Local and Sparse Kernel Operator Learning for Incompressible Flows: An Operator Lagrange Formulation

# Local and Sparse Kernel Operator Learning for Incompressible Flows: An Operator Lagrange Formulation
Source: [https://arxiv.org/html/2610.29009](https://arxiv.org/html/2610.29009)
Matthew Lowerymlowery@cs\.utah\.eduHouman Owhadiowhadi@caltech\.eduVarun Shankarshankar@cs\.utah\.eduorganization=Kahlert School of Computing, University of Utah,state=UT, country=USAorganization=Department of Computing and Mathematical Sciences, California Institute of Technology,state=CA, country=USA

###### Abstract

We present a local and sparse kernel method for learning the solution operators of incompressible flows\. It builds on our earlier property\-preserving kernel method, whose global output reconstruction and dense operator regression scale poorly\. Our method replaces the global output reconstruction by local divergence\-free interpolants with divergence\-free polynomials of any degree, solved in parallel on the GPU\. These interpolants are divergence free, periodic on periodic domains, and, with a multiscale kernel, approach Kolmogorov statistics under refinement\. A compactly supported kernel, positive semidefinite in any input dimension, makes the operator regression sparse\. Its support radius, and hence the sparsity, is fitted to data, and optional affine polynomials in the input capture global trends\. Operator Lagrange weights join the two\. They cost one sparse solve per new input and act on the training fields before the output reconstruction, so their local coefficients are never formed\. We show that, unlike direct evaluation, Lagrange evaluation does not amplify solve errors by the cancellation among spatial coefficients\. On ten fluid benchmarks, validation selects sparse matrices nearly everywhere, with densities that mostly decrease as the training set grows, and affine augmentation lowers the error by more than an order of magnitude for forced isotropic turbulence\. These sparse models are at least as accurate as our earlier dense method and more accurate than neural operators\. Lagrange evaluation is as accurate as direct evaluation with local output reconstructions, and up to nearly three orders of magnitude more accurate with global ones\. We also present GPU timings and storage\.

###### keywords

operator learning ,incompressible flow ,divergence\-free kernels ,compactly supported kernels ,RBF\-FD ,sparse Cholesky ,GPU computing\\MSC\[2020\] 65D12 ,65D15 ,46E22 ,65F50 ,76D05

††corresponding:Corresponding author## 1Introduction

Uncertainty quantification, design, and inverse problems for incompressible flows require the velocity field for many initial conditions, boundary conditions, or forcing functions, and the repeated solution of the Navier–Stokes equations then dominates the cost\[[60](https://arxiv.org/html/2610.29009#bib.bib31),[43](https://arxiv.org/html/2610.29009#bib.bib32),[59](https://arxiv.org/html/2610.29009#bib.bib33),[61](https://arxiv.org/html/2610.29009#bib.bib34)\]\. One remedy is to learn the solution operator, the map from the problem data to the velocity, from a collection of simulations\[[7](https://arxiv.org/html/2610.29009#bib.bib13)\]\. Reduced\-order models have long served this purpose\[[42](https://arxiv.org/html/2610.29009#bib.bib35),[5](https://arxiv.org/html/2610.29009#bib.bib36)\], and neural operators such as DeepONets, Fourier and graph neural operators, transformer\-based models, and kernel neural operators now learn such maps from large data sets\[[37](https://arxiv.org/html/2610.29009#bib.bib9),[33](https://arxiv.org/html/2610.29009#bib.bib5),[34](https://arxiv.org/html/2610.29009#bib.bib6),[32](https://arxiv.org/html/2610.29009#bib.bib7),[70](https://arxiv.org/html/2610.29009#bib.bib12),[36](https://arxiv.org/html/2610.29009#bib.bib8)\]\. We seek an operator learner for incompressible flows that satisfies four criteria\. First, every prediction should be divergence free, and periodic on periodic domains, however accurately the operator itself is learned\. Second, the cost of the output representation should grow linearly with the number of output points, so that fine three\-dimensional outputs remain affordable\. Third, the regression should handle ten thousand or more training functions without factoring a dense matrix\. Fourth, every stage should run in parallel on a graphics processing unit \(GPU\)\.

Kernel methods offer an attractive alternative to neural operators\[[2](https://arxiv.org/html/2610.29009#bib.bib1),[38](https://arxiv.org/html/2610.29009#bib.bib15),[29](https://arxiv.org/html/2610.29009#bib.bib14)\]\. Batlle et al\.\[[2](https://arxiv.org/html/2610.29009#bib.bib1)\]learn an operator from point samples in two steps\. An operator regression, a kernel regression on the space of sampled inputs, maps the samples of the input function to the samples of the output function\. An output reconstruction, a kernel interpolant of these samples in space \(or spacetime\), then recovers the output function\. Since both steps are kernel interpolation or regression problems, the learned operator has a closed form, few parameters, and an error analysis\. For flows, the output representation is the natural place to impose physics\[[45](https://arxiv.org/html/2610.29009#bib.bib29),[30](https://arxiv.org/html/2610.29009#bib.bib30)\]\. In our previous work\[[53](https://arxiv.org/html/2610.29009#bib.bib10)\], we therefore regressed the coefficients of a divergence\-free kernel interpolant of the output instead of its point samples, so that every prediction is exactly divergence free regardless of the accuracy of the operator regression\. We called this the property\-preserving kernel method \(PPKM\)\. PPKM satisfies the first criterion, but both of its kernel problems are global\. Its operator regression factors one dense kernel matrix on allNNtraining inputs, at a cost ofO⁡\(N3\)O\(N^\{3\}\)operations andO⁡\(N2\)O\(N^\{2\}\)storage, and its output reconstruction solves one dense system on all output points\. Both limits bind for three\-dimensional turbulence, where a training set has ten thousand or more functions and an output has thousands of points\. Meeting the other criteria poses three tasks\. The output reconstruction must become local without losing the properties of the global one, the operator regression must become sparse, and the two must be joined without forming the local coefficients of every training field\.

The first task draws on radial basis function\-generated finite differences \(RBF\-FD\)\. RBF\-FD methods replace a global radial basis function \(RBF\) interpolant, whose dense system couples all nodes, by small interpolants on the stencil of each node, so that every linear system has a fixed size and is independent of the others\[[62](https://arxiv.org/html/2610.29009#bib.bib51),[55](https://arxiv.org/html/2610.29009#bib.bib52),[69](https://arxiv.org/html/2610.29009#bib.bib53),[21](https://arxiv.org/html/2610.29009#bib.bib56),[20](https://arxiv.org/html/2610.29009#bib.bib57)\]\. Over the last two decades, RBF\-FD methods have been stabilized for convection\-dominated problems\[[22](https://arxiv.org/html/2610.29009#bib.bib54),[48](https://arxiv.org/html/2610.29009#bib.bib62)\], applied to the Navier–Stokes equations\[[55](https://arxiv.org/html/2610.29009#bib.bib52),[18](https://arxiv.org/html/2610.29009#bib.bib59)\], extended to overlapped stencils, surfaces, and moving domains\[[52](https://arxiv.org/html/2610.29009#bib.bib61),[50](https://arxiv.org/html/2610.29009#bib.bib64),[51](https://arxiv.org/html/2610.29009#bib.bib65)\], and formulated in the least\-squares sense\[[63](https://arxiv.org/html/2610.29009#bib.bib66)\]\. They also have error bounds\[[10](https://arxiv.org/html/2610.29009#bib.bib58)\], node generators for irregular domains\[[49](https://arxiv.org/html/2610.29009#bib.bib63)\], and GPU implementations\[[6](https://arxiv.org/html/2610.29009#bib.bib55)\]\. In particular, polyharmonic splines \(PHS\) augmented with polynomials converge at the order of the polynomial degree and have no shape parameter\[[19](https://arxiv.org/html/2610.29009#bib.bib3),[3](https://arxiv.org/html/2610.29009#bib.bib60)\]\. For flows, the local interpolants must also be divergence free\. Divergence\-free kernels, obtained by applying−Δ​I\+∇∇𝖳\-\\Delta I\+\\nabla\\nabla^\{\\mathsf\{T\}\}to a scalar kernel\[[40](https://arxiv.org/html/2610.29009#bib.bib18),[11](https://arxiv.org/html/2610.29009#bib.bib19),[39](https://arxiv.org/html/2610.29009#bib.bib20),[25](https://arxiv.org/html/2610.29009#bib.bib21),[66](https://arxiv.org/html/2610.29009#bib.bib22)\], have this property by construction and have been used both globally and in partition\-of\-unity form\[[24](https://arxiv.org/html/2610.29009#bib.bib4),[12](https://arxiv.org/html/2610.29009#bib.bib23)\], and Yelnyk\[[71](https://arxiv.org/html/2610.29009#bib.bib24)\]combined divergence\-free PHS with polynomials in local Hermite interpolation for Stokes problems\. However, the high order of PHS\-based RBF\-FD comes from the polynomials, so a local divergence\-free interpolant of high order needs divergence\-free polynomials of any degree, in two and three dimensions, that can be evaluated anywhere on the stencil\.

Our approach is to build these polynomials by differentiating products of Legendre polynomials, select an independent subset on each stencil by a column\-pivoted QR factorization, and combine them with a divergence\-free PHS on the stencils of RBF\-FD, used for interpolation instead of differentiation\. Every local interpolant is then exactly divergence free, and building the stencils from periodic images of the nodes makes the output reconstruction periodic\. For turbulence, we replace the scale\-free PHS by the multiscale Matérn kernel of[Sharma et al\. \[53\]](https://arxiv.org/html/2610.29009#bib.bib10), whose weights encode the Kolmogorov scaling of the velocity increments, so that the output reconstruction approaches prescribed Kolmogorov statistics as the nodes are refined\. Matérn kernels and PHS are Green’s functions of closely related operators, and a Matérn kernel tends to a PHS as its length grows, so this change adds length scales without changing the construction\. Since the local systems are independent and of fixed size, we assemble, factor, and evaluate them in batches on the GPU, and our output reconstruction is embarrassingly parallel\.

For the second task, the dense kernel matrix of the operator regression can be approximated by low\-rank, Nyström, and random\-feature methods, inducing points, structured kernel interpolation, or sparse approximate factors of its inverse\[[13](https://arxiv.org/html/2610.29009#bib.bib25),[26](https://arxiv.org/html/2610.29009#bib.bib26),[57](https://arxiv.org/html/2610.29009#bib.bib28),[44](https://arxiv.org/html/2610.29009#bib.bib27),[72](https://arxiv.org/html/2610.29009#bib.bib39),[17](https://arxiv.org/html/2610.29009#bib.bib37),[1](https://arxiv.org/html/2610.29009#bib.bib38),[47](https://arxiv.org/html/2610.29009#bib.bib43),[56](https://arxiv.org/html/2610.29009#bib.bib41),[68](https://arxiv.org/html/2610.29009#bib.bib40)\]\. In contrast, a compactly supported kernel gives a sparse matrix that can be factored exactly by sparse Cholesky\. Its obstacle is the dimension of the input\. A sampled input function has thousands of coordinates, but no compactly supported radial kernel is positive definite in every dimension\[[65](https://arxiv.org/html/2610.29009#bib.bib16)\]\. Our kernel therefore determines its support from a few principal coordinates of the input, in which a compactly supported function is positive definite, and treats the remaining coordinates with a Matérn factor, which is positive definite in every dimension\. The product is positive semidefinite in any input dimension\. Its support radius sets the sparsity of the kernel matrix, and we choose it from the data instead of fixing it in advance\. A restricted likelihood criterion, with a smooth count of the interacting training pairs as an optional penalty, fits the shape of the kernel and the regularization, and the error on validation fields then selects the radius, so that nearly dense matrices remain among the candidates when accuracy requires them\.

However, a compactly supported kernel returns to zero away from the training inputs, so it represents trends in the input poorly, although many operators depend nearly linearly on part of their input\. We therefore augment the operator regression with affine polynomials in the leading principal coordinates of the input, which it reproduces exactly, as in polynomially augmented kernel interpolation\. The resulting saddle\-point system is solved by a sparse Cholesky factorization of the kernel matrix followed by a QR factorization of the whitened polynomial columns, which avoids forming the Schur complement\[[4](https://arxiv.org/html/2610.29009#bib.bib46)\]\. JAX has no sparse direct solver that keeps a factorization between calls, so we factor the kernel matrix on the GPU with cuDSS, through a primitive of our KernelPack\-JAX framework that exposes cuDSS to compiled JAX programs\[[8](https://arxiv.org/html/2610.29009#bib.bib47)\]\.

The third task leads to operator Lagrange weights\. Since the same kernel regresses every output coefficient, a prediction is a linear combination of the training outputs, with weights that depend only on the new input\. These weights cost one sparse solve per new input, regardless of the size of the output, and they give the coefficients on every stencil at once \(Figure[1](https://arxiv.org/html/2610.29009#S1.F1)\)\. Because the output reconstruction is linear, the weights can also be applied to the sampled training fields before the output reconstruction, so the local coefficients of the training fields, which would take tens of times the storage of the fields themselves, are never formed\. Weights and coefficients give the same prediction in exact arithmetic but round differently\. We show that direct evaluation through coefficients amplifies solve errors by a factor that measures the cancellation among the spatial coefficients, while Lagrange evaluation does not\.

We test our method in two steps\. On scattered nodes, our local output reconstruction converges at close to orders22,44, and66, keeps the divergence and the periodic mismatch at rounding level, and approaches prescribed Kolmogorov statistics under refinement at the separations that the nodes resolve\. Beyond a few thousand output points, it is also cheaper to set up than a global output reconstruction, and its memory grows only linearly with their number\. We then compare kernels with a fixed and a fitted shape, each with and without affine augmentation, on ten fluid benchmarks with up to ten thousand training functions\. Validation selects sparse kernel matrices on all benchmarks except forced isotropic turbulence and the unaugmented Taylor–Green models, and on most of them the selected density decreases as the training set grows\. With these sparse matrices, the test errors decrease steadily with the number of training functions\. At the training sizes of our earlier study\[[53](https://arxiv.org/html/2610.29009#bib.bib10)\], our models are at least as accurate as dense PPKM and the kernel method of[Batlle et al\. \[2\]](https://arxiv.org/html/2610.29009#bib.bib1)on every data set, and more accurate than the neural operators reported there\. Affine augmentation reduces the test error for forced isotropic turbulence by more than an order of magnitude, and together with the fitted shape it does the same for turbulent species transport\. In contrast, the five flows whose training inputs vary along a single direction are insensitive to both choices\. Finally, Lagrange evaluation is as accurate as direct evaluation with our local output reconstruction, whose amplification factors are at most about10410^\{4\}, and up to nearly three orders of magnitude more accurate with global output reconstructions, whose amplification factors reach6×1096\\times 10^\{9\}\.

The remainder of this paper is organized as follows\. Section[2](https://arxiv.org/html/2610.29009#S2)constructs the local property\-preserving output reconstruction, Section[3](https://arxiv.org/html/2610.29009#S3)the sparse operator regression and its affine augmentation, and Section[4](https://arxiv.org/html/2610.29009#S4)the operator Lagrange weights, their rounding errors, and the GPU implementation\. Section[5](https://arxiv.org/html/2610.29009#S5)presents the numerical experiments, and Section[6](https://arxiv.org/html/2610.29009#S6)our conclusions\.

### 1\.1Problem formulation

We consider an incompressible fluid of constant density, governed by

∂𝐮∂t\+\(𝐮⋅∇\)𝐮\\displaystyle\\frac\{\\partial\\mathbf\{u\}\}\{\\partial t\}\+\(\\mathbf\{u\}\\cdot\\nabla\)\\mathbf\{u\}=ν∇2𝐮−1ρ∇p\+1ρ𝐟,inΩ×\(0,T\],\\displaystyle=\\nu\\nabla^\{2\}\\mathbf\{u\}\-\\frac\{1\}\{\\rho\}\\nabla p\+\\frac\{1\}\{\\rho\}\\mathbf\{f\},\\quad\\text\{in \}\\Omega\\times\(0,T\],\(1\)∇⋅𝐮\\displaystyle\\nabla\\cdot\\mathbf\{u\}=0,in​Ω×\(0,T\],\\displaystyle=0,\\quad\\text\{in \}\\Omega\\times\(0,T\],\(2\)𝐮⁡\(𝐱,0\)\\displaystyle\\mathbf\{u\}\(\\mathbf\{x\},0\)=𝐮0​\(𝐱\),in​Ω,\\displaystyle=\\mathbf\{u\}\_\{0\}\(\\mathbf\{x\}\),\\quad\\text\{in \}\\Omega,\(3\)ℬ​𝐮\\displaystyle\\mathcal\{B\}\\mathbf\{u\}=𝐠,on​∂Ω×\(0,T\],\\displaystyle=\\mathbf\{g\},\\quad\\text\{on \}\\partial\\Omega\\times\(0,T\],\(4\)where𝐮\\mathbf\{u\}is the velocity,ppthe pressure,ν\\nuthe kinematic viscosity,ρ\\rhothe density,𝐟\\mathbf\{f\}the body force, andℬ\\mathcal\{B\}the boundary operator\. Our operators map initial data \(a velocity𝐮0\\mathbf\{u\}\_\{0\}, a vorticity, or, for a buoyancy\-driven flow, a temperature\), the boundary data𝐠\\mathbf\{g\}, or the forcing𝐟\\mathbf\{f\}to the velocity\. The training data are simulations of \([1](https://arxiv.org/html/2610.29009#S1.E1)\) from[Sharma et al\. \[53\]](https://arxiv.org/html/2610.29009#bib.bib10)\(Appendix[D\.1](https://arxiv.org/html/2610.29009#A4.SS1)\)\.

Let𝒜\\mathcal\{A\}and𝒱\\mathcal\{V\}be separable Banach spaces of input and output functions, and let𝒢:𝒜→𝒱\\mathcal\{G\}:\\mathcal\{A\}\\rightarrow\\mathcal\{V\}be the operator we wish to learn\. An inputa∈𝒜a\\in\\mathcal\{A\}and its outputv∈𝒱v\\in\\mathcal\{V\}are defined on domainsΩa\\Omega\_\{a\}andΩv\\Omega\_\{v\}, which may be spatial, spacetime, or parametric, with

a:Ωa→ℝda,v:Ωv→ℝd\.\\displaystyle a:\\Omega\_\{a\}\\rightarrow\\mathbb\{R\}^\{d\_\{a\}\},\\quad v:\\Omega\_\{v\}\\rightarrow\\mathbb\{R\}^\{d\}\.\(5\)Heredad\_\{a\}andddare the numbers of input and output components, andDaD\_\{a\}andDvD\_\{v\}are the dimensions of the domains\. We sample inputs atX=\{x1,…,xn\}⊂Ωa⊂ℝDaX=\\\{x\_\{1\},\\ldots,x\_\{n\}\\\}\\subset\\Omega\_\{a\}\\subset\\mathbb\{R\}^\{D\_\{a\}\}, which gives vectors𝐚∈ℝn​da\\mathbf\{a\}\\in\\mathbb\{R\}^\{nd\_\{a\}\}\. For spatial outputs,Dv=dD\_\{v\}=d, and we sample atY=\{yj\}j=1m⊂Ωv⊂ℝdY=\\\{y\_\{j\}\\\}\_\{j=1\}^\{m\}\\subset\\Omega\_\{v\}\\subset\\mathbb\{R\}^\{d\}, which gives vectors𝐯∈ℝm​d\\mathbf\{v\}\\in\\mathbb\{R\}^\{md\}stacked in component blocks, each holding one velocity component at all points ofYY, as in[Sharma et al\. \[53\]](https://arxiv.org/html/2610.29009#bib.bib10)\. For spacetime outputs,Dv=d\+1D\_\{v\}=d\+1, and sampling themmspatial sites atNtN\_\{t\}times gives vectors inℝd​m​Nt\\mathbb\{R\}^\{dmN\_\{t\}\}\(Section[2\.4](https://arxiv.org/html/2610.29009#S2.SS4)\)\.

GivenNNtraining pairs\{\(ai,vi\)\}i=1N\\\{\(a\_\{i\},v\_\{i\}\)\\\}\_\{i=1\}^\{N\}with sampled vectors\{\(𝐚i,𝐯i\)\}i=1N\\\{\(\\mathbf\{a\}\_\{i\},\\mathbf\{v\}\_\{i\}\)\\\}\_\{i=1\}^\{N\}, we write the learned operator, as in[Sharma et al\. \[53\]](https://arxiv.org/html/2610.29009#bib.bib10), as

v≈𝒢~​\(a\)=χ∘f~∘φ⁡\(a\),\\displaystyle v\\approx\\tilde\{\\mathcal\{G\}\}\(a\)=\\chi\\circ\\tilde\{f\}\\circ\\varphi\(a\),\(6\)whereφ\\varphisamples the input, the operator regressionf~\\tilde\{f\}maps the samples to the coefficients of the output, and the output reconstructionχ\\chimaps these coefficients to a function in space \(or spacetime\)\. This structure follows the formulation of[Batlle et al\. \[2\]](https://arxiv.org/html/2610.29009#bib.bib1), in whichf~\\tilde\{f\}maps input samples to output samples, with the difference thatf~\\tilde\{f\}predicts the coefficients of a divergence\-free interpolant\. In PPKM\[[53](https://arxiv.org/html/2610.29009#bib.bib10)\],χ\\chiis a global divergence\-free interpolant andf~\\tilde\{f\}a dense operator regression\. Here,χ\\chiis a collection of local divergence\-free interpolants \(Section[2](https://arxiv.org/html/2610.29009#S2)\), andf~\\tilde\{f\}is a sparse operator regression \(Section[3](https://arxiv.org/html/2610.29009#S3)\)\. Figure[1](https://arxiv.org/html/2610.29009#S1.F1)shows how operator Lagrange weights join the two \(Section[4](https://arxiv.org/html/2610.29009#S4)\)\.

Sampled inputs,φ\\varphi𝐚1\\mathbf\{a\}\_\{1\}𝐚2\\mathbf\{a\}\_\{2\}𝐚3\\mathbf\{a\}\_\{3\}𝐚N\\mathbf\{a\}\_\{N\}⋮\\vdots𝐚∗\\mathbf\{a\}\_\{\*\}new inputSparse operator regression,f~\\tilde\{f\}KU​Uθ=\[λ⁡\(𝐚i,𝐚i′\)\]\+θ​IK\_\{UU\}^\{\\theta\}=\\bigl\[\\lambda\(\\mathbf\{a\}\_\{i\},\\mathbf\{a\}\_\{i^\{\\prime\}\}\)\\bigr\]\+\\theta Iℓ⁡\(𝐚∗\)\\ell\(\\mathbf\{a\}\_\{\*\}\)==kU​\(𝐚∗\)k\_\{U\}\(\\mathbf\{a\}\_\{\*\}\)one sparse Cholesky solveper new inputLocal output reconstruction,χ\\chi𝐛∗j1=Bj1𝖳​ℓ​\(𝐚∗\)\\mathbf\{b\}\_\{\*\}^\{j\_\{1\}\}=B\_\{j\_\{1\}\}^\{\\mathsf\{T\}\}\\ell\(\\mathbf\{a\}\_\{\*\}\)𝐛∗j2=Bj2𝖳​ℓ​\(𝐚∗\)\\mathbf\{b\}\_\{\*\}^\{j\_\{2\}\}=B\_\{j\_\{2\}\}^\{\\mathsf\{T\}\}\\ell\(\\mathbf\{a\}\_\{\*\}\)𝐛∗j3=Bj3𝖳​ℓ​\(𝐚∗\)\\mathbf\{b\}\_\{\*\}^\{j\_\{3\}\}=B\_\{j\_\{3\}\}^\{\\mathsf\{T\}\}\\ell\(\\mathbf\{a\}\_\{\*\}\)every stencilSjS\_\{j\},j=1,…,mj=1,\\ldots,m, has its owncoefficients, all from the sameℓ⁡\(𝐚∗\)\\ell\(\\mathbf\{a\}\_\{\*\}\)ℓ⁡\(𝐚∗\)\\ell\(\\mathbf\{a\}\_\{\*\}\)

Figure 1:Overview of our method for a new input𝐚∗\\mathbf\{a\}\_\{\*\}\. The compactly supported kernelλ\\lambdaof Section[3](https://arxiv.org/html/2610.29009#S3)gives a sparse matrixKU​UθK\_\{UU\}^\{\\theta\}on the sampled training inputs \(left and center\)\. One sparse solve with the kernel valueskU​\(𝐚∗\)k\_\{U\}\(\\mathbf\{a\}\_\{\*\}\)gives the operator Lagrange weightsℓ⁡\(𝐚∗\)\\ell\(\\mathbf\{a\}\_\{\*\}\)of Section[4](https://arxiv.org/html/2610.29009#S4), shown here without their affine correction\. The same weights give the coefficients𝐛∗j=Bj𝖳​ℓ​\(𝐚∗\)\\mathbf\{b\}\_\{\*\}^\{j\}=B\_\{j\}^\{\\mathsf\{T\}\}\\ell\(\\mathbf\{a\}\_\{\*\}\)of the local divergence\-free interpolant of Section[2](https://arxiv.org/html/2610.29009#S2)on every stencilSjS\_\{j\}of the output \(right\), where the rows ofBjB\_\{j\}are the coefficients of the training fields onSjS\_\{j\}\. In practice, we apply the weights to the sampled training fields before the output reconstruction, which gives the same prediction \(Section[4\.2](https://arxiv.org/html/2610.29009#S4.SS2)\)\.We use the indexiifor training functions,jjfor stencils,kkfor components and coordinates,qqfor new \(query\) inputs and query points, andttfor time\. Each stencil hasnsn\_\{s\}nodes\. A subscript∗\*marks a quantity of a new input, a superscript⋆\\stara quantity at query points, and a hat a computed quantity in the error analysis\. The superscript𝖳\\mathsf\{T\}denotes the transpose,IIan identity matrix, and∥⋅∥F\\\|\\cdot\\\|\_\{F\}the Frobenius norm\.

## 2Local property\-preserving output reconstruction

We now construct the output reconstruction mapχ\\chiof \([6](https://arxiv.org/html/2610.29009#S1.E6)\)\. In PPKM\[[53](https://arxiv.org/html/2610.29009#bib.bib10)\],χ\\chiis a single divergence\-free kernel interpolant on allmmoutput points, so its linear system grows with the output resolution\. We replace it by local interpolants, one on the stencil of each output point\. Each local interpolant combines a divergence\-free kernel with divergence\-free polynomials, so that it is divergence free for any coefficient vector and reproduces low\-degree divergence\-free fields\. Its system has a fixed size asmmgrows\. We first describe the kernel, the polynomials, and the local system, and then the extensions to turbulent fields, periodic domains, and spacetime outputs\. Nothing in this section depends on the operator regression of Section[3](https://arxiv.org/html/2610.29009#S3)\.

### 2\.1Divergence\-free interpolation on stencils

#### 2\.1\.1The divergence\-free kernel

The output pointsYYserve as the nodes of the output reconstruction\. For each nodeyj∈Yy\_\{j\}\\in Y, let the stencilSj=\{y~j,1,…,y~j,ns\}S\_\{j\}=\\\{\\tilde\{y\}\_\{j,1\},\\ldots,\\tilde\{y\}\_\{j,n\_\{s\}\}\\\}consist of itsnsn\_\{s\}nearest nodes\. In the tests of Section[5\.1](https://arxiv.org/html/2610.29009#S5.SS1)on bounded domains, we instead use the boundary\-aware stencils of KernelPack\-JAX\[[8](https://arxiv.org/html/2610.29009#bib.bib47)\], which keep only nodes visible from the center inside the domain\. In periodic domains, some of these are translated images of nodes \(Section[2\.3](https://arxiv.org/html/2610.29009#S2.SS3)\)\. For smooth outputs, we use the cubic PHSϕ⁡\(x,y\)=ϕ3​\(‖x−y‖2\)\\phi\(x,y\)=\\phi\_\{3\}\(\\\|x\-y\\\|\_\{2\}\), with radial function

ϕ3​\(r\)=r3,\\displaystyle\\phi\_\{3\}\(r\)=r^\{3\},\(7\)which has no shape parameter to choose \(Section[2\.2](https://arxiv.org/html/2610.29009#S2.SS2)relates it to the Matérn kernels that we use when a length scale is wanted\)\. For the sixth\-order tests of Section[5\.1](https://arxiv.org/html/2610.29009#S5.SS1), we useϕ⁡\(x,y\)=‖x−y‖25\\phi\(x,y\)=\\\|x\-y\\\|\_\{2\}^\{5\}in the same way\. Applying the operator−Δ​I\+∇∇𝖳\-\\Delta I\+\\nabla\\nabla^\{\\mathsf\{T\}\}to a scalar kernel gives a matrix\-valued kernel whose columns are divergence free\[[40](https://arxiv.org/html/2610.29009#bib.bib18),[24](https://arxiv.org/html/2610.29009#bib.bib4)\],

ΦDF​\(x,y\)=\(−Δx​I\+∇x∇x𝖳\)​ϕ​\(x,y\)\.\\displaystyle\\Phi\_\{\\rm DF\}\(x,y\)=\\left\(\-\\Delta\_\{x\}I\+\\nabla\_\{x\}\\nabla\_\{x\}^\{\\mathsf\{T\}\}\\right\)\\phi\(x,y\)\.\(8\)For the cubic PHS, withz=x−yz=x\-yandr=‖z‖2\>0r=\\\|z\\\|\_\{2\}\>0,

ΦDF​\(x,y\)=−3​d​r​I\+3​z​z𝖳/r,ΦDF​\(y,y\)=0\.\\Phi\_\{\\rm DF\}\(x,y\)=\-3drI\+3zz^\{\\mathsf\{T\}\}/r,\\qquad\\Phi\_\{\\rm DF\}\(y,y\)=0\.\(9\)Each column ofΦDF\\Phi\_\{\\rm DF\}has zero divergence in the distributional sense, and zero classical divergence away from its center\[[24](https://arxiv.org/html/2610.29009#bib.bib4)\]\. At the center the kernel is continuous but not differentiable\. For example, alongz=t​𝐞kz=t\\mathbf\{e\}\_\{k\}, with𝐞k\\mathbf\{e\}\_\{k\}thekkth coordinate vector, itskkth diagonal entry is−3​\(d−1\)​\|t\|\-3\(d\-1\)\|t\|\.

#### 2\.1\.2Divergence\-free polynomials of any degree

A PHS interpolant converges at the order set by the degree of its polynomial augmentation\[[19](https://arxiv.org/html/2610.29009#bib.bib3),[3](https://arxiv.org/html/2610.29009#bib.bib60)\]\. A divergence\-free local interpolant of high order therefore needs divergence\-free vector polynomials of any degreeϖ\\varpi, in 2D and 3D, that can be evaluated at any point and not only at the stencil nodes\. Our approach is to differentiate scalar potentials\. OnSjS\_\{j\}, letξ=\(y−y¯j\)/ρj\\xi=\(y\-\\bar\{y\}\_\{j\}\)/\\rho\_\{j\}be centered and scaled coordinates, wherey¯j=ns−1​∑ly~j,l\\bar\{y\}\_\{j\}=n\_\{s\}^\{\-1\}\\sum\_\{l\}\\tilde\{y\}\_\{j,l\}is the centroid of the stencil andρj=maxl⁡‖y~j,l−y¯j‖2\\rho\_\{j\}=\\max\_\{l\}\\\|\\tilde\{y\}\_\{j,l\}\-\\bar\{y\}\_\{j\}\\\|\_\{2\}its radius\. From the productsJα​\(ξ\)=∏kPαk​\(ξk\)J\_\{\\alpha\}\(\\xi\)=\\prod\_\{k\}P\_\{\\alpha\_\{k\}\}\(\\xi\_\{k\}\)of normalized Legendre polynomialsPnP\_\{n\}, of total degree\|α\|≤ϖ\+1\|\\alpha\|\\leq\\varpi\+1, we form the rotated gradients in 2D and the curls of the potentials times the coordinate vectors in 3D,

gα​\(y\)\\displaystyle g\_\{\\alpha\}\(y\)=∇ξ⟂​Jα​\(ξ\),\\displaystyle=\\nabla\_\{\\xi\}^\{\\perp\}J\_\{\\alpha\}\(\\xi\),d=2,\\displaystyle d=2,\(10\)gα,k​\(y\)\\displaystyle g\_\{\\alpha,k\}\(y\)=∇ξ×\(Jα​\(ξ\)​𝐞k\),\\displaystyle=\\nabla\_\{\\xi\}\\times\\bigl\(J\_\{\\alpha\}\(\\xi\)\\mathbf\{e\}\_\{k\}\\bigr\),d=3,k=1,2,3,\\displaystyle d=3,\\quad k=1,2,3,\(11\)whereα\\alphais a multi\-index,∇ξ⟂=\(∂ξ2,−∂ξ1\)𝖳\\nabla\_\{\\xi\}^\{\\perp\}=\(\\partial\_\{\\xi\_\{2\}\},\-\\partial\_\{\\xi\_\{1\}\}\)^\{\\mathsf\{T\}\}\. These candidates have degree at mostϖ\\varpiand are divergence free inξ\\xi, and hence inyy, since the scaling is uniform\. Together, they span the space of divergence\-free vector polynomials of degree at mostϖ\\varpi, which has dimensiond​\(ϖ\+dd\)−\(ϖ\+d−1d\)d\\binom\{\\varpi\+d\}\{d\}\-\\binom\{\\varpi\+d\-1\}\{d\}\(Appendix[C\.1](https://arxiv.org/html/2610.29009#A3.SS1)\)\.

The candidates are not independent\. Constant potentials give zero fields, different potentials can give the same field, and a particular stencil can add further dependencies\. Letg1,…,gncand,jg\_\{1\},\\ldots,g\_\{n\_\{\{\\rm cand\},j\}\}be the candidates, letGj\(y\)=\[g1\(y\)⋯gncand,j\(y\)\]∈ℝd×ncand,jG\_\{j\}\(y\)=\[g\_\{1\}\(y\)\\ \\cdots\\ g\_\{n\_\{\{\\rm cand\},j\}\}\(y\)\]\\in\\mathbb\{R\}^\{d\\times n\_\{\{\\rm cand\},j\}\}hold their values atyy, and letGj∈ℝd​ns×ncand,jG\_\{j\}\\in\\mathbb\{R\}^\{dn\_\{s\}\\times n\_\{\{\\rm cand\},j\}\}hold their values at the stencil nodes in component block order\. We select an independent subset by column\-pivoted QR,

Gj​Πj=Qj​Rj,\\displaystyle G\_\{j\}\\Pi\_\{j\}=Q\_\{j\}R\_\{j\},\(12\)and retain the firstnP,jn\_\{P,j\}pivoted columns, wherenP,jn\_\{P,j\}counts the diagonal entries ofRjR\_\{j\}larger than10−1210^\{\-12\}times the first\. LetIjI\_\{j\}hold the indices of the retained candidates,Gj,IG\_\{j,I\}the corresponding columns ofGjG\_\{j\},Qj,1Q\_\{j,1\}the firstnP,jn\_\{P,j\}columns ofQjQ\_\{j\}, andRj,11∈ℝnP,j×nP,jR\_\{j,11\}\\in\\mathbb\{R\}^\{n\_\{P,j\}\\times n\_\{P,j\}\}the leading block ofRjR\_\{j\}\. Then

Gj,I=Qj,1​Rj,11,Gj,I​Rj,11−1=Qj,1\.\\displaystyle G\_\{j,I\}=Q\_\{j,1\}R\_\{j,11\},\\qquad G\_\{j,I\}R\_\{j,11\}^\{\-1\}=Q\_\{j,1\}\.\(13\)The orthonormal columns ofQj,1Q\_\{j,1\}are only values at the nodes\. To obtain functions, we apply the same triangular transformation to the retained candidate functionsGj,I​\(y\)=\[gc​\(y\)\]c∈IjG\_\{j,I\}\(y\)=\[g\_\{c\}\(y\)\]\_\{c\\in I\_\{j\}\},

𝐏j​\(y\)=Gj,I​\(y\)​Rj,11−1\.\\displaystyle\\mathbf\{P\}\_\{j\}\(y\)=G\_\{j,I\}\(y\)R\_\{j,11\}^\{\-1\}\.\(14\)By \([13](https://arxiv.org/html/2610.29009#S2.E13)\),𝐏j=Qj,1\\mathbf\{P\}\_\{j\}=Q\_\{j,1\}at the stencil nodes, so the polynomial block of the local system is well conditioned\. Every column of𝐏j​\(y\)\\mathbf\{P\}\_\{j\}\(y\)is a fixed combination of divergence\-free fields, and is thus divergence free everywhere\. At a new point, we evaluate the retained candidates and applyRj,11−1R\_\{j,11\}^\{\-1\}by a triangular solve\. If the stencil geometry forces the rank below the full dimension, the interpolant reproduces only the retained subspace\.

We batch this construction over the stencils\. On stencils that are certainly of full rank, pivoting is unnecessary, and we use an unpivoted QR factorization instead \(Appendix[A\.2](https://arxiv.org/html/2610.29009#A1.SS2)\)\.

#### 2\.1\.3The local system

OnSjS\_\{j\}, we determine the kernel and polynomial coefficients together, from the interpolation conditions and the polynomial moment conditions\. The saddle\-point matrix

𝐀j\\displaystyle\\mathbf\{A\}\_\{j\}=\[𝚽j𝐏j𝐏j𝖳0\]∈ℝrj×rj,\\displaystyle=\\begin\{bmatrix\}\\mathbf\{\\Phi\}\_\{j\}&\\mathbf\{P\}\_\{j\}\\\\ \\mathbf\{P\}\_\{j\}^\{\\mathsf\{T\}\}&0\\end\{bmatrix\}\\in\\mathbb\{R\}^\{r\_\{j\}\\times r\_\{j\}\},\(15\)has sizerj=d​ns\+nP,jr\_\{j\}=dn\_\{s\}\+n\_\{P,j\}, where𝚽j∈ℝd​ns×d​ns\\mathbf\{\\Phi\}\_\{j\}\\in\\mathbb\{R\}^\{dn\_\{s\}\\times dn\_\{s\}\}is the Gram matrix of the divergence\-free kernel onSjS\_\{j\}\. For training fieldii, the coefficient vector

𝐛ij=\[𝐛i,Φj𝐛i,Pj\]∈ℝrj\\displaystyle\\mathbf\{b\}\_\{i\}^\{j\}=\\begin\{bmatrix\}\\mathbf\{b\}\_\{i,\\Phi\}^\{j\}\\\\ \\mathbf\{b\}\_\{i,P\}^\{j\}\\end\{bmatrix\}\\in\\mathbb\{R\}^\{r\_\{j\}\}holds kernel coefficients𝐛i,Φj∈ℝd​ns\\mathbf\{b\}\_\{i,\\Phi\}^\{j\}\\in\\mathbb\{R\}^\{dn\_\{s\}\}and polynomial coefficients𝐛i,Pj∈ℝnP,j\\mathbf\{b\}\_\{i,P\}^\{j\}\\in\\mathbb\{R\}^\{n\_\{P,j\}\}, and defines the local interpolant

χij​\(y\)=∑l=1nsΦDF​\(y,y~j,l\)​𝐛i,Φj,l\+𝐏j​\(y\)​𝐛i,Pj,\\displaystyle\\chi\_\{i\}^\{j\}\(y\)=\\sum\_\{l=1\}^\{n\_\{s\}\}\\Phi\_\{\\rm DF\}\(y,\\tilde\{y\}\_\{j,l\}\)\\mathbf\{b\}\_\{i,\\Phi\}^\{j,l\}\+\\mathbf\{P\}\_\{j\}\(y\)\\mathbf\{b\}\_\{i,P\}^\{j\},\(16\)where𝐛i,Φj,l∈ℝd\\mathbf\{b\}\_\{i,\\Phi\}^\{j,l\}\\in\\mathbb\{R\}^\{d\}holds the kernel coefficients of nodell\. Interpolation of the training field at the stencil nodes \(with the value of the original node at a periodic image\) and orthogonality of the kernel coefficients to the polynomials give

𝐀j​𝐛ij=\[𝐯i\|Sj0\],\\displaystyle\\mathbf\{A\}\_\{j\}\\mathbf\{b\}\_\{i\}^\{j\}=\\begin\{bmatrix\}\\left\.\\mathbf\{v\}\_\{i\}\\right\|\_\{S\_\{j\}\}\\\\ 0\\end\{bmatrix\},\(17\)where𝐯i\|Sj∈ℝd​ns\\left\.\\mathbf\{v\}\_\{i\}\\right\|\_\{S\_\{j\}\}\\in\\mathbb\{R\}^\{dn\_\{s\}\}holds the sampled velocities on the stencil\. Since𝐏j\\mathbf\{P\}\_\{j\}has full column rank,𝐀j\\mathbf\{A\}\_\{j\}is nonsingular whenever𝚽j\\mathbf\{\\Phi\}\_\{j\}is positive definite on the null space of𝐏j𝖳\\mathbf\{P\}\_\{j\}^\{\\mathsf\{T\}\}\. Conditional positive definiteness of the scalar PHS does not guarantee this for the matrix kernel on a particular stencil, so we check the solves numerically \(Appendix[A\.2](https://arxiv.org/html/2610.29009#A1.SS2)\)\. We factor all local systems by batched LU on the GPU\.

##### Evaluation

To reconstruct a prediction at query pointsyq⋆∈Y⋆y\_\{q\}^\{\\star\}\\in Y^\{\\star\},q=1,…,m⋆q=1,\\ldots,m^\{\\star\}, we assign each query point to a stencilSj⁡\(q\)S\_\{j\(q\)\}and evaluate that stencil’s basis there\. Let𝚽j⁡\(q\),q,comp⋆∈ℝd×d​ns\\mathbf\{\\Phi\}\_\{j\(q\),q,\\mathrm\{comp\}\}^\{\\star\}\\in\\mathbb\{R\}^\{d\\times dn\_\{s\}\}hold the kernel values in the component block order of the coefficients \(Appendix[A\.1](https://arxiv.org/html/2610.29009#A1.SS1)\) and𝐏j⁡\(q\),q⋆∈ℝd×nP,j⁡\(q\)\\mathbf\{P\}\_\{j\(q\),q\}^\{\\star\}\\in\\mathbb\{R\}^\{d\\times n\_\{P,j\(q\)\}\}the polynomial values, and let

𝐀q⋆=\[𝚽j⁡\(q\),q,comp⋆𝐏j⁡\(q\),q⋆\]∈ℝd×rj⁡\(q\)\.\\displaystyle\\mathbf\{A\}\_\{q\}^\{\\star\}=\\begin\{bmatrix\}\\mathbf\{\\Phi\}\_\{j\(q\),q,\\mathrm\{comp\}\}^\{\\star\}&\\mathbf\{P\}\_\{j\(q\),q\}^\{\\star\}\\end\{bmatrix\}\\in\\mathbb\{R\}^\{d\\times r\_\{j\(q\)\}\}\.\(18\)For coefficients𝐛∗j\\mathbf\{b\}\_\{\*\}^\{j\}predicted by the operator regression \(Section[4\.1](https://arxiv.org/html/2610.29009#S4.SS1)\), the reconstructed velocity is then

𝐯~q⋆=𝐀q⋆​𝐛∗j⁡\(q\)∈ℝd,\\displaystyle\\tilde\{\\mathbf\{v\}\}\_\{q\}^\{\\star\}=\\mathbf\{A\}\_\{q\}^\{\\star\}\\mathbf\{b\}\_\{\*\}^\{j\(q\)\}\\in\\mathbb\{R\}^\{d\},\(19\)and𝐯~⋆\\tilde\{\\mathbf\{v\}\}^\{\\star\}stacks the𝐯~q⋆\\tilde\{\\mathbf\{v\}\}\_\{q\}^\{\\star\}in component blocks\. In the operator benchmarks, the query points are the nodes themselves, each with its own stencil\. Appendix[A\.3](https://arxiv.org/html/2610.29009#A1.SS3)gives the assignment of other points\.

The output reconstruction is divergence free on each stencil, for any coefficients\. However, the local interpolants are not coupled, so the reconstructed field can jump where the assigned stencil changes, and its divergence is controlled only within each stencil\.

On the other hand, every system has the fixed sizerjr\_\{j\}, independent of the number of output pointsmmand of the number of training functionsNN\. Refining the output adds systems but does not enlarge them\. The systems are also independent of each other, so their assembly, factorization, and evaluation are embarrassingly parallel, and we process them in batches on the GPU with no communication between stencils\. In contrast, a global output reconstruction couples all output points in one dense system withd​mdmunknowns, whose factorization costsO⁡\(\(d​m\)3\)O\(\(dm\)^\{3\}\)operations and does not decompose into independent parts\. Section[5\.1\.5](https://arxiv.org/html/2610.29009#S5.SS1.SSS5)measures both costs\. Section[4\.2](https://arxiv.org/html/2610.29009#S4.SS2)shows how the same systems serve every prediction\.

### 2\.2Multiscale Matérn output reconstruction for turbulence

Turbulent velocity fields vary over many length scales, which a kernel can represent only if it has length scales of its own\. Polyharmonic splines have none, since they are homogeneous \(dilating the argument ofr3r^\{3\}orr5r^\{5\}multiplies it by a constant, and so does the derived matrix kernel\), and sums of dilated copies add nothing\. Matérn kernels have one length each, and they are close relatives of the PHS\. The Matérn kernelsM3/2​\(r/σ\)M\_\{3/2\}\(r/\\sigma\)andM5/2​\(r/σ\)M\_\{5/2\}\(r/\\sigma\)of lengthσ\\sigma, withM3/2​\(r\)=\(1\+r\)​e−rM\_\{3/2\}\(r\)=\(1\+r\)e^\{\-r\}andM5/2M\_\{5/2\}given in \([22](https://arxiv.org/html/2610.29009#S2.E22)\) below, have Fourier transforms proportional to\(σ−2\+∥𝐤∥22\)−\(3\+d\)/2\(\\sigma^\{\-2\}\+\\\|\\mathbf\{k\}\\\|\_\{2\}^\{2\}\)^\{\-\(3\+d\)/2\}and\(σ−2\+∥𝐤∥22\)−\(5\+d\)/2\(\\sigma^\{\-2\}\+\\\|\\mathbf\{k\}\\\|\_\{2\}^\{2\}\)^\{\-\(5\+d\)/2\}in the wave vector𝐤\\mathbf\{k\}\. They are therefore, up to constants, the Green’s functions of\(σ−2−Δ\)\(3\+d\)/2\(\\sigma^\{\-2\}\-\\Delta\)^\{\(3\+d\)/2\}and\(σ−2−Δ\)\(5\+d\)/2\(\\sigma^\{\-2\}\-\\Delta\)^\{\(5\+d\)/2\}inℝd\\mathbb\{R\}^\{d\}\[[67](https://arxiv.org/html/2610.29009#bib.bib67),[58](https://arxiv.org/html/2610.29009#bib.bib71),[35](https://arxiv.org/html/2610.29009#bib.bib70),[15](https://arxiv.org/html/2610.29009#bib.bib69)\]\. Lettingσ→∞\\sigma\\to\\inftyremoves the length and leaves\(−Δ\)\(3\+d\)/2\(\-\\Delta\)^\{\(3\+d\)/2\}and\(−Δ\)\(5\+d\)/2\(\-\\Delta\)^\{\(5\+d\)/2\}, whose fundamental solutions are multiples ofr3r^\{3\}andr5r^\{5\}\[[14](https://arxiv.org/html/2610.29009#bib.bib68),[65](https://arxiv.org/html/2610.29009#bib.bib16)\]\. The two Matérn kernels are thus the counterparts, with a length, of the PHSr3r^\{3\}andr5r^\{5\}\. For example,

M5/2​\(rσ\)=1−r26​σ2\+r424​σ4−r545​σ5\+O⁡\(r6σ6\)\.\\displaystyle M\_\{5/2\}\\\!\\left\(\\frac\{r\}\{\\sigma\}\\right\)=1\-\\frac\{r^\{2\}\}\{6\\sigma^\{2\}\}\+\\frac\{r^\{4\}\}\{24\\sigma^\{4\}\}\-\\frac\{r^\{5\}\}\{45\\sigma^\{5\}\}\+O\\\!\\left\(\\frac\{r^\{6\}\}\{\\sigma^\{6\}\}\\right\)\.\(20\)The even terms are polynomials inx−yx\-y, and after the operator−Δ​I\+∇∇𝖳\-\\Delta I\+\\nabla\\nabla^\{\\mathsf\{T\}\}their contribution to an interpolant is a divergence\-free polynomial of degree at most two, which the polynomial part of the local interpolant absorbs whenϖ≥2\\varpi\\geq 2\. Since rescaling the kernel does not change the interpolant, the local interpolant withM5/2​\(r/σ\)M\_\{5/2\}\(r/\\sigma\)thus tends to the one withr5r^\{5\}asσ→∞\\sigma\\to\\infty\. We use the PHS when no length scale is wanted, for smooth outputs whose accuracy is set by the polynomial degree, and Matérn kernels when lengths are wanted\. A mixture of Matérn kernels can have many lengths\. This motivates localizing the multiscale Matérn construction of our earlier work\[[53](https://arxiv.org/html/2610.29009#bib.bib10)\], which follows[Owhadi \[41\]](https://arxiv.org/html/2610.29009#bib.bib45)\. OnSjS\_\{j\}, we sum five Matérn kernels with dyadic lengths and weights

σj,κ=σj,02κ,σj,0=cσρj,ϑj,κ=\(σj,κσj,0\)8/3,κ=1,…,5,\\displaystyle\\sigma\_\{j,\\kappa\}=\\frac\{\\sigma\_\{j,0\}\}\{2^\{\\kappa\}\},\\qquad\\sigma\_\{j,0\}=c\_\{\\sigma\}\\rho\_\{j\},\\qquad\\vartheta\_\{j,\\kappa\}=\\left\(\\frac\{\\sigma\_\{j,\\kappa\}\}\{\\sigma\_\{j,0\}\}\\right\)^\{8/3\},\\qquad\\kappa=1,\\ldots,5,\(21\)whereρj\\rho\_\{j\}is the stencil radius andcσc\_\{\\sigma\}a fixed factor\. Each scale uses the scalar Matérn function

M5/2​\(r\)=\(1\+r\+r23\)​e−r\\displaystyle M\_\{5/2\}\(r\)=\\left\(1\+r\+\\frac\{r^\{2\}\}\{3\}\\right\)e^\{\-r\}\(22\)at the argumentr/σj,κr/\\sigma\_\{j,\\kappa\}\. The weights are applied to the scalar kernels, before the divergence\-free operator, whose two derivatives account for the22in the exponent8/3=2\+2/38/3=2\+2/3\. The remaining2/32/3is the Kolmogorov exponent of the velocity increments in three dimensions\. The local multiscale kernel is

ΦjM​\(x,y\)=1cj​\(−Δx​I\+∇x∇x𝖳\)​∑κ=153​ϑj,κ​M5/2​\(‖x−y‖2σj,κ\),cj=\(d−1\)​∑κ=15ϑj,κ​σj,κ−2,\\displaystyle\\Phi\_\{j\}^\{\\rm M\}\(x,y\)=\\frac\{1\}\{c\_\{j\}\}\\left\(\-\\Delta\_\{x\}I\+\\nabla\_\{x\}\\nabla\_\{x\}^\{\\mathsf\{T\}\}\\right\)\\sum\_\{\\kappa=1\}^\{5\}3\\vartheta\_\{j,\\kappa\}M\_\{5/2\}\\\!\\left\(\\frac\{\\\|x\-y\\\|\_\{2\}\}\{\\sigma\_\{j,\\kappa\}\}\\right\),\\qquad c\_\{j\}=\(d\-1\)\\sum\_\{\\kappa=1\}^\{5\}\\vartheta\_\{j,\\kappa\}\\sigma\_\{j,\\kappa\}^\{\-2\},\(23\)normalized so thatΦjM​\(y,y\)=I\\Phi\_\{j\}^\{\\rm M\}\(y,y\)=I\. The normalization rescales the coefficients and does not change the interpolant\. We replaceΦDF\\Phi\_\{\\rm DF\}byΦjM\\Phi\_\{j\}^\{\\rm M\}in \([15](https://arxiv.org/html/2610.29009#S2.E15)\) and \([18](https://arxiv.org/html/2610.29009#S2.E18)\) and keep the polynomials, so each stencil still solves one system withrjr\_\{j\}unknowns for the whole mixture\. Unlike the cubic kernel, the mixture is smooth at its centers, so the reconstructed fields are classically divergence free everywhere on the stencil\.

The weights build the prescribed increment statistics into the kernel\[[41](https://arxiv.org/html/2610.29009#bib.bib45)\]\. Whether an interpolant on finite stencils reproduces those statistics also depends on the stencil extent and the node spacing, and we test this under refinement in Section[5\.1](https://arxiv.org/html/2610.29009#S5.SS1)and for the turbulent benchmarks in Appendix[A\.5](https://arxiv.org/html/2610.29009#A1.SS5)\.

### 2\.3Periodic stencils

In a periodic domain, the nearest neighbors of a node near one face of the box can lie near the opposite face\. Our stencils are therefore built from translated images of the nodes\. If coordinatekkhas periodΛk\\Lambda\_\{k\}, a nodeyly\_\{l\}has imagesyl\+∑kνk​Λk​𝐞ky\_\{l\}\+\\sum\_\{k\}\\nu\_\{k\}\\Lambda\_\{k\}\\mathbf\{e\}\_\{k\}, with integersνk\\nu\_\{k\}over the periodic coordinates, which carry its index and its sampled velocity,

vi​\(yl\+∑kνk​Λk​𝐞k\)=vi​\(yl\)\.\\displaystyle v\_\{i\}\\Bigl\(y\_\{l\}\+\\sum\_\{k\}\\nu\_\{k\}\\Lambda\_\{k\}\\mathbf\{e\}\_\{k\}\\Bigr\)=v\_\{i\}\(y\_\{l\}\)\.\(24\)For a stencil centeryjy\_\{j\}, we take the displacementsδl=yl−yj\\delta\_\{l\}=y\_\{l\}\-y\_\{j\}and shift every periodic component to its nearest representative,

δl,k←δl,k−Λk​round⁡\(δl,k/Λk\)\.\\delta\_\{l,k\}\\leftarrow\\delta\_\{l,k\}\-\\Lambda\_\{k\}\\operatorname\{round\}\(\\delta\_\{l,k\}/\\Lambda\_\{k\}\)\.\(25\)Since all periodic coordinates are shifted together, this also finds the right images near the edges and corners of the box, where a node’s neighbors are shifted in two or three coordinates \(Figure[2](https://arxiv.org/html/2610.29009#S2.F2)\)\.

Λ1\\Lambda\_\{1\}Λ2\\Lambda\_\{2\}

Figure 2:Periodic stencils in a box with periodsΛ1\\Lambda\_\{1\}andΛ2\\Lambda\_\{2\}\. Gray dots are nodes \(dark\) and their translated images \(light\)\. A stencil near a face \(blue\) and a stencil near a corner \(red\) are built from the nearest images given by \([25](https://arxiv.org/html/2610.29009#S2.E25)\)\. Dashed arrows link each image in a stencil to its node \(circled\), whose sampled velocity it carries\.The stencil consists of thensn\_\{s\}nearest of the pointsyj\+δly\_\{j\}\+\\delta\_\{l\}, with one representative of each node, and the kernel and polynomials are evaluated in these translated coordinates\. Translation does not change the divergence\. Appendix[A\.4](https://arxiv.org/html/2610.29009#A1.SS4)gives the periods used for each benchmark\.

### 2\.4Spacetime outputs

For outputs onΩv=Ω×Γ\\Omega\_\{v\}=\\Omega\\times\\Gamma, withΓ\\Gammaa time interval, PPKM\[[53](https://arxiv.org/html/2610.29009#bib.bib10)\]used a global product kernel

ΦST​\(\(y,t\),\(y′,t′\)\)=ΦDFMat​\(y,y′\)​Φt​\(t,t′\),\\displaystyle\\Phi\_\{\\rm ST\}\(\(y,t\),\(y^\{\\prime\},t^\{\\prime\}\)\)=\\Phi\_\{\\rm DF\}^\{\\rm Mat\}\(y,y^\{\\prime\}\)\\Phi\_\{t\}\(t,t^\{\\prime\}\),\(26\)with a divergence\-free Matérn kernel in space and a scalar Matérn kernelΦt\\Phi\_\{t\}in time, andd​m​NtdmN\_\{t\}coefficients in one system\. We instead keep each stencil and its spatial basis fixed over the sampled times𝒯=\{ts\}s=1Nt\\mathcal\{T\}=\\\{t\_\{s\}\\\}\_\{s=1\}^\{N\_\{t\}\}and interpolate in time\. Let𝐛ij,s∈ℝrj\\mathbf\{b\}\_\{i\}^\{j,s\}\\in\\mathbb\{R\}^\{r\_\{j\}\}be the coefficients of training fieldiionSjS\_\{j\}at timetst\_\{s\}\. Every snapshot solves \([17](https://arxiv.org/html/2610.29009#S2.E17)\) with the same matrix𝐀j\\mathbf\{A\}\_\{j\}, and the target of the operator regression is the stacked vector

𝐛i,𝒯j=\[\(𝐛ij,1\)𝖳,…,\(𝐛ij,Nt\)𝖳\]𝖳∈ℝNt​rj\.\\displaystyle\\mathbf\{b\}\_\{i,\\mathcal\{T\}\}^\{j\}=\[\(\\mathbf\{b\}\_\{i\}^\{j,1\}\)^\{\\mathsf\{T\}\},\\ldots,\(\\mathbf\{b\}\_\{i\}^\{j,N\_\{t\}\}\)^\{\\mathsf\{T\}\}\]^\{\\mathsf\{T\}\}\\in\\mathbb\{R\}^\{N\_\{t\}r\_\{j\}\}\.\(27\)Between the sampled times, we interpolate with the Lagrange polynomials

ωs\(t\)=∏s′≠st−ts′ts−ts′,s=1,…,Nt,\\displaystyle\\omega\_\{s\}\(t\)=\\prod\_\{s^\{\\prime\}\\neq s\}\\frac\{t\-t\_\{s^\{\\prime\}\}\}\{t\_\{s\}\-t\_\{s^\{\\prime\}\}\},\\qquad s=1,\\ldots,N\_\{t\},\(28\)which need no temporal kernel parameter and no further linear system\. With𝐀j⋆​\(y\)\\mathbf\{A\}\_\{j\}^\{\\star\}\(y\)the evaluation of the spatial basis onSjS\_\{j\}as in \([18](https://arxiv.org/html/2610.29009#S2.E18)\), the spacetime output reconstruction of a prediction is

χ∗j​\(y,t\)=∑s=1Ntωs​\(t\)​𝐀j⋆​\(y\)​𝐛∗j,s\.\\displaystyle\\chi\_\{\*\}^\{j\}\(y,t\)=\\sum\_\{s=1\}^\{N\_\{t\}\}\\omega\_\{s\}\(t\)\\mathbf\{A\}\_\{j\}^\{\\star\}\(y\)\\mathbf\{b\}\_\{\*\}^\{j,s\}\.\(29\)Its polynomial part is the tensor product of the retained spatial polynomials with polynomials of degree at mostNt−1N\_\{t\}\-1in time\. Since the weightsωs\\omega\_\{s\}do not depend onyy, the spacetime output reconstruction is divergence free wherever the spatial one is\. The local representation hasNt​∑jrjN\_\{t\}\\sum\_\{j\}r\_\{j\}coefficients, but each system keeps the sizerjr\_\{j\}\. Another sampled time adds only a right\-hand side\.

## 3Sparse operator regression

Having fixed the output representation, we now learn its coefficients as functions of the input\. Each output coefficient is a function on the space of sampled inputs, and we approximate all of them by kernel regression with the same kernel\. PPKM\[[53](https://arxiv.org/html/2610.29009#bib.bib10)\]used a kernel with global support, whoseN×NN\\times Nkernel matrix is dense, so that its Cholesky factorization costsO⁡\(N3\)O\(N^\{3\}\)operations andO⁡\(N2\)O\(N^\{2\}\)storage forNNtraining functions\. We now construct a compactly supported kernel that is positive semidefinite on inputs of any dimension, so that the kernel matrix is sparse\. The rest of the section defines the operator interpolant built from this kernel and its affine augmentation, derives their linear systems, factors them, and chooses the kernel parameters\.

### 3\.1A compactly supported kernel in any input dimension

To make the kernel matrix sparse, the kernel must vanish for inputs that are far apart, and it must remain positive definite on the space of sampled inputs, whose dimensionn​dand\_\{a\}can be in the thousands\. These two requirements conflict for radial kernels\. By Schoenberg’s theorem, a radial function that is positive definite in every dimension is a scale mixture of Gaussians and therefore never compactly supported\[[65](https://arxiv.org/html/2610.29009#bib.bib16)\]\. Every compactly supported radial kernel\[[64](https://arxiv.org/html/2610.29009#bib.bib11),[27](https://arxiv.org/html/2610.29009#bib.bib48)\]is thus positive definite only up to some dimensionDmaxD\_\{\\max\}\. For the Wendland functions, the polynomial degree grows linearly withDmaxD\_\{\\max\}at fixed smoothness\[[64](https://arxiv.org/html/2610.29009#bib.bib11)\], so a function valid in thousands of dimensions would have a degree in the thousands\. Tapering\[[23](https://arxiv.org/html/2610.29009#bib.bib49)\], which multiplies a globally supported kernel by a compactly supported one, inherits the same restriction from its taper\. Low numerical rank of the training inputs does not remove the restriction either, since a new input need not lie in their span\.

Our approach is to separate the two requirements\. We let a low\-dimensional projection of the input determine the support, and use a compactly supported radial function only there, in a dimension in which it is positive definite\. The remaining coordinates enter through a Matérn factor, which is positive definite in every dimension but not compactly supported\. The product is positive semidefinite in any input dimension and has the support of its first factor\. In this paper, we use theC4C^\{4\}Wendland function

w⁡\(t\)=\(1−t\)\+6​\(1\+6​t\+353​t2\),w\(t\)=\(1\-t\)\_\{\+\}^\{6\}\\left\(1\+6t\+\\frac\{35\}\{3\}t^\{2\}\\right\),\(30\)which is positive definite in up to three dimensions\[[64](https://arxiv.org/html/2610.29009#bib.bib11)\], so we project onto at most three coordinates\. Another member of the Wendland family would allow a projection of higher dimension\.

Let𝐚¯\\bar\{\\mathbf\{a\}\}be the sampled input after centering and scaling each coordinate by its training mean and standard deviation \(coordinates with zero training variance are set to zero\)\. Let the columns ofVzV\_\{z\}be the firstnzn\_\{z\}principal directions of the standardized training inputs, wherenzn\_\{z\}is the smaller of three and their numerical rank \(Appendix[B\.1](https://arxiv.org/html/2610.29009#A2.SS1)\), and split

z⁡\(𝐚\)=Vz𝖳​𝐚¯,ζ⁡\(𝐚\)=\(I−Vz​Vz𝖳\)​𝐚¯\.z\(\\mathbf\{a\}\)=V\_\{z\}^\{\\mathsf\{T\}\}\\bar\{\\mathbf\{a\}\},\\qquad\\zeta\(\\mathbf\{a\}\)=\(I\-V\_\{z\}V\_\{z\}^\{\\mathsf\{T\}\}\)\\bar\{\\mathbf\{a\}\}\.\(31\)For a support radiusϱ\>0\\varrho\>0, a residual scaleα⟂\>0\\alpha\_\{\\perp\}\>0, and a positive diagonal matrixD∥D\_\{\\parallel\}, our kernel is

λ⁡\(𝐚,𝐚′\)=\\displaystyle\\lambda\(\\mathbf\{a\},\\mathbf\{a\}^\{\\prime\}\)=\{\}η⁡\(𝐚\)​η​\(𝐚′\)​w​\(‖D∥​\(z⁡\(𝐚\)−z⁡\(𝐚′\)\)‖2ϱ\)​M5/2​\(α⟂​‖ζ⁡\(𝐚\)−ζ⁡\(𝐚′\)‖2δ\)\.\\displaystyle\\eta\(\\mathbf\{a\}\)\\eta\(\\mathbf\{a\}^\{\\prime\}\)\\,w\\\!\\left\(\\frac\{\\\|D\_\{\\parallel\}\(z\(\\mathbf\{a\}\)\-z\(\\mathbf\{a\}^\{\\prime\}\)\)\\\|\_\{2\}\}\{\\varrho\}\\right\)M\_\{5/2\}\\\!\\left\(\\frac\{\\alpha\_\{\\perp\}\\\|\\zeta\(\\mathbf\{a\}\)\-\\zeta\(\\mathbf\{a\}^\{\\prime\}\)\\\|\_\{2\}\}\{\\delta\}\\right\)\.\(32\)HereM5/2M\_\{5/2\}is the Matérn function \([22](https://arxiv.org/html/2610.29009#S2.E22)\),δ\>0\\delta\>0is a residual length, andη\\etais a gate\. The Wendland factor sets the support, since two inputs interact only if their principal coordinates are closer thanϱ\\varrhoin the metricD∥D\_\{\\parallel\}\. The Matérn factor separates inputs with the same principal coordinates\. Note that the kernel is not radial in the full input\.

The gate switches the kernel off for inputs whose residualζ⁡\(𝐚\)\\zeta\(\\mathbf\{a\}\)is much larger than those of the training inputs\. Withζi=ζ⁡\(𝐚i\)\\zeta\_\{i\}=\\zeta\(\\mathbf\{a\}\_\{i\}\), we set

δ=max⁡\{0\.01​maxi​‖𝐚¯i‖2,2\.2​maxi​‖ζi‖2,10−8\},\\delta=\\max\\left\\\{0\.01\\max\_\{i\}\\\|\\bar\{\\mathbf\{a\}\}\_\{i\}\\\|\_\{2\},\\ 2\.2\\max\_\{i\}\\\|\\zeta\_\{i\}\\\|\_\{2\},\\ 10^\{\-8\}\\right\\\},\(33\)and, with the polynomialH⁡\(t\)=126​t5−420​t6\+540​t7−315​t8\+70​t9H\(t\)=126t^\{5\}\-420t^\{6\}\+540t^\{7\}\-315t^\{8\}\+70t^\{9\}, which increases fromH⁡\(0\)=0H\(0\)=0toH⁡\(1\)=1H\(1\)=1sinceH′​\(t\)=630​t4​\(1−t\)4H^\{\\prime\}\(t\)=630t^\{4\}\(1\-t\)^\{4\},

η⁡\(𝐚\)=η^​\(‖ζ⁡\(𝐚\)‖2/δ\),η^​\(r\)=\{1,0≤r≤1/2,H​\(2​\(1−r\)\),1/2<r<1,0,r≥1\.\\eta\(\\mathbf\{a\}\)=\\hat\{\\eta\}\(\\\|\\zeta\(\\mathbf\{a\}\)\\\|\_\{2\}/\\delta\),\\qquad\\hat\{\\eta\}\(r\)=\\begin\{cases\}1,&0\\leq r\\leq 1/2,\\\\ H\(2\(1\-r\)\),&1/2<r<1,\\\\ 0,&r\\geq 1\.\\end\{cases\}\(34\)The gate isC4C^\{4\}and positive for‖ζ⁡\(𝐚\)‖2<δ\\\|\\zeta\(\\mathbf\{a\}\)\\\|\_\{2\}<\\delta\. Since‖ζi‖2≤δ/2\.2\\\|\\zeta\_\{i\}\\\|\_\{2\}\\leq\\delta/2\.2, it equals one on every training input\. A new input need not lie in the span of the principal directionsVzV\_\{z\}, and its gate decreases to zero as its residual grows\.

###### Proposition 1\(Positive definiteness in any input dimension\)\.

For any finite sampled\-input dimension, the kernel \([32](https://arxiv.org/html/2610.29009#S3.E32)\) is positive semidefinite\. Its Gram matrix is strictly positive definite on distinct standardized inputs with positive gates\. Consequently, adding a positive multiple of the identity gives a positive\-definite kernel matrix, including when sampled inputs coincide or gates vanish\.

###### Proof sketch\.

Both factors are positive definite functions composed with linear maps, and products and gate multiplications preserve positive semidefiniteness\. Distinct inputs differ in their principal or their residual coordinates\. Appendix[C\.2](https://arxiv.org/html/2610.29009#A3.SS2)gives the proof\. ∎

The proposition holds for any orthogonal projection, so it needs no estimate of an intrinsic dimension\. The principal components only choose the directions along which the support is cut off\.

##### Fixed and fitted shapes

We compare two choices of the shape parameters\. The fixed\-shape kernel usesD∥=ID\_\{\\parallel\}=Iandα⟂=1\\alpha\_\{\\perp\}=1\. The fitted\-shape kernel also fitsα⟂\\alpha\_\{\\perp\}andD∥=diag⁡\(eυ1,…,eυnz\)D\_\{\\parallel\}=\\operatorname\{diag\}\(e^\{\\upsilon\_\{1\}\},\\ldots,e^\{\\upsilon\_\{n\_\{z\}\}\}\), with∑k=1nzυk=0\\sum\_\{k=1\}^\{n\_\{z\}\}\\upsilon\_\{k\}=0so thatϱ\\varrhoalone sets the overall support and theυk\\upsilon\_\{k\}only redistribute it among the principal directions\. Note that whennz=1n\_\{z\}=1the constraint forcesD∥=\[1\]D\_\{\\parallel\}=\[1\], and when the residual coordinates are negligible we also keepα⟂=1\\alpha\_\{\\perp\}=1\. The fitted\-shape kernel then has the shape of the fixed\-shape one and differs from it only through its radius and regularization \(Section[5\.2\.1](https://arxiv.org/html/2610.29009#S5.SS2.SSS1)lists these cases\)\.

### 3\.2The operator interpolant

We now use the kernelλ\\lambdato approximate the map from a sampled input to the local coefficients of its output\. Let𝐛i∈ℝmrhs\\mathbf\{b\}\_\{i\}\\in\\mathbb\{R\}^\{m\_\{\\rm rhs\}\}stack the local coefficients𝐛ij\\mathbf\{b\}\_\{i\}^\{j\}of training fieldiion all stencils \(and at all sampled times\), wheremrhs=∑jrjm\_\{\\rm rhs\}=\\sum\_\{j\}r\_\{j\}\(timesNtN\_\{t\}for spacetime outputs\)\. Every one of these coefficients is a function of the same input, so we approximate all of them with the same scalar kernel, i\.e\., with the matrix\-valued kernelλ⁡\(𝐚,𝐚′\)​I\\lambda\(\\mathbf\{a\},\\mathbf\{a\}^\{\\prime\}\)I\. The operator regressionf~\\tilde\{f\}of \([6](https://arxiv.org/html/2610.29009#S1.E6)\) is the operator interpolant

f~​\(𝐚\)=∑i=1Nλ⁡\(𝐚,𝐚i\)​𝐜i,\\tilde\{f\}\(\\mathbf\{a\}\)=\\sum\_\{i=1\}^\{N\}\\lambda\(\\mathbf\{a\},\\mathbf\{a\}\_\{i\}\)\\mathbf\{c\}\_\{i\},\(35\)with coefficients𝐜i∈ℝmrhs\\mathbf\{c\}\_\{i\}\\in\\mathbb\{R\}^\{m\_\{\\rm rhs\}\}determined by the regularized interpolation conditions

f~\(𝐚i\)\+θ𝐜i=𝐛i,i=1,…,N,\\tilde\{f\}\(\\mathbf\{a\}\_\{i\}\)\+\\theta\\mathbf\{c\}\_\{i\}=\\mathbf\{b\}\_\{i\},\\qquad i=1,\\ldots,N,\(36\)whereθ\>0\\theta\>0is a regularization parameter\. In the limitθ→0\\theta\\to 0,f~\\tilde\{f\}interpolates the training coefficients, and forθ\>0\\theta\>0it is their kernel ridge regression\. Sinceχ\\chiis linear, the learned operator \([6](https://arxiv.org/html/2610.29009#S1.E6)\) is

𝒢~​\(a\)=∑i=1Nλ⁡\(𝐚,𝐚i\)​χ​\(𝐜i\),\\tilde\{\\mathcal\{G\}\}\(a\)=\\sum\_\{i=1\}^\{N\}\\lambda\(\\mathbf\{a\},\\mathbf\{a\}\_\{i\}\)\\chi\(\\mathbf\{c\}\_\{i\}\),\(37\)a combination ofNNfixed fields with weights that depend on the input\. Every one of these fields is a local divergence\-free output reconstruction, so𝒢~​\(a\)\\tilde\{\\mathcal\{G\}\}\(a\)is divergence free on each stencil for every input, regardless of the accuracy of the operator regression\.

### 3\.3Affine augmentation

Operators often depend nearly linearly on part of their input\. A compactly supported kernel regression represents such a trend poorly, since it returns to zero away from the training inputs\. As in polynomially augmented interpolation\[[4](https://arxiv.org/html/2610.29009#bib.bib46)\], we augment the operator regression with affine functions of up to1515principal coordinates of the input \(more than the at most three that set the support\), which it reproduces exactly, and let the kernel represent the remainder\. These polynomials in the input are unrelated to the divergence\-free polynomials in space of Section[2](https://arxiv.org/html/2610.29009#S2), which are part of the output representation in every model\.

Let the columns ofVpV\_\{p\}be up to1515principal directions of the training inputs, letςk\\varsigma\_\{k\}be the training standard deviation along directionkk, and set

p⁡\(𝐚\)=\(1,\(Vp𝖳​𝐚¯\)1ς1,…,\(Vp𝖳​𝐚¯\)naff−1ςnaff−1\)𝖳,p\(\\mathbf\{a\}\)=\\left\(1,\\frac\{\(V\_\{p\}^\{\\mathsf\{T\}\}\\bar\{\\mathbf\{a\}\}\)\_\{1\}\}\{\\varsigma\_\{1\}\},\\ldots,\\frac\{\(V\_\{p\}^\{\\mathsf\{T\}\}\\bar\{\\mathbf\{a\}\}\)\_\{n\_\{\\rm aff\}\-1\}\}\{\\varsigma\_\{n\_\{\\rm aff\}\-1\}\}\\right\)^\{\\mathsf\{T\}\},\(38\)withnaff−1≤15n\_\{\\rm aff\}\-1\\leq 15limited by the rank of the training inputs\. The augmented operator interpolant is

f~​\(𝐚\)=∑i=1Nλ⁡\(𝐚,𝐚i\)​𝐜i\+𝜷𝖳​p​\(𝐚\),\\tilde\{f\}\(\\mathbf\{a\}\)=\\sum\_\{i=1\}^\{N\}\\lambda\(\\mathbf\{a\},\\mathbf\{a\}\_\{i\}\)\\mathbf\{c\}\_\{i\}\+\\boldsymbol\{\\beta\}^\{\\mathsf\{T\}\}p\(\\mathbf\{a\}\),\(39\)with kernel coefficients𝐜i∈ℝmrhs\\mathbf\{c\}\_\{i\}\\in\\mathbb\{R\}^\{m\_\{\\rm rhs\}\}and affine coefficients𝜷∈ℝnaff×mrhs\\boldsymbol\{\\beta\}\\in\\mathbb\{R\}^\{n\_\{\\rm aff\}\\times m\_\{\\rm rhs\}\}\. Its coefficients are determined by the regularized interpolation conditions of \([36](https://arxiv.org/html/2610.29009#S3.E36)\), now for the augmentedf~\\tilde\{f\}, together with moment conditions on the kernel coefficients,

f~\(𝐚i\)\+θ𝐜i=𝐛i,i=1,…,N,∑i=1Np\(𝐚i\)𝐜i𝖳=0\.\\tilde\{f\}\(\\mathbf\{a\}\_\{i\}\)\+\\theta\\mathbf\{c\}\_\{i\}=\\mathbf\{b\}\_\{i\},\\quad i=1,\\ldots,N,\\qquad\\sum\_\{i=1\}^\{N\}p\(\\mathbf\{a\}\_\{i\}\)\\mathbf\{c\}\_\{i\}^\{\\mathsf\{T\}\}=0\.\(40\)The moment conditions make \([39](https://arxiv.org/html/2610.29009#S3.E39)\) reproduce every affine function of the principal coordinates exactly, for anyθ\\theta\. If𝐛i=T𝖳​p​\(𝐚i\)\\mathbf\{b\}\_\{i\}=T^\{\\mathsf\{T\}\}p\(\\mathbf\{a\}\_\{i\}\)for some matrixTT, then𝐜i=0\\mathbf\{c\}\_\{i\}=0and𝜷=T\\boldsymbol\{\\beta\}=Tsatisfy \([40](https://arxiv.org/html/2610.29009#S3.E40)\)\. As in \([37](https://arxiv.org/html/2610.29009#S3.E37)\), the augmented learned operator is

𝒢~\(a\)=∑i=1Nλ\(𝐚,𝐚i\)χ\(𝐜i\)\+∑k=1naffpk\(𝐚\)χ\(𝜷k,:\),\\tilde\{\\mathcal\{G\}\}\(a\)=\\sum\_\{i=1\}^\{N\}\\lambda\(\\mathbf\{a\},\\mathbf\{a\}\_\{i\}\)\\chi\(\\mathbf\{c\}\_\{i\}\)\+\\sum\_\{k=1\}^\{n\_\{\\rm aff\}\}p\_\{k\}\(\\mathbf\{a\}\)\\chi\(\\boldsymbol\{\\beta\}\_\{k,:\}\),\(41\)a combination ofN\+naffN\+n\_\{\\rm aff\}fixed divergence\-free fields\. Without augmentation,naff=0n\_\{\\rm aff\}=0, the affine terms and the moment conditions disappear, and \([39](https://arxiv.org/html/2610.29009#S3.E39)\)–\([41](https://arxiv.org/html/2610.29009#S3.E41)\) reduce to \([35](https://arxiv.org/html/2610.29009#S3.E35)\)–\([37](https://arxiv.org/html/2610.29009#S3.E37)\)\.

Both interpolants have a variational characterization\. The functionf~=fℋ\+𝜷𝖳​p\\tilde\{f\}=f\_\{\\mathcal\{H\}\}\+\\boldsymbol\{\\beta\}^\{\\mathsf\{T\}\}pminimizes

∑i=1N‖fℋ​\(𝐚i\)\+𝜷𝖳​p​\(𝐚i\)−𝐛i‖22\+θ​‖fℋ‖ℋ2\\sum\_\{i=1\}^\{N\}\\bigl\\\|f\_\{\\mathcal\{H\}\}\(\\mathbf\{a\}\_\{i\}\)\+\\boldsymbol\{\\beta\}^\{\\mathsf\{T\}\}p\(\\mathbf\{a\}\_\{i\}\)\-\\mathbf\{b\}\_\{i\}\\bigr\\\|\_\{2\}^\{2\}\+\\theta\\\|f\_\{\\mathcal\{H\}\}\\\|\_\{\\mathcal\{H\}\}^\{2\}\(42\)over the affine coefficients𝜷\\boldsymbol\{\\beta\}and the functionsfℋf\_\{\\mathcal\{H\}\}in the reproducing kernel Hilbert spaceℋ\\mathcal\{H\}ofλ⁡\(𝐚,𝐚′\)​I\\lambda\(\\mathbf\{a\},\\mathbf\{a\}^\{\\prime\}\)I, since by the representer theorem the minimizingfℋf\_\{\\mathcal\{H\}\}has the form∑iλ⁡\(⋅,𝐚i\)​𝐜i\\sum\_\{i\}\\lambda\(\\cdot,\\mathbf\{a\}\_\{i\}\)\\mathbf\{c\}\_\{i\}, and \([40](https://arxiv.org/html/2610.29009#S3.E40)\) makes the gradient of the convex objective vanish \(Appendix[C\.3](https://arxiv.org/html/2610.29009#A3.SS3)\)\. Without augmentation,𝜷\\boldsymbol\{\\beta\}is absent and \([42](https://arxiv.org/html/2610.29009#S3.E42)\) is the objective of kernel ridge regression\.

### 3\.4Linear systems

We now write the conditions \([36](https://arxiv.org/html/2610.29009#S3.E36)\) and \([40](https://arxiv.org/html/2610.29009#S3.E40)\) as linear systems\. For training inputsU=\{𝐚i\}i=1NU=\\\{\\mathbf\{a\}\_\{i\}\\\}\_\{i=1\}^\{N\}andN∗N\_\{\*\}query inputsU∗=\{𝐚∗\(q\)\}q=1N∗U\_\{\*\}=\\\{\\mathbf\{a\}\_\{\*\}^\{\(q\)\}\\\}\_\{q=1\}^\{N\_\{\*\}\}, we define the kernel matrices

\(KU​U\)i​i′=λ⁡\(𝐚i,𝐚i′\),\(KU∗​U\)q​i=λ⁡\(𝐚∗\(q\),𝐚i\),KU​Uθ=KU​U\+θ​I,\(K\_\{UU\}\)\_\{ii^\{\\prime\}\}=\\lambda\(\\mathbf\{a\}\_\{i\},\\mathbf\{a\}\_\{i^\{\\prime\}\}\),\\qquad\(K\_\{U\_\{\*\}U\}\)\_\{qi\}=\\lambda\(\\mathbf\{a\}\_\{\*\}^\{\(q\)\},\\mathbf\{a\}\_\{i\}\),\\qquad K\_\{UU\}^\{\\theta\}=K\_\{UU\}\+\\theta I,\(43\)withθ\\thetathe regularization of \([36](https://arxiv.org/html/2610.29009#S3.E36)\), and the affine matricesPUP\_\{U\}andPU∗P\_\{U\_\{\*\}\}with rows\(PU\)i,:=p\(𝐚i\)𝖳\(P\_\{U\}\)\_\{i,:\}=p\(\\mathbf\{a\}\_\{i\}\)^\{\\mathsf\{T\}\}and\(PU∗\)q,:=p\(𝐚∗\(q\)\)𝖳\(P\_\{U\_\{\*\}\}\)\_\{q,:\}=p\(\\mathbf\{a\}\_\{\*\}^\{\(q\)\}\)^\{\\mathsf\{T\}\}\. We keep every nonzero entry ofKU​UK\_\{UU\}, with no neighbors discarded by an approximate search\. Its densitynnz⁡\(KU​U\)/N2\\operatorname\{nnz\}\(K\_\{UU\}\)/N^\{2\}, wherennz\\operatorname\{nnz\}counts the nonzero entries, is thus set byϱ\\varrhoand the data, and the storage of its Cholesky factor also depends on the fill\.

Since the kernelλ⁡\(𝐚,𝐚′\)​I\\lambda\(\\mathbf\{a\},\\mathbf\{a\}^\{\\prime\}\)Idoes not couple different output coefficients, the conditions decouple into one system per stencil\. On stenciljj, letBj∈ℝN×rjB\_\{j\}\\in\\mathbb\{R\}^\{N\\times r\_\{j\}\}hold the𝐛ij\\mathbf\{b\}\_\{i\}^\{j\}as rows, and letCj∈ℝN×rjC\_\{j\}\\in\\mathbb\{R\}^\{N\\times r\_\{j\}\}andβj∈ℝnaff×rj\\beta\_\{j\}\\in\\mathbb\{R\}^\{n\_\{\\rm aff\}\\times r\_\{j\}\}hold the corresponding entries of the𝐜i\\mathbf\{c\}\_\{i\}and the columns of𝜷\\boldsymbol\{\\beta\}\. Evaluating \([39](https://arxiv.org/html/2610.29009#S3.E39)\) at the training inputs turns \([40](https://arxiv.org/html/2610.29009#S3.E40)\) into the first equation of

\[KU​UθPUPU𝖳0\]​\[Cjβj\]=\[Bj0\],B∗,j=KU∗​U​Cj\+PU∗​βj,\\begin\{bmatrix\}K\_\{UU\}^\{\\theta\}&P\_\{U\}\\\\ P\_\{U\}^\{\\mathsf\{T\}\}&0\\end\{bmatrix\}\\begin\{bmatrix\}C\_\{j\}\\\\ \\beta\_\{j\}\\end\{bmatrix\}=\\begin\{bmatrix\}B\_\{j\}\\\\ 0\\end\{bmatrix\},\\qquad B\_\{\*,j\}=K\_\{U\_\{\*\}U\}C\_\{j\}\+P\_\{U\_\{\*\}\}\\beta\_\{j\},\(44\)and evaluating it at the query inputs gives the second, where rowqqofB∗,jB\_\{\*,j\}holds the coefficients thatf~\\tilde\{f\}predicts on stenciljjfor query inputqq\. The blockPU𝖳​Cj=0P\_\{U\}^\{\\mathsf\{T\}\}C\_\{j\}=0is the moment condition\. Without augmentation, the same steps applied to \([35](https://arxiv.org/html/2610.29009#S3.E35)\) and \([36](https://arxiv.org/html/2610.29009#S3.E36)\) give

KU​Uθ​Cj=Bj,B∗,j=KU∗​U​Cj\.K\_\{UU\}^\{\\theta\}C\_\{j\}=B\_\{j\},\\qquad B\_\{\*,j\}=K\_\{U\_\{\*\}U\}C\_\{j\}\.\(45\)

### 3\.5Sparse Cholesky and whitened QR

The system \([44](https://arxiv.org/html/2610.29009#S3.E44)\) is an indefinite saddle\-point system\. Forming the Schur complementPU𝖳​\(KU​Uθ\)−1​PUP\_\{U\}^\{\\mathsf\{T\}\}\(K\_\{UU\}^\{\\theta\}\)^\{\-1\}P\_\{U\}squares the condition number of the whitened polynomial columns, as in the normal equations of least squares\. We instead use the Cholesky–QR elimination of our unified kernel and polynomial interpolation\[[4](https://arxiv.org/html/2610.29009#bib.bib46)\]\. SinceKU​UθK\_\{UU\}^\{\\theta\}is sparse and positive definite, we factor it by sparse Cholesky, whiten the polynomial columns, and factor them by thin QR\. Suppressing the fill\-reducing permutation, we obtain

KU​Uθ=LK​LK𝖳,LK−1​PU=QP​RP,ℰU=RP−1​QP𝖳​LK−1\.K\_\{UU\}^\{\\theta\}=L\_\{K\}L\_\{K\}^\{\\mathsf\{T\}\},\\qquad L\_\{K\}^\{\-1\}P\_\{U\}=Q\_\{P\}R\_\{P\},\\qquad\\mathcal\{E\}\_\{U\}=R\_\{P\}^\{\-1\}Q\_\{P\}^\{\\mathsf\{T\}\}L\_\{K\}^\{\-1\}\.\(46\)HereQPQ\_\{P\}has orthonormal columns andRPR\_\{P\}is a nonsingularnaff×naffn\_\{\\rm aff\}\\times n\_\{\\rm aff\}triangular matrix, and we applyℰU\\mathcal\{E\}\_\{U\}by triangular solves without formingLK−1L\_\{K\}^\{\-1\}\. Block elimination of \([44](https://arxiv.org/html/2610.29009#S3.E44)\) then gives

βj=ℰU​Bj,KU​Uθ​Cj=Bj−PU​βj\.\\beta\_\{j\}=\\mathcal\{E\}\_\{U\}B\_\{j\},\\qquad K\_\{UU\}^\{\\theta\}C\_\{j\}=B\_\{j\}\-P\_\{U\}\\beta\_\{j\}\.\(47\)Without augmentation,QPQ\_\{P\},RPR\_\{P\}, andℰU\\mathcal\{E\}\_\{U\}are empty and the correction disappears\. The same factors serve the parameter selection below and the prediction of Section[4\.1](https://arxiv.org/html/2610.29009#S4.SS1), which never forms theCjC\_\{j\}\.

The size of the Cholesky factor, and the speed of its triangular solves, depend on the ordering of the training inputs\. Since two inputs interact only if their principal coordinates are close, the sparsity pattern ofKU​UK\_\{UU\}resembles that of a discretization on a mesh in at most three dimensions, and we order it by nested dissection\. For inputs of numerical rank one \(nz=1n\_\{z\}=1\), sorting the inputs byzzeven makesKU​UK\_\{UU\}a band matrix \(the matrix of an interval graph\), whose Cholesky factor has no entries outside the band\. However, the elimination of a band matrix is a single chain, which leaves the triangular solves of a prediction little parallelism on the GPU, while nested dissection gives a balanced elimination tree\. Thus, nested dissection is our default, and the band ordering is preferable when memory is more limited than prediction time \(Section[5\.2\.5](https://arxiv.org/html/2610.29009#S5.SS2.SSS5)\)\. The ordering changes neither the operator regression nor its predictions beyond rounding\.

### 3\.6Choosing the kernel parameters

The support radiusϱ\\varrho, the regularizationθ\\theta, and, for the fitted\-shape kernel, the shape parameters remain to be chosen\. The support radius is the most important of these, since it trades accuracy against sparsity\. We use two criteria in turn\. For the fitted\-shape kernel, a restricted likelihood criterion fits the shape and the regularization from the training data alone\. The error on validation fields then selects the radius and refines the regularization\.

LetF∈ℝN×nvF\\in\\mathbb\{R\}^\{N\\times n\_\{v\}\}hold all sampled training outputs, withnv=d​mn\_\{v\}=dmvalues per field, ord​m​NtdmN\_\{t\}for spacetime outputs, after the amplitude normalization \([72](https://arxiv.org/html/2610.29009#S5.E72)\) where it applies, letQ0Q\_\{0\}be an orthonormal basis of the range ofPUP\_\{U\},E0=\(I−Q0​Q0𝖳\)​FE\_\{0\}=\(I\-Q\_\{0\}Q\_\{0\}^\{\\mathsf\{T\}\}\)F, andE=N​E0/‖E0‖FE=\\sqrt\{N\}E\_\{0\}/\\\|E\_\{0\}\\\|\_\{F\}, and let

ΠU=\(KU​Uθ\)−1−\(KU​Uθ\)−1​PU​\[PU𝖳​\(KU​Uθ\)−1​PU\]−1​PU𝖳​\(KU​Uθ\)−1,\\Pi\_\{U\}=\(K\_\{UU\}^\{\\theta\}\)^\{\-1\}\-\(K\_\{UU\}^\{\\theta\}\)^\{\-1\}P\_\{U\}\\left\[P\_\{U\}^\{\\mathsf\{T\}\}\(K\_\{UU\}^\{\\theta\}\)^\{\-1\}P\_\{U\}\\right\]^\{\-1\}P\_\{U\}^\{\\mathsf\{T\}\}\(K\_\{UU\}^\{\\theta\}\)^\{\-1\},\(48\)which we apply by sparse solves and never store\. Withnres=N−naffn\_\{\\rm res\}=N\-n\_\{\\rm aff\}, we minimize

𝒥cd=\\displaystyle\\mathcal\{J\}\_\{c\_\{d\}\}=\{\}12​\[logdetKU​Uθ\+2log\|detRP\|nres\+log⁡\(tr⁡\(E𝖳​ΠU​E\)nres\)\]\+cd​d~,\\displaystyle\\frac\{1\}\{2\}\\left\[\\frac\{\\log\\det K\_\{UU\}^\{\\theta\}\+2\\log\|\\det R\_\{P\}\|\}\{n\_\{\\rm res\}\}\+\\log\\\!\\left\(\\frac\{\\operatorname\{tr\}\(E^\{\\mathsf\{T\}\}\\Pi\_\{U\}E\)\}\{n\_\{\\rm res\}\}\\right\)\\right\]\+c\_\{d\}\\widetilde\{d\},\(49\)d~=\\displaystyle\\widetilde\{d\}=\{\}1N⁡\(N−1\)​∑i≠i′sigmoid⁡\(1−‖D∥​\(zi−zi′\)‖2/ϱ0\.03\)\.\\displaystyle\\frac\{1\}\{N\(N\-1\)\}\\sum\_\{i\\neq i^\{\\prime\}\}\\operatorname\{sigmoid\}\\\!\\left\(\\frac\{1\-\\\|D\_\{\\parallel\}\(z\_\{i\}\-z\_\{i^\{\\prime\}\}\)\\\|\_\{2\}/\\varrho\}\{0\.03\}\\right\)\.\(50\)The bracket can be interpreted in two ways\. In kernel terms,tr⁡\(E𝖳​ΠU​E\)\\operatorname\{tr\}\(E^\{\\mathsf\{T\}\}\\Pi\_\{U\}E\)is the squared norm, in the native space of the regularized kernel, of the augmented interpolants of the training outputs after their affine part is removed, and the log\-determinants penalize kernels whose matrices have a large determinant, which are the more flexible ones\. The bracket thus favors kernels that represent the outputs with a small norm, while penalizing overly flexible kernels\. In probabilistic terms, if the outputs are modeled as samples of a Gaussian process with covarianceλ\\lambda, independent noise of varianceθ\\theta, and an affine mean with unknown coefficients, the bracket is, up to constants, the negative log restricted likelihood with the variance profiled out \(withdetRP=1\\det R\_\{P\}=1without augmentation\)\. We borrow the criterion from this setting, but use it only to select kernel parameters, not for uncertainty quantification\.

The termd~\\widetilde\{d\}, withzi=z⁡\(𝐚i\)z\_\{i\}=z\(\\mathbf\{a\}\_\{i\}\)and the logistic functionsigmoid⁡\(x\)=1/\(1\+e−x\)\\operatorname\{sigmoid\}\(x\)=1/\(1\+e^\{\-x\}\), is a smooth count of the interacting pairs, and its weightcd∈\{0,1\}c\_\{d\}\\in\\\{0,1\\\}trades accuracy against sparsity\. We fit one candidate with each weight\. All terms of𝒥cd\\mathcal\{J\}\_\{c\_\{d\}\}come from the sparse factors \(Appendix[B\.3](https://arxiv.org/html/2610.29009#A2.SS3)\)\.

The criterion measures how well the kernel represents the training outputs, but our goal is the error of the predicted velocity for new inputs\. Thus, we discard the radius that minimizes the likelihood criterion and choose the radius by validation alone\. We first evaluate the five radii that give the densities0\.100\.10,0\.250\.25,0\.500\.50,0\.750\.75, and0\.900\.90, with the regularization from the likelihood\. We then take gradient steps in the radius and the regularization on the mean relative velocity error on validation fields, from four starts \(half and a quarter of the largest of these radii, each withθ=10−6\\theta=10^\{\-6\}and10−310^\{\-3\}\), while the fitted shape stays unchanged\. Among all candidates with density at most0\.900\.90, we keep the one with the smallest validation error\. The fixed\-shape kernel goes through the same search without the likelihood step, withθ=10−6\\theta=10^\{\-6\}at the five radii\. All choices are made before the test fields are used \(Appendix[B\.2](https://arxiv.org/html/2610.29009#A2.SS2)\)\.

We thus compare four models, the fixed\-shape and the fitted\-shape kernel, each with and without affine augmentation\. A comparison at the same augmentation isolates the effect of fitting the shape\. Since all parameters are refit when the augmentation changes, a comparison with and without it measures the whole procedure\.

## 4Operator Lagrange prediction

We now combine the operator regression of Section[3](https://arxiv.org/html/2610.29009#S3)with the output reconstruction of Section[2](https://arxiv.org/html/2610.29009#S2)\. The direct way to predict is to solve \([44](https://arxiv.org/html/2610.29009#S3.E44)\) for the coefficientsCjC\_\{j\}andβj\\beta\_\{j\}of every stencil\. This needs one operator right\-hand side for every spatial coefficient, and the local coefficients of every training field\. However, an operator regression with the same kernel for all outputs is a weighted combination of the training outputs, and the weights depend only on the new input\. It suffices to solve once per new input for these weights and apply them to the whole output at once\. Since the output reconstruction is linear, we can apply them even before the output reconstruction, to the sampled training fields\.

### 4\.1Operator Lagrange weights

We first define the weights as functions on the input space\. For an input𝐚\\mathbf\{a\}, letkU​\(𝐚\)=\[λ⁡\(𝐚i,𝐚\)\]i=1Nk\_\{U\}\(\\mathbf\{a\}\)=\[\\lambda\(\\mathbf\{a\}\_\{i\},\\mathbf\{a\}\)\]\_\{i=1\}^\{N\}\. We define its operator Lagrange weightsℓ⁡\(𝐚\)∈ℝN\\ell\(\\mathbf\{a\}\)\\in\\mathbb\{R\}^\{N\}by

\[KU​UθPUPU𝖳0\]​\[ℓ⁡\(𝐚\)μ⁡\(𝐚\)\]=\[kU​\(𝐚\)p⁡\(𝐚\)\],\\begin\{bmatrix\}K\_\{UU\}^\{\\theta\}&P\_\{U\}\\\\ P\_\{U\}^\{\\mathsf\{T\}\}&0\\end\{bmatrix\}\\begin\{bmatrix\}\\ell\(\\mathbf\{a\}\)\\\\ \\mu\(\\mathbf\{a\}\)\\end\{bmatrix\}=\\begin\{bmatrix\}k\_\{U\}\(\\mathbf\{a\}\)\\\\ p\(\\mathbf\{a\}\)\\end\{bmatrix\},\(51\)where the multiplierμ⁡\(𝐚\)\\mu\(\\mathbf\{a\}\)enforces the reproduction of affine functions\. Block elimination with the factors \([46](https://arxiv.org/html/2610.29009#S3.E46)\) gives

ℓ⁡\(𝐚\)=\(I−ℰU𝖳​PU𝖳\)​\(KU​Uθ\)−1​kU​\(𝐚\)\+ℰU𝖳​p​\(𝐚\),\\ell\(\\mathbf\{a\}\)=\\bigl\(I\-\\mathcal\{E\}\_\{U\}^\{\\mathsf\{T\}\}P\_\{U\}^\{\\mathsf\{T\}\}\\bigr\)\(K\_\{UU\}^\{\\theta\}\)^\{\-1\}k\_\{U\}\(\\mathbf\{a\}\)\+\\mathcal\{E\}\_\{U\}^\{\\mathsf\{T\}\}p\(\\mathbf\{a\}\),\(52\)so every weightℓi\\ell\_\{i\}is a combination of the kernel functionsλ⁡\(⋅,𝐚i′\)\\lambda\(\\cdot,\\mathbf\{a\}\_\{i^\{\\prime\}\}\)and the affine functions, of the same form as \([39](https://arxiv.org/html/2610.29009#S3.E39)\)\. Since the matrix of \([51](https://arxiv.org/html/2610.29009#S4.E51)\) is the symmetric matrix of \([44](https://arxiv.org/html/2610.29009#S3.E44)\), the operator interpolant \([39](https://arxiv.org/html/2610.29009#S3.E39)\) and the learned operator \([41](https://arxiv.org/html/2610.29009#S3.E41)\) take the Lagrange form

f~​\(𝐚\)=∑i=1Nℓi​\(𝐚\)​𝐛i,𝒢~​\(a\)=∑i=1Nℓi​\(𝐚\)​χ​\(𝐛i\),\\tilde\{f\}\(\\mathbf\{a\}\)=\\sum\_\{i=1\}^\{N\}\\ell\_\{i\}\(\\mathbf\{a\}\)\\mathbf\{b\}\_\{i\},\\qquad\\tilde\{\\mathcal\{G\}\}\(a\)=\\sum\_\{i=1\}^\{N\}\\ell\_\{i\}\(\\mathbf\{a\}\)\\chi\(\\mathbf\{b\}\_\{i\}\),\(53\)a combination of the reconstructed training fields with weights that depend only on the input \(Appendix[C\.4](https://arxiv.org/html/2610.29009#A3.SS4)\)\. Without augmentation,ℓ⁡\(𝐚\)=\(KU​Uθ\)−1​kU​\(𝐚\)\\ell\(\\mathbf\{a\}\)=\(K\_\{UU\}^\{\\theta\}\)^\{\-1\}k\_\{U\}\(\\mathbf\{a\}\), and \([53](https://arxiv.org/html/2610.29009#S4.E53)\) is the Lagrange form of \([35](https://arxiv.org/html/2610.29009#S3.E35)\) and \([37](https://arxiv.org/html/2610.29009#S3.E37)\)\. The weights are regularized Lagrange \(cardinal\) functions of the training inputs\[[46](https://arxiv.org/html/2610.29009#bib.bib42),[65](https://arxiv.org/html/2610.29009#bib.bib16),[16](https://arxiv.org/html/2610.29009#bib.bib17)\]\. Forθ=0\\theta=0, they are cardinal at the training inputs\. Forθ\>0\\theta\>0, they are not, but they still reproduce the affine functions exactly\.

###### Proposition 2\(Reproduction and preservation under shared weights\)\.

AssumeKU​UθK\_\{UU\}^\{\\theta\}is positive definite andPUP\_\{U\}has full column rank\. Then \([52](https://arxiv.org/html/2610.29009#S4.E52)\) is the unique weight vector in \([51](https://arxiv.org/html/2610.29009#S4.E51)\), andPU𝖳​ℓ​\(𝐚\)=p⁡\(𝐚\)P\_\{U\}^\{\\mathsf\{T\}\}\\ell\(\\mathbf\{a\}\)=p\(\\mathbf\{a\}\)\. In particular, the augmented model reproduces constants and the scaled principal coordinates inp⁡\(𝐚\)p\(\\mathbf\{a\}\)\. Letχ\\chibe any common linear output reconstruction of the training coefficient vectors, and let𝒟\\mathcal\{D\}be a linear constraint with𝒟​χ​\(𝐛i\)=0\\mathcal\{D\}\\chi\(\\mathbf\{b\}\_\{i\}\)=0for every training field\. Then

χ⁡\(∑iℓi​\(𝐚\)​𝐛i\)=∑iℓi​\(𝐚\)​χ​\(𝐛i\),𝒟​χ​\(∑iℓi​\(𝐚\)​𝐛i\)=0\.\\chi\\\!\\left\(\\sum\_\{i\}\\ell\_\{i\}\(\\mathbf\{a\}\)\\mathbf\{b\}\_\{i\}\\right\)=\\sum\_\{i\}\\ell\_\{i\}\(\\mathbf\{a\}\)\\chi\(\\mathbf\{b\}\_\{i\}\),\\qquad\\mathcal\{D\}\\chi\\\!\\left\(\\sum\_\{i\}\\ell\_\{i\}\(\\mathbf\{a\}\)\\mathbf\{b\}\_\{i\}\\right\)=0\.\(54\)

###### Proof sketch\.

The factors \([46](https://arxiv.org/html/2610.29009#S3.E46)\) giveℰU​PU=I\\mathcal\{E\}\_\{U\}P\_\{U\}=I, from which reproduction follows by substitution, and uniqueness follows from the block structure\. The identities \([54](https://arxiv.org/html/2610.29009#S4.E54)\) are linearity\. Appendix[C\.4](https://arxiv.org/html/2610.29009#A3.SS4)gives the details\. ∎

With𝒟\\mathcal\{D\}the divergence on a stencil, the proposition shows that every prediction is divergence free on each stencil, for both kernels and with or without augmentation\. Since the temporal weights \([28](https://arxiv.org/html/2610.29009#S2.E28)\) do not depend on the input, the operator weights also commute with interpolation in time\.

We now turn to the discrete form\. For a new input𝐚∗\\mathbf\{a\}\_\{\*\}, restricting the first identity of \([53](https://arxiv.org/html/2610.29009#S4.E53)\) to stenciljjgives the predicted coefficients

𝐛∗j=Bj𝖳​ℓ​\(𝐚∗\),\\mathbf\{b\}\_\{\*\}^\{j\}=B\_\{j\}^\{\\mathsf\{T\}\}\\ell\(\\mathbf\{a\}\_\{\*\}\),\(55\)and, with the weights ofN∗N\_\{\*\}new inputs as the rows ofL∈ℝN∗×NL\\in\\mathbb\{R\}^\{N\_\{\*\}\\times N\}, the second equation of \([44](https://arxiv.org/html/2610.29009#S3.E44)\) becomes

B∗,j=L​Bj\.B\_\{\*,j\}=LB\_\{j\}\.\(56\)Figure[1](https://arxiv.org/html/2610.29009#S1.F1)illustrates \([55](https://arxiv.org/html/2610.29009#S4.E55)\)\. One sparse solve gives the weights of a new input, and these weights produce the coefficients on every stencil\. The number of operator solves now equals the number of new inputs and no longer depends on the size of the output representation\. To compute \([52](https://arxiv.org/html/2610.29009#S4.E52)\), we first solveKU​Uθ​ℓ0​\(𝐚\)=kU​\(𝐚\)K\_\{UU\}^\{\\theta\}\\ell\_\{0\}\(\\mathbf\{a\}\)=k\_\{U\}\(\\mathbf\{a\}\)and then correct for the affine augmentation,

ℓ⁡\(𝐚\)=ℓ0​\(𝐚\)\+ℰU𝖳​\[p⁡\(𝐚\)−PU𝖳​ℓ0​\(𝐚\)\]\.\\ell\(\\mathbf\{a\}\)=\\ell\_\{0\}\(\\mathbf\{a\}\)\+\\mathcal\{E\}\_\{U\}^\{\\mathsf\{T\}\}\\left\[p\(\\mathbf\{a\}\)\-P\_\{U\}^\{\\mathsf\{T\}\}\\ell\_\{0\}\(\\mathbf\{a\}\)\\right\]\.\(57\)Without augmentation,L=KU∗​U​\(KU​Uθ\)−1L=K\_\{U\_\{\*\}U\}\(K\_\{UU\}^\{\\theta\}\)^\{\-1\}, so that

KU​Uθ​L𝖳=KU∗​U𝖳\.K\_\{UU\}^\{\\theta\}L^\{\\mathsf\{T\}\}=K\_\{U\_\{\*\}U\}^\{\\mathsf\{T\}\}\.\(58\)

### 4\.2Applying the weights before the output reconstruction

With𝐛i\\mathbf\{b\}\_\{i\}as in Section[3\.2](https://arxiv.org/html/2610.29009#S3.SS2), letℛ\\mathcal\{R\}be the linear map from a sampled output𝐯i\\mathbf\{v\}\_\{i\}to its local output reconstructionχ⁡\(𝐛i\)\\chi\(\\mathbf\{b\}\_\{i\}\)\. Since the same stencils, polynomial bases, and temporal weights serve every training field, linearity gives

𝒢~​\(a\)\\displaystyle\\tilde\{\\mathcal\{G\}\}\(a\)=χ⁡\(∑i=1Nℓi​\(𝐚\)​𝐛i\)=∑i=1Nℓi​\(𝐚\)​χ​\(𝐛i\)\\displaystyle=\\chi\\\!\\left\(\\sum\_\{i=1\}^\{N\}\\ell\_\{i\}\(\\mathbf\{a\}\)\\mathbf\{b\}\_\{i\}\\right\)=\\sum\_\{i=1\}^\{N\}\\ell\_\{i\}\(\\mathbf\{a\}\)\\chi\(\\mathbf\{b\}\_\{i\}\)\(59\)=ℛ⁡\(∑i=1Nℓi​\(𝐚\)​𝐯i\)\.\\displaystyle=\\mathcal\{R\}\\\!\\left\(\\sum\_\{i=1\}^\{N\}\\ell\_\{i\}\(\\mathbf\{a\}\)\\mathbf\{v\}\_\{i\}\\right\)\.Our GPU implementation uses the last form\. It combines the sampled training fields with the weights and then appliesℛ\\mathcal\{R\}once per new input\. At setup, adjoint solves with the local matrices𝐀j\\mathbf\{A\}\_\{j\}turnℛ\\mathcal\{R\}into compact evaluation weights on each stencil\. Prediction then stores the sampled training fields, the evaluation weights, and the stencil indices, but none of theNNsets of local coefficients\. For the Taylor–Green benchmarks, whose responses we divide by the input amplitude before fitting \(Section[5\.2\.1](https://arxiv.org/html/2610.29009#S5.SS2.SSS1)\), the same identity holds for the normalized responses\.

### 4\.3Rounding errors of direct and Lagrange evaluation

The forms in \([59](https://arxiv.org/html/2610.29009#S4.E59)\) agree in exact arithmetic, but they round differently\. The difference is important when a moderate velocity is the result of cancellation among large spatial coefficients, which can happen for a global interpolant\. A solve error then enters the velocity amplified by the output reconstruction\. We now quantify this for the model without augmentation and for evaluation through the coefficients, which is the setting of our comparison in Section[5\.3](https://arxiv.org/html/2610.29009#S5.SS3)\. The analysis does not cover the affine correction \([57](https://arxiv.org/html/2610.29009#S4.E57)\) or the order of operations of Section[4\.2](https://arxiv.org/html/2610.29009#S4.SS2)\.

AssumeKU​UθK\_\{UU\}^\{\\theta\}is symmetric and nonsingular\. We fix one stencil and drop its index, writingr=rjr=r\_\{j\},B=Bj∈ℝN×rB=B\_\{j\}\\in\\mathbb\{R\}^\{N\\times r\}, andC=Cj∈ℝN×rC=C\_\{j\}\\in\\mathbb\{R\}^\{N\\times r\}, so that each row ofBBholds the spatial coefficients of one training field\. Let𝐀⋆∈ℝneval×r\\mathbf\{A\}^\{\\star\}\\in\\mathbb\{R\}^\{n\_\{\\rm eval\}\\times r\}stack the evaluation matrices \([18](https://arxiv.org/html/2610.29009#S2.E18)\) of thenevaln\_\{\\rm eval\}velocity values reconstructed from this stencil, so thatB​\(𝐀⋆\)𝖳B\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}holds the reconstructed training velocities\. The direct coefficientsCCand the Lagrange weightsLLsatisfy

KU​Uθ​C=B,L​KU​Uθ=KU∗​U,K\_\{UU\}^\{\\theta\}C=B,\\qquad LK\_\{UU\}^\{\\theta\}=K\_\{U\_\{\*\}U\},\(60\)and both give the predictionV∗=KU∗​U​C​\(𝐀⋆\)𝖳=L​B​\(𝐀⋆\)𝖳V\_\{\*\}=K\_\{U\_\{\*\}U\}C\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}=LB\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\. The same notation describes a global output reconstruction, with a single stencil\.

###### Theorem 3\(Solve errors and multiplication rounding\)\.

For computed approximationsC^\\widehat\{C\}andL^\\widehat\{L\}, define the solve residuals by evaluating their defining equations in exact arithmetic,

Rdir=B−KU​Uθ​C^,RLag=KU∗​U−L^​KU​Uθ\.R\_\{\\rm dir\}=B\-K\_\{UU\}^\{\\theta\}\\widehat\{C\},\\qquad R\_\{\\rm Lag\}=K\_\{U\_\{\*\}U\}\-\\widehat\{L\}K\_\{UU\}^\{\\theta\}\.\(61\)Before subsequent multiplication rounding, the prediction errors satisfy

KU∗​U​C^​\(𝐀⋆\)𝖳−V∗\\displaystyle K\_\{U\_\{\*\}U\}\\widehat\{C\}\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\-V\_\{\*\}=−L​Rdir​\(𝐀⋆\)𝖳,\\displaystyle=\-LR\_\{\\rm dir\}\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\},\(62\)L^​B​\(𝐀⋆\)𝖳−V∗\\displaystyle\\widehat\{L\}B\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\-V\_\{\*\}=−RLag​\(KU​Uθ\)−1​B​\(𝐀⋆\)𝖳\.\\displaystyle=\-R\_\{\\rm Lag\}\(K\_\{UU\}^\{\\theta\}\)^\{\-1\}B\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\.\(63\)Define, withfl⁡\(⋅\)\\operatorname\{fl\}\(\\cdot\)denoting evaluation in floating point,V^∗,dir=fl⁡\(fl⁡\(KU∗​U​C^\)​\(𝐀⋆\)𝖳\)\\widehat\{V\}\_\{\*,\\rm dir\}=\\operatorname\{fl\}\(\\operatorname\{fl\}\(K\_\{U\_\{\*\}U\}\\widehat\{C\}\)\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\)andV^∗,Lag=fl⁡\(fl⁡\(L^​B\)​\(𝐀⋆\)𝖳\)\\widehat\{V\}\_\{\*,\\rm Lag\}=\\operatorname\{fl\}\(\\operatorname\{fl\}\(\\widehat\{L\}B\)\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\)\.

Suppose the coefficient products and the output reconstruction use standard floating\-point dot products, possibly split intoddcomponent blocks followed by summation\.

Assume no overflow or underflow, and setγk=k​umach/\(1−k​umach\)\\gamma\_\{k\}=ku\_\{\\rm mach\}/\(1\-ku\_\{\\rm mach\}\), whereumachu\_\{\\rm mach\}is unit roundoff andmax⁡\{N,r\+d\}​umach<1\\max\\\{N,r\+d\\\}u\_\{\\rm mach\}<1\. Withγmult=γN\+γr\+d\+γN​γr\+d\\gamma\_\{\\rm mult\}=\\gamma\_\{N\}\+\\gamma\_\{r\+d\}\+\\gamma\_\{N\}\\gamma\_\{r\+d\}, we have

‖V^∗,dir−V∗‖F\\displaystyle\\\|\\widehat\{V\}\_\{\*,\\rm dir\}\-V\_\{\*\}\\\|\_\{F\}≤‖L‖F​‖Rdir‖F​‖\(𝐀⋆\)𝖳‖F\+γmult​‖\|KU∗​U​‖C^‖​\(𝐀⋆\)𝖳\|‖F,\\displaystyle\\leq\\\|L\\\|\_\{F\}\\\|R\_\{\\rm dir\}\\\|\_\{F\}\\\|\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\\\|\_\{F\}\+\\gamma\_\{\\rm mult\}\\big\\\|\\,\|K\_\{U\_\{\*\}U\}\|\\,\|\\widehat\{C\}\|\\,\|\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\|\\,\\big\\\|\_\{F\},\(64\)‖V^∗,Lag−V∗‖F\\displaystyle\\\|\\widehat\{V\}\_\{\*,\\rm Lag\}\-V\_\{\*\}\\\|\_\{F\}≤‖RLag‖F​‖\(KU​Uθ\)−1​B​\(𝐀⋆\)𝖳‖F\+γmult​‖\|L^​‖B‖​\(𝐀⋆\)𝖳\|‖F\.\\displaystyle\\leq\\\|R\_\{\\rm Lag\}\\\|\_\{F\}\\\|\(K\_\{UU\}^\{\\theta\}\)^\{\-1\}B\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\\\|\_\{F\}\+\\gamma\_\{\\rm mult\}\\big\\\|\\,\|\\widehat\{L\}\|\\,\|B\|\\,\|\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\|\\,\\big\\\|\_\{F\}\.\(65\)Here absolute values are taken entrywise\. The residual identities require symmetry and nonsingularity, but not positive definiteness\.

###### Proof sketch\.

Substituting the residuals into the two predictions gives \([62](https://arxiv.org/html/2610.29009#S4.E62)\)–\([63](https://arxiv.org/html/2610.29009#S4.E63)\)\. The standard bound for floating\-point dot products then bounds the multiplications\. Appendix[C\.5](https://arxiv.org/html/2610.29009#A3.SS5)gives the proof\. ∎

The bounds become clearer with the amplification factor

αspatial=‖B‖F​‖\(𝐀⋆\)𝖳‖F‖B​\(𝐀⋆\)𝖳‖F≥1\\alpha\_\{\\rm spatial\}=\\frac\{\\\|B\\\|\_\{F\}\\\|\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\\\|\_\{F\}\}\{\\\|B\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\\\|\_\{F\}\}\\geq 1\(66\)which measures how much the coefficients cancel in the output reconstruction of the training velocities\. If‖Rdir‖F≤ηdir​‖B‖F\\\|R\_\{\\rm dir\}\\\|\_\{F\}\\leq\\eta\_\{\\rm dir\}\\\|B\\\|\_\{F\}and‖RLag‖F≤ηLag​‖KU∗​U‖F\\\|R\_\{\\rm Lag\}\\\|\_\{F\}\\leq\\eta\_\{\\rm Lag\}\\\|K\_\{U\_\{\*\}U\}\\\|\_\{F\}, the solve errors satisfy

‖KU∗​U​C^​\(𝐀⋆\)𝖳−V∗‖F\\displaystyle\\\|K\_\{U\_\{\*\}U\}\\widehat\{C\}\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\-V\_\{\*\}\\\|\_\{F\}≤ηdir​‖L‖F​αspatial​‖B​\(𝐀⋆\)𝖳‖F,\\displaystyle\\leq\\eta\_\{\\rm dir\}\\\|L\\\|\_\{F\}\\alpha\_\{\\rm spatial\}\\\|B\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\\\|\_\{F\},\(67a\)‖L^​B​\(𝐀⋆\)𝖳−V∗‖F\\displaystyle\\\|\\widehat\{L\}B\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\-V\_\{\*\}\\\|\_\{F\}≤ηLag​‖KU∗​U‖F​‖\(KU​Uθ\)−1‖F​‖B​\(𝐀⋆\)𝖳‖F\.\\displaystyle\\leq\\eta\_\{\\rm Lag\}\\\|K\_\{U\_\{\*\}U\}\\\|\_\{F\}\\\|\(K\_\{UU\}^\{\\theta\}\)^\{\-1\}\\\|\_\{F\}\\\|B\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\\\|\_\{F\}\.\(67b\)In direct evaluation, the relative solve errorηdir\\eta\_\{\\rm dir\}is amplified byαspatial\\alpha\_\{\\rm spatial\}\. In Lagrange evaluation, the coefficients have already been combined into velocities, andαspatial\\alpha\_\{\\rm spatial\}does not appear\. A global interpolant with a smooth kernel can have very largeαspatial\\alpha\_\{\\rm spatial\}, since its coefficients are large and cancel, while a local interpolant on a small stencil has moderate coefficients\. The rounding of the multiplications, however, contains the entrywise product with\|\(𝐀⋆\)𝖳\|\|\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\|in both bounds \(of\|C^\|\|\\widehat\{C\}\|in direct and of\|B\|\|B\|in Lagrange evaluation\), so cancellation in the output reconstruction also enters the rounding, and the theorem does not make Lagrange evaluation more accurate in general\. We test these predictions in Section[5\.3](https://arxiv.org/html/2610.29009#S5.SS3)\.

### 4\.4GPU implementation

Algorithm[1](https://arxiv.org/html/2610.29009#alg1)summarizes our implementation\. Every floating\-point computation runs on the GPU in double precision, from the normalization of the inputs and the stencil selection to the factorizations and the output reconstruction, using the routines of KernelPack\-JAX\[[8](https://arxiv.org/html/2610.29009#bib.bib47)\]\. Only the integer bookkeeping of the cell\-based stencil search for large non\-periodic node sets \(Section[5\.1\.5](https://arxiv.org/html/2610.29009#S5.SS1.SSS5)\) runs on the host\. The kernel is assembled pair by pair over its support, in passes of bounded size, so that no dense kernel or distance matrix is ever formed\.

However, JAX has no sparse direct solver that keeps a factorization between calls\. Our implementation therefore factorsKU​UθK\_\{UU\}^\{\\theta\}with NVIDIA’s cuDSS through the primitivekernelpack\.sparse\.cudssof KernelPack\-JAX, which exposes cuDSS to XLA, the compiler behind JAX\. We use cuDSS in its symmetric positive\-definite mode, with its nested\-dissection ordering \(Section[3\.5](https://arxiv.org/html/2610.29009#S3.SS5)\)\. The factors stay on the GPU, and the right\-hand sides of a batch of new inputs are solved as the columns of one call\. A change of the support \(i\.e\., ofϱ\\varrho\) requires a new ordering and symbolic analysis, while a change of the values only requires a numerical refactorization\. Appendix[B\.4](https://arxiv.org/html/2610.29009#A2.SS4)describes the batching and the checks of the timed runs\.

Algorithm 1Local and sparse kernel operator learning on the GPU\.1:training and validation pairs, new inputs, output points, spatial kernel and polynomial degree

2:

3:select the stencils, with periodic images where needed \(Section[2\.3](https://arxiv.org/html/2610.29009#S2.SS3)\)

4:build the divergence\-free polynomials and the local systems, and prepare

ℛ\\mathcal\{R\}\(Section[2\.1](https://arxiv.org/html/2610.29009#S2.SS1)\)

5:for spacetime outputs, add the temporal weights \([28](https://arxiv.org/html/2610.29009#S2.E28)\)

6:

7:standardize the training inputs and compute their principal directions

8:for each choice of augmentation, keep the fixed shape or fit it by \([49](https://arxiv.org/html/2610.29009#S3.E49)\)

9:select the radius and regularization on the validation fields

10:assemble

KU​UθK\_\{UU\}^\{\\theta\}in sparse form and factor it with cuDSS

11:with augmentation, form the whitened QR correction \([46](https://arxiv.org/html/2610.29009#S3.E46)\)

12:

13:solve for the Lagrange weights \([57](https://arxiv.org/html/2610.29009#S4.E57)\) of a batch of new inputs

14:combine the sampled training fields and apply

ℛ\\mathcal\{R\}by \([59](https://arxiv.org/html/2610.29009#S4.E59)\)

15:check the solve residuals and the reproduction of affine functions

### 4\.5Work and storage

Table[1](https://arxiv.org/html/2610.29009#S4.T1)lists the work and storage of each step\. Let𝒩K=nnz⁡\(KU​U\)\\mathcal\{N\}\_\{K\}=\\operatorname\{nnz\}\(K\_\{UU\}\)and𝒩fac=nnz⁡\(LK\)\\mathcal\{N\}\_\{\\rm fac\}=\\operatorname\{nnz\}\(L\_\{K\}\), whereLKL\_\{K\}is the Cholesky factor ofKU​UθK\_\{UU\}^\{\\theta\}in \([46](https://arxiv.org/html/2610.29009#S3.E46)\)\. Withℒi\\mathcal\{L\}\_\{i\}the set of rows in which columniiofLKL\_\{K\}is nonzero, so that∑i=1N\|ℒi\|=𝒩fac\\sum\_\{i=1\}^\{N\}\|\\mathcal\{L\}\_\{i\}\|=\\mathcal\{N\}\_\{\\rm fac\}, the sparse factorization takes, up to a constant factor,

Fchol=∑i=1N\|ℒi\|2,𝒩fac2N≤Fchol≤N​𝒩fac,F\_\{\\rm chol\}=\\sum\_\{i=1\}^\{N\}\|\\mathcal\{L\}\_\{i\}\|^\{2\},\\qquad\\frac\{\\mathcal\{N\}\_\{\\rm fac\}^\{2\}\}\{N\}\\leq F\_\{\\rm chol\}\\leq N\\mathcal\{N\}\_\{\\rm fac\},\(68\)floating\-point operations\[[9](https://arxiv.org/html/2610.29009#bib.bib2)\], where the lower bound follows from the Cauchy–Schwarz inequality and the upper bound from\|ℒi\|≤N\|\\mathcal\{L\}\_\{i\}\|\\leq N\. For a dense factor,\|ℒi\|=N−i\+1\|\\mathcal\{L\}\_\{i\}\|=N\-i\+1andFchol≈N3/3F\_\{\\rm chol\}\\approx N^\{3\}/3\. Letbqb\_\{q\}be the number of new inputs in a batch, letmrhsm\_\{\\rm rhs\}be the number of spatial coefficients, and letnvn\_\{v\}be the number of sampled output values per field \(nv=d​mn\_\{v\}=dm, ord​m​NtdmN\_\{t\}for spacetime outputs\)\. We count the work of a triangular solve as proportional to the number of nonzeros of the factor\.

Table 1:Work and storage\. Here𝒩K\\mathcal\{N\}\_\{K\}and𝒩fac\\mathcal\{N\}\_\{\\rm fac\}are the numbers of nonzeros of the kernel matrix and of its Cholesky factor,FcholF\_\{\\rm chol\}the work \([68](https://arxiv.org/html/2610.29009#S4.E68)\) of the factorization,bqb\_\{q\}the number of new inputs in a batch,naffn\_\{\\rm aff\}the number of affine columns,nvn\_\{v\}the number of sampled output values per field,m⋆m^\{\\star\}the number of query points,rjr\_\{j\}the size of the local system on stenciljj, andmrhs=∑jrjm\_\{\\rm rhs\}=\\sum\_\{j\}r\_\{j\}\. The last two rows belong to direct evaluation through coefficients, which our predictions avoid\. For spacetime outputs, the counts of these two rows are multiplied byNtN\_\{t\}\.StepWorkStorageKernel factorizationFcholF\_\{\\rm chol\}𝒩K\\mathcal\{N\}\_\{K\},𝒩fac\\mathcal\{N\}\_\{\\rm fac\}Affine correction \(naffn\_\{\\rm aff\}columns\)solves withnaffn\_\{\\rm aff\}right\-hand sides,O⁡\(N​naff2\)O\(Nn\_\{\\rm aff\}^\{2\}\)QRO⁡\(N​naff\)O\(Nn\_\{\\rm aff\}\)Lagrange weights,bqb\_\{q\}queriesO⁡\(bq​𝒩fac\)\+O⁡\(bq​N​naff\)O\(b\_\{q\}\\mathcal\{N\}\_\{\\rm fac\}\)\+O\(b\_\{q\}Nn\_\{\\rm aff\}\)bq​Nb\_\{q\}N\(dense\)Combining training fieldsO⁡\(bq​N​nv\)O\(b\_\{q\}Nn\_\{v\}\)N​nvNn\_\{v\}Local systems \(setup\)O⁡\(rj3\)O\(r\_\{j\}^\{3\}\)per stencilLocal evaluation,m⋆m^\{\\star\}pointsO⁡\(bq​m⋆​d2​ns\)O\(b\_\{q\}m^\{\\star\}d^\{2\}n\_\{s\}\)per snapshotO⁡\(m⋆​d2​ns\)O\(m^\{\\star\}d^\{2\}n\_\{s\}\)weights,O⁡\(m⋆​ns\)O\(m^\{\\star\}n\_\{s\}\)indicesTraining coefficientsO⁡\(N​∑jrj2\)O\(N\\sum\_\{j\}r\_\{j\}^\{2\}\)N​∑jrjN\\sum\_\{j\}r\_\{j\}Applying weights to coefficientsO⁡\(bq​N​mrhs\)O\(b\_\{q\}Nm\_\{\\rm rhs\}\)Two observations follow from the table\. First, the local output reconstruction costsO⁡\(rj3\)O\(r\_\{j\}^\{3\}\)per stencil, withrj=d​ns\+nP,jr\_\{j\}=dn\_\{s\}\+n\_\{P,j\}fixed by the stencil size and the polynomial degree, so its cost grows linearly with the number of output points instead of cubically\. Second, the cost of the operator regression is governed by the factor ofKU​UθK\_\{UU\}^\{\\theta\}, whose size depends on the fill as well as on the density ofKU​UK\_\{UU\}\[[9](https://arxiv.org/html/2610.29009#bib.bib2)\]\. At a fixed density, the kernel matrix and its factor still grow asN2N^\{2\}\. For this reason, we report density, factor storage, and prediction time separately in Section[5](https://arxiv.org/html/2610.29009#S5)\.

Note that the Lagrange weights are dense even whenKU​UK\_\{UU\}is sparse\. Direct evaluation instead solvesmrhsm\_\{\\rm rhs\}right\-hand sides once and can reuse its coefficients for any number of new inputs, so the cheaper route depends on how many predictions are needed\. Appendix[B\.3](https://arxiv.org/html/2610.29009#A2.SS3)gives the storage accounting of our experiments\.

## 5Numerical experiments

We now test our method in three steps\. We first test the local output reconstruction alone, with exact fields in place of predictions, for convergence and for the three properties it preserves \(Section[5\.1](https://arxiv.org/html/2610.29009#S5.SS1)\)\. We then compare four operator models on ten fluid benchmarks as the training set grows, in accuracy, sparsity, and storage \(Section[5\.2](https://arxiv.org/html/2610.29009#S5.SS2)\)\. Finally, we compare direct and Lagrange evaluation in floating point and measure the cost of prediction \(Section[5\.3](https://arxiv.org/html/2610.29009#S5.SS3)\)\.

### 5\.1Local output reconstruction

Exact velocity fields in place of predictions separate the error of the output reconstruction from that of the operator regression\. We use these tests to answer three questions\. Does it converge at the order set by its polynomial degree on scattered nodes? Are its divergence and its periodic mismatch at rounding level? And does the multiscale Matérn kernel reproduce prescribed turbulence statistics as the nodes are refined?

#### 5\.1\.1Convergence on scattered nodes

We interpolate manufactured divergence\-free fields in the smooth domainsΩd=\{Fd\>0\}\\Omega\_\{d\}=\\\{F\_\{d\}\>0\\\},d=2,3d=2,3, bounded by a quartic curve and a quartic surface\. Each field is the curl of a trigonometric stream function \(d=2d=2\) or vector potential \(d=3d=3\) multiplied byFd2F\_\{d\}^\{2\}, so it vanishes on the boundary \(Appendix[D\.2](https://arxiv.org/html/2610.29009#A4.SS2)givesFdF\_\{d\}and the potentials\)\. We use divergence\-free polynomials of degreeϖ=1\\varpi=1,33, and55for target orders22,44, and66\. Each stencil hasns=2​⌈nP/d⌉\+1n\_\{s\}=2\\lceil n\_\{P\}/d\\rceil\+1nodes, wherenPn\_\{P\}is the dimension of the polynomial space, and both the PHS \(r3r^\{3\}for degrees11and33, andr5r^\{5\}for degree55\) and the multiscale Matérn kernel \(cσ=16c\_\{\\sigma\}=16, which sets the Matérn lengths relative to the stencil radius in \([21](https://arxiv.org/html/2610.29009#S2.E21)\)\) use the same stencils and polynomials\. The node sets are four refinements generated by the node generator of KernelPack\-JAX\[[8](https://arxiv.org/html/2610.29009#bib.bib47)\], with172172to19071907nodes in 2D and197197to25702570in 3D, and the stencils are boundary aware \(Section[2\.1](https://arxiv.org/html/2610.29009#S2.SS1)\)\.

Letvhv\_\{h\}be the output reconstruction of the exact fieldvvfrom the nodesYYof a node set with spacinghh\. We measure its relative error on a fixed setQQof uniformly random interior evaluation points \(40964096in 2D and3276832768in 3D\), against the fill distance sampled on the same set,

Esp\\displaystyle E\_\{\\rm sp\}=\(∑y∈Q‖vh​\(y\)−v⁡\(y\)‖22∑y∈Q‖v⁡\(y\)‖22\)1/2,\\displaystyle=\\left\(\\frac\{\\sum\_\{y\\in Q\}\\\|v\_\{h\}\(y\)\-v\(y\)\\\|\_\{2\}^\{2\}\}\{\\sum\_\{y\\in Q\}\\\|v\(y\)\\\|\_\{2\}^\{2\}\}\\right\)^\{1/2\},hQ\\displaystyle h\_\{Q\}=maxy∈Q⁡miny′∈Y​‖y−y′‖2,\\displaystyle=\\max\_\{y\\in Q\}\\min\_\{y^\{\\prime\}\\in Y\}\\\|y\-y^\{\\prime\}\\\|\_\{2\},\(69\)where each evaluation point uses the stencil of its nearest node\.

Figure[3](https://arxiv.org/html/2610.29009#S5.F3)shows the results\. In 2D, both kernels converge at close to the target orders, with least\-squares rates of2\.12\.1,3\.93\.9, and5\.85\.8\. In 3D, the rates over all four node sets are lower,1\.71\.7–1\.91\.9,3\.43\.4–3\.73\.7, and5\.05\.0, but the sixth\-order rate rises to5\.55\.5over the three finest node sets\. The 3D deficit is likely due to the coarsest node sets, which are not yet in the asymptotic range\. The two kernels give nearly the same errors on smooth fields, so the multiscale kernel does not reduce the accuracy on smooth fields\.

Figure 3:Convergence of the local output reconstruction on scattered nodes in the domainsΩ2\\Omega\_\{2\}\(left\) andΩ3\\Omega\_\{3\}\(right\), with PHS and multiscale Matérn kernels and divergence\-free polynomials\. Relative errorEspE\_\{\\rm sp\}at random evaluation points that are not nodes, against the sampled fill distancehQh\_\{Q\}of \([69](https://arxiv.org/html/2610.29009#S5.E69)\)\. Dotted lines:hQ2h\_\{Q\}^\{2\},hQ4h\_\{Q\}^\{4\},hQ6h\_\{Q\}^\{6\}\.
#### 5\.1\.2Divergence and periodicity

Since every basis function is divergence free, the divergence of the output reconstruction should vanish up to rounding, however large its approximation error\. Differentiating each local expansion on its own stencil, we find a normalized root\-mean\-square \(RMS\) divergence at rounding level for both kernels, all degrees, and all node sets \(Appendix[D\.2](https://arxiv.org/html/2610.29009#A4.SS2)\)\. We test periodic stencils on scattered nodes in\[0,2π\)d\[0,2\\pi\)^\{d\}, comparing the reconstructed velocities and their Jacobians at periodically equivalent points on faces, edges, and corners\. The differences are again at rounding level for both kernels and all degrees, while the approximation error varies with degree and resolution as expected\.

#### 5\.1\.3Kolmogorov statistics

The multiscale kernel builds a Kolmogorov law into its weights\. We now test whether interpolation on finite stencils preserves this law\. We reconstruct a three\-dimensional solenoidal Fourier ensemble whose velocity increments follow the Kolmogorov lawS2​\(r\)∝r2/3S\_\{2\}\(r\)\\propto r^\{2/3\}over the separations0\.03≤r≤0\.80\.03\\leq r\\leq 0\.8\. The ensemble is a sum of15141514Fourier modes with Gaussian coefficients perpendicular to their wave vectors and a−5/3\-5/3energy spectrum, restricted toΩ3\\Omega\_\{3\}\(Appendix[D\.2\.3](https://arxiv.org/html/2610.29009#A4.SS2.SSS3)\)\. Belowr=0\.03r=0\.03, its finite bandwidth makes the increments smooth\. For pairs of pointsxk​\(r\)x\_\{k\}\(r\)andyk​\(r\)y\_\{k\}\(r\)at distancerralong eight fixed directionsk=1,…,8k=1,\\ldots,8, we compute the second\-order structure function exactly from the covariance of the ensemble,

S2,k​\(r\)\\displaystyle S\_\{2,k\}\(r\)=𝔼⁡\[‖v⁡\(xk​\(r\)\)−v⁡\(yk​\(r\)\)‖22\],‖xk​\(r\)−yk​\(r\)‖2=r,\\displaystyle=\\mathbb\{E\}\\bigl\[\\\|v\(x\_\{k\}\(r\)\)\-v\(y\_\{k\}\(r\)\)\\\|\_\{2\}^\{2\}\\bigr\],\\qquad\\\|x\_\{k\}\(r\)\-y\_\{k\}\(r\)\\\|\_\{2\}=r,S2​\(r\)\\displaystyle S\_\{2\}\(r\)=18​∑k=18S2,k​\(r\),\\displaystyle=\\frac\{1\}\{8\}\\sum\_\{k=1\}^\{8\}S\_\{2,k\}\(r\),\(70\)both for the ensemble and, withvhv\_\{h\}in place ofvv, for its output reconstruction \(S2,k,hS\_\{2,k,h\}\)\. We measure the worst relative error over the directions,

ES2​\(r\)=max1≤k≤8⁡\|S2,k,h​\(r\)−S2,k​\(r\)\|S2,k​\(r\)\.E\_\{S\_\{2\}\}\(r\)=\\max\_\{1\\leq k\\leq 8\}\\frac\{\|S\_\{2,k,h\}\(r\)\-S\_\{2,k\}\(r\)\|\}\{S\_\{2,k\}\(r\)\}\.\(71\)We use the multiscale Matérn kernel with polynomial degrees11,33, and55on six node sets, with node spacingshhfrom0\.260\.26to0\.050\.05\(128128to1356713567nodes\)\.

Figure[4](https://arxiv.org/html/2610.29009#S5.F4)shows the structure functions approaching the reference under refinement\. At the largest separation,r=0\.8r=0\.8, the error decays likeh0\.64h^\{0\.64\},h0\.73h^\{0\.73\}, andh0\.67h^\{0\.67\}for the three degrees\. These are fits over a finite range ofhh, not asymptotic rates\. For degree five, the fitted slope ofS2S\_\{2\}over the upper part of the range moves toward the reference slope\. However, the error at small separations stays large even on the finest node set, since these separations are below the node spacing\. To see whether locality is to blame, we compare local and global interpolation on one node set, with the same kernel and polynomials \(Appendix[D\.2\.3](https://arxiv.org/html/2610.29009#A4.SS2.SSS3)\)\. Both have nearly the same error, atr=0\.8r=0\.8and over the whole range, so the small\-separation error is likely due to the resolution of the nodes and not to the stencils\.

Figure 4:Kolmogorov statistics of the local multiscale Matérn output reconstruction\. Left: structure functions for degree five on three of the six node sets, and the exact reference\. Right: worst relative errorES2​\(0\.8\)E\_\{S\_\{2\}\}\(0\.8\)of \([71](https://arxiv.org/html/2610.29009#S5.E71)\) of the reconstructed structure function against the node spacinghh, with anh2/3h^\{2/3\}guide\.
#### 5\.1\.4Output reconstruction in the fluid benchmarks

We now test the output reconstruction on the predictions of the operator benchmarks \(Section[5\.2\.1](https://arxiv.org/html/2610.29009#S5.SS2.SSS1)\)\. The data sets hold the velocity at more mesh points than themmoutput points that we use as nodes, and the remaining points lie between the nodes\. We reconstruct the predicted nodal velocities of the model with the smallest validation error atN=10000N=10000with the global divergence\-free Matérn interpolant of our earlier work and with our local output reconstruction, and measure the error at these points \(Table[2](https://arxiv.org/html/2610.29009#S5.T2), Appendix[A\.5](https://arxiv.org/html/2610.29009#A1.SS5)\)\. Neither output reconstruction is uniformly more accurate\. The global one is more accurate for both cylinder flows, the backward\-facing step, and Taylor–Green vortices \(space\), by factors between1\.061\.06and2\.72\.7\. The local one is more accurate for the other six, by factors between1\.041\.04\(turbulent species transport\) and1010\(buoyancy\-driven cavity\), although each local system has at most6262unknowns in 2D and262262in 3D, against up to2100021000in the global system\. For several flows, the errors between the nodes are much larger than those at the nodes, so they are set by the resolution of themmnodes and not by the operator regression\. For both turbulent flows, the multiscale Matérn kernel has a lower error than the PHS, and validation selectscσ=16c\_\{\\sigma\}=16for both\. Almost all stencils keep the full polynomial space, except in the species\-transport mixer, whose geometry truncates the rank on more than half the stencils \(Table[7](https://arxiv.org/html/2610.29009#A1.T7)in Appendix[A\.2](https://arxiv.org/html/2610.29009#A1.SS2)\)\.

Table 2:Test error between the nodes\. We reconstruct the predicted nodal velocities of the 100 test inputs \(model with the smallest validation error atN=10000N=10000\) and measure the mean relative velocity error at the evaluation points, the mesh points of each data set that are not among itsmmnodes\. Global: the divergence\-free Matérn interpolant of[Sharma et al\. \[53\]](https://arxiv.org/html/2610.29009#bib.bib10)on all nodes\. Local: our output reconstruction with the cubic PHS, or with the multiscale Matérn kernel \(cσ=16c\_\{\\sigma\}=16\) where marked \(Appendix[A\.5](https://arxiv.org/html/2610.29009#A1.SS5)compares it with the PHS\)\. Bold: smaller error in each row\.Data setNodesmmEvaluation pointsGlobalLocalForced isotropic turbulence700030001\.88×10−21\.88\\times 10^\{\-2\}1\.65×𝟏𝟎−𝟐\\mathbf\{1\.65\\times 10^\{\-2\}\}\(multiscale\)Turbulent species transport700043881\.52×10−11\.52\\times 10^\{\-1\}1\.47×𝟏𝟎−𝟏\\mathbf\{1\.47\\times 10^\{\-1\}\}\(multiscale\)Merging vortices50069471\.88×10−11\.88\\times 10^\{\-1\}1\.35×𝟏𝟎−𝟏\\mathbf\{1\.35\\times 10^\{\-1\}\}Lid\-driven cavity100085666\.15×10−16\.15\\times 10^\{\-1\}8\.71×𝟏𝟎−𝟐\\mathbf\{8\.71\\times 10^\{\-2\}\}Cylinder flow \(no shedding\)100085208\.67×𝟏𝟎−𝟐\\mathbf\{8\.67\\times 10^\{\-2\}\}9\.19×10−29\.19\\times 10^\{\-2\}Cylinder flow \(vortex shedding\)100085209\.65×𝟏𝟎−𝟐\\mathbf\{9\.65\\times 10^\{\-2\}\}1\.12×10−11\.12\\times 10^\{\-1\}Backward\-facing step100063595\.84×𝟏𝟎−𝟑\\mathbf\{5\.84\\times 10^\{\-3\}\}1\.04×10−21\.04\\times 10^\{\-2\}Buoyancy\-driven cavity500055149\.42×10−29\.42\\times 10^\{\-2\}9\.31×𝟏𝟎−𝟑\\mathbf\{9\.31\\times 10^\{\-3\}\}Taylor–Green vortices \(space\)50069472\.30×𝟏𝟎−𝟒\\mathbf\{2\.30\\times 10^\{\-4\}\}6\.20×10−46\.20\\times 10^\{\-4\}Taylor–Green vortices \(spacetime\)50069471\.99×10−31\.99\\times 10^\{\-3\}6\.20×𝟏𝟎−𝟒\\mathbf\{6\.20\\times 10^\{\-4\}\}
#### 5\.1\.5Cost of the local output reconstruction

We now compare the setup cost of our local output reconstruction with that of a global divergence\-free output reconstruction, as in PPKM, on scattered nodes in the unit square and the unit cube\. The global setup assembles the densed​m×d​mdm\\times dmGram matrix of a divergence\-free Matérn kernel and factors it by Cholesky\. The local setup selects the stencils, builds the polynomial bases and the local systems, factors them, and forms the evaluation weights of Section[4\.2](https://arxiv.org/html/2610.29009#S4.SS2), with the degree\-four output reconstruction of the operator benchmarks\. Both run on the GPU in double precision, and we time them after compilation\. Figure[5](https://arxiv.org/html/2610.29009#S5.F5)shows the results\.

Figure 5:Setup time and memory of our local output reconstruction \(degree four, cubic PHS\) and of a global divergence\-free Matérn output reconstruction \(dense Gram matrix and Cholesky factor\) against the numbermmof output points, on scattered nodes in the unit square \(2D\) and cube \(3D\)\. Dotted: the global cost extrapolated as\(d​m\)3\(dm\)^\{3\}in time and\(d​m\)2\(dm\)^\{2\}in memory beyond the largest size that a single dense factorization fits on our GPU\. All times are measured after compilation\.The local setup grows at most linearly inmm, in both time and memory, as Table[1](https://arxiv.org/html/2610.29009#S4.T1)predicts\. The global setup grows like\(d​m\)3\(dm\)^\{3\}in time and\(d​m\)2\(dm\)^\{2\}in memory, and a single dense factorization no longer fits on our GPU beyondd​m≈2\.5×104dm\\approx 2\.5\\times 10^\{4\}\. It could be factored in blocks, at the same cubic cost\. On small node sets, the dense global factorization is faster, since its work is done by a few large and efficient GPU kernels\. The local setup becomes the faster one nearm=3×103m=3\\times 10^\{3\}in 2D andm=5×103m=5\\times 10^\{3\}in 3D, and its memory is smaller at every size we tested, by a factor that grows linearly withmm\.

Two choices keep the local setup linear inmm, and neither changes the output reconstruction beyond rounding\. The polynomial bases are built by QR factorizations instead of singular value decompositions \(Appendix[A\.2](https://arxiv.org/html/2610.29009#A1.SS2)\), and non\-periodic node sets of more than40964096nodes find their stencils by an exact search on a grid of cells, whose work grows linearly withmm\. Together, they make the setup of the output reconstruction a small part of the offline cost of every operator benchmark \(Appendix[D\.6\.1](https://arxiv.org/html/2610.29009#A4.SS6.SSS1)\)\.

### 5\.2Sparse operator regression

#### 5\.2\.1Benchmarks and models

We use the ten data sets of our previous work\[[53](https://arxiv.org/html/2610.29009#bib.bib10)\], listed in Table[3](https://arxiv.org/html/2610.29009#S5.T3)\. Eight are two\-dimensional laminar flows, two of them the analytical Taylor–Green vortices, and two are three\-dimensional turbulent flows, a Reynolds\-averaged simulation of a static mixer and a large\-eddy simulation of forced isotropic turbulence\. In each, the input is an initial velocity, temperature, or vorticity, an inflow, or a body force, sampled at points of the domain or of its inflow boundary, and the output is the velocity at a fixed later time \(or, for Taylor–Green vortices \(spacetime\), at four times\), sampled atmmoutput points\. The output points are approximate Fekete points, i\.e\., well\-spaced points selected from the mesh\[[53](https://arxiv.org/html/2610.29009#bib.bib10)\]\. Appendix[D\.1](https://arxiv.org/html/2610.29009#A4.SS1)gives the geometries, parameter distributions, and simulations\.

Each data set has1000010000training pairs,100100validation pairs, and100100test pairs\. We fit every model on the firstN∈\{500,1000,5000,10000\}N\\in\\\{500,1000,5000,10000\\\}training pairs, so the training sets are nested, and the validation and test pairs are the same for everyNN\. The validation and test pairs were also used in our previous study and during the development of the method, so our comparisons are retrospective\. Freezing all choices before the final test runs prevents test\-based selection in this study but does not make the test set new\. Our error measure, the test error, is the relative velocity error‖𝐯~⋆−𝐯⋆‖2/‖𝐯⋆‖2\\\|\\tilde\{\\mathbf\{v\}\}^\{\\star\}\-\\mathbf\{v\}^\{\\star\}\\\|\_\{2\}/\\\|\\mathbf\{v\}^\{\\star\}\\\|\_\{2\}of the prediction𝐯~⋆\\tilde\{\\mathbf\{v\}\}^\{\\star\}of a test field𝐯⋆\\mathbf\{v\}^\{\\star\}at the output points, averaged over the100100test fields \(and, for Taylor–Green vortices \(spacetime\), over the four times\)\. It measures the prediction for inputs that are used neither for fitting nor for validation\.

Table 3:The ten benchmarks: input and output of the operator, number of sampled input coordinates, numbernzn\_\{z\}of principal input coordinates \(the numerical rank of the training inputs, capped at three\), spatial dimensiondd, numbermmof output points, and spatial kernel\. Taylor–Green vortices \(spacetime\) has spacetime outputs, at the same500500points at four times\.Data setInput→\\tooutputInput coord\.nzn\_\{z\}ddmmSpatial kernelForced isotropic turbulence𝐟→𝐮⁡\(⋅,0\.5\)\\mathbf\{f\}\\to\\mathbf\{u\}\(\\cdot,0\.5\)21000337000Five\-scale MatérnTurbulent species transportinflow→𝐮⁡\(⋅,0\.5\)\\to\\mathbf\{u\}\(\\cdot,0\.5\)1388237000Five\-scale MatérnMerging vorticesω⁡\(⋅,0\)→𝐮⁡\(⋅,0\.4\)\\omega\(\\cdot,0\)\\to\\mathbf\{u\}\(\\cdot,0\.4\)50032500Cubic PHSLid\-driven cavity𝐮⁡\(⋅,0\)→𝐮⁡\(⋅,5\)\\mathbf\{u\}\(\\cdot,0\)\\to\\mathbf\{u\}\(\\cdot,5\)2000121000Cubic PHSCylinder flow \(no shedding\)𝐮⁡\(⋅,0\)→𝐮⁡\(⋅,10\)\\mathbf\{u\}\(\\cdot,0\)\\to\\mathbf\{u\}\(\\cdot,10\)2000121000Cubic PHSCylinder flow \(vortex shedding\)𝐮⁡\(⋅,0\)→𝐮⁡\(⋅,10\)\\mathbf\{u\}\(\\cdot,0\)\\to\\mathbf\{u\}\(\\cdot,10\)2000121000Cubic PHSBackward\-facing stepinflow→𝐮⁡\(⋅,5\)\\to\\mathbf\{u\}\(\\cdot,5\)11121000Cubic PHSBuoyancy\-driven cavityT⁡\(⋅,0\)→𝐮⁡\(⋅,5\)T\(\\cdot,0\)\\to\\mathbf\{u\}\(\\cdot,5\)5000125000Cubic PHSTaylor–Green vortices \(space\)𝐮⁡\(⋅,0\)→𝐮⁡\(⋅,1\)\\mathbf\{u\}\(\\cdot,0\)\\to\\mathbf\{u\}\(\\cdot,1\)100012500Cubic PHSTaylor–Green vortices \(spacetime\)𝐮⁡\(⋅,0\)→𝐮⁡\(⋅,t\)\\mathbf\{u\}\(\\cdot,0\)\\to\\mathbf\{u\}\(\\cdot,t\),t∈\[0\.7,1\]t\\in\[0\.7,1\]100012500Cubic PHSThe benchmarks differ less in their flows than in their inputs, and this affects how we interpret the results\. In three of them, forced isotropic turbulence, turbulent species transport, and merging vortices, the inputs vary in at least two independent directions, and the operator is a multivariate map\. In the other seven, the training inputs have numerical rank one\. These inputs are multiples of one input function \(e\.g\., the speed of the lid or the amplitude of the inflow\), so the operator restricted to the data is a curve in the output space, and the operator regression is one\-dimensional\. Two of these, the Taylor–Green vortices, are degenerate in a further sense\. Their outputs are proportional to the input amplitude, so we divide the responses by the amplitude before fitting and multiply by it after prediction\. With𝒜i\>0\\mathcal\{A\}\_\{i\}\>0the amplitude of training inputiirelative to the median training amplitude and𝒜⁡\(𝐚\)\\mathcal\{A\}\(\\mathbf\{a\}\)that of a new input, we fit the normalized responses𝐯i/𝒜i\\mathbf\{v\}\_\{i\}/\\mathcal\{A\}\_\{i\}and predict

𝐯~​\(𝐚\)=𝒜⁡\(𝐚\)​∑i=1Nℓi​\(𝐚\)​𝐯i𝒜i\.\\tilde\{\\mathbf\{v\}\}\(\\mathbf\{a\}\)=\\mathcal\{A\}\(\\mathbf\{a\}\)\\sum\_\{i=1\}^\{N\}\\ell\_\{i\}\(\\mathbf\{a\}\)\\frac\{\\mathbf\{v\}\_\{i\}\}\{\\mathcal\{A\}\_\{i\}\}\.\(72\)The normalized operator is then constant, which affine augmentation reproduces, and Proposition[2](https://arxiv.org/html/2610.29009#Thmproposition2)applies to the normalized responses\. All four models use this normalization for both Taylor–Green data sets, and the other eight data sets use unnormalized responses\. Taylor–Green vortices \(spacetime\) thus tests our spacetime output reconstruction but not the operator regression\.

We compare four models \(Section[3\.6](https://arxiv.org/html/2610.29009#S3.SS6)\)\. The*fixed\-shape*kernel usesD∥=ID\_\{\\parallel\}=Iandα⟂=1\\alpha\_\{\\perp\}=1in \([32](https://arxiv.org/html/2610.29009#S3.E32)\), an isotropic metric on the principal coordinates and a unit scale for the residual coordinates\. The*fitted\-shape*kernel has its shape fitted by the restricted likelihood\. Each is used with and without affine augmentation, and each chooses its support radius and regularization by validation\. For inputs of rank one, the shape has no free parameters, so the fitted\-shape kernel differs from the fixed\-shape one only through the regularization that the likelihood proposes, which enters the validation search\.

All four models share one local output reconstruction, with divergence\-free polynomials of degreeϖ=4\\varpi=4\(dimension2020in 2D and8585in 3D\) on stencils ofns=21n\_\{s\}=21and5959nodes, the cubic PHS for the smooth flows, and the multiscale Matérn kernel withcσ=16c\_\{\\sigma\}=16\(which sets its lengthsσj,κ=cσ​ρj/2κ\\sigma\_\{j,\\kappa\}=c\_\{\\sigma\}\\rho\_\{j\}/2^\{\\kappa\}relative to the stencil radiusρj\\rho\_\{j\}in \([21](https://arxiv.org/html/2610.29009#S2.E21)\)\) for the two turbulent flows\. Section[5\.1\.4](https://arxiv.org/html/2610.29009#S5.SS1.SSS4)compares these choices with alternatives\. All computations run in double precision on one NVIDIA RTX 5090\.

#### 5\.2\.2Sparsity and convergence inNN

We first ask whether a sparse kernel matrix is compatible with accurate prediction\. Validation chooses the support radius, and with it the densitydK=nnz⁡\(KU​U\)/N2d\_\{K\}=\\mathrm\{nnz\}\(K\_\{UU\}\)/N^\{2\}of the kernel matrix, among candidates with densities up to90%90\\%\. If accuracy required a dense matrix, validation would select the largest candidates\. Figure[6](https://arxiv.org/html/2610.29009#S5.F6)\(right\) shows the densities selected atN=10000N=10000for all ten benchmarks and all four models of Section[5\.2\.1](https://arxiv.org/html/2610.29009#S5.SS2.SSS1)\. Validation selects10%10\\%for the lid\-driven cavity, both cylinder flows, and the buoyancy\-driven cavity, about33%33\\%for the backward\-facing step, and between9%9\\%and25%25\\%for turbulent species transport and merging vortices\. Nearly dense matrices are selected only for forced isotropic turbulence and for Taylor–Green without augmentation\. The value10%10\\%is the sparsest of the five prescribed densities that validation evaluates, although its gradient refinement can also reach sparser matrices\.

Figure[7](https://arxiv.org/html/2610.29009#S5.F7)shows how the selected density changes withNN\. For most benchmarks, it decreases asNNgrows, from between20%20\\%and90%90\\%atN=500N=500to10%10\\%atN=10000N=10000for the lid\-driven cavity and both cylinder flows, so the sparsity of the operator regression increases with the size of the training set\. Forced isotropic turbulence stays at90%90\\%for everyNN, and the backward\-facing step at about33%33\\%\.

Figure 6:All benchmarks atN=10000N=10000\. Left: test error of the fixed\-shape kernel \(D∥=ID\_\{\\parallel\}=I,α⟂=1\\alpha\_\{\\perp\}=1\) without augmentation, divided by the test error of each other model \(values above one are improvements\)\. Right: densitydKd\_\{K\}of the selected kernel matrix\.Figure 7:Selected densitydKd\_\{K\}of the kernel matrix against the numberNNof training functions for the three multivariate flows and the lid\-driven cavity\. The affine columns are not counted\. Figure[21](https://arxiv.org/html/2610.29009#A4.F21)shows the other benchmarks\.With these sparse matrices, the test errors decrease steadily as the training set grows\. Figure[8](https://arxiv.org/html/2610.29009#S5.F8)shows the errors for the three multivariate flows and for the lid\-driven cavity, a representative flow with rank\-one inputs\. Least\-squares fits give errors proportional toN−τN^\{\-\\tau\}withτ\\taubetween0\.40\.4and1\.31\.3for every benchmark and model \(Table[9](https://arxiv.org/html/2610.29009#A4.T9)in Appendix[D\.6](https://arxiv.org/html/2610.29009#A4.SS6)\), with two exceptions\. With augmentation, the backward\-facing step decays only likeN−0\.15N^\{\-0\.15\}\. For Taylor–Green, the augmented models reproduce the constant normalized operator to rounding error, while the error of the unaugmented models grows slowly withNN\.

Figure 8:Test error \(mean relative velocity error of the predictions for the 100 test inputs\) against the numberNNof training functions for the three multivariate flows and the lid\-driven cavity\. Dashed: with affine augmentation\. Figure[20](https://arxiv.org/html/2610.29009#A4.F20)shows the other benchmarks\.
#### 5\.2\.3Kernel shape and affine augmentation

We next compare the four models\. Figure[6](https://arxiv.org/html/2610.29009#S5.F6)\(left\) shows, for each benchmark atN=10000N=10000, the error of the fixed\-shape kernel without augmentation divided by the error of each other model\. On the five rank\-one flows other than Taylor–Green, these ratios lie between0\.980\.98and1\.361\.36, so neither the fitted shape nor the augmentation changes the error by more than about a quarter\. This is consistent with the operator regression being one\-dimensional on these data\.

On the multivariate flows, however, the ratios are larger and depend on the flow\. For forced isotropic turbulence, augmentation reduces the error by a factor of2929, and by3939together with the fitted shape, at the same density\. For turbulent species transport, the fitted shape reduces it by a factor of66, and by1212together with augmentation, while doubling the selected density\. For merging vortices, all four errors lie within12%12\\%of each other\. Augmentation also lowers the selected density for merging vortices and for both Taylor–Green benchmarks\. Neither choice is uniformly better, so we select both by validation\.

#### 5\.2\.4Comparison with earlier results

We now compare our models with the results that[Sharma et al\. \[53\]](https://arxiv.org/html/2610.29009#bib.bib10)report on the same data sets for dense PPKM, for the kernel method of[Batlle et al\. \[2\]](https://arxiv.org/html/2610.29009#bib.bib1)\(VKM\), which maps the samples of the input to the samples of the output, and for two neural operators, Geo\-FNO\[[32](https://arxiv.org/html/2610.29009#bib.bib7)\]and Transolver\[[70](https://arxiv.org/html/2610.29009#bib.bib12)\], each trained with and without a divergence penalty\. Table[4](https://arxiv.org/html/2610.29009#S5.T4)lists these errors at the training sizes used there, together with the error of our model that has the smallest validation error among the four\. The comparison has two caveats\. The earlier errors are averages over all200200held\-out pairs, which include our100100validation pairs, and the neural operators were trained in single precision\.

Table 4:Test error \(mean relative velocity error of the predictions for held\-out inputs\) of our sparse local model \(the one of the four with the smallest validation error\) and of the methods reported by[Sharma et al\. \[53\]](https://arxiv.org/html/2610.29009#bib.bib10)at the same training sizeNN: dense PPKM, the kernel method of[Batlle et al\. \[2\]](https://arxiv.org/html/2610.29009#bib.bib1)on output samples \(VKM\), and Geo\-FNO and Transolver, each with the better of its errors with and without a divergence penalty\. The earlier errors are averages over all200200held\-out pairs and ours over the100100test pairs\. Our models of the cylinder flow without shedding atN=100N=100were fitted for this comparison, with the settings of the other training sizes\. Bold: smallest test error in each row\.Data setNNOursPPKMVKMGeo\-FNOTransolverForced isotropic turbulence100007\.1×𝟏𝟎−𝟓\\mathbf\{7\.1\\times 10^\{\-5\}\}3\.6×10−43\.6\\times 10^\{\-4\}1\.6×10−41\.6\\times 10^\{\-4\}1\.5×10−41\.5\\times 10^\{\-4\}1\.2×10−31\.2\\times 10^\{\-3\}Turbulent species transport100003\.7×𝟏𝟎−𝟓\\mathbf\{3\.7\\times 10^\{\-5\}\}2\.4×10−42\.4\\times 10^\{\-4\}2\.4×10−42\.4\\times 10^\{\-4\}5\.0×10−45\.0\\times 10^\{\-4\}2\.7×10−32\.7\\times 10^\{\-3\}Merging vortices5001\.9×𝟏𝟎−𝟔\\mathbf\{1\.9\\times 10^\{\-6\}\}1\.1×10−41\.1\\times 10^\{\-4\}1\.1×10−41\.1\\times 10^\{\-4\}1\.1×10−31\.1\\times 10^\{\-3\}3\.4×10−33\.4\\times 10^\{\-3\}Lid\-driven cavity100004\.7×𝟏𝟎−𝟕\\mathbf\{4\.7\\times 10^\{\-7\}\}1\.4×10−61\.4\\times 10^\{\-6\}1\.5×10−61\.5\\times 10^\{\-6\}1\.5×10−41\.5\\times 10^\{\-4\}2\.1×10−42\.1\\times 10^\{\-4\}Cylinder flow \(no shedding\)1002\.8×𝟏𝟎−𝟓\\mathbf\{2\.8\\times 10^\{\-5\}\}4\.0×10−54\.0\\times 10^\{\-5\}5\.3×10−55\.3\\times 10^\{\-5\}1\.6×10−41\.6\\times 10^\{\-4\}5\.7×10−35\.7\\times 10^\{\-3\}Cylinder flow \(vortex shedding\)100003\.0×𝟏𝟎−𝟕\\mathbf\{3\.0\\times 10^\{\-7\}\}3\.1×10−73\.1\\times 10^\{\-7\}2\.3×10−62\.3\\times 10^\{\-6\}3\.7×10−53\.7\\times 10^\{\-5\}1\.5×10−41\.5\\times 10^\{\-4\}Backward\-facing step5009\.5×𝟏𝟎−𝟖\\mathbf\{9\.5\\times 10^\{\-8\}\}1\.5×10−61\.5\\times 10^\{\-6\}2\.9×10−62\.9\\times 10^\{\-6\}9\.3×10−59\.3\\times 10^\{\-5\}7\.2×10−47\.2\\times 10^\{\-4\}Buoyancy\-driven cavity100004\.5×𝟏𝟎−𝟔\\mathbf\{4\.5\\times 10^\{\-6\}\}6\.3×10−66\.3\\times 10^\{\-6\}6\.1×10−66\.1\\times 10^\{\-6\}1\.4×10−41\.4\\times 10^\{\-4\}1\.9×10−41\.9\\times 10^\{\-4\}Taylor–Green vortices \(space\)50006\.0×𝟏𝟎−𝟏𝟏\\mathbf\{6\.0\\times 10^\{\-11\}\}8\.8×10−108\.8\\times 10^\{\-10\}1\.7×10−91\.7\\times 10^\{\-9\}4\.2×10−44\.2\\times 10^\{\-4\}6\.1×10−56\.1\\times 10^\{\-5\}Taylor–Green vortices \(spacetime\)50005\.4×𝟏𝟎−𝟏𝟏\\mathbf\{5\.4\\times 10^\{\-11\}\}8\.7×10−108\.7\\times 10^\{\-10\}8\.7×10−108\.7\\times 10^\{\-10\}7\.8×10−47\.8\\times 10^\{\-4\}2\.2×10−42\.2\\times 10^\{\-4\}Our models are at least as accurate as dense PPKM on all ten data sets, by factors between1\.061\.06\(cylinder flow with vortex shedding\) and about5656\(merging vortices\), and the same holds against VKM\. Against the better of the two neural operators, the factor ranges from22for forced isotropic turbulence to more than10610^\{6\}for Taylor–Green\. The gains over dense PPKM are likely due to the affine augmentation, the fitted kernel shape, and the validation of the radius and regularization, and for Taylor–Green also to the amplitude normalization\. Thus, at these training sizes, the sparse operator regression is no less accurate than the dense one\.

In addition, the neural\-operator predictions in[Sharma et al\. \[53\]](https://arxiv.org/html/2610.29009#bib.bib10)have maximum pointwise divergences between about0\.10\.1and2020, while ours are divergence free on each stencil\. The training times reported there, from seconds to about a minute and a half for the kernel methods with fixed shape parameters and from minutes to almost a day for the neural operators, are not comparable to our selection times, which include the search over radii and regularizations \(Appendix[D\.6\.1](https://arxiv.org/html/2610.29009#A4.SS6.SSS1)\)\.

#### 5\.2\.5Storage

The storage of a model consists mainly of the Cholesky factor of its kernel matrix and the sampled training fields, which the prediction \([59](https://arxiv.org/html/2610.29009#S4.E59)\) uses in place of their local coefficients\. Figure[9](https://arxiv.org/html/2610.29009#S5.F9)shows the storage againstNN\. Over our range ofNN, the training fields dominate, and the storage of most models grows roughly linearly inNN, with least\-squares exponents between0\.90\.9and1\.11\.1\. The exceptions are merging vortices, the backward\-facing step, and the unaugmented Taylor–Green models, with exponents between1\.21\.2and1\.71\.7, likely because the factor is a larger share of their storage \(small outputs, or densities that do not decrease\)\. The Cholesky factor itself grows likeN1\.4N^\{1\.4\}toN2\.4N^\{2\.4\}, since even a decreasing density leaves a number of nonzeros that grows faster thanNN, and it would eventually dominate for larger training sets\.

The size of the factor depends on the ordering as well as on the density\. With the nested\-dissection ordering of Section[3\.5](https://arxiv.org/html/2610.29009#S3.SS5), the factors atN=10000N=10000have between11and4\.34\.3times fewer entries \(a median of2\.82\.8\) than with the approximate\-minimum\-degree ordering, with identical predictions\. Forced isotropic turbulence gains nothing, since its nearly dense kernel matrix has a dense factor in any ordering\. The fill also makes storage not proportional to the density\. For example, the fitted\-shape augmented species model has2\.42\.4times the density of the fixed\-shape augmented one, but only1\.41\.4times as many factor entries\. For this reason, we report storage separately from density \(Table[11](https://arxiv.org/html/2610.29009#A4.T11)\)\.

Figure 9:Accounted storage of the fitted models \(stored arrays and factors, not measured peak memory\) againstNNfor the three multivariate flows and the lid\-driven cavity\. Figure[22](https://arxiv.org/html/2610.29009#A4.F22)shows the other benchmarks\.

### 5\.3Operator Lagrange weights

We now turn to the operator Lagrange weights of Section[4](https://arxiv.org/html/2610.29009#S4)\. Their main purpose is computational\. They make a prediction cost one sparse solve per new input, regardless of the size of the output representation, and they let us combine the training fields before the output reconstruction\. We first measure these savings\. We then test the prediction of Theorem[3](https://arxiv.org/html/2610.29009#Thmproposition3)that the weights are also more accurate than direct evaluation, but only when the output reconstruction amplifies solve errors\.

#### 5\.3\.1Storage of the training data

Since the weights are applied to the sampled training fields, a prediction needs theN​nvNn\_\{v\}sampled values, withnv=d​mn\_\{v\}=dm, and not theN​∑jrjN\\sum\_\{j\}r\_\{j\}local coefficients of the training fields\. With one stencil per output point, the ratio of the two is∑jrj/\(d​m\)=ns\+nP/d\\sum\_\{j\}r\_\{j\}/\(dm\)=n\_\{s\}\+n\_\{P\}/dwhen every stencil keeps the full polynomial space, of dimensionnPn\_\{P\}\. For our degree\-four output reconstructions, the local coefficients would thus take3131times the storage of the training fields in 2D and8787times in 3D\. For forced isotropic turbulence atN=10000N=10000, for example, this is the difference between1\.71\.7and147147GB in double precision\.

#### 5\.3\.2Prediction cost

Figure[10](https://arxiv.org/html/2610.29009#S5.F10)shows the time to predict all100100test fields once the model is set up, including the evaluation of the kernel, the sparse solves for the weights, and the local output reconstruction\. For the lid\-driven cavity, a twentyfold increase inNNraises this time seven\- to eightfold, so prediction grows sublinearly inNNover this range\. The turbulent flows cost more atN=10000N=10000, about three times the time of the lid\-driven cavity for forced isotropic turbulence and1\.71\.7–2\.02\.0times for turbulent species transport, since they select denser kernels with more fill\.

The ordering also affects the prediction time\. Nested dissection makes the median prediction1\.31\.3times faster atN=10000N=10000than approximate minimum degree, since its smaller factors also give more parallel triangular solves\. Fitting the kernel shape has no consistent effect on prediction time, but it adds offline cost, since a fitted\-shape model takes several times longer to select than a fixed\-shape one \(Appendix[D\.6\.1](https://arxiv.org/html/2610.29009#A4.SS6.SSS1)\)\.

Figure 10:Median time to predict all100100test fields after setup, including the kernel evaluation, the sparse solves, and the local output reconstruction\. Fitting and setup are excluded \(Appendix[D\.6\.1](https://arxiv.org/html/2610.29009#A4.SS6.SSS1)\)\.
#### 5\.3\.3Accuracy of direct and Lagrange evaluation

Recall the amplification factorαspatial=‖B‖F​‖\(𝐀⋆\)𝖳‖F/‖B​\(𝐀⋆\)𝖳‖F\\alpha\_\{\\rm spatial\}=\\\|B\\\|\_\{F\}\\\|\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\\\|\_\{F\}/\\\|B\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\\\|\_\{F\}of \([66](https://arxiv.org/html/2610.29009#S4.E66)\), which compares the size of the spatial coefficientsBBof the training fields and of the evaluation matrix𝐀⋆\\mathbf\{A\}^\{\\star\}with the size of the reconstructed velocities\. It is large when large coefficients cancel in the output reconstruction\. By \([67](https://arxiv.org/html/2610.29009#S4.E67)\), a solve error enters direct evaluation multiplied byαspatial\\alpha\_\{\\rm spatial\}, and Lagrange evaluation without this factor\. The theorem therefore predicts a gain from Lagrange evaluation whereαspatial\\alpha\_\{\\rm spatial\}is large, and none where it is small\. We test this prediction in two steps\.

First, we computeαspatial\\alpha\_\{\\rm spatial\}for all ten benchmarks, from10001000training fields, for the global divergence\-free Matérn output reconstruction of our earlier work and for our local output reconstruction\. For the local output reconstruction, which has oneBjB\_\{j\}and one𝐀j⋆\\mathbf\{A\}\_\{j\}^\{\\star\}per stencil, we combine the stencils as\(∑j‖Bj‖F2​‖𝐀j⋆‖F2\)1/2/‖B​\(𝐀⋆\)𝖳‖F\(\\sum\_\{j\}\\\|B\_\{j\}\\\|\_\{F\}^\{2\}\\\|\\mathbf\{A\}\_\{j\}^\{\\star\}\\\|\_\{F\}^\{2\}\)^\{1/2\}/\\\|B\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\\\|\_\{F\}, which bounds the amplification of the solve errors summed over all stencils and reduces toαspatial\\alpha\_\{\\rm spatial\}for a single global stencil\. Table[5](https://arxiv.org/html/2610.29009#S5.T5)shows that the coefficients of the global output reconstructions are large and cancel, withαspatial\\alpha\_\{\\rm spatial\}above10710^\{7\}, up to6×1096\\times 10^\{9\}, on all benchmarks except Taylor–Green vortices \(spacetime\), whose kernel is less flat\. Our local output reconstructions haveαspatial\\alpha\_\{\\rm spatial\}between66and150150in 2D,396396for forced isotropic turbulence, and about10410^\{4\}for turbulent species transport, where more than half the stencils have a truncated polynomial rank \(Table[7](https://arxiv.org/html/2610.29009#A1.T7)\)\.

Table 5:Amplification factorαspatial=‖B‖F​‖\(𝐀⋆\)𝖳‖F/‖B​\(𝐀⋆\)𝖳‖F\\alpha\_\{\\rm spatial\}=\\\|B\\\|\_\{F\}\\\|\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\\\|\_\{F\}/\\\|B\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\\\|\_\{F\}of \([66](https://arxiv.org/html/2610.29009#S4.E66)\), which measures the cancellation among the spatial coefficients, forN=1000N=1000training fields, for the global and for our local output reconstruction \(stencils combined as in Section[5\.3](https://arxiv.org/html/2610.29009#S5.SS3)\), and the largest ratio of the direct to the Lagrange error with global output reconstruction from Table[6](https://arxiv.org/html/2610.29009#S5.T6)\. Bold: smaller amplification factor in each row\.Data setGlobalαspatial\\alpha\_\{\\rm spatial\}Localαspatial\\alpha\_\{\\rm spatial\}Largest gain \(global\)Forced isotropic turbulence3\.7×1093\.7\\times 10^\{9\}𝟑𝟗𝟔\\mathbf\{396\}1\.21\.2Turbulent species transport2\.9×1092\.9\\times 10^\{9\}𝟗𝟖𝟖𝟐\\mathbf\{9882\}7\.17\.1Merging vortices4\.8×1094\.8\\times 10^\{9\}𝟕𝟑\\mathbf\{73\}868\.1868\.1Lid\-driven cavity5\.8×1095\.8\\times 10^\{9\}𝟏𝟑\\mathbf\{13\}34\.534\.5Cylinder flow \(no shedding\)5\.1×1075\.1\\times 10^\{7\}𝟏𝟒𝟗\\mathbf\{149\}4\.84\.8Cylinder flow \(vortex shedding\)5\.2×1075\.2\\times 10^\{7\}𝟏𝟒𝟗\\mathbf\{149\}2\.02\.0Backward\-facing step5\.3×1085\.3\\times 10^\{8\}𝟑𝟖\\mathbf\{38\}215\.3215\.3Buoyancy\-driven cavity5\.9×1095\.9\\times 10^\{9\}𝟔\\mathbf\{6\}14\.814\.8Taylor–Green vortices \(space\)1\.6×1071\.6\\times 10^\{7\}𝟕𝟑\\mathbf\{73\}6\.16\.1Taylor–Green vortices \(spacetime\)1\.9×1021\.9\\times 10^\{2\}𝟏𝟒𝟓\\mathbf\{145\}1\.01\.0Second, we evaluate the same operator models both ways, with everything else held fixed\. These are the unaugmented models of Section[5\.2](https://arxiv.org/html/2610.29009#S5.SS2), which the theorem covers: the fixed\-shape kernel at the radii of the five prescribed densities and at its validated radius, and the fitted\-shape kernel at its validated parameters, forNNfrom500500to1000010000\(Appendix[D\.4](https://arxiv.org/html/2610.29009#A4.SS4)\)\. Table[6](https://arxiv.org/html/2610.29009#S5.T6)lists the ratio of the direct error to the Lagrange error on all ten benchmarks\.

Table 6:Ratio of the test error of direct evaluation to that of Lagrange evaluation, for the same unaugmented operator models \(the fixed\-shape kernel of Section[3\.1](https://arxiv.org/html/2610.29009#S3.SS1)at the radii of the five prescribed densities and at its validated radius, and the fitted\-shape kernel at its validated radius, forN=500N=500,10001000,50005000, and1000010000\) with global and local output reconstruction\. Values above one favor Lagrange evaluation\. Over all 560 pairs, the smallest ratio is0\.999870\.99987\.Global output reconstructionLocal output reconstructionData setModelsMedianMaxModelsMax\|\|ratio−1\|\-1\|Forced isotropic turbulence281\.001\.2285\.9×10−115\.9\\times 10^\{\-11\}Turbulent species transport281\.007\.1288\.6×10−78\.6\\times 10^\{\-7\}Merging vortices2848\.96868\.1284\.2×10−84\.2\\times 10^\{\-8\}Lid\-driven cavity285\.0734\.5281\.4×10−71\.4\\times 10^\{\-7\}Cylinder flow \(no shedding\)281\.054\.8287\.0×10−87\.0\\times 10^\{\-8\}Cylinder flow \(vortex shedding\)281\.012\.0282\.4×10−82\.4\\times 10^\{\-8\}Backward\-facing step2816\.51215\.3282\.0×10−82\.0\\times 10^\{\-8\}Buoyancy\-driven cavity281\.4714\.8281\.9×10−81\.9\\times 10^\{\-8\}Taylor–Green vortices \(space\)281\.056\.1281\.7×10−61\.7\\times 10^\{\-6\}Taylor–Green vortices \(spacetime\)281\.001\.0282\.4×10−62\.4\\times 10^\{\-6\}The results agree with this prediction\. With global output reconstruction, Lagrange evaluation is more accurate by up to a factor of868868\(merging vortices\), with median gains above55for merging vortices, the lid\-driven cavity, and the backward\-facing step\. With our local output reconstruction, the two evaluations agree to within a relative difference of3×10−63\\times 10^\{\-6\}for every model, as the smallαspatial\\alpha\_\{\\rm spatial\}implies\. A largeαspatial\\alpha\_\{\\rm spatial\}allows a gain but does not guarantee one, since the gain also requires a solve residual that is not negligible\. Forced isotropic turbulence, for example, hasαspatial≈4×109\\alpha\_\{\\rm spatial\}\\approx 4\\times 10^\{9\}with its global output reconstruction but a gain of at most1\.21\.2\.

Over all560560pairs, Lagrange evaluation is never less accurate than direct evaluation by more than0\.02%0\.02\\%\. Appendix[D\.4](https://arxiv.org/html/2610.29009#A4.SS4)follows one global model of the lid\-driven cavity asNNgrows \(Table[8](https://arxiv.org/html/2610.29009#A4.T8)\)\. There, the gain grows from1\.31\.3to5\.15\.1together with the norm of the spatial coefficients, and solving the direct system for the sampled velocities instead of the coefficients removes it, which indicates that the large coefficients are the cause\.

Lagrange evaluation thus serves our local output reconstruction through its cost and storage, and it would also protect a global or otherwise ill\-conditioned output reconstruction, such as that of our earlier method, from the amplification of solve errors\.

## 6Conclusions

In this article, we presented a local and sparse kernel method for learning the solution operators of incompressible flows\. It keeps the guarantee of our earlier property\-preserving method\[[53](https://arxiv.org/html/2610.29009#bib.bib10)\], that every prediction is divergence free, and removes the two costs of that method that grow with the problem size\. Local divergence\-free interpolants replace the global output reconstruction, so the output can be refined without enlarging any linear system, and the local systems are independent and solved in parallel on the GPU\. The output reconstruction is divergence free, periodic on periodic domains, and, with the multiscale Matérn kernel, approaches Kolmogorov statistics under refinement at the separations that the nodes resolve\. A compactly supported kernel that is positive semidefinite in any input dimension replaces the dense operator regression, so the kernel matrix is sparse\. Its support radius, and with it the sparsity, is fitted to the data, and optional affine polynomials in the input, which the operator regression reproduces exactly, represent trends that a compactly supported kernel cannot\. Operator Lagrange weights join the two, so each new input costs one sparse solve and the local coefficients of the training fields are never formed\.

The local output reconstruction converges at close to the order of its polynomial degree on scattered nodes, is about as accurate between the nodes as the global output reconstruction \(more accurate on six of ten benchmarks and less accurate on the other four\) with systems that are smaller by about two orders of magnitude, is cheaper to set up than a global output reconstruction beyond a few thousand output points, with memory that grows linearly with their number, keeps the divergence and the periodic mismatch at rounding level, and, with the multiscale Matérn kernel, approaches prescribed Kolmogorov statistics at the separations that the nodes resolve\.

In the operator benchmarks, the test errors decrease roughly likeN−τN^\{\-\\tau\}withτ\\taubetween0\.40\.4and1\.31\.3in all but two degenerate cases, and the most accurate kernels are sparse on all benchmarks but one, with densities that mostly decrease as the training set grows\. At the training sizes of our earlier study, these sparse models are at least as accurate as dense PPKM on every data set, and more accurate than the neural operators reported there by factors from22to more than10610^\{6\}\. Over our range of training sizes, the storage of most models grows roughly linearly inNN\. The benchmarks with rank\-one inputs, apart from Taylor–Green, are insensitive to the choice of kernel and augmentation\. The benchmarks with multivariate inputs are not, and the best choice differs between them\. Affine augmentation gives the largest gain for forced isotropic turbulence, a fitted kernel shape for turbulent species transport, and neither helps for merging vortices\. Our recommendation is therefore to select these choices by validation and to report error, density, and storage together\.

Lagrange evaluation removes the need to store the local coefficients of the training fields, which would takens\+nP/dn\_\{s\}\+n\_\{P\}/dtimes \(the stencil size plus the number of polynomials per velocity component\) the storage of the training fields themselves\. Its accuracy agrees with our rounding analysis\. With our local output reconstruction, whose amplification factors are at most about10410^\{4\}, it is as accurate as direct evaluation\. With global output reconstructions, whose amplification factors reach6×1096\\times 10^\{9\}, it is up to nearly three orders of magnitude more accurate\.

Several extensions are natural\. First, benchmarks with richer input families, and a comparison with dense operator regression and with neural operators on identical test sets and hardware, would test these conclusions beyond our ten benchmarks\. Second, our local interpolants are divergence free on each stencil but are not coupled across stencils\. Blending them with a partition of unity would give a continuous field, but the blending weights introduce divergence\. The divergence\-free partition\-of\-unity method of[Drake et al\. \[12\]](https://arxiv.org/html/2610.29009#bib.bib23)avoids this by blending potentials instead of velocities\. Third, at a fixed density, the storage of the kernel matrix still grows quadratically with the number of training functions, which hierarchical or multilevel sparse factorizations could reduce\. Finally, our rounding analysis covers the model without augmentation and evaluation through coefficients\. Extending it to the affine correction and to evaluation before the output reconstruction would complete the analysis\.

## Credit authorship contribution statement

Ramansh Sharma:conceptualization, methodology, software, investigation, validation, visualization, writing – original draft, review, editing\. Matthew Lowery:methodology, software Houman Owhadi:conceptualization, methodology, supervision\. Varun Shankar:conceptualization, funding acquisition, methodology, project administration, resources, software, supervision, validation, visualization, writing\.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper\.

## Data availability

Our implementation, including the scripts that build the tables and figures of this paper, is available in KernelPack\-JAX at[https://github\.com/VarShankar/kernelpack\-jax](https://github.com/VarShankar/kernelpack-jax)\. The ten benchmark data sets are those of[Sharma et al\. \[53\]](https://arxiv.org/html/2610.29009#bib.bib10)and are publicly available on Zenodo\[[54](https://arxiv.org/html/2610.29009#bib.bib72)\]\.

## Acknowledgements

RS, ML, and VS were supported by the National Science Foundation under DMS 2505986, Cooperative Agreement 2421782, and the Simons Foundation grant MPS\-AI\-00010515 awarded to the NSF\-Simons AI Institute for Cosmic Origins — CosmicAI,[https://www\.cosmicai\.org](https://www.cosmicai.org/)\. VS was additionally supported by the Air Force Office of Scientific Research \(AFOSR\) under LRIR award FA9550\-25\-1\-0042\. HO acknowledges support from the Air Force Office of Scientific Research under MURI award number FOA\-AFRL\-AFOSR\-2023\-0004 \(Mathematics of Digital Twins\), the Department of Energy under award number DE\-SC0023163 \(SEA\-CROGS: Scalable, Efficient, and Accelerated Causal Reasoning Operators, Graphs and Spikes for Earth and Embedded Systems\) and the DoD Vannevar Bush Faculty Fellowship Program under ONR award number N00014\-18\-1\-2363\.

## Declaration of generative AI and AI\-assisted technologies in the writing process

During the preparation of this work, the authors used both Codex \(OpenAI\) and Claude \(Anthropic\) to help clean up the manuscript and implement parts of the software\. After using this tool, the authors reviewed and edited the content as needed and take full responsibility for the content of the publication\.

## Appendix ADetails of the local output reconstruction

### A\.1Ordering of vector coefficients

The coefficient vectors of \([17](https://arxiv.org/html/2610.29009#S2.E17)\) are ordered by component, with all values of one component before those of the next, but the kernel evaluation at a query pointyq⋆y\_\{q\}^\{\\star\}is naturally ordered by stencil node,

𝚽j⁡\(q\),q,pt⋆=\[ΦDF​\(yq⋆,y~j⁡\(q\),1\)⋯ΦDF​\(yq⋆,y~j⁡\(q\),ns\)\]∈ℝd×d​ns\.\\displaystyle\\mathbf\{\\Phi\}\_\{j\(q\),q,\\mathrm\{pt\}\}^\{\\star\}=\\begin\{bmatrix\}\\Phi\_\{\\rm DF\}\(y\_\{q\}^\{\\star\},\\tilde\{y\}\_\{j\(q\),1\}\)&\\cdots&\\Phi\_\{\\rm DF\}\(y\_\{q\}^\{\\star\},\\tilde\{y\}\_\{j\(q\),n\_\{s\}\}\)\\end\{bmatrix\}\\in\\mathbb\{R\}^\{d\\times dn\_\{s\}\}\.\(73\)LetΠd,ns\\Pi\_\{d,n\_\{s\}\}be the permutation withΠd,ns​𝐛Φ,comp=𝐛Φ,pt\\Pi\_\{d,n\_\{s\}\}\\mathbf\{b\}\_\{\\Phi,\\mathrm\{comp\}\}=\\mathbf\{b\}\_\{\\Phi,\\mathrm\{pt\}\}, where𝐛Φ,pt\\mathbf\{b\}\_\{\\Phi,\\mathrm\{pt\}\}groups theddcomponents of each node\. The evaluation matrix in \([18](https://arxiv.org/html/2610.29009#S2.E18)\) is then

𝚽j⁡\(q\),q,comp⋆\\displaystyle\\mathbf\{\\Phi\}\_\{j\(q\),q,\\mathrm\{comp\}\}^\{\\star\}=𝚽j⁡\(q\),q,pt⋆​Πd,ns\.\\displaystyle=\\mathbf\{\\Phi\}\_\{j\(q\),q,\\mathrm\{pt\}\}^\{\\star\}\\Pi\_\{d,n\_\{s\}\}\.\(74\)

### A\.2Polynomial basis and local solves on the GPU

We first remove the dependencies among the candidates that do not depend on the stencil, once, before sampling\. A singular value decomposition of the fixed map from candidates to polynomial coefficients gives an orthonormal basisΞ\\Xiof its row space, shared by all stencils of one dimension and degree, with

Gj=Gj​Ξ​Ξ𝖳,Ξ𝖳​Ξ=I,G\_\{j\}=G\_\{j\}\\Xi\\Xi^\{\\mathsf\{T\}\},\\qquad\\Xi^\{\\mathsf\{T\}\}\\Xi=I,\(75\)in exact arithmetic\. ThusGj​ΞG\_\{j\}\\Xihas the nonzero singular values ofGjG\_\{j\}, and we factor it by thin QR,Gj​Ξ=Q~j​R~jG\_\{j\}\\Xi=\\widetilde\{Q\}\_\{j\}\\widetilde\{R\}\_\{j\}\. On most stencils,R~j\\widetilde\{R\}\_\{j\}is far from singular, and the polynomial basis is

𝐏j​\(y\)=Gj​\(y\)​Ξ​R~j−1,\\mathbf\{P\}\_\{j\}\(y\)=G\_\{j\}\(y\)\\Xi\\widetilde\{R\}\_\{j\}^\{\-1\},\(76\)whose values at the stencil nodes are the orthonormal columns ofQ~j\\widetilde\{Q\}\_\{j\}, as in \([14](https://arxiv.org/html/2610.29009#S2.E14)\)\. We use \([76](https://arxiv.org/html/2610.29009#A1.E76)\) when the eigenvalues ofR~j𝖳​R~j\\widetilde\{R\}\_\{j\}^\{\\mathsf\{T\}\}\\widetilde\{R\}\_\{j\}certify that its smallest singular value is at least10−610^\{\-6\}times its largest, far above the rank threshold of10−1210^\{\-12\}, since these eigenvalues carry absolute errors of the order of the unit roundoff times the largest one\. On the remaining stencils, the column\-pivoted QR factorizationR~j​Πj=Q^j​R^j\\widetilde\{R\}\_\{j\}\\Pi\_\{j\}=\\widehat\{Q\}\_\{j\}\\widehat\{R\}\_\{j\}determines the rank as in \([12](https://arxiv.org/html/2610.29009#S2.E12)\), and the basis is \([14](https://arxiv.org/html/2610.29009#S2.E14)\) withGj,IG\_\{j,I\}replaced by the retained columns ofGj​Ξ​ΠjG\_\{j\}\\Xi\\Pi\_\{j\}\. Both factorizations are batched over the stencils, and the columns of different stencils may differ, since every system is padded to the same width \(below\)\. A singular value decomposition on every stencil would decide the rank as well, but its batched form is slow on the GPU for these matrices \(85×8585\\times 85in 3D at degree four, about1010ms each\)\. On the species\-transport node set, where most stencils are rank deficient, both rank decisions agree on every stencil, and the output reconstructions between the nodes agree to10−1210^\{\-12\}\.

To keep the batches of fixed size, every system carries all2020\(2D\) or8585\(3D\) polynomial columns at degree four\. The columns of discarded modes are set to zero and given a unit diagonal, which decouples their coefficients without changing the interpolant\. Every system therefore has the padded width6262\(2D\) or262262\(3D\), and our costs and storage include the padding\.

If the LU solution of a local system is not finite, we use the pseudoinverse of that system, also for its evaluation weights\. The resulting field is still divergence free, but it need not interpolate or reproduce the full polynomial space\. The pseudoinverse was never needed\. The LU solutions are finite on every node stencil of all ten operator benchmarks, including the rank\-deficient stencils of Table[7](https://arxiv.org/html/2610.29009#A1.T7), and in the manufactured tests of Section[5\.1](https://arxiv.org/html/2610.29009#S5.SS1)every stencil kept the full polynomial space\.

Table 7:Retained divergence\-free polynomial ranks of the local systems on the nodes of each benchmark, at degree four and with periodic images, from the construction above\. The full ranks are 20 in 2D and 85 in 3D\. Taylor–Green vortices \(spacetime\) has the stencils of Taylor–Green vortices \(space\)\.Data setStencilsRetained rankTruncated stencilsTurbulent species transport700045–854338Forced isotropic turbulence7000850Buoyancy\-driven cavity5000200Lid\-driven cavity1000200Cylinder flow \(no shedding\)1000200Cylinder flow \(vortex shedding\)1000200Backward\-facing step1000200Taylor–Green vortices \(space\)50019–201Merging vortices50019–201
### A\.3Evaluation away from the nodes

The operator benchmarks evaluate predictions at the nodes\. To evaluate at another point, our GPU implementation selects a stencil centered at that point and builds its basis there\. The comparisons of Table[2](https://arxiv.org/html/2610.29009#S5.T2)use this rule\. The manufactured tests of Section[5\.1](https://arxiv.org/html/2610.29009#S5.SS1)instead use the stencil of the nearest node, and their results between the nodes refer to that rule\.

### A\.4Periodic stencils

The Taylor–Green flows and the merging vortices have period2​π2\\piin each direction\. Forced isotropic turbulence has period2​π2\\pion the box\[0,2​π\]3\[0,2\\pi\]^\{3\}, which we supply explicitly, since the cell centers of its grid do not reach the faces of the box\.

### A\.5Output reconstruction between the nodes

Table[2](https://arxiv.org/html/2610.29009#S5.T2)reconstructs the test predictions at the nodes of the model with the smallest validation error among the four of Section[5\.2\.1](https://arxiv.org/html/2610.29009#S5.SS2.SSS1), atN=10000N=10000, so that the operator regression is that of Section[5\.2](https://arxiv.org/html/2610.29009#S5.SS2)and only the output reconstruction changes\. The evaluation points of Table[2](https://arxiv.org/html/2610.29009#S5.T2)are all mesh or grid points of the data set that are not among themmoutput points\. The local output reconstruction selects a stencil centered at each of them \(Appendix[A\.3](https://arxiv.org/html/2610.29009#A1.SS3)\)\. The global output reconstruction is the divergence\-free interpolant of[Sharma et al\. \[53\]](https://arxiv.org/html/2610.29009#bib.bib10)with the scalar kernel3​M5/2​\(ε​r\)3M\_\{5/2\}\(\\varepsilon r\)on all nodes, with the shape parameters of that work \(ε=0\.3\\varepsilon=0\.3for the lid\-driven cavity,0\.06360\.0636for both cylinder flows,0\.060\.06for the backward\-facing step,0\.80\.8for the buoyancy\-driven cavity,0\.020\.02for merging vortices and Taylor–Green vortices \(space\), and1\.51\.5for Taylor–Green vortices \(spacetime\)\), applied at each sampled time for Taylor–Green vortices \(spacetime\)\. For the periodic data sets, it uses one representative of each set of periodically equivalent nodes\.

We select the scale factorcσc\_\{\\sigma\}of the multiscale kernel on the100100validation fields, by reconstructing their exact nodal values and measuring the error between the nodes\. For turbulent species transport,cσ=2c\_\{\\sigma\}=2,44,88, and1616give errors of0\.1660\.166,0\.1580\.158,0\.1500\.150, and0\.1470\.147, and the PHS gives0\.1690\.169\. For forced isotropic turbulence, they give0\.02280\.0228,0\.02060\.0206,0\.01780\.0178, and0\.01650\.0165, and the PHS gives0\.02270\.0227\. Both selectcσ=16c\_\{\\sigma\}=16, which the operator benchmarks use\. For turbulent species transport, the error between the nodes remains much larger than at the nodes, so an accurate prediction at the nodes does not guarantee one everywhere\.

We also compare the second\-order structure functions

S2​\(r\)=⟨‖v⁡\(x\)−v⁡\(y\)‖22⟩‖x−y‖≈r\\displaystyle S\_\{2\}\(r\)=\\left\\langle\\\|v\(x\)\-v\(y\)\\\|\_\{2\}^\{2\}\\right\\rangle\_\{\\\|x\-y\\\|\\approx r\}\(77\)of the output reconstructions and of the reference fields on the same pairs of points between the nodes \(256256anchors with up to four partners in each of1212logarithmic distance bins, with periodic distances for forced isotropic turbulence\), averaged over the100100test fields \(Figure[11](https://arxiv.org/html/2610.29009#A1.F11)\)\. The relative discrepancy of the binned structure function is1\.7%1\.7\\%for the PHS and2\.6%2\.6\\%for the multiscale kernel for turbulent species transport, and0\.40%0\.40\\%and0\.20%0\.20\\%for forced isotropic turbulence\. The multiscale kernel thus lowers the pointwise error between the nodes for both flows, but improves the binned increments only for forced isotropic turbulence\. Since the reference fields come from Reynolds\-averaged \(RANS\) and coarse large\-eddy \(LES\) simulations, this comparison measures agreement with the reference, not with physical turbulence\.

0\.0030\.0030\.010\.010\.030\.030\.10\.1200200300300500500800800rrS2​\(r\)S\_\{2\}\(r\)Turbulent species transportReferenceLocal PHSLocal multiscale

0\.70\.71122330\.50\.50\.70\.7111\.41\.4rrS2​\(r\)S\_\{2\}\(r\)Forced isotropic turbulenceReferenceLocal PHSLocal multiscale

Figure 11:Second\-order structure functions between the nodes of the reference fields and of the local PHS and multiscale Matérn output reconstructions of the test predictions atN=10000N=10000\.

## Appendix BDetails of the operator regression

### B\.1Normalization and principal directions

At everyNN, we compute the normalization and the principal directions from the firstNNtraining inputs alone\. Each coordinate is divided by its standard deviation, or set to zero if that vanishes, and the numerical rank uses the thresholdmax⁡\{10−10,2​umach​max⁡\(N,n​da\)\}\\max\\\{10^\{\-10\},2u\_\{\\rm mach\}\\max\(N,nd\_\{a\}\)\\\}relative to the largest singular value, withumachu\_\{\\rm mach\}the unit roundoff\. The four models at oneNNshare the normalization, the principal directions, and the gate \([34](https://arxiv.org/html/2610.29009#S3.E34)\), and the two models with augmentation share the affine matrix\.

### B\.2Fitting and validation

The fitted\-shape kernel starts from the radiusϱ0\\varrho\_\{0\}that gives25%25\\%density and from the regularizationθ=10−6\\theta=10^\{\-6\}, and takes6060steps of the Adam optimizer\[[31](https://arxiv.org/html/2610.29009#bib.bib50)\]with learning rate0\.030\.03for each ofcd=0c\_\{d\}=0andcd=1c\_\{d\}=1\. Its log\-radius is confined to\[log⁡\(ϱ0/8\),log⁡\(4​ϱ0\)\]\[\\log\(\\varrho\_\{0\}/8\),\\log\(4\\varrho\_\{0\}\)\]and its regularization toθ=etθ\+10−12\\theta=e^\{t\_\{\\theta\}\}\+10^\{\-12\}withtθ∈\[log⁡\(10−12\),log⁡\(0\.1\)\]t\_\{\\theta\}\\in\[\\log\(10^\{\-12\}\),\\log\(0\.1\)\]\. The anisotropy usesnz−1n\_\{z\}\-1free parametersbk∈\[−2,2\]b\_\{k\}\\in\[\-2,2\], completed by a zero, throughυ~k=log⁡\(2\)​tanh⁡\(bk\)\\tilde\{\\upsilon\}\_\{k\}=\\log\(2\)\\tanh\(b\_\{k\}\)andυk=υ~k−nz−1​∑k′=1nzυ~k′\\upsilon\_\{k\}=\\tilde\{\\upsilon\}\_\{k\}\-n\_\{z\}^\{\-1\}\\sum\_\{k^\{\\prime\}=1\}^\{n\_\{z\}\}\\tilde\{\\upsilon\}\_\{k^\{\\prime\}\}\. The residual scale lies in\[1/4,4\]\[1/4,4\]\. It is inactive for the seven benchmarks withnz=1n\_\{z\}=1and stays at one for turbulent species transport\.

Validation evaluates the radii of the five density budgets of Section[3\.6](https://arxiv.org/html/2610.29009#S3.SS6)\. Withϱmax\\varrho\_\{\\max\}the largest of these, it also searches radii in\[ϱmax/8,ϱmax\]\[\\varrho\_\{\\max\}/8,\\varrho\_\{\\max\}\]from four starts \(radiiϱmax/2\\varrho\_\{\\max\}/2andϱmax/4\\varrho\_\{\\max\}/4, regularizations10−610^\{\-6\}and10−310^\{\-3\}\), each with3535Adam steps on all100100validation fields\. A fitted\-shape model repeats this for both of its fitted shapes, so its search is twice as long\. We keep the model with the smallest validation error among those with density at most0\.900\.90, and break ties by density\.

### B\.3Work, storage, and timing

We assemble the kernel matrix by visiting all pairs of training inputs in blocks of bounded size and keeping those within the support\. This keeps the memory bounded, but the work of the search is still quadratic inNN, since compact support alone does not give a faster neighbor search\. The likelihood and its gradient use the factors through

ΠU=LK−𝖳​\(I−QP​QP𝖳\)​LK−1,\\Pi\_\{U\}=L\_\{K\}^\{\-\\mathsf\{T\}\}\(I\-Q\_\{P\}Q\_\{P\}^\{\\mathsf\{T\}\}\)L\_\{K\}^\{\-1\},\(78\)with the entries of the inverse needed for the trace derivatives computed only on the support\. When the outputs are wide, we replaceEEby the triangular factor of a thin QR factorizationE𝖳=QE​REE^\{\\mathsf\{T\}\}=Q\_\{E\}R\_\{E\}, sinceE​E𝖳=RE𝖳​REEE^\{\\mathsf\{T\}\}=R\_\{E\}^\{\\mathsf\{T\}\}R\_\{E\}\.

The storage we report is the sum of the stored model arrays \(including the training fields\), the sparse matrix arrays, cuDSS’s estimate of the memory of its factors, and the solve buffers\. It is computed from the stored arrays instead of measured at its peak, and it excludes compiled code and temporary workspace\. A kernel with𝒩K\\mathcal\{N\}\_\{K\}nonzeros in compressed sparse row form takes12​𝒩K\+4​\(N\+1\)12\\mathcal\{N\}\_\{K\}\+4\(N\+1\)bytes\.

We report the setup of the output reconstruction, which all four models share, separately from the fitting and validation of each model\. The warm prediction time of Figure[10](https://arxiv.org/html/2610.29009#S5.F10)is the median of1111repetitions of predicting all100100test fields after the model is set up and compiled\. The four models of a benchmark run in one process and share compiled code, so small differences between their times are not significant\.

### B\.4Batches and checks

The100100test inputs are solved in four batches of3232right\-hand sides, the last one padded by repeated inputs\. Repeated solves replay CUDA graphs \(recorded sequences of GPU kernel launches\)\. After every factorization, we check the residual of the solves, the reproduction of affine functions, and the agreement of the batched and compiled predictions with reference predictions\. Every reported model passed these checks\.

## Appendix CProofs

### C\.1Completeness of the polynomial candidates

In 2D, every divergence\-free vector polynomial of degree at mostϖ\\varpihas a polynomial stream function of degree at mostϖ\+1\\varpi\+1, so the rotated gradients of the scalar basis span the space\. In 3D, split a divergence\-free polynomial into homogeneous partsvjv\_\{j\}of degreejj, each of which is divergence free\. The identity

∇ξ×\(vj×ξ\)=\(j\+2\)​vj\\nabla\_\{\\xi\}\\times\(v\_\{j\}\\times\\xi\)=\(j\+2\)v\_\{j\}\(79\)shows thatvjv\_\{j\}is the curl of the potential\(vj×ξ\)/\(j\+2\)\(v\_\{j\}\\times\\xi\)/\(j\+2\)of degreej\+1j\+1, and expanding the components of this potential in the scalar basis gives a combination of the candidates \([11](https://arxiv.org/html/2610.29009#S2.E11)\)\. For the dimension, the divergence maps the vector polynomials of degree at mostϖ\\varpionto the scalar polynomials of degree at mostϖ−1\\varpi\-1\(integrate in one coordinate to find a preimage\), and rank–nullity givesd​\(ϖ\+dd\)−\(ϖ\+d−1d\)d\\binom\{\\varpi\+d\}\{d\}\-\\binom\{\\varpi\+d\-1\}\{d\}\.

### C\.2Positive definiteness of the kernel

###### Proof of Proposition[1](https://arxiv.org/html/2610.29009#Thmproposition1)\.

The scaled principal coordinatesD∥​zD\_\{\\parallel\}zlie inℝnz\\mathbb\{R\}^\{n\_\{z\}\}withnz≤3n\_\{z\}\\leq 3, so the Wendland factor is a positive semidefinite kernel\. The Matérn factor is positive semidefinite on the residual space in every finite dimension\. The Gram matrix of their product is the Schur product of two positive semidefinite matrices, and multiplying it on both sides by the diagonal matrix of gates keeps it positive semidefinite\.

For strictness, group distinct standardized inputs by their distinct principal coordinatesz\(1\),…,z\(J\)z^\{\(1\)\},\\ldots,z^\{\(J\)\}, and letWWbe the Wendland Gram matrix on these, which is positive definite\. LetΨ⟂\\Psi\_\{\\perp\}be a feature map of the residual Matérn kernel, and for coefficientscic\_\{i\}let

fι=∑i:zi=z\(ι\)ciη\(i\)Ψ⟂\(ζi\)\.f\_\{\\iota\}=\\sum\_\{i:z\_\{i\}=z^\{\(\\iota\)\}\}c\_\{i\}\\eta\(\_\{i\}\)\\Psi\_\{\\perp\}\(\\zeta\_\{i\}\)\.\(80\)The quadratic form of the product Gram matrix is∑ι,ι′Wι​ι′​⟨fι,fι′⟩≥λmin​\(W\)​∑ι‖fι‖2\\sum\_\{\\iota,\\iota^\{\\prime\}\}W\_\{\\iota\\iota^\{\\prime\}\}\\langle f\_\{\\iota\},f\_\{\\iota^\{\\prime\}\}\\rangle\\geq\\lambda\_\{\\min\}\(W\)\\sum\_\{\\iota\}\\\|f\_\{\\iota\}\\\|^\{2\}\. Inputs in the same group differ in their residuals, so strict positive definiteness of the Matérn kernel givesfι≠0f\_\{\\iota\}\\neq 0whenever a coefficient of the group is nonzero and the gates are positive\. \(If the residual space is trivial, distinct inputs lie in distinct groups\.\) Finally,c𝖳​\(KU​U\+θ​I\)​c≥θ​‖c‖22\>0c^\{\\mathsf\{T\}\}\(K\_\{UU\}\+\\theta I\)c\\geq\\theta\\\|c\\\|\_\{2\}^\{2\}\>0for everyc≠0c\\neq 0andθ\>0\\theta\>0\. ∎

The proof does not assume that the inputs lie in a low\-dimensional affine subspace\. Note that standardization can make two inputs equal if they differ only in coordinates with zero training variance\. The regularization keeps the matrix positive definite in that case\.

### C\.3Variational form of the operator regression

Letfℋ∈ℋf\_\{\\mathcal\{H\}\}\\in\\mathcal\{H\}, and splitfℋ=f∥\+f⟂f\_\{\\mathcal\{H\}\}=f\_\{\\parallel\}\+f\_\{\\perp\}, wheref∥f\_\{\\parallel\}lies in the span of the functionsλ\(⋅,i\)𝐞k\\lambda\(\\cdot,\_\{i\}\)\\mathbf\{e\}\_\{k\}andf⟂f\_\{\\perp\}is orthogonal to it\. By the reproducing property,f⟂\(i\)=0f\_\{\\perp\}\(\_\{i\}\)=0for everyii, sof⟂f\_\{\\perp\}leaves the data term of \([42](https://arxiv.org/html/2610.29009#S3.E42)\) unchanged and only increases the norm\. A minimizer therefore has the formfℋ=∑iλ\(⋅,i\)𝐜if\_\{\\mathcal\{H\}\}=\\sum\_\{i\}\\lambda\(\\cdot,\_\{i\}\)\\mathbf\{c\}\_\{i\}, with‖fℋ‖ℋ2=tr⁡\(C𝖳​KU​U​C\)\\\|f\_\{\\mathcal\{H\}\}\\\|\_\{\\mathcal\{H\}\}^\{2\}=\\operatorname\{tr\}\(C^\{\\mathsf\{T\}\}K\_\{UU\}C\), whereCChas the rows𝐜i𝖳\\mathbf\{c\}\_\{i\}^\{\\mathsf\{T\}\}\. WithBBthe matrix with rows𝖳i\{\}\_\{i\}^\{\\mathsf\{T\}\}andP=PUP=P\_\{U\}, the objective becomes the convex quadratic

J⁡\(C,𝜷\)=‖KU​U​C\+P​𝜷−B‖F2\+θ​tr⁡\(C𝖳​KU​U​C\),J\(C,\\boldsymbol\{\\beta\}\)=\\\|K\_\{UU\}C\+P\\boldsymbol\{\\beta\}\-B\\\|\_\{F\}^\{2\}\+\\theta\\operatorname\{tr\}\(C^\{\\mathsf\{T\}\}K\_\{UU\}C\),\(81\)with gradients

∇CJ=2​KU​U​\(KU​U​C\+P​𝜷−B\+θ​C\),∇𝜷J=2​P𝖳​\(KU​U​C\+P​𝜷−B\)\.\\nabla\_\{C\}J=2K\_\{UU\}\\bigl\(K\_\{UU\}C\+P\\boldsymbol\{\\beta\}\-B\+\\theta C\\bigr\),\\qquad\\nabla\_\{\\boldsymbol\{\\beta\}\}J=2P^\{\\mathsf\{T\}\}\\bigl\(K\_\{UU\}C\+P\\boldsymbol\{\\beta\}\-B\\bigr\)\.\(82\)The conditions \([40](https://arxiv.org/html/2610.29009#S3.E40)\) state thatKU​U​C\+P​𝜷−B=−θ​CK\_\{UU\}C\+P\\boldsymbol\{\\beta\}\-B=\-\\theta CandP𝖳​C=0P^\{\\mathsf\{T\}\}C=0\. Both gradients therefore vanish, and every solution of \([40](https://arxiv.org/html/2610.29009#S3.E40)\) minimizesJJ, and hence \([42](https://arxiv.org/html/2610.29009#S3.E42)\)\. Without augmentation, the same argument applies withPPand𝜷\\boldsymbol\{\\beta\}removed\.

### C\.4Block elimination and reproduction

###### Proof of Proposition[2](https://arxiv.org/html/2610.29009#Thmproposition2)\.

WriteK=KU​Uθ=LK​LK𝖳K=K\_\{UU\}^\{\\theta\}=L\_\{K\}L\_\{K\}^\{\\mathsf\{T\}\}andP=PUP=P\_\{U\}\. SincePPhas full column rank,LK−1​P=QP​RPL\_\{K\}^\{\-1\}P=Q\_\{P\}R\_\{P\}withRPR\_\{P\}invertible, and

ℰU​P=RP−1​QP𝖳​LK−1​P=I,K​ℰU𝖳=P​RP−1​RP−𝖳\.\\mathcal\{E\}\_\{U\}P=R\_\{P\}^\{\-1\}Q\_\{P\}^\{\\mathsf\{T\}\}L\_\{K\}^\{\-1\}P=I,\\qquad K\\mathcal\{E\}\_\{U\}^\{\\mathsf\{T\}\}=PR\_\{P\}^\{\-1\}R\_\{P\}^\{\-\\mathsf\{T\}\}\.\(83\)Letℓ0=K−1​kU​\(\)\\ell\_\{0\}=K^\{\-1\}k\_\{U\}\(\)andp~=p⁡\(\)−P𝖳​ℓ0\\tilde\{p\}=p\(\)\-P^\{\\mathsf\{T\}\}\\ell\_\{0\}\. Thenℓ=ℓ0\+ℰU𝖳​p~\\ell=\\ell\_\{0\}\+\\mathcal\{E\}\_\{U\}^\{\\mathsf\{T\}\}\\tilde\{p\}satisfies

P𝖳​ℓ=P𝖳​ℓ0\+p~=p⁡\(\),K​ℓ=kU​\(\)\+P​RP−1​RP−𝖳​p~,P^\{\\mathsf\{T\}\}\\ell=P^\{\\mathsf\{T\}\}\\ell\_\{0\}\+\\tilde\{p\}=p\(\),\\qquad K\\ell=k\_\{U\}\(\)\+PR\_\{P\}^\{\-1\}R\_\{P\}^\{\-\\mathsf\{T\}\}\\tilde\{p\},\(84\)so \([51](https://arxiv.org/html/2610.29009#S4.E51)\) holds withμ=−RP−1​RP−𝖳​p~\\mu=\-R\_\{P\}^\{\-1\}R\_\{P\}^\{\-\\mathsf\{T\}\}\\tilde\{p\}\. If\(v,ω\)\(v,\\omega\)solves the homogeneous system, thenP𝖳​v=0P^\{\\mathsf\{T\}\}v=0andv𝖳​K​v=0v^\{\\mathsf\{T\}\}Kv=0, sov=0v=0, and full column rank givesω=0\\omega=0\. The solution is unique\.

For targets in the space of affine functions,Bj=PU​TjB\_\{j\}=P\_\{U\}T\_\{j\}, we getBj𝖳​ℓ=Tj𝖳​p​\(\)B\_\{j\}^\{\\mathsf\{T\}\}\\ell=T\_\{j\}^\{\\mathsf\{T\}\}p\(\), which is reproduction\. The coefficients of \([44](https://arxiv.org/html/2610.29009#S3.E44)\) are

Cj=ΠU​Bj,βj=ℰU​Bj,kU​\(\)𝖳​Cj\+p​\(\)𝖳​βj=ℓ​\(\)𝖳​Bj,C\_\{j\}=\\Pi\_\{U\}B\_\{j\},\\qquad\\beta\_\{j\}=\\mathcal\{E\}\_\{U\}B\_\{j\},\\qquad k\_\{U\}\(\)^\{\\mathsf\{T\}\}C\_\{j\}\+p\(\)^\{\\mathsf\{T\}\}\\beta\_\{j\}=\\ell\(\)^\{\\mathsf\{T\}\}B\_\{j\},\(85\)where the last identity follows by transposing the augmented system\. Direct and Lagrange evaluation thus agree in exact arithmetic\. Finally, linearity ofχ\\chiand𝒟\\mathcal\{D\}gives \([54](https://arxiv.org/html/2610.29009#S4.E54)\)\. For a nonzero common constraint𝒟χ\(i\)=f\\mathcal\{D\}\\chi\(\_\{i\}\)=f, the affine model also gives𝒟χ\(∑iℓi\)i=f\\mathcal\{D\}\\chi\(\\sum\_\{i\}\\ell\_\{i\}\{\}\_\{i\}\)=f, since its constant column enforces∑iℓi=1\\sum\_\{i\}\\ell\_\{i\}=1\. ∎

Without augmentation,ℓ=\(KU​Uθ\)−1​kU​\(\)\\ell=\(K\_\{UU\}^\{\\theta\}\)^\{\-1\}k\_\{U\}\(\)\. Ifθ=0\\theta=0, thenkU\(i\)k\_\{U\}\(\_\{i\}\)is theiith column ofKU​UK\_\{UU\}, soℓ\(i\)=𝐞i\\ell\(\_\{i\}\)=\\mathbf\{e\}\_\{i\}and the weights are cardinal\. Withθ\>0\\theta\>0, the vectorkUk\_\{U\}excludes the regularization, and the weights are no longer cardinal\.

### C\.5Rounding errors of direct and Lagrange evaluation

###### Proof of Theorem[3](https://arxiv.org/html/2610.29009#Thmproposition3)\.

The first residual equation givesC^=\(KU​Uθ\)−1​\(B−Rdir\)\\widehat\{C\}=\(K\_\{UU\}^\{\\theta\}\)^\{\-1\}\(B\-R\_\{\\rm dir\}\), and multiplying byKU∗​UK\_\{U\_\{\*\}U\}and\(𝐀⋆\)𝖳\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}gives \([62](https://arxiv.org/html/2610.29009#S4.E62)\)\. The second givesL^=\(KU∗​U−RLag\)​\(KU​Uθ\)−1\\widehat\{L\}=\(K\_\{U\_\{\*\}U\}\-R\_\{\\rm Lag\}\)\(K\_\{UU\}^\{\\theta\}\)^\{\-1\}, and hence \([63](https://arxiv.org/html/2610.29009#S4.E63)\)\.

For the rounding, letZ=fl⁡\(M1​M2\)Z=\\operatorname\{fl\}\(M\_\{1\}M\_\{2\}\)with\(M1,M2\)=\(KU∗​U,C^\)\(M\_\{1\},M\_\{2\}\)=\(K\_\{U\_\{\*\}U\},\\widehat\{C\}\)or\(L^,B\)\(\\widehat\{L\},B\)\. The standard bound for matrix products\[[28](https://arxiv.org/html/2610.29009#bib.bib44), Chapter 3\]gives\|Z−M1​M2\|≤γN​\|M1\|​\|M2\|\|Z\-M\_\{1\}M\_\{2\}\|\\leq\\gamma\_\{N\}\|M\_\{1\}\|\|M\_\{2\}\|, and the product with\(𝐀⋆\)𝖳\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}satisfies\|fl⁡\(Z​\(𝐀⋆\)𝖳\)−Z​\(𝐀⋆\)𝖳\|≤γr\+d​\|Z\|​\|\(𝐀⋆\)𝖳\|\|\\operatorname\{fl\}\(Z\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\)\-Z\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\|\\leq\\gamma\_\{r\+d\}\|Z\|\|\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\|, also when it is computed in component blocks \(dot products of length at mostrrfollowed by at mostd−1d\-1additions\)\. Together,

\|fl⁡\(fl⁡\(M1​M2\)​\(𝐀⋆\)𝖳\)−M1​M2​\(𝐀⋆\)𝖳\|≤γmult​\|M1​‖M2‖​\(𝐀⋆\)𝖳\|\.\\big\|\\operatorname\{fl\}\(\\operatorname\{fl\}\(M\_\{1\}M\_\{2\}\)\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\)\-M\_\{1\}M\_\{2\}\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\\big\|\\leq\\gamma\_\{\\rm mult\}\|M\_\{1\}\|\|M\_\{2\}\|\|\(\\mathbf\{A\}^\{\\star\}\)^\{\\mathsf\{T\}\}\|\.\(86\)The triangle inequality and submultiplicativity of the Frobenius norm then give \([64](https://arxiv.org/html/2610.29009#S4.E64)\)–\([65](https://arxiv.org/html/2610.29009#S4.E65)\)\. ∎

## Appendix DDetails of the numerical experiments

### D\.1Benchmarks

The ten data sets are those of our previous work\[[53](https://arxiv.org/html/2610.29009#bib.bib10)\], where their simulations are described in full\. Each figure shows the first simulation of a data set\. The color fields interpolate the simulation samples linearly \(Appendix[D\.3](https://arxiv.org/html/2610.29009#A4.SS3)\)\.

#### D\.1\.1Flow past a cylinder

Two\-dimensional flow passes a cylinder of radius0\.80\.8centered at\(1\.5,7\)\(1\.5,7\)in the channelΩa=Ωv=\[0,20\]×\[0,14\]\\Omega\_\{a\}=\\Omega\_\{v\}=\[0,20\]\\times\[0,14\], with an inflow on the left, freestream conditions at the top and bottom, no slip on the cylinder, and zero pressure at the outlet\. The constant initial velocity varies within the vortex\-shedding regime in one data set and below the onset of vortex shedding in the other\. We learn𝒢:𝐮⁡\(x,0\)→𝐮⁡\(y,10\)\\mathcal\{G\}:\\mathbf\{u\}\(x,0\)\\rightarrow\\mathbf\{u\}\(y,10\)onm=1000m=1000approximate Fekete points \(well\-spaced points selected from the mesh as in[Sharma et al\. \[53\]](https://arxiv.org/html/2610.29009#bib.bib10)\) of the95209520mesh points\.

![Refer to caption](https://arxiv.org/html/2610.29009v1/fastol_fields_flow_cylinder_shedding.png)Figure 12:Cylinder flow\. \(A\) Initial speed\. \(B\) Speed att=10t=10, on the same color scale\.
#### D\.1\.2Taylor–Green vortices

The analytical Taylor–Green vortices decay on the periodic domainΩa=Ωv=\[0,2​π\]2\\Omega\_\{a\}=\\Omega\_\{v\}=\[0,2\\pi\]^\{2\}, sampled at500500Fekete points\. For the spatial data set, we learn𝒢:𝐮⁡\(x,0\)→𝐮⁡\(y,1\)\\mathcal\{G\}:\\mathbf\{u\}\(x,0\)\\rightarrow\\mathbf\{u\}\(y,1\)\. For the spacetime data set, we learn the map from𝐮⁡\(x,0\)\\mathbf\{u\}\(x,0\)to the velocity onΩ×\[0\.7,1\]\\Omega\\times\[0\.7,1\], with snapshots at𝒯=\{0\.7,0\.8,0\.9,1\.0\}\\mathcal\{T\}=\\\{0\.7,0\.8,0\.9,1\.0\\\}and cubic interpolation in time \(Section[2\.4](https://arxiv.org/html/2610.29009#S2.SS4)\)\. Both use the amplitude normalization \([72](https://arxiv.org/html/2610.29009#S5.E72)\)\.

![Refer to caption](https://arxiv.org/html/2610.29009v1/fastol_fields_taylor_green.png)Figure 13:Taylor–Green vortices\. \(A\) Initial speed\. \(B\) Speed att=1t=1, on the same color scale\.
#### D\.1\.3Lid\-driven cavity

In the cavityΩa=Ωv=\[0,1\]2\\Omega\_\{a\}=\\Omega\_\{v\}=\[0,1\]^\{2\}, the speed of the lid is drawn from a truncated normal distribution, and the fluid starts at rest, so the initial velocity is nonzero only on the lid\. We learn𝒢:𝐮⁡\(x,0\)→𝐮⁡\(y,5\)\\mathcal\{G\}:\\mathbf\{u\}\(x,0\)\\rightarrow\\mathbf\{u\}\(y,5\)onm=1000m=1000Fekete points selected from the95669566mesh points\.

![Refer to caption](https://arxiv.org/html/2610.29009v1/fastol_fields_lid_cavity_flow.png)Figure 14:Lid\-driven cavity\. \(A\) Initial speed\. \(B\) Speed att=5t=5, on the same color scale\.
#### D\.1\.4Backward\-facing step

A parabolic inflow, scaled by a factor drawn from a truncated normal distribution, drives the flow over a step into the channelΩa=Ωv=\[0,15\]×\[−0\.5,0\.5\]\\Omega\_\{a\}=\\Omega\_\{v\}=\[0,15\]\\times\[\-0\.5,0\.5\]\. We learn𝒢:𝐮⁡\(x,0\)→𝐮⁡\(y,5\)\\mathcal\{G\}:\\mathbf\{u\}\(x,0\)\\rightarrow\\mathbf\{u\}\(y,5\)onm=1000m=1000Fekete points selected from the73597359mesh points\. The input has only1111coordinates, the samples of the inflow profile\.

![Refer to caption](https://arxiv.org/html/2610.29009v1/fastol_fields_backward_facing_step.png)Figure 15:Backward\-facing step\. \(A\) Inflow profile\. \(B\) Speed att=5t=5, with the transverse direction expanded\.
#### D\.1\.5Buoyancy\-driven cavity

In the cavityΩa=Ωv=\[0,1\]2\\Omega\_\{a\}=\\Omega\_\{v\}=\[0,1\]^\{2\}, an initial temperature fieldT⁡\(x,0\)T\(x,0\)drives the flow by buoyancy\. We learn the map fromT⁡\(x,0\)T\(x,0\)to the velocity att=5t=5onm=5000m=5000Fekete points selected from the1051410514mesh points\.

![Refer to caption](https://arxiv.org/html/2610.29009v1/fastol_fields_buoyancy_cavity_flow.png)Figure 16:Buoyancy\-driven cavity\. \(A\) Initial temperature\. \(B\) Speed att=5t=5\.
#### D\.1\.6Merging vortices

Two vortices merge on the periodic domainΩa=Ωv=\[0,2​π\]2\\Omega\_\{a\}=\\Omega\_\{v\}=\[0,2\\pi\]^\{2\}, and the width of their cores varies across the data set\. We learn𝒢:ω⁡\(x,0\)→𝐮⁡\(y,0\.4\)\\mathcal\{G\}:\\omega\(x,0\)\\rightarrow\\mathbf\{u\}\(y,0\.4\)from the initial vorticity on500500Fekete points selected from the74477447mesh points\.

![Refer to caption](https://arxiv.org/html/2610.29009v1/fastol_fields_merge_vortices_easier.png)Figure 17:Merging vortices\. \(A\) Initial vorticity\. \(B\) Speed att=0\.4t=0\.4\.
#### D\.1\.7Turbulent species transport

Air and methane mix turbulently in a static Kenics mixer, whose three blades twist along thezzaxis, at Reynolds numbers between10510^\{5\}and10610^\{6\}\. We learn the map from the inlet velocity \(13881388samples\) to the velocity att=0\.5t=0\.5on70007000Fekete points selected from1138811388locations\.

AInlet velocity![Refer to caption](https://arxiv.org/html/2610.29009v1/figures/species_transport/input.png)
BSpeed att=0\.5t=0\.5![Refer to caption](https://arxiv.org/html/2610.29009v1/figures/species_transport/output.png)

Figure 18:Turbulent species transport\. \(A\) Inlet velocity on the planez=0z=0\. \(B\) Speed att=0\.5t=0\.5on the whole domain\.
#### D\.1\.8Forced homogeneous isotropic turbulence

Forced isotropic turbulence evolves in the periodic boxΩa=Ωv=\[0,2​π\]3\\Omega\_\{a\}=\\Omega\_\{v\}=\[0,2\\pi\]^\{3\}, simulated by large\-eddy simulation with the Smagorinsky model on32332^\{3\}cells, withΔ​t=0\.002\\Delta t=0\.002, final timeT=0\.5T=0\.5, and Reynolds number6×1066\\times 10^\{6\}\. Every simulation starts from the divergence\-free field

ux​\(x,0\)\\displaystyle u\_\{x\}\(x,0\)=sin⁡\(2​x3\)\+cos⁡\(2​x2\)\+12​\(sin⁡\(3​x3\)\+cos⁡\(3​x2\)\)\+14​\(sin⁡\(5​x3\)\+cos⁡\(5​x2\)\),\\displaystyle=\\sin\(2x\_\{3\}\)\+\\cos\(2x\_\{2\}\)\+\\tfrac\{1\}\{2\}\\bigl\(\\sin\(3x\_\{3\}\)\+\\cos\(3x\_\{2\}\)\\bigr\)\+\\tfrac\{1\}\{4\}\\bigl\(\\sin\(5x\_\{3\}\)\+\\cos\(5x\_\{2\}\)\\bigr\),uy​\(x,0\)\\displaystyle u\_\{y\}\(x,0\)=sin⁡\(2​x1\)\+cos⁡\(2​x3\)\+12​\(sin⁡\(3​x1\)\+cos⁡\(3​x3\)\)\+14​\(sin⁡\(5​x1\)\+cos⁡\(5​x3\)\),\\displaystyle=\\sin\(2x\_\{1\}\)\+\\cos\(2x\_\{3\}\)\+\\tfrac\{1\}\{2\}\\bigl\(\\sin\(3x\_\{1\}\)\+\\cos\(3x\_\{3\}\)\\bigr\)\+\\tfrac\{1\}\{4\}\\bigl\(\\sin\(5x\_\{1\}\)\+\\cos\(5x\_\{3\}\)\\bigr\),uz​\(x,0\)\\displaystyle u\_\{z\}\(x,0\)=sin⁡\(2​x2\)\+cos⁡\(2​x1\)\+12​\(sin⁡\(3​x2\)\+cos⁡\(3​x1\)\)\+14​\(sin⁡\(5​x2\)\+cos⁡\(5​x1\)\),\\displaystyle=\\sin\(2x\_\{2\}\)\+\\cos\(2x\_\{1\}\)\+\\tfrac\{1\}\{2\}\\bigl\(\\sin\(3x\_\{2\}\)\+\\cos\(3x\_\{1\}\)\\bigr\)\+\\tfrac\{1\}\{4\}\\bigl\(\\sin\(5x\_\{2\}\)\+\\cos\(5x\_\{1\}\)\\bigr\),\(87\)and is driven by the steady divergence\-free forcing

𝐟⁡\(x\)=\[c1​\(sin⁡\(q1​x3\)\+cos⁡\(q2​x2\)\)c2​\(sin⁡\(q1​x1\)\+cos⁡\(q2​x3\)\)c3​\(sin⁡\(q1​x2\)\+cos⁡\(q2​x1\)\)\],\\mathbf\{f\}\(x\)=\\begin\{bmatrix\}c\_\{1\}\\bigl\(\\sin\(q\_\{1\}x\_\{3\}\)\+\\cos\(q\_\{2\}x\_\{2\}\)\\bigr\)\\\\ c\_\{2\}\\bigl\(\\sin\(q\_\{1\}x\_\{1\}\)\+\\cos\(q\_\{2\}x\_\{3\}\)\\bigr\)\\\\ c\_\{3\}\\bigl\(\\sin\(q\_\{1\}x\_\{2\}\)\+\\cos\(q\_\{2\}x\_\{1\}\)\\bigr\)\\end\{bmatrix\},\(88\)whose amplitudesc1,c2,c3c\_\{1\},c\_\{2\},c\_\{3\}are drawn from a normal distribution with standard deviation0\.150\.15truncated to\[−0\.3,0\.3\]\[\-0\.3,0\.3\], and whose distinct wavenumbersq1,q2q\_\{1\},q\_\{2\}are drawn from\{1,2,3\}\\\{1,2,3\\\}\. We learn the map from the sampled forcing to the velocity att=0\.5t=0\.5on70007000Fekete points of the grid\.

AForcing![Refer to caption](https://arxiv.org/html/2610.29009v1/figures/forced_turb/input.png)
BSpeed att=0\.5t=0\.5![Refer to caption](https://arxiv.org/html/2610.29009v1/figures/forced_turb/output.png)

Figure 19:Forced isotropic turbulence\. \(A\) Forcing magnitude\. \(B\) Speed att=0\.5t=0\.5\.

### D\.2Manufactured tests of the output reconstruction

#### D\.2\.1Domains and fields

The convergence tests of Section[5\.1](https://arxiv.org/html/2610.29009#S5.SS1)use the domainsΩd=\{Fd\>0\}\\Omega\_\{d\}=\\\{F\_\{d\}\>0\\\}with

F2​\(x,y\)\\displaystyle F\_\{2\}\(x,y\)=1−x2−1\.4​y2−0\.15​x4\\displaystyle=1\-x^\{2\}\-1\.4y^\{2\}\-0\.15x^\{4\}−0\.10​x2​y2−0\.20​y4\+0\.10​x3,\\displaystyle\\quad\-0\.10x^\{2\}y^\{2\}\-0\.20y^\{4\}\+0\.10x^\{3\},\(89\)F3​\(x,y,z\)\\displaystyle F\_\{3\}\(x,y,z\)=1−x2−1\.3​y2−1\.7​z2−0\.10​\(x4\+y4\+z4\)\\displaystyle=1\-x^\{2\}\-1\.3y^\{2\}\-1\.7z^\{2\}\-0\.10\(x^\{4\}\+y^\{4\}\+z^\{4\}\)−0\.08​x2​y2−0\.06​y2​z2\+0\.10​x​y​z\.\\displaystyle\\quad\-0\.08x^\{2\}y^\{2\}\-0\.06y^\{2\}z^\{2\}\+0\.10xyz\.\(90\)In 2D, the field isv=\(∂yψ,−∂xψ\)v=\(\\partial\_\{y\}\\psi,\-\\partial\_\{x\}\\psi\)with

ψ=F22​\[sin⁡\(π​x\)​sin⁡\(2​π​y\)\+0\.2​cos⁡\(3​π​x\)​sin⁡\(π​y\)\],\\displaystyle\\psi=F\_\{2\}^\{2\}\\bigl\[\\sin\(\\pi x\)\\sin\(2\\pi y\)\+0\.2\\cos\(3\\pi x\)\\sin\(\\pi y\)\\bigr\],\(91\)and in 3D,v=∇×Av=\\nabla\\times Awith

A=F32​\(sin⁡\(π​y\)​sin⁡\(2​π​z\),sin⁡\(π​z\)​sin⁡\(2​π​x\),sin⁡\(π​x\)​sin⁡\(2​π​y\)\)𝖳\.\\displaystyle A=F\_\{3\}^\{2\}\\bigl\(\\sin\(\\pi y\)\\sin\(2\\pi z\),\\sin\(\\pi z\)\\sin\(2\\pi x\),\\sin\(\\pi x\)\\sin\(2\\pi y\)\\bigr\)^\{\\mathsf\{T\}\}\.\(92\)Both fields are divergence free and vanish on the boundary\.

The node generator of KernelPack\-JAX\[[8](https://arxiv.org/html/2610.29009#bib.bib47)\]produced four node sets in each domain, at spacings0\.1030\.103,0\.06750\.0675,0\.04450\.0445, and0\.02950\.0295in 2D \(172172to19071907nodes\) and0\.220\.22,0\.160\.16,0\.1180\.118, and0\.0880\.088in 3D \(197197to25702570nodes\)\. The errors use fixed sets of40964096\(2D\) and3276832768\(3D\) uniformly random interior points\. For degreesϖ=1,3,5\\varpi=1,3,5, the divergence\-free polynomial spaces have dimensionsnP=5,14,27n\_\{P\}=5,14,27in 2D and11,50,13311,50,133in 3D, and every stencil keeps the full space\. At the finest level, the sixth\-order errors are about4\.5×10−64\.5\\times 10^\{\-6\}in 2D and1\.6×10−31\.6\\times 10^\{\-3\}in 3D for both kernels\.

#### D\.2\.2Divergence and periodicity

On the evaluation points where the output reconstruction is differentiable \(i\.e\., away from the centers of the cubic kernel\), we measure the divergence relative to the size of the velocity gradient,

Drms=\(\|QD\|−1​∑y∈QD\|∇⋅vh​\(y\)\|2\)1/2max⁡\{1,\(\|QD\|−1​∑y∈QD‖D​vh​\(y\)‖F2\)1/2\},\\displaystyle D\_\{\\rm rms\}=\\frac\{\\bigl\(\|Q\_\{D\}\|^\{\-1\}\\sum\_\{y\\in Q\_\{D\}\}\|\\nabla\\\!\\cdot v\_\{h\}\(y\)\|^\{2\}\\bigr\)^\{1/2\}\}\{\\max\\\{1,\\bigl\(\|Q\_\{D\}\|^\{\-1\}\\sum\_\{y\\in Q\_\{D\}\}\\\|Dv\_\{h\}\(y\)\\\|\_\{F\}^\{2\}\\bigr\)^\{1/2\}\\\}\},\(93\)with the derivative of each local expansion taken on its own stencil\. Its largest value over all smooth tests is1\.5×10−131\.5\\times 10^\{\-13\}\.

The periodic tests on\[0,2π\)d\[0,2\\pi\)^\{d\}use the potentials

ψ\\displaystyle\\psi=sin⁡x​cos⁡\(2​y\)\+0\.2​cos⁡\(3​x\)​sin⁡y,d=2,\\displaystyle=\\sin x\\cos\(2y\)\+0\.2\\cos\(3x\)\\sin y,\\qquad d=2,\(94\)A\\displaystyle A=\(sin⁡y​sin⁡\(2​z\),sin⁡z​sin⁡\(2​x\),sin⁡x​sin⁡\(2​y\)\)𝖳,d=3,\\displaystyle=\(\\sin y\\sin\(2z\),\\sin z\\sin\(2x\),\\sin x\\sin\(2y\)\)^\{\\mathsf\{T\}\},\\qquad d=3,\(95\)on657657\(2D\) and10421042\(3D\) scattered nodes\. We compare the reconstructed velocities and Jacobians at points that differ by integer multiples of the period, on the faces, edges, and corners of the box and at128128interior points\. The largest differences are1\.4×10−121\.4\\times 10^\{\-12\}for the velocities and1\.6×10−121\.6\\times 10^\{\-12\}for the Jacobians\.

#### D\.2\.3The turbulent ensemble

The statistical test of Section[5\.1](https://arxiv.org/html/2610.29009#S5.SS1)uses the solenoidal Fourier ensemble

v⁡\(y\)=∑𝐤∈𝒦\[𝐜𝐤​cos⁡\(𝐤⋅y\)\+𝐬𝐤​sin⁡\(𝐤⋅y\)\],𝔼⁡\[𝐜𝐤​𝐜𝐤𝖳\]=𝔼⁡\[𝐬𝐤​𝐬𝐤𝖳\]=w𝐤​\(I−𝐤𝐤𝖳‖𝐤‖22\),\\displaystyle v\(y\)=\\sum\_\{\\mathbf\{k\}\\in\\mathcal\{K\}\}\\bigl\[\\mathbf\{c\}\_\{\\mathbf\{k\}\}\\cos\(\\mathbf\{k\}\\cdot y\)\+\\mathbf\{s\}\_\{\\mathbf\{k\}\}\\sin\(\\mathbf\{k\}\\cdot y\)\\bigr\],\\qquad\\mathbb\{E\}\[\\mathbf\{c\}\_\{\\mathbf\{k\}\}\\mathbf\{c\}\_\{\\mathbf\{k\}\}^\{\\mathsf\{T\}\}\]=\\mathbb\{E\}\[\\mathbf\{s\}\_\{\\mathbf\{k\}\}\\mathbf\{s\}\_\{\\mathbf\{k\}\}^\{\\mathsf\{T\}\}\]=w\_\{\\mathbf\{k\}\}\\left\(I\-\\frac\{\\mathbf\{k\}\\mathbf\{k\}^\{\\mathsf\{T\}\}\}\{\\\|\\mathbf\{k\}\\\|\_\{2\}^\{2\}\}\\right\),\(96\)with independent Gaussian coefficients perpendicular to their wave vectors,15141514modes up to wavenumber10241024, and weightsw𝐤w\_\{\\mathbf\{k\}\}that integrate a∥𝐤∥2−5/3\\\|\\mathbf\{k\}\\\|\_\{2\}^\{\-5/3\}energy spectrum over radial shells, normalized so that2​∑𝐤w𝐤=12\\sum\_\{\\mathbf\{k\}\}w\_\{\\mathbf\{k\}\}=1\. Its covariance is divergence free, and its structure function approximately follows the Kolmogorov law over the separations0\.03≤r≤0\.80\.03\\leq r\\leq 0\.8, which we fixed before any output reconstruction\. At smaller separations, the finite bandwidth makes the increments smooth\. We restrict the ensemble toΩ3\\Omega\_\{3\}and reconstruct it on six node sets with spacings0\.260\.26,0\.190\.19,0\.140\.14,0\.1010\.101,0\.0750\.075, and0\.050\.05\(128128to1356713567nodes\)\. The structure functions \([70](https://arxiv.org/html/2610.29009#S5.E70)\) use3232separations from10−510^\{\-5\}to0\.80\.8in eight fixed directions and are computed exactly from the covariance, not from samples\. For degree five,ES2​\(0\.8\)E\_\{S\_\{2\}\}\(0\.8\)falls from14%14\\%on the coarsest set to4\.6%4\.6\\%on the finest, while the worst error over the whole range remains64%64\\%\.

For the comparison of local and global output reconstruction, we use the node set with17701770nodes and one fixed five\-scale Matérn kernel for both, with the same polynomials\. The global system uses the covariance anchored at a pointaa,K⁡\(x,y\)−K⁡\(x,a\)−K⁡\(a,y\)\+K⁡\(a,a\)K\(x,y\)\-K\(x,a\)\-K\(a,y\)\+K\(a,a\), which is positive definite and gives the same interpolant, since the constant polynomials absorb the difference\. For degree five, the errorsES2​\(0\.8\)E\_\{S\_\{2\}\}\(0\.8\)are9\.9%9\.9\\%\(local\) and10\.6%10\.6\\%\(global\), the normalized mean square errors of the increments are0\.1680\.168and0\.1660\.166, and the worst errors over the whole range are about87%87\\%for both\.

### D\.3Field figures

The field figures show the first simulation of each data set and are drawn by the MATLAB scripts in our repository\. In two dimensions, the speed‖𝐮‖2\\\|\\mathbf\{u\}\\\|\_\{2\}is interpolated linearly from the samples\. Panels of the same quantity share a color scale, and the transverse direction of the backward\-facing step is expanded\.

### D\.4Direct and Lagrange evaluation

Table[6](https://arxiv.org/html/2610.29009#S5.T6)evaluates seven unaugmented operator models for each data set and eachN∈\{500,1000,5000,10000\}N\\in\\\{500,1000,5000,10000\\\}: the fixed\-shape kernel at the radii of the five prescribed densities of Section[3\.6](https://arxiv.org/html/2610.29009#S3.SS6)\(from10%10\\%to90%90\\%, withθ=10−6\\theta=10^\{\-6\}\) and at its validated radius and regularization, and the fitted\-shape kernel at its validated parameters\. We rebuild each model from its selected parameters and check that its Lagrange predictions at the nodes agree with those of Section[5\.2](https://arxiv.org/html/2610.29009#S5.SS2)\(largest relative difference1\.0×10−101\.0\\times 10^\{\-10\}\)\. Each model is then evaluated through its coefficients and through its Lagrange weights, with the same normalization, factorization \(cuDSS with nested dissection\), support, and regularization, and with the same spatial basis\. With the local output reconstruction, direct evaluation solves the kernel system for the coefficients of all stencils, in blocks of stencils, and Lagrange evaluation applies the weights to the same coefficients\. The global output reconstruction is that of Appendix[A\.5](https://arxiv.org/html/2610.29009#A1.SS5), on the nodes\. The test errors are measured at the nodes on the100100test fields\.

Table 8:Test error at the nodes \(mean relative velocity error over the 100 test inputs\) of direct and Lagrange evaluation of the validated unaugmented fixed\-shape model of the lid\-driven cavity with global output reconstruction, of direct evaluation applied to the sampled velocities instead of the coefficients, and the Frobenius norm of the spatial coefficients of the training fields\. Bold: smallest test error in each row\.NNDirectLagrangeDirect/LagrangeDirect on samples‖B‖F\\\|B\\\|\_\{F\}5001\.54×10−51\.54\\times 10^\{\-5\}1\.20×𝟏𝟎−𝟓\\mathbf\{1\.20\\times 10^\{\-5\}\}1\.281\.20×𝟏𝟎−𝟓\\mathbf\{1\.20\\times 10^\{\-5\}\}1\.58×1091\.58\\times 10^\{9\}10008\.39×10−68\.39\\times 10^\{\-6\}4\.25×10−64\.25\\times 10^\{\-6\}1\.974\.24×𝟏𝟎−𝟔\\mathbf\{4\.24\\times 10^\{\-6\}\}2\.24×1092\.24\\times 10^\{9\}50003\.19×10−63\.19\\times 10^\{\-6\}9\.54×10−79\.54\\times 10^\{\-7\}3\.349\.32×𝟏𝟎−𝟕\\mathbf\{9\.32\\times 10^\{\-7\}\}5\.00×1095\.00\\times 10^\{9\}100003\.42×10−63\.42\\times 10^\{\-6\}6\.64×10−76\.64\\times 10^\{\-7\}5\.146\.42×𝟏𝟎−𝟕\\mathbf\{6\.42\\times 10^\{\-7\}\}7\.08×1097\.08\\times 10^\{9\}Table[8](https://arxiv.org/html/2610.29009#A4.T8)follows the validated fixed\-shape model of the lid\-driven cavity with global output reconstruction asNNgrows\. The Frobenius norm of the spatial coefficients of the training fields grows from1\.6×1091\.6\\times 10^\{9\}to7\.1×1097\.1\\times 10^\{9\}, which shows the cancellation that Theorem[3](https://arxiv.org/html/2610.29009#Thmproposition3)warns about, and the ratio of the direct to the Lagrange error grows from1\.31\.3to5\.15\.1\. Direct evaluation applied to the sampled velocities instead of the coefficients is as accurate as Lagrange evaluation, so the gain arises in the output reconstruction, through the large coefficients\.

### D\.5Software

Our implementation and the scripts of this paper are part of KernelPack\-JAX \([https://github\.com/VarShankar/kernelpack\-jax](https://github.com/VarShankar/kernelpack-jax)\)\. The operator models of Section[5\.2](https://arxiv.org/html/2610.29009#S5.SS2)were fitted withscripts/run\_fastol\_gpu\_campaign\.py\. The comparisons of Section[5\.1\.4](https://arxiv.org/html/2610.29009#S5.SS1.SSS4), Section[5\.3](https://arxiv.org/html/2610.29009#S5.SS3), and Appendices[A\.5](https://arxiv.org/html/2610.29009#A1.SS5)and[D\.4](https://arxiv.org/html/2610.29009#A4.SS4)rebuild these models with the released code, whose predictions agree with those of the original fits to1\.0×10−101\.0\\times 10^\{\-10\}\(Appendix[D\.4](https://arxiv.org/html/2610.29009#A4.SS4)\), and run on the same GPU withscripts/fastol\_gpu\_evaluation\_study\.py\. The directoryFastOLPaperlists the scripts that produce every result and holds the scripts that build the tables and figures from them\.

### D\.6All operator benchmarks

This section collects the results of all ten benchmarks for every training size\. Figures[20](https://arxiv.org/html/2610.29009#A4.F20)and[23](https://arxiv.org/html/2610.29009#A4.F23)complete Figures[8](https://arxiv.org/html/2610.29009#S5.F8)and[10](https://arxiv.org/html/2610.29009#S5.F10), Figures[21](https://arxiv.org/html/2610.29009#A4.F21)and[22](https://arxiv.org/html/2610.29009#A4.F22)complete Figures[7](https://arxiv.org/html/2610.29009#S5.F7)and[9](https://arxiv.org/html/2610.29009#S5.F9), and Table[9](https://arxiv.org/html/2610.29009#A4.T9)gives the rates at which the errors decrease withNN\. Tables[10](https://arxiv.org/html/2610.29009#A4.T10)and[11](https://arxiv.org/html/2610.29009#A4.T11)collect the errors, storage, and prediction times atN=10000N=10000, and Tables[12](https://arxiv.org/html/2610.29009#A4.T12)–[21](https://arxiv.org/html/2610.29009#A4.T21)list all160160fitted models\. Two entries show why we report storage separately from density\. The fixed\-shape and fitted\-shape augmented species models have densities of9\.1%9\.1\\%and21\.7%21\.7\\%but2626and3535million factor entries, and the factor of the backward\-facing step, at33%33\\%density, has3838million entries, three quarters of a dense one\. All cost columns use the nested\-dissection ordering of Section[3\.5](https://arxiv.org/html/2610.29009#S3.SS5), measured by a replay of the selected models \(scripts/fastol\_gpu\_cost\_replay\.py\)\. The test errors were computed with the approximate\-minimum\-degree ordering, and the predictions with the two orderings agree to1\.2×10−101\.2\\times 10^\{\-10\}\.

Table 9:Least\-squares exponentτ\\tauof the test error∝N−τ\\propto N^\{\-\\tau\}overN∈\{500,1000,5000,10000\}N\\in\\\{500,1000,5000,10000\\\}\. Bold: fastest decay in each row\.Data setFixedFittedFixed \+ affineFitted \+ affineForced isotropic turbulence0\.680\.690\.760\.66Turbulent species transport0\.851\.091\.031\.26Merging vortices0\.770\.730\.570\.53Lid\-driven cavity0\.971\.090\.961\.08Cylinder flow \(no shedding\)0\.910\.870\.850\.84Cylinder flow \(vortex shedding\)0\.540\.530\.440\.47Backward\-facing step0\.390\.420\.140\.15Buoyancy\-driven cavity0\.820\.910\.920\.96Taylor–Green vortices \(space\)−\-0\.56−\-1\.062\.813\.24Taylor–Green vortices \(spacetime\)−\-0\.58−\-1\.032\.793\.22Table 10:Test error \(mean relative velocity error of the predictions for the 100 test inputs\) atN=10000N=10000\. The two augmented models use the same affine columns\. Each model is fitted and validated separately, and all four use the same local output reconstruction\. Both Taylor–Green cases use the amplitude normalization \([72](https://arxiv.org/html/2610.29009#S5.E72)\)\. Bold: smallest test error in each row\.Data setFixedFittedFixed \+ affineFitted \+ affineForced isotropic turbulence2\.80×10−32\.80\\times 10^\{\-3\}1\.62×10−31\.62\\times 10^\{\-3\}9\.77×10−59\.77\\times 10^\{\-5\}7\.14×𝟏𝟎−𝟓\\mathbf\{7\.14\\times 10^\{\-5\}\}Turbulent species transport4\.34×10−44\.34\\times 10^\{\-4\}6\.86×10−56\.86\\times 10^\{\-5\}1\.89×10−41\.89\\times 10^\{\-4\}3\.69×𝟏𝟎−𝟓\\mathbf\{3\.69\\times 10^\{\-5\}\}Merging vortices3\.02×𝟏𝟎−𝟕\\mathbf\{3\.02\\times 10^\{\-7\}\}3\.07×10−73\.07\\times 10^\{\-7\}3\.39×10−73\.39\\times 10^\{\-7\}3\.19×10−73\.19\\times 10^\{\-7\}Lid\-driven cavity6\.42×10−76\.42\\times 10^\{\-7\}4\.76×10−74\.76\\times 10^\{\-7\}6\.42×10−76\.42\\times 10^\{\-7\}4\.72×𝟏𝟎−𝟕\\mathbf\{4\.72\\times 10^\{\-7\}\}Cylinder flow \(no shedding\)2\.02×10−72\.02\\times 10^\{\-7\}1\.94×10−71\.94\\times 10^\{\-7\}1\.70×𝟏𝟎−𝟕\\mathbf\{1\.70\\times 10^\{\-7\}\}1\.73×10−71\.73\\times 10^\{\-7\}Cylinder flow \(vortex shedding\)3\.19×10−73\.19\\times 10^\{\-7\}2\.95×𝟏𝟎−𝟕\\mathbf\{2\.95\\times 10^\{\-7\}\}3\.19×10−73\.19\\times 10^\{\-7\}2\.96×10−72\.96\\times 10^\{\-7\}Taylor–Green vortices \(space\)2\.82×10−62\.82\\times 10^\{\-6\}2\.11×10−62\.11\\times 10^\{\-6\}2\.42×10−102\.42\\times 10^\{\-10\}5\.22×𝟏𝟎−𝟏𝟏\\mathbf\{5\.22\\times 10^\{\-11\}\}Taylor–Green vortices \(spacetime\)2\.46×10−62\.46\\times 10^\{\-6\}1\.86×10−61\.86\\times 10^\{\-6\}2\.18×10−102\.18\\times 10^\{\-10\}4\.70×𝟏𝟎−𝟏𝟏\\mathbf\{4\.70\\times 10^\{\-11\}\}Backward\-facing step6\.33×10−86\.33\\times 10^\{\-8\}6\.33×10−86\.33\\times 10^\{\-8\}5\.86×𝟏𝟎−𝟖\\mathbf\{5\.86\\times 10^\{\-8\}\}5\.86×𝟏𝟎−𝟖\\mathbf\{5\.86\\times 10^\{\-8\}\}Buoyancy\-driven cavity5\.38×10−65\.38\\times 10^\{\-6\}4\.47×𝟏𝟎−𝟔\\mathbf\{4\.47\\times 10^\{\-6\}\}5\.38×10−65\.38\\times 10^\{\-6\}4\.62×10−64\.62\\times 10^\{\-6\}Table 11:Accounted storage in decimal GB / median warm prediction time in ms for 100 fields atN=10000N=10000\. Storage includes the stored model arrays, the solve buffers, and cuDSS’s estimate of the memory of its factors\. It is not measured peak GPU memory\. Times include operator Lagrange evaluation and local output reconstruction\. Bold: smallest storage and smallest time in each row\.Data setFixedFittedFixed \+ affineFitted \+ affineForced isotropic turbulence10\.680/ 235\.210\.680/ 234\.010\.685 / 244\.110\.682 /233\.4Turbulent species transport2\.662 / 126\.63\.130 / 126\.02\.630/123\.33\.137 / 137\.4Merging vortices1\.397 / 79\.61\.029 / 62\.40\.772 /37\.70\.767/ 39\.0Lid\-driven cavity1\.135/ 75\.91\.135/ 71\.51\.136 /70\.21\.136 / 70\.3Cylinder flow \(no shedding\)1\.141/ 73\.81\.141/73\.71\.142 / 75\.71\.142 / 74\.0Cylinder flow \(vortex shedding\)1\.140/ 76\.21\.140/ 75\.31\.141 / 76\.21\.141 /74\.2Taylor–Green vortices \(space\)3\.606 / 117\.04\.334 / 119\.00\.899/52\.30\.899/ 54\.6Taylor–Green vortices \(spacetime\)3\.766 / 119\.84\.494 / 120\.11\.059/59\.01\.059/ 61\.7Backward\-facing step1\.679 / 72\.41\.679 / 68\.81\.657/ 66\.31\.657/65\.3Buoyancy\-driven cavity2\.504/ 189\.42\.504/ 184\.92\.505 / 177\.02\.505 /173\.4Figure 20:Test error of the other six benchmarks\. Dashed: with affine augmentation\.Figure 21:Selected density of the other six benchmarks\. With augmentation, the Taylor–Green models select about10%10\\%density at the two largestNN, against75%75\\%or90%90\\%without it\.Figure 22:Accounted storage of the other six benchmarks \(Appendix[B\.3](https://arxiv.org/html/2610.29009#A2.SS3)\)\.Figure 23:Median warm time to predict all100100test fields for the other six benchmarks\.Table 12:Forced isotropic turbulence: all models\. Test error is the mean relative velocity error of the predictions for the 100 test inputs\. Density isdK=nnz⁡\(KU​U\)/N2d\_\{K\}=\\mathrm\{nnz\}\(K\_\{UU\}\)/N^\{2\}in percent\. Factor is the number of entries of the cuDSS Cholesky factor in millions\. GB is the accounted storage and ms the median warm time for all 100 fields\. Bold: smallest test error at eachNN\.NNOperator modelTest errorDensity \(%\)Factor \(10610^\{6\}\)GBms500Fixed shape0\.021490\.000\.1250\.46560\.9500Fitted shape0\.01390\.000\.1250\.46564\.3500Fixed shape \+ affine9\.37×10−49\.37\\times 10^\{\-4\}90\.000\.1250\.46758\.5500Fitted shape \+ affine4\.90×𝟏𝟎−𝟒\\mathbf\{4\.90\\times 10^\{\-4\}\}90\.000\.1260\.46756\.51000Fixed shape0\.015790\.000\.9900\.75064\.81000Fitted shape0\.013990\.000\.9910\.75070\.21000Fixed shape \+ affine6\.05×10−46\.05\\times 10^\{\-4\}90\.000\.9900\.75265\.21000Fitted shape \+ affine3\.43×𝟏𝟎−𝟒\\mathbf\{3\.43\\times 10^\{\-4\}\}90\.000\.9910\.75265\.55000Fixed shape5\.16×10−35\.16\\times 10^\{\-3\}90\.0024\.7224\.390126\.85000Fitted shape4\.69×10−34\.69\\times 10^\{\-3\}90\.0024\.7484\.390127\.25000Fixed shape \+ affine1\.67×10−41\.67\\times 10^\{\-4\}90\.0024\.7224\.394126\.95000Fitted shape \+ affine1\.10×𝟏𝟎−𝟒\\mathbf\{1\.10\\times 10^\{\-4\}\}90\.0024\.7514\.394126\.110000Fixed shape2\.80×10−32\.80\\times 10^\{\-3\}90\.0098\.89910\.680235\.210000Fitted shape1\.62×10−31\.62\\times 10^\{\-3\}90\.0098\.98410\.680234\.010000Fixed shape \+ affine9\.77×10−59\.77\\times 10^\{\-5\}90\.0098\.89910\.685244\.110000Fitted shape \+ affine7\.14×𝟏𝟎−𝟓\\mathbf\{7\.14\\times 10^\{\-5\}\}90\.0098\.65110\.682233\.4Table 13:Turbulent species transport: all models\. Test error is the mean relative velocity error of the predictions for the 100 test inputs\. Density isdK=nnz⁡\(KU​U\)/N2d\_\{K\}=\\mathrm\{nnz\}\(K\_\{UU\}\)/N^\{2\}in percent\. Factor is the number of entries of the cuDSS Cholesky factor in millions\. GB is the accounted storage and ms the median warm time for all 100 fields\. Bold: smallest test error at eachNN\.NNOperator modelTest errorDensity \(%\)Factor \(10610^\{6\}\)GBms500Fixed shape4\.39×10−34\.39\\times 10^\{\-3\}63\.940\.1160\.22654\.3500Fitted shape2\.50×10−32\.50\\times 10^\{\-3\}90\.000\.1290\.22852\.8500Fixed shape \+ affine3\.35×10−33\.35\\times 10^\{\-3\}40\.980\.0970\.22453\.1500Fitted shape \+ affine2\.48×𝟏𝟎−𝟑\\mathbf\{2\.48\\times 10^\{\-3\}\}44\.600\.0960\.22454\.01000Fixed shape1\.25×10−31\.25\\times 10^\{\-3\}89\.711\.0000\.27857\.11000Fitted shape9\.53×10−49\.53\\times 10^\{\-4\}85\.620\.9620\.27657\.81000Fixed shape \+ affine1\.49×10−31\.49\\times 10^\{\-3\}22\.920\.3130\.25055\.61000Fitted shape \+ affine8\.75×𝟏𝟎−𝟒\\mathbf\{8\.75\\times 10^\{\-4\}\}75\.000\.9880\.27359\.15000Fixed shape2\.02×10−42\.02\\times 10^\{\-4\}69\.0624\.7431\.85898\.25000Fitted shape2\.95×10−42\.95\\times 10^\{\-4\}90\.0024\.7442\.036101\.75000Fixed shape \+ affine1\.96×𝟏𝟎−𝟒\\mathbf\{1\.96\\times 10^\{\-4\}\}69\.5624\.6891\.86298\.55000Fitted shape \+ affine2\.55×10−42\.55\\times 10^\{\-4\}66\.7124\.7461\.83898\.310000Fixed shape4\.34×10−44\.34\\times 10^\{\-4\}10\.4024\.4312\.662126\.610000Fitted shape6\.86×10−56\.86\\times 10^\{\-5\}21\.9533\.6133\.130126\.010000Fixed shape \+ affine1\.89×10−41\.89\\times 10^\{\-4\}9\.1225\.7402\.630123\.310000Fitted shape \+ affine3\.69×𝟏𝟎−𝟓\\mathbf\{3\.69\\times 10^\{\-5\}\}21\.7135\.3853\.137137\.4Table 14:Merging vortices: all models\. Test error is the mean relative velocity error of the predictions for the 100 test inputs\. Density isdK=nnz⁡\(KU​U\)/N2d\_\{K\}=\\mathrm\{nnz\}\(K\_\{UU\}\)/N^\{2\}in percent\. Factor is the number of entries of the cuDSS Cholesky factor in millions\. GB is the accounted storage and ms the median warm time for all 100 fields\. Bold: smallest test error at eachNN\.NNOperator modelTest errorDensity \(%\)Factor \(10610^\{6\}\)GBms500Fixed shape2\.85×10−62\.85\\times 10^\{\-6\}75\.060\.1040\.0256\.0500Fitted shape2\.51×10−62\.51\\times 10^\{\-6\}75\.000\.1060\.0256\.8500Fixed shape \+ affine1\.94×10−61\.94\\times 10^\{\-6\}37\.360\.0650\.0217\.9500Fitted shape \+ affine1\.66×𝟏𝟎−𝟔\\mathbf\{1\.66\\times 10^\{\-6\}\}33\.640\.0600\.0216\.91000Fixed shape1\.41×10−61\.41\\times 10^\{\-6\}63\.810\.3860\.0607\.31000Fitted shape1\.45×10−61\.45\\times 10^\{\-6\}60\.310\.3870\.0588\.61000Fixed shape \+ affine8\.60×10−78\.60\\times 10^\{\-7\}29\.730\.2250\.0479\.11000Fitted shape \+ affine8\.46×𝟏𝟎−𝟕\\mathbf\{8\.46\\times 10^\{\-7\}\}29\.040\.2170\.0477\.45000Fixed shape3\.54×𝟏𝟎−𝟕\\mathbf\{3\.54\\times 10^\{\-7\}\}24\.586\.4040\.43324\.95000Fitted shape3\.91×10−73\.91\\times 10^\{\-7\}25\.006\.3560\.43625\.25000Fixed shape \+ affine3\.66×10−73\.66\\times 10^\{\-7\}24\.586\.4040\.43424\.95000Fitted shape \+ affine4\.08×10−74\.08\\times 10^\{\-7\}18\.334\.8920\.36921\.710000Fixed shape3\.02×𝟏𝟎−𝟕\\mathbf\{3\.02\\times 10^\{\-7\}\}24\.6726\.3531\.39779\.610000Fitted shape3\.07×10−73\.07\\times 10^\{\-7\}15\.8518\.0091\.02962\.410000Fixed shape \+ affine3\.39×10−73\.39\\times 10^\{\-7\}10\.0010\.7030\.77237\.710000Fitted shape \+ affine3\.19×10−73\.19\\times 10^\{\-7\}10\.0010\.0470\.76739\.0Table 15:Lid\-driven cavity: all models\. Test error is the mean relative velocity error of the predictions for the 100 test inputs\. Density isdK=nnz⁡\(KU​U\)/N2d\_\{K\}=\\mathrm\{nnz\}\(K\_\{UU\}\)/N^\{2\}in percent\. Factor is the number of entries of the cuDSS Cholesky factor in millions\. GB is the accounted storage and ms the median warm time for all 100 fields\. Bold: smallest test error at eachNN\.NNOperator modelTest errorDensity \(%\)Factor \(10610^\{6\}\)GBms500Fixed shape1\.20×10−51\.20\\times 10^\{\-5\}22\.850\.0530\.0469\.2500Fitted shape1\.20×10−51\.20\\times 10^\{\-5\}22\.850\.0530\.04610\.8500Fixed shape \+ affine1\.15×𝟏𝟎−𝟓\\mathbf\{1\.15\\times 10^\{\-5\}\}21\.490\.0480\.04610\.4500Fitted shape \+ affine1\.15×𝟏𝟎−𝟓\\mathbf\{1\.15\\times 10^\{\-5\}\}25\.000\.0550\.0479\.21000Fixed shape4\.24×10−64\.24\\times 10^\{\-6\}19\.990\.1900\.09610\.61000Fitted shape4\.11×10−64\.11\\times 10^\{\-6\}25\.000\.2230\.09811\.41000Fixed shape \+ affine4\.13×10−64\.13\\times 10^\{\-6\}20\.000\.1890\.09610\.81000Fitted shape \+ affine4\.10×𝟏𝟎−𝟔\\mathbf\{4\.10\\times 10^\{\-6\}\}25\.000\.2230\.09811\.15000Fixed shape9\.32×10−79\.32\\times 10^\{\-7\}10\.002\.6000\.53834\.45000Fitted shape6\.73×𝟏𝟎−𝟕\\mathbf\{6\.73\\times 10^\{\-7\}\}10\.002\.6000\.53834\.65000Fixed shape \+ affine9\.32×10−79\.32\\times 10^\{\-7\}10\.002\.6000\.53934\.25000Fitted shape \+ affine6\.80×10−76\.80\\times 10^\{\-7\}10\.002\.6000\.53934\.710000Fixed shape6\.42×10−76\.42\\times 10^\{\-7\}10\.0011\.1981\.13575\.910000Fitted shape4\.76×10−74\.76\\times 10^\{\-7\}10\.0011\.1981\.13571\.510000Fixed shape \+ affine6\.42×10−76\.42\\times 10^\{\-7\}10\.0011\.1981\.13670\.210000Fitted shape \+ affine4\.72×𝟏𝟎−𝟕\\mathbf\{4\.72\\times 10^\{\-7\}\}10\.0011\.1981\.13670\.3Table 16:Cylinder flow \(no shedding\): all models\. Test error is the mean relative velocity error of the predictions for the 100 test inputs\. Density isdK=nnz⁡\(KU​U\)/N2d\_\{K\}=\\mathrm\{nnz\}\(K\_\{UU\}\)/N^\{2\}in percent\. Factor is the number of entries of the cuDSS Cholesky factor in millions\. GB is the accounted storage and ms the median warm time for all 100 fields\. Bold: smallest test error at eachNN\.NNOperator modelTest errorDensity \(%\)Factor \(10610^\{6\}\)GBms500Fixed shape2\.91×10−62\.91\\times 10^\{\-6\}89\.110\.1290\.0539\.8500Fitted shape2\.42×10−62\.42\\times 10^\{\-6\}90\.000\.1280\.0539\.6500Fixed shape \+ affine2\.27×10−62\.27\\times 10^\{\-6\}85\.520\.1280\.05210\.6500Fitted shape \+ affine2\.20×𝟏𝟎−𝟔\\mathbf\{2\.20\\times 10^\{\-6\}\}90\.000\.1280\.05310\.91000Fixed shape9\.21×10−79\.21\\times 10^\{\-7\}54\.740\.6740\.11113\.51000Fitted shape9\.21×10−79\.21\\times 10^\{\-7\}54\.740\.6740\.11113\.31000Fixed shape \+ affine6\.80×10−76\.80\\times 10^\{\-7\}45\.100\.5810\.10713\.41000Fitted shape \+ affine6\.45×𝟏𝟎−𝟕\\mathbf\{6\.45\\times 10^\{\-7\}\}50\.000\.3940\.10712\.65000Fixed shape1\.97×10−71\.97\\times 10^\{\-7\}16\.224\.5550\.60726\.05000Fitted shape1\.97×10−71\.97\\times 10^\{\-7\}16\.224\.5550\.60727\.25000Fixed shape \+ affine1\.93×10−71\.93\\times 10^\{\-7\}10\.002\.7260\.54020\.55000Fitted shape \+ affine1\.81×𝟏𝟎−𝟕\\mathbf\{1\.81\\times 10^\{\-7\}\}16\.224\.5550\.60826\.310000Fixed shape2\.02×10−72\.02\\times 10^\{\-7\}10\.0011\.9811\.14173\.810000Fitted shape1\.94×10−71\.94\\times 10^\{\-7\}10\.0011\.9811\.14173\.710000Fixed shape \+ affine1\.70×𝟏𝟎−𝟕\\mathbf\{1\.70\\times 10^\{\-7\}\}10\.0011\.9811\.14275\.710000Fitted shape \+ affine1\.73×10−71\.73\\times 10^\{\-7\}10\.0011\.9811\.14274\.0Table 17:Cylinder flow \(vortex shedding\): all models\. Test error is the mean relative velocity error of the predictions for the 100 test inputs\. Density isdK=nnz⁡\(KU​U\)/N2d\_\{K\}=\\mathrm\{nnz\}\(K\_\{UU\}\)/N^\{2\}in percent\. Factor is the number of entries of the cuDSS Cholesky factor in millions\. GB is the accounted storage and ms the median warm time for all 100 fields\. Bold: smallest test error at eachNN\.NNOperator modelTest errorDensity \(%\)Factor \(10610^\{6\}\)GBms500Fixed shape1\.35×10−61\.35\\times 10^\{\-6\}70\.210\.1220\.05112\.2500Fitted shape1\.35×10−61\.35\\times 10^\{\-6\}70\.210\.1220\.0519\.7500Fixed shape \+ affine1\.11×10−61\.11\\times 10^\{\-6\}68\.610\.1220\.05112\.6500Fitted shape \+ affine1\.09×𝟏𝟎−𝟔\\mathbf\{1\.09\\times 10^\{\-6\}\}75\.000\.1250\.05112\.01000Fixed shape1\.19×10−61\.19\\times 10^\{\-6\}27\.920\.2620\.09911\.61000Fitted shape8\.54×10−78\.54\\times 10^\{\-7\}25\.000\.2310\.09811\.91000Fixed shape \+ affine8\.31×𝟏𝟎−𝟕\\mathbf\{8\.31\\times 10^\{\-7\}\}16\.610\.1640\.09412\.61000Fitted shape \+ affine8\.31×𝟏𝟎−𝟕\\mathbf\{8\.31\\times 10^\{\-7\}\}16\.610\.1640\.09410\.15000Fixed shape3\.53×10−73\.53\\times 10^\{\-7\}10\.002\.7100\.53923\.85000Fitted shape3\.30×𝟏𝟎−𝟕\\mathbf\{3\.30\\times 10^\{\-7\}\}10\.002\.7100\.53925\.65000Fixed shape \+ affine3\.53×10−73\.53\\times 10^\{\-7\}10\.002\.7100\.54024\.35000Fitted shape \+ affine3\.31×10−73\.31\\times 10^\{\-7\}10\.002\.7100\.54024\.410000Fixed shape3\.19×10−73\.19\\times 10^\{\-7\}10\.0011\.8651\.14076\.210000Fitted shape2\.95×𝟏𝟎−𝟕\\mathbf\{2\.95\\times 10^\{\-7\}\}10\.0011\.8651\.14075\.310000Fixed shape \+ affine3\.19×10−73\.19\\times 10^\{\-7\}10\.0011\.8651\.14176\.210000Fitted shape \+ affine2\.96×10−72\.96\\times 10^\{\-7\}10\.0011\.8651\.14174\.2Table 18:Taylor–Green vortices \(space\): all models\. Test error is the mean relative velocity error of the predictions for the 100 test inputs\. Density isdK=nnz⁡\(KU​U\)/N2d\_\{K\}=\\mathrm\{nnz\}\(K\_\{UU\}\)/N^\{2\}in percent\. Factor is the number of entries of the cuDSS Cholesky factor in millions\. GB is the accounted storage and ms the median warm time for all 100 fields\. Bold: smallest test error at eachNN\.NNOperator modelTest errorDensity \(%\)Factor \(10610^\{6\}\)GBms500Fixed shape9\.41×10−79\.41\\times 10^\{\-7\}46\.170\.0940\.0286\.5500Fitted shape7\.72×𝟏𝟎−𝟕\\mathbf\{7\.72\\times 10^\{\-7\}\}50\.000\.0970\.0297\.7500Fixed shape \+ affine8\.76×10−78\.76\\times 10^\{\-7\}40\.070\.0830\.0287\.7500Fitted shape \+ affine8\.83×10−78\.83\\times 10^\{\-7\}50\.000\.0970\.0296\.71000Fixed shape3\.31×10−73\.31\\times 10^\{\-7\}51\.760\.4220\.0687\.61000Fitted shape7\.86×𝟏𝟎−𝟗\\mathbf\{7\.86\\times 10^\{\-9\}\}50\.000\.4140\.0678\.61000Fixed shape \+ affine3\.63×10−83\.63\\times 10^\{\-8\}32\.160\.2880\.0608\.01000Fitted shape \+ affine1\.03×10−81\.03\\times 10^\{\-8\}50\.000\.4140\.0678\.25000Fixed shape2\.45×10−62\.45\\times 10^\{\-6\}75\.0022\.9231\.05638\.95000Fitted shape2\.51×10−62\.51\\times 10^\{\-6\}90\.0024\.8891\.19941\.55000Fixed shape \+ affine2\.67×10−102\.67\\times 10^\{\-10\}10\.002\.6440\.33914\.25000Fitted shape \+ affine6\.00×𝟏𝟎−𝟏𝟏\\mathbf\{6\.00\\times 10^\{\-11\}\}10\.002\.6440\.33914\.710000Fixed shape2\.82×10−62\.82\\times 10^\{\-6\}75\.0072\.6773\.606117\.010000Fitted shape2\.11×10−62\.11\\times 10^\{\-6\}90\.0099\.8004\.334119\.010000Fixed shape \+ affine2\.42×10−102\.42\\times 10^\{\-10\}10\.0011\.7280\.89952\.310000Fitted shape \+ affine5\.22×𝟏𝟎−𝟏𝟏\\mathbf\{5\.22\\times 10^\{\-11\}\}10\.0011\.7280\.89954\.6Table 19:Taylor–Green vortices \(spacetime\): all models\. Test error is the mean relative velocity error of the predictions for the 100 test inputs\. Density isdK=nnz⁡\(KU​U\)/N2d\_\{K\}=\\mathrm\{nnz\}\(K\_\{UU\}\)/N^\{2\}in percent\. Factor is the number of entries of the cuDSS Cholesky factor in millions\. GB is the accounted storage and ms the median warm time for all 100 fields\. Bold: smallest test error at eachNN\.NNOperator modelTest errorDensity \(%\)Factor \(10610^\{6\}\)GBms500Fixed shape8\.01×10−78\.01\\times 10^\{\-7\}46\.230\.0930\.0526\.4500Fitted shape6\.58×𝟏𝟎−𝟕\\mathbf\{6\.58\\times 10^\{\-7\}\}50\.000\.0970\.0536\.3500Fixed shape \+ affine7\.38×10−77\.38\\times 10^\{\-7\}39\.510\.0820\.0528\.1500Fitted shape \+ affine7\.65×10−77\.65\\times 10^\{\-7\}50\.000\.0970\.0536\.81000Fixed shape2\.71×10−72\.71\\times 10^\{\-7\}51\.580\.4190\.1167\.91000Fitted shape9\.30×10−99\.30\\times 10^\{\-9\}50\.000\.4140\.1159\.81000Fixed shape \+ affine3\.34×10−83\.34\\times 10^\{\-8\}32\.270\.2840\.10811\.01000Fitted shape \+ affine8\.61×𝟏𝟎−𝟗\\mathbf\{8\.61\\times 10^\{\-9\}\}50\.000\.4140\.1157\.75000Fixed shape2\.14×10−62\.14\\times 10^\{\-6\}75\.0022\.9231\.13641\.95000Fitted shape2\.19×10−62\.19\\times 10^\{\-6\}90\.0024\.8891\.27940\.25000Fixed shape \+ affine2\.41×10−102\.41\\times 10^\{\-10\}10\.002\.6440\.41916\.85000Fitted shape \+ affine5\.40×𝟏𝟎−𝟏𝟏\\mathbf\{5\.40\\times 10^\{\-11\}\}10\.002\.6440\.41916\.610000Fixed shape2\.46×10−62\.46\\times 10^\{\-6\}75\.0072\.6773\.766119\.810000Fitted shape1\.86×10−61\.86\\times 10^\{\-6\}90\.0099\.8004\.494120\.110000Fixed shape \+ affine2\.18×10−102\.18\\times 10^\{\-10\}10\.0011\.7281\.05959\.010000Fitted shape \+ affine4\.70×𝟏𝟎−𝟏𝟏\\mathbf\{4\.70\\times 10^\{\-11\}\}10\.0011\.7281\.05961\.7Table 20:Backward\-facing step: all models\. Test error is the mean relative velocity error of the predictions for the 100 test inputs\. Density isdK=nnz⁡\(KU​U\)/N2d\_\{K\}=\\mathrm\{nnz\}\(K\_\{UU\}\)/N^\{2\}in percent\. Factor is the number of entries of the cuDSS Cholesky factor in millions\. GB is the accounted storage and ms the median warm time for all 100 fields\. Bold: smallest test error at eachNN\.NNOperator modelTest errorDensity \(%\)Factor \(10610^\{6\}\)GBms500Fixed shape2\.43×10−72\.43\\times 10^\{\-7\}36\.410\.0730\.0248\.8500Fitted shape2\.43×10−72\.43\\times 10^\{\-7\}36\.410\.0730\.0248\.6500Fixed shape \+ affine9\.46×𝟏𝟎−𝟖\\mathbf\{9\.46\\times 10^\{\-8\}\}34\.940\.0680\.0249\.6500Fitted shape \+ affine9\.46×𝟏𝟎−𝟖\\mathbf\{9\.46\\times 10^\{\-8\}\}34\.940\.0680\.0248\.61000Fixed shape9\.31×10−89\.31\\times 10^\{\-8\}49\.590\.3500\.05911\.61000Fitted shape1\.14×10−71\.14\\times 10^\{\-7\}25\.000\.2360\.0508\.41000Fixed shape \+ affine6\.68×𝟏𝟎−𝟖\\mathbf\{6\.68\\times 10^\{\-8\}\}47\.150\.3420\.0589\.91000Fitted shape \+ affine7\.44×10−87\.44\\times 10^\{\-8\}25\.000\.2360\.0508\.45000Fixed shape6\.77×10−86\.77\\times 10^\{\-8\}29\.087\.7110\.50425\.85000Fitted shape6\.77×10−86\.77\\times 10^\{\-8\}29\.087\.7110\.50426\.15000Fixed shape \+ affine5\.98×𝟏𝟎−𝟖\\mathbf\{5\.98\\times 10^\{\-8\}\}28\.907\.7850\.50326\.55000Fitted shape \+ affine5\.98×𝟏𝟎−𝟖\\mathbf\{5\.98\\times 10^\{\-8\}\}28\.907\.7850\.50326\.210000Fixed shape6\.33×10−86\.33\\times 10^\{\-8\}32\.9840\.5061\.67972\.410000Fitted shape6\.33×10−86\.33\\times 10^\{\-8\}32\.9840\.5061\.67968\.810000Fixed shape \+ affine5\.86×𝟏𝟎−𝟖\\mathbf\{5\.86\\times 10^\{\-8\}\}32\.8838\.2281\.65766\.310000Fitted shape \+ affine5\.86×𝟏𝟎−𝟖\\mathbf\{5\.86\\times 10^\{\-8\}\}32\.8838\.2281\.65765\.3Table 21:Buoyancy\-driven cavity: all models\. Test error is the mean relative velocity error of the predictions for the 100 test inputs\. Density isdK=nnz⁡\(KU​U\)/N2d\_\{K\}=\\mathrm\{nnz\}\(K\_\{UU\}\)/N^\{2\}in percent\. Factor is the number of entries of the cuDSS Cholesky factor in millions\. GB is the accounted storage and ms the median warm time for all 100 fields\. Bold: smallest test error at eachNN\.NNOperator modelTest errorDensity \(%\)Factor \(10610^\{6\}\)GBms500Fixed shape5\.63×𝟏𝟎−𝟓\\mathbf\{5\.63\\times 10^\{\-5\}\}64\.680\.1180\.15430\.7500Fitted shape5\.63×𝟏𝟎−𝟓\\mathbf\{5\.63\\times 10^\{\-5\}\}64\.680\.1180\.15429\.6500Fixed shape \+ affine8\.18×10−58\.18\\times 10^\{\-5\}10\.000\.0240\.14829\.1500Fitted shape \+ affine7\.39×10−57\.39\\times 10^\{\-5\}10\.000\.0240\.14828\.61000Fixed shape3\.03×𝟏𝟎−𝟓\\mathbf\{3\.03\\times 10^\{\-5\}\}17\.330\.1700\.29832\.51000Fitted shape3\.03×𝟏𝟎−𝟓\\mathbf\{3\.03\\times 10^\{\-5\}\}17\.330\.1700\.29832\.21000Fixed shape \+ affine3\.04×10−53\.04\\times 10^\{\-5\}17\.270\.1660\.29836\.41000Fitted shape \+ affine3\.04×10−53\.04\\times 10^\{\-5\}17\.270\.1660\.29832\.65000Fixed shape6\.65×10−66\.65\\times 10^\{\-6\}10\.002\.8381\.143111\.55000Fitted shape5\.02×𝟏𝟎−𝟔\\mathbf\{5\.02\\times 10^\{\-6\}\}10\.002\.8381\.143104\.15000Fixed shape \+ affine6\.65×10−66\.65\\times 10^\{\-6\}10\.002\.8381\.144107\.75000Fitted shape \+ affine5\.35×10−65\.35\\times 10^\{\-6\}10\.002\.8381\.144109\.910000Fixed shape5\.38×10−65\.38\\times 10^\{\-6\}10\.0011\.9472\.504189\.410000Fitted shape4\.47×𝟏𝟎−𝟔\\mathbf\{4\.47\\times 10^\{\-6\}\}10\.0011\.9472\.504184\.910000Fixed shape \+ affine5\.38×10−65\.38\\times 10^\{\-6\}10\.0011\.9472\.505177\.010000Fitted shape \+ affine4\.62×10−64\.62\\times 10^\{\-6\}10\.0011\.9472\.505173\.4#### D\.6\.1Offline cost

Table[22](https://arxiv.org/html/2610.29009#A4.T22)gives the offline costs atN=10000N=10000\. Fitting and validating a fitted\-shape model takes1111–2727minutes, against22–77minutes for a fixed\-shape model\. Once the parameters are chosen, rebuilding the factorization takes about one to five seconds for most benchmarks and about4343seconds for the dense forced\-turbulence models\. The setup of the output reconstruction, which the four models share, takes0\.030\.03–2\.92\.9seconds after compilation, and a first setup, which also compiles, takes1\.61\.6–2424seconds\.

Table 22:Offline and initialization costs atN=10000N=10000\. Selection includes kernel fitting, where applicable, and validation\. Rebuild assembles and factors the selected operator\. Initialization includes two synchronized 100\-field prediction passes, compilation, and CUDA graph preparation\. The four models share the setup of the output reconstruction and may reuse compiled code\. The dash marks the selection time of the fixed\-shape backward\-facing\-step model, which contains an external pause\. Its raw time is in the released data\. Bold: smallest time in each column for each data set\.Data setOperator modelSelection \(min\)Rebuild \(s\)Initialization \(s\)Forced isotropic turbulenceFixed shape3\.8343\.273\.71Fitted shape26\.3241\.633\.28Fixed shape \+ affine6\.4543\.704\.15Fitted shape \+ affine27\.0142\.773\.05Turbulent species transportFixed shape3\.241\.484\.17Fitted shape25\.894\.643\.70Fixed shape \+ affine5\.791\.766\.05Fitted shape \+ affine24\.014\.933\.79Merging vorticesFixed shape4\.810\.913\.35Fitted shape15\.262\.923\.20Fixed shape \+ affine4\.941\.124\.94Fitted shape \+ affine11\.700\.813\.26Lid\-driven cavityFixed shape2\.120\.862\.37Fitted shape12\.970\.872\.20Fixed shape \+ affine3\.060\.913\.54Fitted shape \+ affine10\.900\.852\.26Cylinder flow \(no shedding\)Fixed shape2\.320\.872\.58Fitted shape12\.770\.862\.54Fixed shape \+ affine4\.280\.923\.77Fitted shape \+ affine14\.530\.902\.50Cylinder flow \(vortex shedding\)Fixed shape2\.120\.852\.47Fitted shape13\.580\.852\.61Fixed shape \+ affine3\.070\.913\.43Fitted shape \+ affine12\.860\.882\.37Taylor–Green vortices \(space\)Fixed shape3\.413\.183\.61Fitted shape12\.805\.543\.43Fixed shape \+ affine5\.321\.255\.47Fitted shape \+ affine15\.300\.683\.54Taylor–Green vortices \(spacetime\)Fixed shape3\.493\.303\.87Fitted shape16\.103\.603\.60Fixed shape \+ affine4\.820\.885\.22Fitted shape \+ affine15\.470\.693\.64Backward\-facing stepFixed shape—0\.951\.48Fitted shape14\.230\.991\.31Fixed shape \+ affine3\.253\.272\.53Fitted shape \+ affine13\.420\.941\.36Buoyancy\-driven cavityFixed shape2\.111\.652\.72Fitted shape17\.671\.482\.53Fixed shape \+ affine3\.161\.644\.06Fitted shape \+ affine17\.001\.512\.63

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