\{r\}\\bm\{\\mu\}\)\\big\]\_\{r=1\}^\{F\},\(76\)whereγG,βG\\gamma\_\{\\mathrm\{G\}\},\\beta\_\{\\mathrm\{G\}\}are MLPs\. Their width scales withkk, so the parameter count of GINO grows with the conditioning dimension — atk=40k=40the norm MLPs alone add roughly1\.51\.5M parameters, which is whyFF\(ada\_in\_frequencies\) is kept small in our configurations\.

##### Decoding\.

A second GNO integrates back from the grid to the query nodes,

𝒴i=1\|𝒩r​\(ωi\)\|​∑g∈𝒩r​\(ωi\)κG′​\(ωi,ξ𝐠\)​𝒴𝒢​\[𝐠\],𝒩r​\(ωi\)=\{g:‖ξ𝐠−ωi‖2≤r\}\.\\mathcal\{Y\}\_\{i\}=\\frac\{1\}\{\|\\mathcal\{N\}\_\{r\}\(\\omega\_\{i\}\)\|\}\\sum\_\{g\\in\\mathcal\{N\}\_\{r\}\(\\omega\_\{i\}\)\}\\kappa^\{\\prime\}\_\{\\mathrm\{G\}\}\\big\(\\omega\_\{i\},\\,\\xi\_\{\\mathbf\{g\}\}\\big\)\\,\\mathcal\{Y\}\_\{\\mathcal\{G\}\}\[\\mathbf\{g\}\],\\qquad\\mathcal\{N\}\_\{r\}\(\\omega\_\{i\}\)=\\\{g:\\\|\\xi\_\{\\mathbf\{g\}\}\-\\omega\_\{i\}\\\|\_\{2\}\\leq r\\\}\.\(77\)Like the spline decoder of[Equation11](https://arxiv.org/html/2610.22446#S3.E11),[Equation77](https://arxiv.org/html/2610.22446#A6.E77)is linear in the grid values: once trained, the prediction at nodeiiis a combination of the\|𝒩r​\(ωi\)\|\|\\mathcal\{N\}\_\{r\}\(\\omega\_\{i\}\)\|grid values inside the ball, with weights that depend only on the positions of the node and the grid points, never on the field itself\. The ball𝒩r\\mathcal\{N\}\_\{r\}plays the role our spline stencil plays, with two differences: its size is set by the radiusrrrather than by the spline degree, and its weightsκG′\\kappa^\{\\prime\}\_\{\\mathrm\{G\}\}are learned without any constraint, whereas the analytic spline weights form a partition of unity\.

## Appendix GConditioning Methods

All conditioning methods receive the physical parameter vector𝝁∈ℝk\\bm\{\\mu\}\\in\\mathbb\{R\}^\{k\}and act at the same place in the pipeline: on the per\-node encoded coordinate features𝐟=φ⁡\(ω¯\)∈ℝBbatch×N×cf\\mathbf\{f\}=\\varphi\(\\bar\{\\omega\}\)\\in\\mathbb\{R\}^\{B\_\{\\mathrm\{batch\}\}\\times N\\times c\_\{f\}\},cf=96c\_\{f\}=96, before the backbone’s input lift \([Equation3](https://arxiv.org/html/2610.22446#S3.E3)\)\. The three normalised coordinatesω¯\\bar\{\\omega\}are excluded from conditioning for every method and every backbone; the conditioned features𝐟~\\tilde\{\\mathbf\{f\}\}are concatenated withω¯\\bar\{\\omega\}and lifted,𝐡i=Pin​\[𝐟~i;ω¯i\]\\mathbf\{h\}\_\{i\}=P\_\{\\mathrm\{in\}\}\[\\tilde\{\\mathbf\{f\}\}\_\{i\};\\bar\{\\omega\}\_\{i\}\]\. No method acts on the width\-cclifted features or on intermediate backbone states in any reported run\.

### G\.1FiLM

Feature\-wise Linear Modulation \(FiLM\)\([Perez et al\., 2018](https://arxiv.org/html/2610.22446#bib.bib47)\)is a lightweight conditioning strategy that affinely transforms each channel of the field representation based on the conditioning parameters\. An MLPHμ:ℝk→ℝ2​cfH\_\{\\mu\}:\\mathbb\{R\}^\{k\}\\to\\mathbb\{R\}^\{2c\_\{f\}\}, composed of the embeddingEμE\_\{\\mu\}and the two headsHγH\_\{\\gamma\},HβH\_\{\\beta\}of[Section3\.1](https://arxiv.org/html/2610.22446#S3.SS1), maps𝝁\\bm\{\\mu\}to per\-channel scale and shift parameters:

\(𝜸μ,𝜷μ\)=Hμ​\(𝝁\),𝜸μ,𝜷μ∈ℝcf\(\\bm\{\\gamma\}\_\{\\mu\},\\bm\{\\beta\}\_\{\\mu\}\)=H\_\{\\mu\}\(\\bm\{\\mu\}\),\\qquad\\bm\{\\gamma\}\_\{\\mu\},\\bm\{\\beta\}\_\{\\mu\}\\in\\mathbb\{R\}^\{c\_\{f\}\}\(78\)
The conditioned field is then given by an element\-wise affine transformation applied uniformly across all spatial locations:

𝐟~=𝜸μ⊙𝐟\+𝜷μ\\tilde\{\\mathbf\{f\}\}=\\bm\{\\gamma\}\_\{\\mu\}\\odot\\mathbf\{f\}\+\\bm\{\\beta\}\_\{\\mu\}\(79\)
where⊙\\odotdenotes element\-wise multiplication broadcast over the spatial dimensions\. The FiLM layer preserves the channel countcfc\_\{f\}and does not alter the spatial structure of the representation\. The scale and shift heads have zero\-initialised weights and biases𝟏\\mathbf\{1\}and𝟎\\mathbf\{0\}, so𝜸μ=𝟏\\bm\{\\gamma\}\_\{\\mu\}=\\mathbf\{1\}and𝜷μ=𝟎\\bm\{\\beta\}\_\{\\mu\}=\\mathbf\{0\}exactly at initialisation and the unconditioned backbone behaviour is recovered\.

### G\.2Concatenation

The simplest conditioning strategy directly augments the input field by appending the conditioning parameters as additional spatial channels\. Each scalar in𝝁\\bm\{\\mu\}is broadcast uniformly across all mesh nodes or grid locations and concatenated to the field:

𝐟~=\[𝐟∥1Ω⊗𝝁\]∈ℝcf\+k\\tilde\{\\mathbf\{f\}\}=\\bigl\[\\mathbf\{f\}\\;\\\|\\;\\mathbf\{1\}\_\{\\Omega\}\\otimes\\bm\{\\mu\}\\bigr\]\\in\\mathbb\{R\}^\{c\_\{f\}\+k\}\(80\)
where𝟏Ω⊗𝝁\\mathbf\{1\}\_\{\\Omega\}\\otimes\\bm\{\\mu\}denotes the spatial broadcast of𝝁\\bm\{\\mu\}to every node or grid point\. This increases the number of feature channels fromcfc\_\{f\}tocf\+kc\_\{f\}\+k, so the lift becomesPin:ℝcf\+k\+3→ℝcP\_\{\\mathrm\{in\}\}:\\mathbb\{R\}^\{c\_\{f\}\+k\+3\}\\to\\mathbb\{R\}^\{c\}\. Concatenation conditioning places no constraint on how the model uses the parameter information, leaving the entire burden of interpreting𝝁\\bm\{\\mu\}to the backbone architecture\.

### G\.3Cross\-Attention

Cross\-attention conditioning enables the model to selectively query physical parameter information at each spatial location through an attention mechanism\. The conditioning parameters𝝁\\bm\{\\mu\}are first embedded into a set ofnτn\_\{\\tau\}learnable parameter tokens:

𝒯=reshape⁡\(Wτ​𝝁\)\+Eτ∈ℝnτ×cτ\\mathcal\{T\}=\\operatorname\{reshape\}\\bigl\(W\_\{\\tau\}\\bm\{\\mu\}\\bigr\)\+E\_\{\\tau\}\\in\\mathbb\{R\}^\{n\_\{\\tau\}\\times c\_\{\\tau\}\}\(81\)
whereWτ∈ℝ\(nτ⋅cτ\)×kW\_\{\\tau\}\\in\\mathbb\{R\}^\{\(n\_\{\\tau\}\\cdot c\_\{\\tau\}\)\\times k\}is a learnable projection andEτ∈ℝnτ×cτE\_\{\\tau\}\\in\\mathbb\{R\}^\{n\_\{\\tau\}\\times c\_\{\\tau\}\}are learnable positional embeddings\. The field features at each spatial location serve as queries, attending to the parameter tokens as keys and values:

Q\\displaystyle Q=WQ​𝐟∈ℝN×cτ,\\displaystyle=W\_\{Q\}\\,\\mathbf\{f\}\\in\\mathbb\{R\}^\{N\\times c\_\{\\tau\}\},\(82\)K=V\\displaystyle K=V=LayerNorm⁡\(𝒯\)∈ℝnτ×cτ,\\displaystyle=\\operatorname\{LayerNorm\}\(\\mathcal\{T\}\)\\in\\mathbb\{R\}^\{n\_\{\\tau\}\\times c\_\{\\tau\}\},\(83\)A\\displaystyle A=softmax⁡\(Q​K⊤cτ\)​V∈ℝN×cτ\\displaystyle=\\operatorname\{softmax\}\\\!\\left\(\\frac\{QK^\{\\top\}\}\{\\sqrt\{c\_\{\\tau\}\}\}\\right\)V\\in\\mathbb\{R\}^\{N\\times c\_\{\\tau\}\}\(84\)
whereNNis the number of spatial locations and multi\-head attention is used in practice\. The output is projected back to thecfc\_\{f\}feature channels and added as a residual:

𝐟~=𝐟\+Wout​A\\tilde\{\\mathbf\{f\}\}=\\mathbf\{f\}\+W\_\{\\mathrm\{out\}\}A\(85\)
The output projectionWoutW\_\{\\mathrm\{out\}\}is initialised to zero so that the module acts as an identity at the start of training\. This preserves the original backbone dynamics and allows the conditioning signal to be introduced gradually\.

This section presents tables showcasing full scale ablations across all of the experiments across all models and conditioning methods\. We find that the learnable Spline\-FNO, on average offers the best performance and compute\-efficiency\. Similarly FiLM conditioning method across the spatial features of the input performs remarkably well compared to concatenation and cross\-attention\.

ModelCond\. Method\# ParametersTrain Time\(hrs:mins\)Rel\.L2L\_\{2\}% \(↓\\downarrow\)Rel\.L1L\_\{1\}% \(↓\\downarrow\)L2/L1L\_\{2\}/L\_\{1\}R2R^\{2\}\(↑\\uparrow\)TransolverNone2,433,4130:410\.05890\.04601\.28040\.9989TransolverConcatenation2,434,4370:410\.04580\.03411\.34490\.9993TransolverCross\-attention2,464,4220:440\.04380\.03301\.32810\.9994TransolverFiLM2,450,5030:410\.03850\.02801\.37420\.9995UPTNone2,252,0970:330\.09070\.07411\.22300\.9972UPTConcatenation2,252,9930:320\.04230\.03081\.37490\.9994UPTCross\-attention2,283,1060:370\.04100\.02951\.38840\.9995UPTFiLM2,269,1870:340\.03980\.02831\.40700\.9995MSPTNone2,190,5291:260\.09750\.08191\.18970\.9963MSPTConcatenation2,192,0651:260\.03990\.02861\.39350\.9995MSPTCross\-attention2,221,4091:290\.03970\.02871\.38160\.9995MSPTFiLM2,207,4891:260\.03680\.02581\.42740\.9996GNOTNone2,192,8411:330\.06740\.05461\.23590\.9985GNOTConcatenation2,193,6091:330\.04480\.03301\.35490\.9994GNOTCross\-attention2,223,7211:370\.04160\.03031\.37230\.9994GNOTFiLM2,209,8011:340\.04200\.03011\.39410\.9995NUNONone2,392,4550:130\.07930\.06061\.30960\.9975NUNOConcatenation2,392,5230:120\.07200\.05341\.34910\.9978NUNOCross\-attention2,423,3350:160\.07980\.05911\.34970\.9970NUNOFiLM2,409,4150:130\.06880\.05021\.37000\.9979GINONone2,326,2671:060\.06710\.05051\.32670\.9984GINOConcatenation2,326,7751:060\.05220\.03731\.39960\.9991GINOCross\-attention2,357,1471:100\.05530\.03971\.39450\.9989GINOFiLM2,343,2271:080\.05180\.03671\.40900\.9991SplineFNONone2,350,8131:160\.08750\.07531\.16200\.9973SplineFNOConcatenation2,350,9411:160\.04830\.03781\.27790\.9992SplineFNOCross\-attention2,381,6931:200\.03860\.02961\.30270\.9995SplineFNOFiLM2,367,7731:160\.03430\.02641\.30270\.9996L\-SplineFNONone2,378,8150:420\.08490\.07221\.17730\.9973L\-SplineFNOConcatenation2,378,9110:420\.03400\.02681\.26560\.9996L\-SplineFNOCross\-attention2,409,6950:450\.03350\.02591\.29280\.9996L\-SplineFNOFiLM2,395,7750:420\.02710\.02131\.26870\.9998

Table 18:Performance metrics of various models and conditioning methods for the Heat Sink \(3D, steady\-state, thermal\) case\. Best value per metric within each architecture in bold\.ModelCond\. Method\# ParametersTrain Time\(hrs:mins\)Rel\.L2L\_\{2\}% \(↓\\downarrow\)Rel\.L1L\_\{1\}% \(↓\\downarrow\)L2/L1L\_\{2\}/L\_\{1\}R2R^\{2\}\(↑\\uparrow\)TransolverNone2,433,4131:4723\.673322\.95901\.03110\.6963TransolverConcatenation2,434,4371:470\.90060\.65261\.37990\.9994TransolverCross\-attention2,464,4221:570\.85730\.64951\.31980\.9995TransolverFiLM2,450,5031:490\.56980\.40591\.40390\.9998UPTNone2,252,0971:2827\.025825\.63781\.05410\.6351UPTConcatenation2,252,9931:281\.91361\.31801\.45190\.9975UPTCross\-attention2,283,1061:400\.82410\.59241\.39120\.9995UPTFiLM2,269,1871:320\.62630\.44341\.41270\.9996MSPTNone2,190,5293:2824\.064523\.34101\.03100\.6848MSPTConcatenation2,192,0653:270\.65460\.46121\.41930\.9997MSPTCross\-attention2,221,4093:380\.62310\.45991\.35490\.9997MSPTFiLM2,207,4893:300\.51170\.34791\.47080\.9998GNOTNone2,192,8414:0622\.487621\.78461\.03230\.7350GNOTConcatenation2,193,6094:060\.63930\.46821\.36560\.9997GNOTCross\-attention2,223,7214:170\.73050\.54061\.35130\.9996GNOTFiLM2,209,8014:090\.49980\.35291\.41640\.9998NUNONone2,392,4550:4022\.507421\.49361\.04720\.7652NUNOConcatenation2,392,5230:402\.54841\.85411\.37440\.9970NUNOCross\-attention2,423,3350:501\.81241\.27751\.41870\.9985NUNOFiLM2,409,4150:422\.00161\.40651\.42310\.9981SplineFNONone2,370,4303:1621\.552020\.43201\.05480\.7872SplineFNOConcatenation2,370,4983:215\.61803\.48941\.61000\.9885SplineFNOCross\-attention2,401,3103:305\.42543\.35001\.61950\.9895SplineFNOFiLM2,387,3903:225\.48673\.38841\.61920\.9893L\-SplineFNONone2,370,7710:5223\.393022\.75001\.02830\.7366L\-SplineFNOConcatenation2,370,8390:542\.20501\.25751\.75350\.9978L\-SplineFNOCross\-attention2,401,6511:031\.92971\.05961\.82120\.9984L\-SplineFNOFiLM2,387,7310:521\.97101\.08171\.82220\.9983

Table 19:Performance metrics of various models and conditioning methods for the Bracket \(3D, steady\-state, elasto\-plastic\) case\. Best value per metric within each architecture in bold\.ModelCond\. Method\# ParametersTrain Time\(hrs:mins\)Rel\.L2L\_\{2\}% \(↓\\downarrow\)Rel\.L1L\_\{1\}% \(↓\\downarrow\)L2/L1L\_\{2\}/L\_\{1\}R2R^\{2\}\(↑\\uparrow\)TransolverNone2,434,4411:2664\.907354\.69291\.18680\.3622TransolverConcatenation2,435,2091:277\.60325\.78021\.31540\.9514TransolverCross\-attention2,464,8741:3614\.821711\.55401\.28280\.8201TransolverFiLM2,451,4671:269\.28766\.97281\.33200\.9273UPTNone2,252,9970:5762\.962452\.74891\.19360\.3524UPTConcatenation2,253,6690:5614\.998211\.61381\.29140\.8207UPTCross\-attention2,283,4301:0714\.727411\.43211\.28820\.8215UPTFiLM2,270,0230:5913\.992310\.76911\.29930\.8446MSPTNone2,191,3016:1363\.266252\.96311\.19450\.4050MSPTConcatenation2,192,4536:147\.62965\.71691\.33460\.9424MSPTCross\-attention2,221,6056:228\.54146\.13881\.39140\.9317MSPTFiLM2,208,1976:157\.50285\.71091\.31380\.9487GNOTNone2,193,2293:2363\.760053\.20951\.19830\.4010GNOTConcatenation2,193,8053:246\.88125\.18391\.32740\.9578GNOTCross\-attention2,223,5333:337\.63225\.53761\.37820\.9453GNOTFiLM2,210,1253:267\.28535\.25631\.38600\.9496NUNONone2,407,9090:1467\.614057\.03821\.18540\.3014NUNOConcatenation2,408,0050:1413\.896210\.41031\.33480\.8482NUNOCross\-attention2,438,2130:2413\.967910\.38161\.34540\.8508NUNOFiLM2,424,8050:1414\.035410\.48921\.33810\.8466SplineFNONone2,341,7990:4370\.294059\.43501\.18270\.2824SplineFNOConcatenation2,341,8770:4321\.666014\.13201\.53320\.8196SplineFNOCross\-attention2,372,1030:5219\.511013\.55501\.43950\.8223SplineFNOFiLM2,358,6950:4219\.037013\.52301\.40770\.8290L\-SplineFNONone2,352,0511:0462\.245051\.39701\.21110\.4520L\-SplineFNOConcatenation2,352,1471:0410\.47806\.72381\.55830\.9412L\-SplineFNOCross\-attention2,382,3551:137\.88065\.57031\.41470\.9589L\-SplineFNOFiLM2,368,9471:059\.39236\.44681\.45690\.9484

Table 20:Performance metrics of various models and conditioning methods for the Aircraft \(3D, steady\-state, compressible flow, 5 fields\) case\. Best value per metric within each architecture in bold\.R2R^\{2\}is the unweighted mean of the per\-variable fieldR2R^\{2\}over the five outputs\.ModelCond\. Method\# ParametersTrain Time\(hrs:mins\)Rel\.L2L\_\{2\}% \(↓\\downarrow\)Rel\.L1L\_\{1\}% \(↓\\downarrow\)L2/L1L\_\{2\}/L\_\{1\}R2R^\{2\}\(↑\\uparrow\)TransolverNone2,433,9271:1114\.690811\.72121\.25340\.4330TransolverConcatenation2,435,7191:117\.77805\.66771\.37230\.9355TransolverCross\-attention2,467,4321:178\.45836\.26341\.35040\.9195TransolverFiLM2,451,2091:127\.46855\.54151\.34770\.9378UPTNone2,252,5470:5416\.738213\.11771\.27600\.4039UPTConcatenation2,254,1150:548\.66756\.23521\.39010\.9217UPTCross\-attention2,286,0521:018\.44156\.14471\.37380\.9226UPTFiLM2,269,8290:567\.32975\.49531\.33380\.9382MSPTNone2,190,9156:1815\.414212\.44511\.23860\.3535MSPTConcatenation2,193,6036:167\.50685\.58201\.34480\.9381MSPTCross\-attention2,224,2916:227\.41615\.46971\.35580\.9372MSPTFiLM2,208,0676:177\.45165\.53791\.34560\.9378GNOTNone2,193,0353:0114\.983311\.95311\.25350\.4415GNOTConcatenation2,194,3793:027\.90325\.82511\.35670\.9311GNOTCross\-attention2,226,4113:088\.53106\.28411\.35760\.9238GNOTFiLM2,210,1873:037\.83895\.85291\.33930\.9317NUNONone2,392,7130:3215\.552812\.90501\.20520\.1954NUNOConcatenation2,392,8320:326\.99045\.17301\.35130\.9446NUNOCross\-attention2,426,0890:376\.99945\.12711\.36520\.9453NUNOFiLM2,409,8650:336\.86655\.12081\.34090\.9428GINONone2,326,3533:0016\.489013\.66091\.20700\.1204GINOConcatenation2,327,2423:017\.21325\.27721\.36690\.9439GINOCross\-attention2,359,7293:087\.13495\.23451\.36300\.9433GINOFiLM2,343,5053:037\.34125\.42241\.35390\.9414SplineFNONone2,357,4771:5415\.626912\.49131\.25100\.3607SplineFNOConcatenation2,358,1491:547\.61615\.69451\.33750\.9284SplineFNOCross\-attention2,390,8532:007\.27285\.46641\.33040\.9342SplineFNOFiLM2,374,6291:597\.63875\.67731\.34550\.9345L\-SplineFNONone2,374,3510:3714\.325411\.69691\.22470\.3515L\-SplineFNOConcatenation2,374,7080:377\.23375\.24471\.37920\.9403L\-SplineFNOCross\-attention2,407,7270:436\.46784\.66101\.38760\.9522L\-SplineFNOFiLM2,391,5030:386\.51064\.79721\.35720\.9519

Table 21:Performance metrics of various models and conditioning methods for the FDS Building Fire \(3D, time\-averaged LES, three output fields\) case\. Single run per entry \(seed 42\); training times on a single NVIDIA H100;N=92,837N=92\{,\}837nodes\. Best value per metric within each architecture in bold\.ModelCond\. Method\# ParametersTrain Time\(hrs:mins\)Rel\.L2L\_\{2\}% \(↓\\downarrow\)Rel\.L1L\_\{1\}% \(↓\\downarrow\)L2/L1L\_\{2\}/L\_\{1\}R2R^\{2\}\(↑\\uparrow\)TransolverNone2,433,4131:1418\.879318\.23111\.03560\.0207TransolverConcatenation2,434,6931:130\.70630\.59261\.19180\.9982TransolverCross\-attention2,465,1261:220\.92520\.83881\.10290\.9954TransolverFiLM2,450,5671:140\.44810\.37041\.20970\.9993UPTNone2,252,0970:4818\.886918\.22961\.03610\.0181UPTConcatenation2,253,2170:481\.16210\.97951\.18640\.9959UPTCross\-attention2,283,8100:580\.93160\.81831\.13850\.9952UPTFiLM2,269,2510:490\.45180\.35231\.28250\.9993MSPTNone2,190,5295:2418\.909418\.25301\.03600\.0255MSPTConcatenation2,192,4495:241\.03360\.88481\.16810\.9959MSPTCross\-attention2,222,1135:390\.70460\.62271\.13150\.9961MSPTFiLM2,207,5535:250\.42530\.34541\.23120\.9994GNOTNone2,192,8412:5118\.903718\.25081\.03580\.0238GNOTConcatenation2,193,8012:521\.47631\.36331\.08280\.9886GNOTCross\-attention2,224,4252:590\.63000\.54381\.15860\.9980GNOTFiLM2,209,8652:520\.60420\.49201\.22800\.9989NUNONone2,392,4550:0418\.898218\.24301\.03590\.0234NUNOConcatenation2,392,5400:040\.89910\.67201\.33810\.9972NUNOCross\-attention2,424,0390:131\.28731\.11011\.15960\.9886NUNOFiLM2,409,4790:051\.01690\.82791\.22820\.9956GINONone2,326,2671:1618\.887018\.24301\.03530\.0158GINOConcatenation2,326,9021:161\.02030\.70041\.45680\.9958GINOCross\-attention2,357,8511:261\.32550\.98861\.34080\.9916GINOFiLM2,343,2911:171\.06310\.75141\.41480\.9955SplineFNONone2,357,2790:1718\.904518\.24961\.03590\.0245SplineFNOConcatenation2,357,7590:160\.86170\.73381\.17440\.9957SplineFNOCross\-attention2,388,8630:250\.79990\.70951\.12750\.9972SplineFNOFiLM2,374,3030:170\.50800\.41851\.21380\.9991L\-SplineFNONone2,371,7270:2618\.899018\.24701\.03570\.0239L\-SplineFNOConcatenation2,371,8120:241\.23061\.05461\.16690\.9958L\-SplineFNOCross\-attention2,403,3110:320\.69240\.62821\.10210\.9963L\-SplineFNOFiLM2,388,7510:250\.43570\.34871\.24940\.9991

Table 22:Performance metrics of various models and conditioning methods for the Heated Duct \(3D, steady\-state, multi\-region conjugate heat transfer\) case\. Best value per metric within each architecture in bold\.ModelCond\. Method\# ParametersTrain Time\(hrs:mins\)Rel\.L2L\_\{2\}% \(↓\\downarrow\)Rel\.L1L\_\{1\}% \(↓\\downarrow\)L2/L1L\_\{2\}/L\_\{1\}R2R^\{2\}\(↑\\uparrow\)TransolverNone2,119,9572:236\.37995\.58411\.14250\.2876TransolverConcatenation2,121,3972:240\.50160\.41671\.20380\.9953TransolverCross\-attention2,219,7662:420\.30520\.24621\.23960\.9983TransolverFiLM2,178,9032:260\.28840\.24631\.17080\.9984UPTNone2,382,6411:156\.32555\.53871\.14200\.2766UPTConcatenation2,384,0331:160\.94080\.82031\.14700\.9844UPTCross\-attention2,482,4501:370\.30340\.25741\.17910\.9985UPTFiLM2,441,5871:200\.19830\.16611\.19420\.9994MSPTNone2,190,5297:396\.33075\.51681\.14750\.2818MSPTConcatenation2,192,8337:390\.42190\.35251\.19700\.9969MSPTCross\-attention2,290,0817:580\.29810\.25611\.16370\.9987MSPTFiLM2,249,2177:400\.28110\.23811\.18040\.9991GNOTNone2,377,4098:486\.36875\.56381\.14470\.2882GNOTConcatenation2,378,6098:460\.65760\.56461\.16480\.9924GNOTCross\-attention2,476,9619:090\.32510\.26361\.23360\.9986GNOTFiLM2,436,0978:500\.21290\.17291\.23190\.9994NUNONone2,374,9390:086\.31305\.50671\.14640\.2771NUNOConcatenation2,375,0950:080\.43930\.34931\.25750\.9953NUNOCross\-attention2,474,4910:250\.46140\.38221\.20740\.9935NUNOFiLM2,433,6270:070\.30530\.24701\.23620\.9980SplineFNONone2,350,8130:196\.36375\.55091\.14640\.2875SplineFNOConcatenation2,351,0050:180\.79420\.67581\.17510\.9848SplineFNOCross\-attention2,450,3650:370\.27200\.22841\.19100\.9979SplineFNOFiLM2,409,5010:190\.21790\.18171\.19890\.9992L\-SplineFNONone2,351,9191:056\.33225\.51511\.14820\.2815L\-SplineFNOConcatenation2,352,1111:060\.72210\.60811\.18760\.9872L\-SplineFNOCross\-attention2,451,4711:230\.42270\.36471\.15920\.9978L\-SplineFNOFiLM2,410,6071:060\.21150\.16991\.24500\.9991

Table 23:Performance metrics of various models and conditioning methods for the HIVE Monoblock \(3D, steady\-state, thermal\) case\. Best value per metric within each architecture in bold\.

## Appendix HScalability

We extend the comparison in[Section4\.3](https://arxiv.org/html/2610.22446#S4.SS3)using synthetic three\-dimensional point clouds with increasing node count, scalar FiLM conditioning, and one output channel\. All runs use a single A100 80GB GPU, FP32 with TF32 disabled, and batch size one\. We retain the HIVE architecture settings and 99 coordinate features, holding each model’s latent resolution fixed as the point cloud becomes denser\. We measure inference without an autograd graph, a forward pass with autograd enabled, a backward pass, and a complete Adam training step\. Backward timing excludes the prerequisite forward pass; its memory includes the retained graph\. Measurements use five repetitions after warm\-up for each of three seeds\. Timing curves report medians of per\-seed timing medians, and memory curves report medians of per\-seed peak PyTorch allocations, including resident tensors but excluding CUDA\-context and non\-PyTorch allocations\. Model and data setup are outside the timed regions\.

[Figure16](https://arxiv.org/html/2610.22446#A8.F16)shows that L\-SplineFNO’s runtime and memory grow approximately linearly over the observed range, while its lower memory cost allows execution on substantially larger point clouds\. Several baselines also exhibit near\-linear trends over their feasible ranges; NUNO is faster in the successful steady\-state trials, so runtime and capacity capture different aspects of scalability\. The approximately level L\-SplineFNO curves in[Figure17](https://arxiv.org/html/2610.22446#A8.F17)make its stable cost per node more apparent, with fixed overhead contributing more at smaller sizes\. These results along with[section4\.3](https://arxiv.org/html/2610.22446#S4.SS3)describe the tested implementations at fixed latent resolution; they do not establish accuracy scaling or a unique asymptotic advantage\.

Figure 16:Runtime \(top\) and peak allocated GPU memory \(bottom\) as node count increases, separated by execution phase\. Both axes are logarithmic\. Points summarise successful seeds only, and lines connect measured sizes without extrapolation\. NUNO’s training result at ten million nodes includes two successful seeds; the remaining seed runs out of memory\. Missing points indicate unsuccessful scaling rather than zero cost\.Figure 17:Runtime and peak allocated memory per million nodes, using the same measurements and aggregation as[Figure16](https://arxiv.org/html/2610.22446#A8.F16)\. Both axes are logarithmic\. A nearly horizontal curve indicates approximately linear growth of the total cost over the measured range; a decreasing curve can reflect amortisation of fixed overhead\. L\-SplineFNO maintains a broadly stable cost per node as the point cloud grows\.
## Appendix IAblations

### I\.1Zero\-shot super\-resolution

Table 24:Zero\-shot super\-resolution with FiLM\-conditioned L\-SplineFNO\.ExperimentNodesRel\.L2L\_\{2\}\(%\)R2R^\{2\}Heat Sink31,3130\.02390\.9996Bracket14,703–15,2103\.51420\.9958Aircraft331,9718\.32370\.9899We evaluate L\-SplineFNO without fine\-tuning on held\-out native meshes containing at least95%95\\%of the maximum available nodes, which is1\.91×1\.91\\times,1\.791\.79–1\.86×1\.86\\times, and2\.53×2\.53\\timesthe training resolution for Heat Sink, Bracket, and Aircraft, respectively\. We chose these tasks to demonstrate super\-resolution because of the inherent mesh variability across their datasets\. As shown in[Table24](https://arxiv.org/html/2610.22446#A9.T24), the model retains high predictive agreement at each resolution\. This zero\-shot transfer is enabled by the coordinate\-dependent spline scatter and decoder, which operate independently of the number and ordering of mesh nodes: L\-SplineFNO can therefore be deployed on substantially finer meshes without retraining while largely preserving its predictive accuracy\.

### I\.2Multiscale depth and spline degree

We isolate two properties of the spline transfer while holding the backbone, FiLM conditioning pathway, optimiser, and training budget fixed: the range of geometric information supplied to the FNO, and the polynomial degree, smoothness, and compact support of the spline basis\. Every configuration is trained on all six benchmark cases at three seeds; a seed redraws the train/test split and the initialisation while the sampled point cloud stays fixed, and the heated duct and building fire use region modebasis\. Because error and runtime differ substantially between benchmarks,[Figures18\(a\)](https://arxiv.org/html/2610.22446#A9.F18.sf1)and[18\(b\)](https://arxiv.org/html/2610.22446#A9.F18.sf2)report changes relative to the single\-scale cubic configuration used for the main results, paired by seed so that every configuration is compared with the baseline on the same test set; the corresponding absolute errors, runtimes, parameter counts, andR2R^\{2\}values are given in[Tables25](https://arxiv.org/html/2610.22446#A9.T25)and[26](https://arxiv.org/html/2610.22446#A9.T26)\.

\(a\)Effect of scatter\-scale depth on predictive error and training cost\.\(b\)Effect of spline degree on predictive error and training cost\.
Figure 18:Accuracy–cost trade\-offs for multiscale scattering and spline degree\. Values are percentage changes relative to the single\-scale and cubic configurations, respectively, computed against the baseline trained at the same seed\. Points are means over three seeds and error bars one standard deviation; spline orders66and88were run at a single seed and carry no bar\. Spline orders22,44,66, and88correspond to degreesp=3p=3,55,77, and99, i\.e\. aCp−1C^\{p\-1\}basis\. By increasing spline order and the scatter length, the model only makes marginal gains against a linear climb of computational costs\. Absolute results are reported in[Tables25](https://arxiv.org/html/2610.22446#A9.T25)and[26](https://arxiv.org/html/2610.22446#A9.T26)\.##### Multiscale scatter\.

The single\-scale L\-SplineFNO scatters the unstructured nodal features onto one fine latent grid\. A scatter scales\>1s\>1adds a second branch over the same physical domain with approximatelysstimes fewer grid intervals along each axis, so its compact spline stencil covers a wider physical neighbourhood; the coarse representation is interpolated back to the fine grid and fused with the other branches by a learned pointwise projection before the FNO blocks\. We compare a fine branch alone \(11\), fine and medium branches \(11–22\), and fine, medium, and coarse branches \(11–22–44\), with the analytic cubic decoder unchanged in all three configurations\. The additional branches add at most0\.13%0\.13\\%to the parameter count\.

Multiscale depth gives no systematic accuracy gain\. Aircraft is the only case it helps: the11–22hierarchy lowers its relativeL2L\_\{2\}error in every seed, by5\.2±5\.7%5\.2\\pm 5\.7\\%, while11–22–44gives3\.7±11\.0%3\.7\\pm 11\.0\\%with no consistent sign\. The heat sink is consistently harmed, with errors6\.1±4\.1%6\.1\\pm 4\.1\\%and6\.9±0\.9%6\.9\\pm 0\.9\\%higher under the two hierarchies\. On the bracket and building fire the paired changes stay below2%2\\%, and on the heated duct and HIVE they are smaller than the seed\-to\-seed spread\. The cost, by contrast, is systematic: training time rises by2121–44%44\\%for11–22and by4040–89%89\\%for11–22–44\. The aircraft results suggest that its data form a thin surface inside a volumetric latent grid, where a fine stencil leaves most knots unreached and a coarser branch carries surface information further; on volumetric meshes the fine branch already covers the grid, and the wider branches mainly add smoothing\. Additional scatter scales are therefore not a reliable source of accuracy at this budget, which is why the main results use a single scale\.

Table 25:Multiscale L\-SplineFNO \(FiLM conditioning\) across the benchmark cases at increasing scatter\-scale depth, all at spline degree 3\. Heated\-duct and FDS run under region modebasis\. Errors,L2/L1L\_\{2\}/L\_\{1\}andR2R^\{2\}are mean±\\pms\.d\. over seeds 42, 43 and 44 where a cell has several \(train/test split and initialisation vary with the seed\); training time is the seed mean\. For the multi\-variable cases \(Aircraft, FDS\)R2R^\{2\}is the mean of the per\-variable scores\.CaseScatter Scales\# ParametersTrain Time\(hrs:mins\)Rel\.L2L\_\{2\}% \(↓\\downarrow\)Rel\.L1L\_\{1\}% \(↓\\downarrow\)L2/L1L\_\{2\}/L\_\{1\}R2R^\{2\}\(↑\\uparrow\)Heat\-sink12,395,7750:430\.02842±\\pm0\.003310\.02201±\\pm0\.002691\.2910\.999681–22,396,9690:510\.03012±\\pm0\.003110\.02303±\\pm0\.002571\.3090\.999661–2–42,397,5630:590\.03036±\\pm0\.003270\.02310±\\pm0\.002431\.3140\.99965Bracket12,387,7310:541\.9825±\\pm0\.02071\.1096±\\pm0\.02831\.7870\.998321–22,388,3441:161\.9734±\\pm0\.02841\.0996±\\pm0\.03161\.7950\.998331–2–42,388,6511:372\.0134±\\pm0\.03281\.1319±\\pm0\.02251\.7790\.99826Aircraft12,368,9471:058\.9469±\\pm0\.52296\.2004±\\pm0\.19861\.4420\.957591–22,371,0451:218\.4801±\\pm0\.66386\.1162±\\pm0\.47411\.3860\.959561–2–42,372,0871:378\.5837±\\pm0\.66415\.8942±\\pm0\.19951\.4550\.96172FDS12,391,4671:026\.6007±\\pm0\.09764\.8465±\\pm0\.04961\.3620\.951931–22,392,0801:276\.6617±\\pm0\.05944\.8770±\\pm0\.03031\.3660\.950201–2–42,392,3871:486\.5833±\\pm0\.06394\.8186±\\pm0\.05081\.3660\.95111Heated\-duct12,391,2310:130\.4291±\\pm0\.04110\.3569±\\pm0\.05401\.2090\.999161–22,391,8440:190\.4519±\\pm0\.09610\.3725±\\pm0\.10091\.2250\.999061–2–42,392,1510:250\.4479±\\pm0\.08250\.3756±\\pm0\.07551\.1950\.99921HIVE12,410,6071:060\.2890±\\pm0\.06920\.2437±\\pm0\.06951\.1980\.998611–22,412,7051:280\.2943±\\pm0\.13410\.2412±\\pm0\.11881\.2330\.998521–2–42,413,7471:500\.3079±\\pm0\.10560\.2587±\\pm0\.09641\.1980\.99848

##### Spline degree\.

Increasing the spline degreeppraises the continuity of the analytic basis fromC2C^\{2\}for the cubic basis toC4C^\{4\},C6C^\{6\}, andC8C^\{8\}for degrees five, seven, and nine, and widens the compact support fromp\+1=4p\+1=4knots per coordinate axis to six, eight, and ten\. We vary this degree while retaining a single scatter scale, so the ablation changes the scatter and decode bases but not the FNO backbone; the parameter count moves by at most nine, through the rank of the partition\-of\-unity subspace of the learnable kernel\. Higher degrees do not improve accuracy\. Degree five, measured at all three seeds, lowers the aircraft error and leaves the building fire unchanged, but raises the error on the bracket, heated duct, HIVE, and heat sink\. At degrees seven and nine, run at a single seed, the error is never meaningfully below the cubic one and grows with degree on the bracket, aircraft, and heated duct\. Aircraft is again the case that benefits from a moderately wider support, in line with the multiscale result, but the extra smoothing of degrees seven and nine outweighs it there too\. This is consistent with a wider, smoother basis offering little for fields governed by local geometric detail and, for the heated duct, by sharp changes in gradient across material interfaces, where excess smoothness mixes information across these features\.

The computational penalty is substantially larger and systematic\. In three dimensions, a degree\-pptensor\-product basis contains\(p\+1\)3\(p\+1\)^\{3\}stencil entries per point —6464,216216,512512, and10001000for degrees three, five, seven, and nine — so the scatter and decode work scales asO⁡\(N​\(p\+1\)3\)O\(N\(p\+1\)^\{3\}\), whereNNis the number of mesh nodes\. Training time accordingly rises exponenially with the degree\. Because the stencil is streamed in memory\-bounded chunks, every degree completes on every mesh, and the cost appears as training time rather than memory\. With the parameter count effectively fixed, this cost comes from basis construction alone, which leaves the cubic basis as the most reliable accuracy–efficiency compromise across the benchmark suite\.

Table 26:Spline degree on single\-scale L\-SplineFNO \(FiLM conditioning\) across the benchmark cases\. Heated\-duct and FDS run under region modebasis\. Errors,L2/L1L\_\{2\}/L\_\{1\}andR2R^\{2\}are mean±\\pms\.d\. over seeds 42, 43 and 44 where a cell has several \(train/test split and initialisation vary with the seed\); training time is the seed mean\. For the multi\-variable cases \(Aircraft, FDS\)R2R^\{2\}is the mean of the per\-variable scores\. Degrees 7 and 9 are a single seed \(42\), so their absolute errors are not comparable with the seed means above them; the bracketed figure is the change against degree 3 at the same seed\.CaseSpline Degree\# ParametersTrain Time\(hrs:mins\)Rel\.L2L\_\{2\}% \(↓\\downarrow\)Rel\.L1L\_\{1\}% \(↓\\downarrow\)L2/L1L\_\{2\}/L\_\{1\}R2R^\{2\}\(↑\\uparrow\)Heat\-sink32,395,7750:420\.02867±\\pm0\.001840\.02215±\\pm0\.001571\.2950\.9996852,395,7691:100\.03425±\\pm0\.007500\.02627±\\pm0\.005511\.3020\.9995772,395,7782:000\.02852 \(\+0\.4%\)0\.021981\.2980\.9997392,395,7693:240\.02903 \(\+2\.2%\)0\.022651\.2820\.99972Bracket32,387,7310:541\.9702±\\pm0\.03151\.0861±\\pm0\.03521\.8150\.9983352,387,7251:401\.9806±\\pm0\.03281\.1317±\\pm0\.02721\.7500\.9983072,387,7342:502\.0421 \(\+4\.2%\)1\.19911\.7030\.9981292,387,7254:442\.1541 \(\+9\.9%\)1\.29391\.6650\.99790Aircraft32,368,9471:059\.0509±\\pm0\.27486\.2016±\\pm0\.18031\.4590\.9572052,368,9412:018\.5180±\\pm0\.64856\.1067±\\pm0\.17761\.3940\.9596872,368,9503:539\.4539 \(\+1\.1%\)6\.37421\.4830\.9479292,368,9416:5510\.121 \(\+8\.2%\)6\.78321\.4920\.94308FDS32,391,4671:026\.6110±\\pm0\.09254\.8578±\\pm0\.04381\.3610\.9517052,391,4612:116\.5946±\\pm0\.15504\.8214±\\pm0\.10681\.3680\.9518772,391,4704:416\.7042 \(−\-0\.2%\)4\.91291\.3650\.9489292,391,4618:306\.6797 \(−\-0\.5%\)4\.87141\.3710\.94975Heated\-duct32,391,2310:130\.4401±\\pm0\.04600\.3693±\\pm0\.05741\.1980\.9991152,391,2250:320\.4510±\\pm0\.05560\.3771±\\pm0\.06611\.2030\.9990772,391,2341:110\.4383 \(\+1\.5%\)0\.36311\.2070\.9991892,391,2252:140\.4604 \(\+6\.6%\)0\.37231\.2370\.99914HIVE32,410,6071:060\.2919±\\pm0\.12590\.2484±\\pm0\.11811\.1900\.9985552,410,6012:380\.3134±\\pm0\.09530\.2682±\\pm0\.09001\.1760\.9983972,410,6105:370\.2339 \(\+12\.1%\)0\.18781\.2460\.9990192,410,60110:360\.2347 \(\+12\.5%\)0\.19581\.1990\.99902

### I\.3Analytic Gradient

Figure 19:Gradient comparisons across autograd vector Jacobian products across the entire model, Decoder queries and the analytic spline across the six benchmark cases\.With the encoded latent grid held fixed, the L\-SplineFNO decoder defines a continuous field over the query coordinates\. For a selected scalar output, its affine readout can be applied to the latent coefficients before interpolation, giving

u^​\(q\)=∑g∈𝒢cg​Φg​\(q\)\+b,∇qu^​\(q\)=∑g∈𝒢cg​∇qΦg​\(q\),cg=w𝖳​hg,\\widehat\{u\}\(q\)=\\sum\_\{g\\in\\mathcal\{G\}\}c\_\{g\}\\,\\Phi\_\{g\}\(q\)\+b,\\qquad\\nabla\_\{q\}\\widehat\{u\}\(q\)=\\sum\_\{g\\in\\mathcal\{G\}\}c\_\{g\}\\,\\nabla\_\{q\}\\Phi\_\{g\}\(q\),\\qquad c\_\{g\}=w^\{\\mathsf\{T\}\}h\_\{g\},\(86\)wherehgh\_\{g\}is a latent\-grid coefficient andΦg\\Phi\_\{g\}is the normalized cubic tensor\-product basis\. The readout bias contributes no derivative\. Boundary normalization must also be differentiated: writingΦg=Bg/S\\Phi\_\{g\}=B\_\{g\}/SwithS=∑hBh\>0S=\\sum\_\{h\}B\_\{h\}\>0on the covered query domain gives∇Φg=\(S∇Bg−Bg∇S\)/S2\\nabla\\Phi\_\{g\}=\(S\\nabla B\_\{g\}\-B\_\{g\}\\nabla S\)/S^\{2\}\. Thus the calculation differentiates the actual normalized decoder, including queries near grid boundaries\.

Each query touches at most4d4^\{d\}grid points indddimensions\. After projecting the latent coefficients onto the requested output channel, the implementation gathers each local coefficient once and reuses it for all first\-derivative components\. Axis\-basis evaluation and material\-specific readouts are batched to reduce repeated operations and temporary arrays\. This givesO⁡\(N​4d\)O\(N4^\{d\}\)query\-evaluation work for a fixed number of output channels, in addition to the grid projection\. For material\-specific readouts, region membership is held fixed, so derivatives are evaluated within each material\. Coordinate and output normalization scales convert the results to physical units\. These expressions remain differentiable with respect to the latent coefficients and readout weights, allowing first\-derivative losses to train the model\.

The two pointwise methods in Table[27](https://arxiv.org/html/2610.22446#A9.T27)evaluate this same fixed\-grid field\. Full\-model autograd VJP instead differentiates the sum of the selected output over all nodes with respect to the input coordinates\. Moving an input node can change the encoded grid and hence predictions elsewhere, so this coordinate sensitivity is a different quantity from the pointwise field gradient\. Decoder autograd starts from a prepared forward graph, whereas the analytic method constructs its basis derivatives and projects the coefficients during the timed call; these operations are not cached outside the measurement\.

Across the six cases, the analytic evaluator is1\.841\.84–3\.67×3\.67\\timesfaster than decoder autograd and uses5252–68%68\\%less additional peak memory, using the unrounded measurements\. The largest relativeL2L\_\{2\}disagreement between their physical gradients is3\.04×10−63\.04\\times 10^\{\-6\}\. Figure[19](https://arxiv.org/html/2610.22446#A9.F19)shows the corresponding per\-case costs\. These results concern first\-derivative evaluation after forward preparation; they do not measure end\-to\-end inference or the memory needed to retain that prepared state\.

Autograd VJPDecoder autogradAnalytic spline\(coord\. sensitivity\)\(pointwise∇u^\\nabla\\widehat\{u\}, latent fixed\)GeometryuuNNmsMiBmsMiBmsMiBHeat sinkTT16 38413\.3235\.84\.7125\.82\.539\.9Bracketσ\\sigma8 19213\.786\.34\.149\.82\.220\.5AircraftPP131 07258\.9423\.114\.2248\.43\.983\.4FDSΔ​T\\Delta T98 49631\.7362\.610\.2237\.43\.784\.3Heated ductTT124 80028\.9343\.910\.8239\.03\.581\.8HIVETT194 69160\.6395\.319\.0251\.55\.2120\.4Table 27:L\-SplineFNO spatial\-derivative cost: time and additional peak memory\. Decoder autograd and the analytic spline compute the same pointwise gradient of fielduuwith the encoded latent grid fixed\. Autograd VJP measures full\-model coordinate sensitivity\. Timings cover only derivative evaluation and unit conversion, excluding encoding and prediction, with a fresh forward state per call\. Values are medians of 20 runs after 3 warm\-ups on an A100 80GB \(batch 1, FP32, TF32 off\)\. Memory is the extra peak CUDA allocation above the prepared state, including any saved autograd graph, not total GPU memory\.