几何—物理混淆机制损害跨变域偏微分方程学习

# Geometry–physics confounding impairs PDE learning across varying domains
Source: [https://arxiv.org/html/2609.38623](https://arxiv.org/html/2609.38623)
Yinghao ChengAffiliation:College of Mechanical & Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, ChinaAffiliation:School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, Belfast, United KingdomGengxiang ChenAffiliation:College of Mechanical & Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, ChinaAffiliation:School of Mechanical and Aerospace Engineering, Queen’s University Belfast, Belfast, United KingdomXu LiuQinglu MengAffiliation:College of Mechanical & Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, ChinaYixin JingAffiliation:College of Mechanical & Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, ChinaXiangguo TangAffiliation:College of Mechanical & Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, ChinaWenping MouAffiliation:College of Mechanical & Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, ChinaLihui WangAffiliation:Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hong Kong, ChinaAffiliation:Department of Production Engineering, KTH Royal Institute of Technology, Stockholm, SwedenYingguang LiAffiliation:College of Mechanical & Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, ChinaAffiliation:Correspondence: liyingguang@nuaa\.edu\.cn\.

###### Abstract

Learning partial differential equation \(PDE\) dynamics across varying domains is central to predictive modelling and data\-driven discovery of governing equations\. However, geometric variation alters both field representation and the governing differential operators, confounding geometric effects with intrinsic physical properties in the observed dynamics\. This work identifies*geometry–physics confounding*as a unified failure mechanism for PDE learning across varying domains\. In forward operator learning, this confounding increases the burden of inferring geometry\-dependent operator changes from finite data, reducing data efficiency and generalisation\. In equation discovery, omitting geometry\-induced operators misspecifies the candidate library, leading to biased parameters, missed governing terms and spurious terms\. We propose a de\-confounding framework that makes the known geometry\-to\-operator transformation explicit\. Geometry\-induced coefficient fields improve prediction and data efficiency across five operator\-learning benchmarks, while geometry\-complete candidate libraries recover the generating equations and reduce held\-out PDE residuals by more than two orders of magnitude in both evolving\-domain systems\. By separating known geometric action from intrinsic physics, the proposed framework supports more reliable and data\-efficient PDE learning across scientific and engineering problems with varying geometries\.

Partial differential equations \(PDEs\) on varying and evolving domains govern a broad spectrum of transport, wave, and deformation phenomena across continuum physics and biology, from turbulent aerodynamics over complex geometries\[[1](https://arxiv.org/html/2609.38623#bib.bib27),[2](https://arxiv.org/html/2609.38623#bib.bib2),[3](https://arxiv.org/html/2609.38623#bib.bib7)\]and patient\-specific cardiac mechanics and forward electrocardiographic modelling\[[4](https://arxiv.org/html/2609.38623#bib.bib24),[5](https://arxiv.org/html/2609.38623#bib.bib29),[6](https://arxiv.org/html/2609.38623#bib.bib33),[7](https://arxiv.org/html/2609.38623#bib.bib34)\]to finite\-strain elastodynamics\[[8](https://arxiv.org/html/2609.38623#bib.bib28)\], morphogenetic tissue growth\[[9](https://arxiv.org/html/2609.38623#bib.bib25)\], and moving phase boundaries\[[10](https://arxiv.org/html/2609.38623#bib.bib31),[11](https://arxiv.org/html/2609.38623#bib.bib22),[12](https://arxiv.org/html/2609.38623#bib.bib15)\]\. Learning PDE dynamics across varying geometries from spatiotemporal observations, encompassing solution\-operator learning for predictive simulation\[[13](https://arxiv.org/html/2609.38623#bib.bib12),[14](https://arxiv.org/html/2609.38623#bib.bib13),[15](https://arxiv.org/html/2609.38623#bib.bib14),[16](https://arxiv.org/html/2609.38623#bib.bib26),[17](https://arxiv.org/html/2609.38623#bib.bib4)\]and data\-driven discovery of governing equations\[[18](https://arxiv.org/html/2609.38623#bib.bib18),[19](https://arxiv.org/html/2609.38623#bib.bib19),[20](https://arxiv.org/html/2609.38623#bib.bib30),[21](https://arxiv.org/html/2609.38623#bib.bib1),[22](https://arxiv.org/html/2609.38623#bib.bib20)\], is an increasingly important computational foundation for scientific discovery and engineering design\. Since geometric variation changes not only the domain on which the state is represented but also the differential operators governing its evolution, capturing how geometry transforms these operators is a central challenge for PDE learning across varying domains\.

Recent advances in neural operators have enabled the learning of PDE solution operators between function spaces\[[13](https://arxiv.org/html/2609.38623#bib.bib12),[14](https://arxiv.org/html/2609.38623#bib.bib13),[15](https://arxiv.org/html/2609.38623#bib.bib14)\]\. For PDEs on complex geometries, early extensions typically mapped irregular physical domains to regular computational domains, using learned deformations to latent grids in Geo\-FNO\[[2](https://arxiv.org/html/2609.38623#bib.bib2)\], graph\-based lifting to Cartesian latent grids in GINO\[[23](https://arxiv.org/html/2609.38623#bib.bib3)\], or diffeomorphic mappings to a common reference domain in DIMON\[[17](https://arxiv.org/html/2609.38623#bib.bib4)\]\. As operator learning has been extended to increasingly complex and unstructured geometries, recent methods have incorporated geometric information directly into the model through coordinates, point clouds, meshes, or continuous geometric representations, as exemplified by neural\-field operators\[[24](https://arxiv.org/html/2609.38623#bib.bib6)\], physics\-aware attention in Transolver\[[3](https://arxiv.org/html/2609.38623#bib.bib7)\], and more recent graph\- and transformer\-based architectures for large\-scale irregular domains\[[25](https://arxiv.org/html/2609.38623#bib.bib8),[26](https://arxiv.org/html/2609.38623#bib.bib9),[27](https://arxiv.org/html/2609.38623#bib.bib11)\]\. These developments have substantially expanded the applicability of neural operators to PDEs on varying geometries\. Their treatment of geometry remains primarily concerned with the representation and processing of the computational domain, without explicitly accounting for how geometric variation alters the governing differential operators\. Consequently, these geometry\-dependent operator changes must still be inferred implicitly from finite training data\.

Learning PDE dynamics from observations collected across varying domains requires correspondence between fields on different geometries, established explicitly through reference maps or implicitly through geometry\-conditioned representations\. A reference map places these fields on a common domain, but the same pull\-back also transforms the spatial differential operators and, for moving domains, introduces transport associated with domain motion\[[28](https://arxiv.org/html/2609.38623#bib.bib21),[29](https://arxiv.org/html/2609.38623#bib.bib23),[11](https://arxiv.org/html/2609.38623#bib.bib22)\]\. Domain\-mapping theory establishes the equivalence between varying\-domain PDEs and their transformed problems on a fixed reference domain\[[30](https://arxiv.org/html/2609.38623#bib.bib16)\], while recent theory for neural shape operators provides well\-posedness, shape regularity and approximation\-rate results for the corresponding shape\-to\-solution maps\[[31](https://arxiv.org/html/2609.38623#bib.bib10)\]\. Related studies have shown that operator\-preserving conformal mappings can improve the learning of Laplace solution operators\[[32](https://arxiv.org/html/2609.38623#bib.bib5)\], while boundary\-adapted coordinates can support the discovery of moving\-boundary dynamics\[[33](https://arxiv.org/html/2609.38623#bib.bib32)\]\. These task\-specific treatments demonstrate the value of respecting geometric structure, but leave unresolved whether a learned or discovered model can distinguish known geometric action from intrinsic physical relations when both enter the transformed dynamics through the same coefficients\.

We refer to this non\-unique physical attribution as*geometry–physics confounding*\. When a PDE is expressed on a common reference domain, geometric transformation changes its differential operators: homogeneous and isotropic physics can appear through spatially varying and anisotropic coefficients, while time\-dependent mappings additionally introduce arbitrary Lagrangian–Eulerian \(ALE\) transport associated with domain motion\[[30](https://arxiv.org/html/2609.38623#bib.bib16),[28](https://arxiv.org/html/2609.38623#bib.bib21),[29](https://arxiv.org/html/2609.38623#bib.bib23),[11](https://arxiv.org/html/2609.38623#bib.bib22),[33](https://arxiv.org/html/2609.38623#bib.bib32)\]\. Physically distinct combinations of geometry and intrinsic properties can therefore generate identical reference\-domain dynamics and cannot be attributed uniquely from those dynamics alone\. This ambiguity is conditional on the information used for attribution: supplying the domain map makes the decomposition available, but only if the learning or discovery procedure represents its PDE\-specific action\. In operator learning, leaving the geometric action implicit adds the finite\-data burden of learning geometry\-dependent operator changes from examples\. In equation discovery, a transformed operator that is absent from the candidate library cannot be selected, leading to missed governing terms and the absorption of geometric effects into biased coefficients or spurious terms\.

Here we establish geometry–physics confounding as a unified failure mechanism in operator prediction and equation discovery across varying domains\. A diffusion example demonstrates exact observational equivalence and shows how implicit geometric action increases the finite\-data learning burden in prediction and causes structural candidate\-library misspecification in equation discovery\. To address these failures, we propose a de\-confounding framework that supplies geometry\-induced coefficient fields for operator learning and geometry\-complete candidate libraries for equation discovery\. The framework modifies the problem representation within existing neural\-operator formulations and retains the sparse\-regression algorithm\. Five operator\-learning benchmarks spanning reference\-domain mapping and geometry\-aware models evaluate prediction and data efficiency\[[17](https://arxiv.org/html/2609.38623#bib.bib4),[2](https://arxiv.org/html/2609.38623#bib.bib2),[3](https://arxiv.org/html/2609.38623#bib.bib7)\], while thermoelastic and free\-boundary systems evaluate equation discovery and closed\-loop transfer on evolving domains\. Across these experiments, making geometric action explicit improves finite\-data prediction across all five operator\-learning benchmarks and enables recovery of the generating equations in both evolving\-domain systems\.

## Geometry–physics confounding and de\-confounding methods

### Problem setup

![Refer to caption](https://arxiv.org/html/2609.38623v1/fig01_confounding_demo.png)Figure 1:Geometry–physics confounding in PDE learning\.a, Problem setup for PDE learning on varying geometries, illustrating sample observations pairing geometry with field data, operator learning of solution mapsFωF\_\{\\omega\}, and equation discovery identifying governing differential laws\.b, Geometry–physics confounding mechanism: a homogeneous isotropic diffusion process on a deformed domain and an equivalent fixed\-domain model with geometry\-induced capacityjgj\_\{g\}and transformed conductivity𝜿^g\\widehat\{\\bm\{\\kappa\}\}\_\{g\}yield identical pulled\-back dynamics\. Fitting a misspecified reference model yields identical apparent diffusion coefficients \(κ^Ref=0\.010\\widehat\{\\kappa\}\_\{\\rm Ref\}=0\.010\) across physically distinct systems \(contour plot, right\)\.c, De\-confounding strategies: geometry\-induced coefficient fields for operator learning \(left\) and geometry\-complete candidate libraries for equation discovery \(right\)\.d, Learning failures and recovery: for operator rate prediction \(left and centre\), coordinate\-based inputs yield high errors, while explicitly incorporating geometry\-induced coefficient fields reduces error by over fivefold, and the geometry\-induced PDE terms yield numerical\-precision predictions \(<10−14<10^\{\-14\}\)\. Prediction curves and shaded bands show means and±1\\pm 1s\.d\. across five optimisation runs\. For equation discovery \(right\), the apparent reference coefficient exhibits a 19\.3% geometric drift and an equation residual of0\.2230\.223, whereas the geometry\-complete candidate library recovers the generating coefficient with numerical\-precision residual \(<10−16<10^\{\-16\}\)\.As shown in Fig\.[1](https://arxiv.org/html/2609.38623#Sx1.F1)a, we consider PDE\-learning samples associated with geometriesg∈𝒢g\\in\\mathcal\{G\}\. For each geometry, we consider a physical domainΩg​\(t\)⊂ℝd\\Omega\_\{g\}\(t\)\\subset\\mathbb\{R\}^\{d\}, a state fielduu, and problem datasscomprising initial and boundary conditions, source terms, PDE coefficients and control inputs\. We represent the PDE problem compactly as follows, with the initial and boundary conditions specified inssunderstood implicitly:

∂tu=𝒩Ωg​\(t\)​\[u;s\],x∈Ωg​\(t\),t∈\(0,T\],\\partial\_\{t\}u=\\mathcal\{N\}\_\{\\Omega\_\{g\}\(t\)\}\[u;s\],\\qquad x\\in\\Omega\_\{g\}\(t\),\\quad t\\in\(0,T\],\(1\)where∂t\\partial\_\{t\}denotes the time derivative associated with the physical description, and𝒩\\mathcal\{N\}denotes the remaining physical\-domain differential operator\. Steady problems satisfy0=𝒩Ωg​\[u;s\]0=\\mathcal\{N\}\_\{\\Omega\_\{g\}\}\[u;s\]\.

In real\-world applications, the geometries may represent a family of related objects under study, such as corresponding blood vessels or organs across different patients or alternative airfoil shapes explored in aircraft design\. They may also evolve over time through a moving\-boundary process, as in glacier melt or tumour growth\. When domain evolution is coupled to the state, a mechanical or kinematic evolution law is required to close the coupled state–geometry problem\.

Assuming that the physical problem is well posed for the prescribed data and geometry, operator learning approximates the geometry\-conditioned solution map

ℱ𝝎:\(s,Ωg​\(⋅\)\)⟼u,\\mathcal\{F\}\_\{\\bm\{\\omega\}\}:\\bigl\(s,\\Omega\_\{g\}\(\\cdot\)\\bigr\)\\longmapsto u,\(2\)where one parameter vector𝝎\\bm\{\\omega\}is shared across the sampled geometries, and the objective is to predict solution fields or trajectories for previously unseen prescribed problem data, geometries, or their combinations\.

Equation discovery, by contrast, starts from observations across geometries and a prespecified library ofppcandidate operators defined on the physical domain, and seeks

\(u,sknown,Ωg​\(⋅\)\)⟼𝒄⋆,\\bigl\(u,s^\{\\rm known\},\\Omega\_\{g\}\(\\cdot\)\\bigr\)\\longmapsto\\bm\{c\}^\{\\star\},\(3\)wheresknowns^\{\\rm known\}denotes the part ofssactually supplied during identification, excluding the candidate coefficients to be inferred\. The vector𝒄⋆∈ℝp\\bm\{c\}^\{\\star\}\\in\\mathbb\{R\}^\{p\}contains the coefficients of the full candidate library, with nonzero entries identifying the active terms\. The coefficient vector𝒄⋆\\bm\{c\}^\{\\star\}is shared across the sampled geometries\. Although the two tasks seek different objects, both concern PDEs across related geometries or on evolving domains, with the same geometric transformations of the governing equations\.

The vessel, organ and airfoil examples above, together with successive domains in tumour growth, naturally form related geometry families\. We consider topology\-preserving families for which each physical domain is related to a fixed reference domainΩ^⊂ℝd\\widehat\{\\Omega\}\\subset\\mathbb\{R\}^\{d\}by a sufficiently regular, orientation\-preserving diffeomorphism

Φg​\(⋅,t\):Ω^→Ωg​\(t\),𝑱g=∇ξΦg,jg=det𝑱g\>0,\\Phi\_\{g\}\(\\cdot,t\):\\widehat\{\\Omega\}\\to\\Omega\_\{g\}\(t\),\\qquad\\bm\{J\}\_\{g\}=\\nabla\_\{\\xi\}\\Phi\_\{g\},\\qquad j\_\{g\}=\\det\\bm\{J\}\_\{g\}\>0,\(4\)with time dependence suppressed for a time\-independent map\. Under the regularity assumptions required by the chosen formulation,Φg\\Phi\_\{g\}induces a pullback from physical\-domain fields to reference\-domain fields\[[30](https://arxiv.org/html/2609.38623#bib.bib16),[31](https://arxiv.org/html/2609.38623#bib.bib10),[34](https://arxiv.org/html/2609.38623#bib.bib17)\]

u^=𝒫g​u,\(𝒫g​u\)​\(ξ,t\):=u⁡\(Φg​\(ξ,t\),t\),s^=𝒫gs​s,\\widehat\{u\}=\\mathcal\{P\}\_\{g\}u,\\qquad\(\\mathcal\{P\}\_\{g\}u\)\(\\xi,t\):=u\\bigl\(\\Phi\_\{g\}\(\\xi,t\),t\\bigr\),\\qquad\\widehat\{s\}=\\mathcal\{P\}\_\{g\}^\{s\}s,\(5\)where𝒫g\\mathcal\{P\}\_\{g\}represents the same state field on the reference domain by evaluating it at corresponding physical points, and𝒫gs\\mathcal\{P\}\_\{g\}^\{s\}denotes the corresponding transformations of the prescribed problem data\. Applying the chain rule and changes of variables to Eq\. \([1](https://arxiv.org/html/2609.38623#Sx1.E1)\) yields the reference\-domain evolution equation

∂tu^\|ξ=𝒩^g,t​\[u^;s^\],ξ∈Ω^,t∈\(0,T\],\\left\.\\partial\_\{t\}\\widehat\{u\}\\right\|\_\{\\xi\}=\\widehat\{\\mathcal\{N\}\}\_\{g,t\}\[\\widehat\{u\};\\widehat\{s\}\],\\qquad\\xi\\in\\widehat\{\\Omega\},\\quad t\\in\(0,T\],\(6\)where𝒩^g,t\\widehat\{\\mathcal\{N\}\}\_\{g,t\}denotes the reference\-domain evolution operator obtained by transforming Eq\. \([1](https://arxiv.org/html/2609.38623#Sx1.E1)\), including its time derivative\. Under the stated regularity assumptions, equivalence follows directly from the invertibility ofΦg\\Phi\_\{g\}\. Specifically,𝒫g\\mathcal\{P\}\_\{g\}and its inverse establish a one\-to\-one correspondence between solutions of the consistently transformed problems\. The two formulations describe the same physical evolution in different coordinates\. AlthoughΩ^\\widehat\{\\Omega\}is fixed, the transformed operator retains geometric dependence through the spatial transformation and any map\-motion contributions required by the adopted time description\. Thus, the same physical\-domain law admits a family of reference\-domain formulations in which geometry enters through the transformed operators and problem data\. We next examine how an incomplete representation of geometric transformations can obscure physical attribution and create distinct difficulties for operator learning and equation discovery\.

### Geometry–physics confounding and failure mechanism

To examine this attribution problem, consider diffusion onΩg\\Omega\_\{g\}with a time\-independent map:

∂tu=κ​Δx​u,x∈Ωg,\\partial\_\{t\}u=\\kappa\\Delta\_\{x\}u,\\qquad x\\in\\Omega\_\{g\},\(7\)whereκ\\kappadenotes the physical\-domain diffusion coefficient\. Pulling this equation back through the time\-independent map toΩ^\\widehat\{\\Omega\}gives

∂tu^=1jg​∇ξ⋅\(𝜿^g​∇ξu^\),𝜿^g=κ​jg​𝑱g−1​𝑱g−𝖳\.\\partial\_\{t\}\\widehat\{u\}=\\frac\{1\}\{j\_\{g\}\}\\nabla\_\{\\xi\}\\\!\\cdot\\\!\\left\(\\widehat\{\\bm\{\\kappa\}\}\_\{g\}\\nabla\_\{\\xi\}\\widehat\{u\}\\right\),\\qquad\\widehat\{\\bm\{\\kappa\}\}\_\{g\}=\\kappa j\_\{g\}\\bm\{J\}\_\{g\}^\{\-1\}\\bm\{J\}\_\{g\}^\{\-\\mathsf\{T\}\}\.\(8\)Thus homogeneous isotropic diffusion in physical coordinates appears on the reference domain with the Jacobian determinantjgj\_\{g\}and transformed diffusion coefficient field𝜿^g\\widehat\{\\bm\{\\kappa\}\}\_\{g\}\. In the equivalent mass or weak form,jgj\_\{g\}acts as a spatially varying capacity\. The same spatial transformation applies at each time for an evolving geometry, while the time\-dependent map additionally introduces transport induced by domain motion \(Methods\)\. For operator learning,𝜿^g\\widehat\{\\bm\{\\kappa\}\}\_\{g\}forms part of the equivalent reference\-domain problem data and is supplied as a complete field\. In the discovery experiment below, candidate coefficients are assigned to the transformed operators and inferred from data\.

Equation \([8](https://arxiv.org/html/2609.38623#Sx1.E8)\) shows how two physically distinct descriptions can produce the same pulled\-back state trajectory when attribution does not use the domain map separately\. A homogeneous material on a deformed domain and a heterogeneous anisotropic material on a fixed domain reduce to the same reference\-domain equation for matched initial and boundary data and therefore generate identical pulled\-back trajectories \(Fig\.[1](https://arxiv.org/html/2609.38623#Sx1.F1)b\)\. Although their geometries and physical\-domain PDE coefficients differ before pull\-back, these quantities enter the trajectory through the same transformed coefficient fields\. We call this non\-unique physical attribution*geometry–physics confounding*\.

The information available for attribution and the chosen model class determine whether this ambiguity remains\. A misspecified reference\-domain model that compresses the transformed operator into a single apparent diffusion coefficient yields identical fitted scalars \(κ^Ref=0\.010\\widehat\{\\kappa\}\_\{\\rm Ref\}=0\.010\) across distinct geometry–coefficient pairs \(Fig\.[1](https://arxiv.org/html/2609.38623#Sx1.F1)b, right\)\. That scalar alone cannot determine how much of the observed rate of change originates from geometric distortion and how much belongs to the material\. Using the domain map’s PDE\-specific action makes this decomposition available\. Figure[1](https://arxiv.org/html/2609.38623#Sx1.F1)c outlines the strategies detailed in the next subsection\.

For operator learning, leaving that action implicit creates an additional finite\-data learning problem\. The rate\-prediction experiments below focus on the geometry\-dependent action that the learned predictor must represent\. We expanded Eq\. \([8](https://arxiv.org/html/2609.38623#Sx1.E8)\) into five state derivatives and five geometry\-induced coefficient fields derived from the Jacobian\. Their products form the five geometry\-induced PDE terms used in the final representation \(Methods\)\. The representation sequence in Fig\.[1](https://arxiv.org/html/2609.38623#Sx1.F1)d begins with state and physical coordinates, followed by exact state derivatives, the geometry\-induced coefficient fieldsqgq\_\{g\}derived from the Jacobian, and the geometry\-induced PDE termsqg⊙𝒟⁡\[u\]q\_\{g\}\\odot\\mathcal\{D\}\[u\]\. The models using state derivatives and geometry\-induced coefficient fields use the same neural architecture and differ only in whether their inputs expose the geometry\-to\-operator action\. At the largest training sets, the latter reduced test error by factors of 5\.4 for pointwise rate prediction and 5\.3 for complete\-field rate prediction at the held\-out geometry \(Fig\.[1](https://arxiv.org/html/2609.38623#Sx1.F1)d, left and centre\)\. Fitting independent linear weights to the five geometry\-induced PDE terms reduced both errors to numerical precision\.

For equation discovery, an implicit geometric action produces structural rather than only statistical failure\. Before considering the evolving\-domain trajectories, the fixed\-deformation experiment in Fig\.[1](https://arxiv.org/html/2609.38623#Sx1.F1)d \(right\) tests spatial candidate\-library misspecification\. A discovery algorithm can select only operators present in its candidate library\. Data were generated with diffusion coefficientκ=0\.01\\kappa=0\.01and fit with the standard reference LaplacianΔξ​u^\\Delta\_\{\\xi\}\\widehat\{u\}\. As deformation increased, the apparent Laplacian coefficient drifted to0\.008070\.00807, producing a19\.3%19\.3\\%bias \(Fig\.[1](https://arxiv.org/html/2609.38623#Sx1.F1)d, top right\)\. Because a single scalar cannot reproduce the spatially varying anisotropic operator, the relative equation residual simultaneously rose to0\.2230\.223\(Fig\.[1](https://arxiv.org/html/2609.38623#Sx1.F1)d, bottom right\)\. The mapped diffusion operator recovered the generating coefficient and reduced the structural residual to numerical precision\. The drift in the apparent Laplacian coefficient measures the geometric effect absorbed by the misspecified term, while the remaining residual measures the part that the library cannot represent\. The fixed\-deformation and evolving\-domain experiments test the same candidate\-library requirement\. Time\-dependent maps additionally introduce the associated domain\-motion operators\.

### De\-confounding through geometry\-induced representations

The preceding analysis suggests a task\-matched use of the known domain correspondence\. We use*geometry\-induced representations*for the map\-dependent quantities that arise when we write a physical\-domain PDE on the reference domain\. The appropriate representation depends on the object being learned\. For operator learning, an equivalent weak\-form pull\-back provides a systematic construction of the transformed problem specification\. For equation discovery, a strong\-form pull\-back maps the prespecified physical library into structured reference\-domain candidate operators\. In both cases, the representation makes the geometry\-to\-operator transformation explicit, so the learning procedure need not reconstruct that transformation implicitly from examples\. Both constructions apply across related and evolving geometries, and a time\-dependent map additionally contributes the corresponding domain\-motion terms\.

#### Weak\-form pull\-back for operator learning\.

Forg∈𝒢g\\in\\mathcal\{G\}, suppose that the physical\-domain problem is written in the weak residual form

ℜΩg​\(t\)​\(u,v,s\)=0for all admissible​v\.\\mathfrak\{R\}\_\{\\Omega\_\{g\}\(t\)\}\(u,v;s\)=0\\quad\\text\{for all admissible \}v\.\(9\)This notation retains any time dependence\. Subject to the regularity and boundary\-condition assumptions of the physical weak problem, pulling back the trial and test fields and applying the change of variables to all volume and boundary terms yields an equivalent problem on reference spaces,

ℜ^​\(u^,v^,s^w\)=0for all​v^∈𝒱^,s^w=𝒲w​\(s,Φg,∂tΦg\)\.\\widehat\{\\mathfrak\{R\}\}\\bigl\(\\widehat\{u\},\\widehat\{v\};\\widehat\{s\}^\{\\rm w\}\\bigr\)=0\\quad\\text\{for all \}\\widehat\{v\}\\in\\widehat\{\\mathcal\{V\}\},\\qquad\\widehat\{s\}^\{\\rm w\}=\\mathcal\{W\}^\{\\rm w\}\(s;\\Phi\_\{g\},\\partial\_\{t\}\\Phi\_\{g\}\)\.\(10\)For operators that admit integration by parts under the imposed boundary conditions, this weak\-form construction also lowers the differentiation burden\. For example, a second\-order divergence\-form term can be evaluated after pull\-back using first derivatives of the trial and test fields together withjgj\_\{g\},𝑱g−1\\bm\{J\}\_\{g\}^\{\-1\}and cofactor\-like fields\. It is then unnecessary to expand the divergence pointwise and differentiate the map\-induced coefficientjg​𝑱g−1​𝑱g−𝖳j\_\{g\}\\bm\{J\}\_\{g\}^\{\-1\}\\bm\{J\}\_\{g\}^\{\-\\mathsf\{T\}\}, an expansion that would generally require second derivatives of the state and higher derivatives of the domain map\. The weak form can therefore avoid these higher\-order evaluations while retaining an equivalent variational problem\. Here𝒲w\\mathcal\{W\}^\{\\rm w\}transforms the prescribed problem specification\. It acts jointly on the initial and boundary conditions, forcing terms, known PDE coefficients and control inputs, as applicable, and includes all coefficient and data fields generated by the changes of variables in the volume and boundary terms\. For a time\-dependent map, these fields also include the map velocity and associated domain\-motion terms\. We can therefore represent the varying\-domain solution family on the common reference domain through the solution operator

ℱ^:s^w⟼u^,ℱ⁡\(s,Φg\)=𝒫g−1​ℱ^​\(𝒲w​\(s,Φg,∂tΦg\)\)\.\\widehat\{\\mathcal\{F\}\}:\\widehat\{s\}^\{\\rm w\}\\longmapsto\\widehat\{u\},\\qquad\\mathcal\{F\}\(s;\\Phi\_\{g\}\)=\\mathcal\{P\}\_\{g\}^\{\-1\}\\widehat\{\\mathcal\{F\}\}\\bigl\(\\mathcal\{W\}^\{\\rm w\}\(s;\\Phi\_\{g\},\\partial\_\{t\}\\Phi\_\{g\}\)\\bigr\)\.\(11\)A physical\-domain PDE coefficient that is constant may therefore become a spatially varying coefficient field after pull\-back\. The operator\-learning formulation does not separate such a field into an invariant internal parameter and a geometric factor\. We treat it directly as part of the transformed problem specifications^w\\widehat\{s\}^\{\\rm w\}\.

#### Strong\-form pull\-back for equation discovery\.

Forg∈𝒢g\\in\\mathcal\{G\}, we specify the candidate library before geometric transformation as

𝓣g,t​\[u;sknown\]=\(𝒯1,Ωg​\(t\)​\[u;sknown\],…,𝒯p,Ωg​\(t\)​\[u;sknown\]\)\.\\bm\{\\mathcal\{T\}\}\_\{g,t\}\[u;s^\{\\rm known\}\]=\\bigl\(\\mathcal\{T\}\_\{1,\\Omega\_\{g\}\(t\)\}\[u;s^\{\\rm known\}\],\\ldots,\\mathcal\{T\}\_\{p,\\Omega\_\{g\}\(t\)\}\[u;s^\{\\rm known\}\]\\bigr\)\.\(12\)Heresknowns^\{\\rm known\}contains only the problem data supplied during identification\. The coefficients assigned to the candidates remain unknown\. Writes^known=𝒫gs​sknown\\widehat\{s\}^\{\\rm known\}=\\mathcal\{P\}\_\{g\}^\{s\}s^\{\\rm known\}\. Let𝒫g,tstr\\mathcal\{P\}\_\{g,t\}^\{\\rm str\}denote the strong\-form change of variables induced byΦg​\(⋅,t\)\\Phi\_\{g\}\(\\cdot,t\)\. It acts on an entire differential expression\. In particular, the standard Eulerian time derivative obeys

∂tu\|x∘Φg=∂tu^\|ξ−𝒜g,t​\[u^\],𝒜g,t​\[u^\]:=\(𝑱g−1​∂tΦg\)⋅∇ξu^,\\left\.\\partial\_\{t\}u\\right\|\_\{x\}\\circ\\Phi\_\{g\}=\\left\.\\partial\_\{t\}\\widehat\{u\}\\right\|\_\{\\xi\}\-\\mathcal\{A\}\_\{g,t\}\[\\widehat\{u\}\],\\qquad\\mathcal\{A\}\_\{g,t\}\[\\widehat\{u\}\]:=\\bigl\(\\bm\{J\}\_\{g\}^\{\-1\}\\partial\_\{t\}\\Phi\_\{g\}\\bigr\)\\cdot\\nabla\_\{\\xi\}\\widehat\{u\},\(13\)where𝒜g,t\\mathcal\{A\}\_\{g,t\}is the arbitrary Lagrangian–Eulerian \(ALE\) transport operator and the displayed sign follows the Eulerian convention\[[28](https://arxiv.org/html/2609.38623#bib.bib21),[29](https://arxiv.org/html/2609.38623#bib.bib23)\]\. An evolving\-domain construction includes this term, while𝒜g,t=0\\mathcal\{A\}\_\{g,t\}=0for a time\-independent map\. More generally, the domain\-motion contribution is determined by the relative velocity between the map and the adopted physical transport, and vanishes when the map follows the material motion\. For a prescribed Eulerian description, the map fixes both the term and its sign, so it can be collected in𝒬^g,t\\widehat\{\\mathcal\{Q\}\}\_\{g,t\}rather than fitted\. A material\-time description may instead absorb the contribution\. In the identification experiments, we retain𝒜g,t\\mathcal\{A\}\_\{g,t\}as a single geometry\-induced candidate in a common transformed library\. This choice tests whether model selection retains explicit domain\-motion transport when it is active and rejects the same candidate when the adopted material description makes its generating coefficient zero\. We then replace each physical candidate with its geometry\-equivalent reference\-domain operator and append the ALE operator directly to the transformed library:

𝒯^k,g,t:=𝒫g,tstr​\[𝒯k,Ωg​\(t\)\],𝚯^g,t:=\[𝒯^1,g,t,…,𝒯^p,g,t,𝒜g,t\]\.\\widehat\{\\mathcal\{T\}\}\_\{k,g,t\}:=\\mathcal\{P\}\_\{g,t\}^\{\\rm str\}\\bigl\[\\mathcal\{T\}\_\{k,\\Omega\_\{g\}\(t\)\}\\bigr\],\\qquad\\widehat\{\\bm\{\\Theta\}\}\_\{g,t\}:=\\bigl\[\\widehat\{\\mathcal\{T\}\}\_\{1,g,t\},\\ldots,\\widehat\{\\mathcal\{T\}\}\_\{p,g,t\},\\mathcal\{A\}\_\{g,t\}\\bigr\]\.\(14\)After applying the same transformation to the full governing balance, we collect the known terms to obtain the reference\-domain identification problem

𝒬^g,t​\[u^;s^known\]=𝚯^g,t​\[u^;s^known\]​𝒄~⋆,g∈𝒢\.\\widehat\{\\mathcal\{Q\}\}\_\{g,t\}\[\\widehat\{u\};\\widehat\{s\}^\{\\rm known\}\]=\\widehat\{\\bm\{\\Theta\}\}\_\{g,t\}\[\\widehat\{u\};\\widehat\{s\}^\{\\rm known\}\]\\widetilde\{\\bm\{c\}\}^\{\\star\},\\qquad g\\in\\mathcal\{G\}\.\(15\)Here𝒄~⋆\\widetilde\{\\bm\{c\}\}^\{\\star\}augments the full candidate coefficient vector𝒄⋆\\bm\{c\}^\{\\star\}with the coefficient of the ALE candidate used in these experiments\. Both vectors retain zero entries for inactive candidates\. Their entries are weights of specified candidate operators, whose relation to physical parameters depends on the chosen representation\. The ALE entry records whether domain\-motion transport remains explicit under the adopted time description, and its coefficient describes the adopted kinematic representation\. Each transformed physical candidate may combine several map\-dependent factors and reference derivatives, but it remains a single structured operator with one fitted coefficient\.

![[Uncaptioned image]](https://arxiv.org/html/2609.38623v1/figures/fig02_operator_learning_DIMON.jpg)Figure 2:Prediction accuracy and data efficiency of DIMON and DIMON\-Geo\.a, Model configurations\.b, Geometry\-induced weak\-form representations for Laplace and reaction–diffusion\.c,e, Test relativeL2L\_\{2\}errors versus training\-set size \(left\) and estimated sample requirements at matched target errors \(right\) for Laplace and reaction–diffusion, respectively\.d, Laplace predictions for two test samples using 500 training samples\.f, Reaction–diffusion predictions for two test samples att=0\.16t=0\.16\(top\) andt=0\.32t=0\.32\(bottom\), using 1500 training samples\. Examples ind,fuse seed 0\. Inc,e, markers and error bars show the mean and±1\\pm 1sample s\.d\. of test\-set mean errors across five seeds\. Curves and shaded bands show the interpolated mean and mean±1\\pm 1sample s\.d\., respectively\. Sample requirements are estimated by interpolation \(Methods\), with percentages indicating savings relative to DIMON\. Ind,f, pointwise error reduction is DIMON absolute error minus DIMON\-Geo absolute error, so positive values favour DIMON\-Geo\. Solid and dashed contours denote ground truth and predictions, respectively\.

## Results

### Geometry\-induced representations improve neural\-operator predictions across varying geometries

We tested whether explicitly representing geometry\-induced operator variation improves prediction within existing neural\-operator formulations\. The five benchmarks comprised the Laplace and nonlinear reaction–diffusion examples from DIMON\[[17](https://arxiv.org/html/2609.38623#bib.bib4)\], and the public pipe\-flow, airfoil and elasticity benchmarks introduced with Geo\-FNO and subsequently evaluated with Transolver\[[2](https://arxiv.org/html/2609.38623#bib.bib2),[3](https://arxiv.org/html/2609.38623#bib.bib7)\]\. They span steady and time\-dependent equations, fluid and solid physics, and structured and unstructured discretizations\. Within each benchmark, the model architectures and training settings were matched as closely as possible between the two variants\. Differences concerned the input representations and the associated adaptations to input processing and network architecture, as detailed in Supplementary Sections S2–S6\. We varied the number of labelled geometry–solution samples and repeated each configuration with five paired random seeds\. Reported errors are the mean test relativeL2L\_\{2\}error±\\pmone sample standard deviation across these runs\. Data\-efficiency estimates refer to labelled solutions\.

#### Reference\-domain mapping models\.

DIMON learns geometry\-dependent solution maps on a common reference domain\. DIMON\-Geo retains its branch–trunk formulation and supplies geometric coefficient fields through dedicated branches alongside the boundary or initial data \(Fig\.[2](https://arxiv.org/html/2609.38623#Sx1.F2)a\)\. The weak\-form motivation is summarised in Fig\.[2](https://arxiv.org/html/2609.38623#Sx1.F2)b and derived in Methods\. The reaction–diffusion variant also changes the initial\-condition encoding and branch capacity \(Supplementary Section S3\)\. DIMON\-Geo achieved lower mean test relativeL2L\_\{2\}errors at every tested training size in both problems \(Fig\.[2](https://arxiv.org/html/2609.38623#Sx1.F2)c,e\)\.

For Laplace, atn=300n=300, DIMON\-Geo reduced the error from0\.0987±0\.01240\.0987\\pm 0\.0124to0\.0448±0\.01050\.0448\\pm 0\.0105\(54\.6%\)\. Atn=3000n=3000, the reduction was smaller, from0\.00864±0\.000900\.00864\\pm 0\.00090to0\.00827±0\.000370\.00827\\pm 0\.00037\(4\.3%\)\. At a target relative error of 2%, interpolation of the mean learning curves estimated 906 training samples for DIMON and 573 for DIMON\-Geo, a 36\.7% reduction\. Estimated sample savings ranged from 13\.8% to 40\.7% across the plotted target errors \(Fig\.[2](https://arxiv.org/html/2609.38623#Sx1.F2)c\)\. Figure[2](https://arxiv.org/html/2609.38623#Sx1.F2)d compares predictions and pointwise absolute\-error reductions for two test samples\.

For reaction–diffusion, the reduction increased from 10\.4% atn=2700n=2700\(0\.0501±0\.00160\.0501\\pm 0\.0016to0\.0449±0\.00450\.0449\\pm 0\.0045\) to 69\.4% atn=500n=500\(0\.3676±0\.01490\.3676\\pm 0\.0149to0\.1127±0\.00780\.1127\\pm 0\.0078\)\. At a target relative error of 10%, the estimated training requirement decreased from 1974 samples for DIMON to 592 for DIMON\-Geo, a 70\.0% reduction\. Estimated sample savings ranged from 47\.2% to 70\.0% across the plotted target errors \(Fig\.[2](https://arxiv.org/html/2609.38623#Sx1.F2)e\)\. Figure[2](https://arxiv.org/html/2609.38623#Sx1.F2)f shows two illustrative test\-sample snapshots\. In both problems, the geometry\-conditioned variant showed its largest relative error reduction at the smallest tested training size\.

![[Uncaptioned image]](https://arxiv.org/html/2609.38623v1/figures/fig03_operator_learning_Transolver.jpg)Figure 3:Prediction accuracy and data efficiency of Transolver and Transolver\-Geo\.a, Model configurations and geometric inputs\.b,d,f, Test relativeL2L\_\{2\}errors versus training\-set size \(left\) and estimated sample requirements at matched target errors \(right\) for pipe flow, airfoil flow and elasticity, respectively\.c,e, Horizontal\-velocity and Mach\-number predictions for pipe and airfoil flow, with full fields and outlet\- or shock\-region enlargements, respectively\.g, Elastic\-stress comparisons, with absolute errors in the two model columns\. Inc,e,g, two test samples per case are shown using 400 training samples and seed 0\. Inb,d,f, markers and error bars show the mean and±1\\pm 1sample s\.d\. of test\-set mean errors across five seeds\. Curves and shaded bands show the interpolated mean and mean±1\\pm 1sample s\.d\., respectively\. Sample requirements are estimated by interpolation \(Methods\), with percentages indicating savings relative to Transolver\. Inc,e,g, pointwise error reduction is Transolver absolute error minus Transolver\-Geo absolute error, so positive values favour Transolver\-Geo\.

#### Geometry\-aware attention models\.

We next evaluated the same representation principle in Transolver \(Fig\.[3](https://arxiv.org/html/2609.38623#Sx2.F3)a\)\. Within each benchmark, the baseline and geometry\-conditioned variants shared the attention backbone and case\-specific optimisation schedule\. Transolver\-Geo replaces physical\-coordinate inputs with fixed reference coordinates and case\-specific geometric channels\. Field definitions are given in Methods and implementation details in Supplementary Sections S4–S6\. The reference meshes use all 1,000 training\-pool geometries, including for smaller labelled subsets\. Under these protocols, Transolver\-Geo achieved lower mean test relativeL2L\_\{2\}errors at every tested training size in all three benchmarks \(Fig\.[3](https://arxiv.org/html/2609.38623#Sx2.F3)b,d,f\)\.

For pipe flow, the relative reduction remained between 19\.8% and 21\.8% across the learning curve\. Atn=200n=200, the error decreased from0\.00988±0\.000620\.00988\\pm 0\.00062to0\.00789±0\.000150\.00789\\pm 0\.00015\. With 600 samples, the augmented model attained a lower error than the baseline with 1000 samples \(0\.00452 versus 0\.00495\)\. Across the target errors plotted in Fig\.[3](https://arxiv.org/html/2609.38623#Sx2.F3)b, the estimated reduction in required training samples ranged from 36\.7% to 49\.8%\. Figure[3](https://arxiv.org/html/2609.38623#Sx2.F3)c compares the full horizontal\-velocity fields and outlet\-region enlargements for two test samples\.

For airfoil flow, Transolver\-Geo reduced the mean error by 36\.5% atn=200n=200\(0\.0160±0\.00380\.0160\\pm 0\.0038to0\.01018±0\.000240\.01018\\pm 0\.00024\), compared with 14\.1% atn=1000n=1000\. The augmented model with 800 samples was also more accurate than the baseline with 1000 samples \(0\.00563 versus 0\.00616\)\. Across the plotted target errors, the estimated sample savings ranged from 31\.6% to 44\.7% \(Fig\.[3](https://arxiv.org/html/2609.38623#Sx2.F3)d\)\. Figure[3](https://arxiv.org/html/2609.38623#Sx2.F3)e compares the full Mach\-number fields and shock\-region enlargements for two test samples\.

For elasticity, the target is the stress field in a unit cell with a varying internal void\. The response depends on the constitutive law and the domain and boundary configuration, whereasCgC\_\{g\}describes the local stress\-flux transformation rather than a complete elasticity operator\. We therefore supplementCgC\_\{g\}with the domain\-map displacementdg=x−ξd\_\{g\}=x\-\\xi, which makes the global shape and boundary positions explicit relative to the reference mesh\. This field describes the input geometry, not the mechanical displacement induced by loading\. The variant using these jointCg\+dgC\_\{g\}\+d\_\{g\}inputs reduced the mean error by 18\.4% atn=200n=200and 24\.1% atn=1000n=1000\. With 800 samples, it outperformed the baseline trained with 1000 samples \(0\.00649 versus 0\.00720\)\. Across the plotted target errors, the estimated sample savings ranged from 20\.7% to 26\.7% \(Fig\.[3](https://arxiv.org/html/2609.38623#Sx2.F3)f\)\. Figure[3](https://arxiv.org/html/2609.38623#Sx2.F3)g shows the ground\-truth stress, the absolute errors of the two models and their pointwise error reduction for two test samples\. The improvement is consistent with the complementary geometric information provided by the domain\-map displacement and the cofactor field\.

Across the five benchmarks, the geometry\-conditioned variants achieved lower mean prediction errors at all tested training sizes under the case\-specific protocols\. Together, the learning curves support the predictive utility and data efficiency of these representations within the tested model families, equations and discretizations\.

### Geometry\-complete libraries enable equation discovery on evolving domains

Equation discovery was evaluated for deformation\-coupled transport and growth\-driven free\-boundary transport\[[9](https://arxiv.org/html/2609.38623#bib.bib25),[12](https://arxiv.org/html/2609.38623#bib.bib15)\]\. In both systems, the geometry trajectory and its evolution closure were known, and the identified state equation was coupled to that closure for held\-out prediction\. We compare a reference\-coordinate library \(Ref\), containing the standard operators defined on the fixed reference domain, with a geometry\-complete library \(Geo\), which retains these terms and adds the mapped diffusion and ALE transport candidates induced by the observed geometry\. The two candidate libraries are compared in Extended Data Fig\. 1\. Extended Data Fig\. 2 shows the training and held\-out geometries\.

#### Heat transport on a deforming solid\.

The thermoelastic case tests whether a single intrinsic conduction law can be identified while thermal expansion changes the physical domain\. LetTTdenote temperature,qqthe applied heat source and𝒅\\bm\{d\}the displacement definingx=ξ\+𝒅⁡\(ξ,t\)x=\\xi\+\\bm\{d\}\(\\xi,t\)\. On reference coordinates, the energy balance is

Tt=0\.01​𝒟d​\[T\]\+q,𝒟d​\[T\]=jd−1​Divξ⁡\(jd​𝑱d−1​𝑱d−𝖳​∇ξT\)\.T\_\{t\}=0\.01\\,\\mathcal\{D\}\_\{d\}\[T\]\+q,\\qquad\\mathcal\{D\}\_\{d\}\[T\]=j\_\{d\}^\{\-1\}\\operatorname\{Div\}\_\{\\xi\}\\left\(j\_\{d\}\\bm\{J\}\_\{d\}^\{\-1\}\\bm\{J\}\_\{d\}^\{\-\\mathsf\{T\}\}\\nabla\_\{\\xi\}T\\right\)\.\(16\)Figure[4](https://arxiv.org/html/2609.38623#Sx2.F4)a shows the displacement magnitude and metric distortion for a representative held\-out realization\. The non\-uniformjdj\_\{d\}field illustrates the spatially varying geometric action contained in𝒟d​\[T\]\\mathcal\{D\}\_\{d\}\[T\]\.

![Refer to caption](https://arxiv.org/html/2609.38623v1/fig04_thermoelastic_discovery_transfer.png)Figure 4:Thermoelastic equation discovery and prediction on a held\-out realization\.a, Domain deformation\|d\|\|d\|and metric distortionjd=detJdj\_\{d\}=\\det J\_\{d\}for a representative held\-out realization\.b, Generating and selected terms and the corresponding equations identified using the reference\-coordinate \(Ref\) and geometry\-complete \(Geo\) candidate libraries\. Filled and open circles denote included and excluded terms, respectively, and the cross marks the generating term missed by Ref\. In the equations, purple denotes the source termqqand its fitted coefficient, blue denotes the mapped\-diffusion term and its fitted coefficient, and black denotes the additional terms selected by Ref\.c, Temperature prediction on a representative held\-out trajectory att=2\.0t=2\.0, comparing ground truth with predictions from the reference\-coordinate and geometry\-complete identified equations\. Dashed outlines mark the true boundary\.d, Coupled von Mises stress prediction computed from the predicted temperature and displacement fields\.e, Relative errors in temperature and von Mises stress over time\.f, Coefficient relativeL2L\_\{2\}error under 0–5% additive state\-field noise\.g, Full\-trajectory relativeL2L\_\{2\}error across four held\-out trajectories for temperatureTT, displacement𝒅\\bm\{d\}, and von Mises stressσv\\sigma\_\{v\}\. The geometry closure is known\.The reference\-coordinate candidate library was structurally misspecified because it did not contain the mapped diffusion operator𝒟d​\[T\]\\mathcal\{D\}\_\{d\}\[T\]\. The selected equation therefore combined reference diffusion with a spurious state\-modulated termT​Δξ​TT\\Delta\_\{\\xi\}T, giving a supportF1F\_\{1\}score of0\.40\.4and a held\-out PDE residual of0\.1710\.171\(Fig\.[4](https://arxiv.org/html/2609.38623#Sx2.F4)b\)\. In contrast, the geometry\-complete candidate library recovered the generating termsqqand𝒟d​\[T\]\\mathcal\{D\}\_\{d\}\[T\], rejected both inactive ALE candidates, and estimated the generating candidate coefficients with a relative error of2\.03×10−52\.03\\times 10^\{\-5\}\. Their rejection is consistent with the material\-time description adopted for this system, which assigns zero generating coefficients to domain\-motion transport\. The held\-out PDE residual was3\.02×10−53\.02\\times 10^\{\-5\}, more than three orders of magnitude lower than that of the reference\-coordinate candidate library\.

Recovering the generating equation also reduced closed\-loop prediction errors\. On a representative held\-out realization, spatial predictions att=2\.0t=2\.0show that the reference\-coordinate model incurs severe localized errors along the deformed inner boundary \(EL2=0\.48E\_\{L\_\{2\}\}=0\.48for temperature, Fig\.[4](https://arxiv.org/html/2609.38623#Sx2.F4)c, and0\.580\.58for von Mises stress, Fig\.[4](https://arxiv.org/html/2609.38623#Sx2.F4)d\), whereas the geometry\-complete equation matches the true boundary profile with relative errors below1\.2×10−51\.2\\times 10^\{\-5\}\. In forward integration, reference errors rise rapidly within the initial steps to order10010^\{0\}, whereas geometry\-complete errors remain bounded below10−510^\{\-5\}over time \(Fig\.[4](https://arxiv.org/html/2609.38623#Sx2.F4)e\)\. Across four held\-out trajectories with shifted heat sources, the geometry\-complete equation consistently reduces full\-trajectory relativeL2L\_\{2\}errors in temperatureTT, displacement𝒅\\bm\{d\}, and von Mises stressσv\\sigma\_\{v\}by more than four orders of magnitude relative to the reference\-coordinate baseline \(Fig\.[4](https://arxiv.org/html/2609.38623#Sx2.F4)g and Extended Data Fig\. 3\)\.

#### Nutrient transport on a growing domain\.

The tumour\-growth case tests the joint identification of mapped diffusion and transport induced by boundary motion\. Letccdenote nutrient concentration,R⁡\(θ,t\)R\(\\theta,t\)the boundary radius,SSnutrient supply andϵ\\epsilonthe metabolic uptake field\. On reference polar coordinates, the transport law is

ct=0\.012​𝒟R​\[c\]\+𝒜R​\[c\]\+S−0\.72​ϵ​c,𝒜R​\[c\]=ρ​RtR​cρ\.c\_\{t\}=0\.012\\,\\mathcal\{D\}\_\{R\}\[c\]\+\\mathcal\{A\}\_\{R\}\[c\]\+S\-0\.72\\,\\epsilon c,\\qquad\\mathcal\{A\}\_\{R\}\[c\]=\\rho\\frac\{R\_\{t\}\}\{R\}c\_\{\\rho\}\.\(17\)The representative trajectory in Fig\.[5](https://arxiv.org/html/2609.38623#Sx2.F5)a shows the increase in domain area from3\.433\.43to4\.054\.05together with the redistribution of nutrient concentration\. Att=1\.2t=1\.2, mapped diffusion and ALE transport have distinct spatial contributions to the nutrient dynamics \(Fig\.[5](https://arxiv.org/html/2609.38623#Sx2.F5)b\)\.

![Refer to caption](https://arxiv.org/html/2609.38623v1/fig05_tumor_discovery_transfer.png)Figure 5:Tumour\-growth equation discovery and prediction on a held\-out realization\.a, Free\-boundary motion and nutrient evolution on a shared colour scale\. The domains are uniformly rescaled within each snapshot to make growth visible\. Arrows indicate the initial direction of boundary motion and labels report physical domain areaAA\.b, Generating mapped\-diffusion \(0\.012​𝒟R​\[c\]0\.012\\mathcal\{D\}\_\{R\}\[c\]\) and ALE\-transport \(𝒜R​\[c\]\\mathcal\{A\}\_\{R\}\[c\]\) PDE contributions att=1\.2t=1\.2\.c, Generating and selected terms and the corresponding identified equations\. Filled and open circles denote included and excluded terms, respectively, and crosses mark the generating terms missed by Ref\. In the equations, purple denotes the source and consumption terms and their fitted coefficients, blue denotes the mapped\-diffusion and ALE\-transport terms and their fitted coefficients, and black denotes the additional terms selected by Ref\.d, Closed\-loop relative errors over time for nutrientEcE\_\{c\}, boundary radiusERE\_\{R\}, and domain areaEAE\_\{A\}\.e, Coefficient relativeL2L\_\{2\}error under 0–5% state\-field noise\.f, Full\-trajectory relativeL2L\_\{2\}error across four held\-out trajectories for nutrientcc, boundaryRR, and areaAA\. The moving\-boundary closure is known\.In this setting, the reference\-coordinate candidate library omitted both mapped diffusion𝒟R​\[c\]\\mathcal\{D\}\_\{R\}\[c\]and ALE transport𝒜R​\[c\]\\mathcal\{A\}\_\{R\}\[c\]\. The selected equation therefore failed to recover the mapped\-diffusion and ALE\-transport terms and included four non\-generating terms, giving anF1F\_\{1\}score of0\.40\.4and an equation residual of0\.1800\.180\(Fig\.[5](https://arxiv.org/html/2609.38623#Sx2.F5)c\)\. The geometry\-complete candidate library recovered the four generating terms with a candidate\-coefficient relative error of4\.25×10−54\.25\\times 10^\{\-5\}and an equation residual of6\.93×10−46\.93\\times 10^\{\-4\}, more than two orders of magnitude lower than the reference\-coordinate residual\.

During closed\-loop evolution, the reference\-coordinate model accumulates errors rapidly, reaching relative errors of order10−110^\{\-1\}in nutrient concentrationEcE\_\{c\}, boundary radiusERE\_\{R\}and domain areaEAE\_\{A\}\(Fig\.[5](https://arxiv.org/html/2609.38623#Sx2.F5)d\)\. By contrast, the geometry\-complete equation maintains accurate tracking throughout the trajectory, keeping all three relative errors bounded between10−610^\{\-6\}and10−510^\{\-5\}\. Across four held\-out growth trajectories with non\-convex, multi\-lobed boundaries, the geometry\-complete equation consistently reduces full\-trajectory relativeL2L\_\{2\}errors in nutrient concentration, boundary radius and total domain area by three to four orders of magnitude relative to the reference\-coordinate baseline \(Fig\.[5](https://arxiv.org/html/2609.38623#Sx2.F5)f and Extended Data Fig\. 4\)\.

#### Recovery under noise and limited trajectory diversity\.

The sensitivity of numerical differentiation to measurement noise was next evaluated after supplying the required candidate operators\. With additive state\-field noise and noise\-free geometry, the thermoelastic system retained exact support at all tested levels up to5%5\\%\. At5%5\\%, the median coefficient error was0\.01410\.0141and the held\-out residual was0\.01430\.0143\(Fig\.[4](https://arxiv.org/html/2609.38623#Sx2.F4)f\)\. The tumour\-growth system retained exact support up to1%1\\%noise, whereas the medianF1F\_\{1\}score decreased to0\.7270\.727at2%2\\%and0\.4440\.444at5%5\\%\(Fig\.[5](https://arxiv.org/html/2609.38623#Sx2.F5)e\)\. This pattern is consistent with the numerical sensitivity of the coupled advection–diffusion operators \(Extended Data Fig\. 5\)\. In the clean\-data trajectory\-subset analysis, one trajectory recovered the generating support in both systems, while pooling trajectories reduced coefficient variability across the enumerated subsets \(Extended Data Fig\. 6\)\. These results separate candidate\-library misspecification from the effects of state\-field noise and training\-trajectory selection in the tested systems\.

## Discussion

Geometry–physics confounding links two apparently different limitations of PDE learning across varying domains\. In operator learning, an implicit geometric action must be approximated from finite examples and therefore increases the representation burden\. In equation discovery, the same omission changes the candidate space itself: an absent transformed operator cannot be recovered by sparse regression\. Domain motion extends this distinction by introducing a transport contribution whose presence depends on the adopted physical description\.

The operator\-learning results show that geometry\-induced coefficient fields can reduce this representation burden within existing solution\-operator formulations\. The improvement across DIMON and Transolver indicates that the useful information is not tied to a particular architecture or discretisation\. Its effect nevertheless depends on how completely the supplied fields describe the transformed PDE\. For example, elasticity requires both the local cofactor field and the displacement field to represent local deformation and global shape\.

For equation discovery, the fixed\-deformation and evolving\-domain results show the corresponding structural effect\. Reference\-coordinate libraries absorbed unrepresented geometric action into biased coefficients and additional terms, whereas the geometry\-complete libraries recovered candidate coefficients shared across training trajectories\. The thermoelastic system rejected inactive ALE candidates under its material\-time description, while the tumour system retained active domain\-motion transport\. The noise results further show that candidate completeness and derivative quality are separate requirements: supplying the correct operators removes library misspecification but does not eliminate sensitivity to noisy state derivatives\.

The present cases primarily involve relatively ideal geometric maps\. Accurate, well\-conditioned mappings are expected to provide the most favourable setting for geometry\-induced representations by limiting errors in the transformed coefficients\. Nevertheless, we also observed benefits for PDE learning in datasets containing some lower\-quality mappings, suggesting that these representations can remain useful without uniformly ideal geometric correspondences\. This motivates further relaxing the requirement for exact geometry\-induced terms and examining how approximate representations affect learning performance\. Future work will assess sensitivity to mapping and representation errors and seek to establish the conditions under which geometry\-induced representations improve PDE learning and the extent of their theoretical advantages through formal analysis\.

## Methods

### Domain maps and geometry\-induced representation

We use the domain correspondencex=Φg​\(ξ,t\)x=\\Phi\_\{g\}\(\\xi,t\), its Jacobian𝑱g\\bm\{J\}\_\{g\}and determinantjg\>0j\_\{g\}\>0defined in Eq\. \([4](https://arxiv.org/html/2609.38623#Sx1.E4)\)\. The pull\-back of a state or test field is denoted by a hat, as inu^=u∘Φg\\widehat\{u\}=u\\circ\\Phi\_\{g\}\. The geometric action is represented by coefficient fields for operator learning and by assembled differential terms for equation discovery\. The constructions below specify these two representations\.

#### Geometry\-induced fields for operator learning\.

The five solution\-operator benchmarks use time\-independent domain maps\. For each case, we give the equivalent weak problem onΩ^\\widehat\{\\Omega\}and identify the geometric fields that enter it\. Trial fields satisfy any strongly imposed boundary conditions, and the corresponding test fields vanish on those boundaries\. Each weak identity holds for all admissible test fields with sufficient regularity for the displayed integrals to be well defined\. Scalar gradients are column vectors\. For vector fields, gradient rows correspond to field components\. Thus\(∇xu\)∘Φg=𝑱g−𝖳​∇ξu^\(\\nabla\_\{x\}u\)\\circ\\Phi\_\{g\}=\\bm\{J\}\_\{g\}^\{\-\\mathsf\{T\}\}\\nabla\_\{\\xi\}\\widehat\{u\},\(∇x𝒖\)∘Φg=\(∇ξ𝒖^\)​𝑱g−1\(\\nabla\_\{x\}\\bm\{u\}\)\\circ\\Phi\_\{g\}=\(\\nabla\_\{\\xi\}\\widehat\{\\bm\{u\}\}\)\\bm\{J\}\_\{g\}^\{\-1\}andd​x=jg​d​ξdx=j\_\{g\}\\,d\\xi\.

Laplace\.For the Laplace equation−Δx​u=0\-\\Delta\_\{x\}u=0with prescribed boundary valuesu=uDu=u\_\{D\}, the equivalent weak problem is to findu^\\widehat\{u\}with boundary traceuD∘Φgu\_\{D\}\\circ\\Phi\_\{g\}such that

∫Ω^\(∇ξv^\)𝖳​𝑲g​∇ξu^​𝑑ξ=0,\\int\_\{\\widehat\{\\Omega\}\}\(\\nabla\_\{\\xi\}\\widehat\{v\}\)^\{\\mathsf\{T\}\}\\bm\{K\}\_\{g\}\\nabla\_\{\\xi\}\\widehat\{u\}\\,d\\xi=0,\(18\)for all scalar test fieldsv^\\widehat\{v\}vanishing on∂Ω^\\partial\\widehat\{\\Omega\}\. Here𝑲g=jg​𝑱g−1​𝑱g−𝖳\\bm\{K\}\_\{g\}=j\_\{g\}\\bm\{J\}\_\{g\}^\{\-1\}\\bm\{J\}\_\{g\}^\{\-\\mathsf\{T\}\}is the geometric diffusion factor\. Geometry enters the interior balance through𝑲g\\bm\{K\}\_\{g\}and the prescribed data through the boundary map\. DIMON\-Geo therefore supplies the three independent entries of this symmetric two\-dimensional tensor and the pulled\-back boundary data through separate branches\.

Reaction–diffusion\.For the the Reaction–diffusion equationut=Δx​u\+r⁡\(u\)u\_\{t\}=\\Delta\_\{x\}u\+r\(u\), wherer⁡\(u\)=u⁡\(1−u2\)r\(u\)=u\(1\-u^\{2\}\)\[[17](https://arxiv.org/html/2609.38623#bib.bib4)\], zero normal diffusive flux gives the equivalent weak balance

∫Ω^jg​∂tu^​v^​𝑑ξ\+∫Ω^\(∇ξv^\)𝖳​𝑲g​∇ξu^​𝑑ξ=∫Ω^jg​r​\(u^\)​v^​𝑑ξ,\\int\_\{\\widehat\{\\Omega\}\}j\_\{g\}\\partial\_\{t\}\\widehat\{u\}\\,\\widehat\{v\}\\,d\\xi\+\\int\_\{\\widehat\{\\Omega\}\}\(\\nabla\_\{\\xi\}\\widehat\{v\}\)^\{\\mathsf\{T\}\}\\bm\{K\}\_\{g\}\\nabla\_\{\\xi\}\\widehat\{u\}\\,d\\xi=\\int\_\{\\widehat\{\\Omega\}\}j\_\{g\}r\(\\widehat\{u\}\)\\widehat\{v\}\\,d\\xi,\(19\)for every admissible scalar test fieldv^\\widehat\{v\}\. The initial condition isu^​\(ξ,0\)=u0∘Φg​\(ξ\)\\widehat\{u\}\(\\xi,0\)=u\_\{0\}\\circ\\Phi\_\{g\}\(\\xi\)\. Accordingly, DIMON\-Geo suppliesjgj\_\{g\}and the three independent entries of𝑲g\\bm\{K\}\_\{g\}, together with the pulled\-back initial condition\. The reaction function retains its form, but its weak integral carries the volume weightjgj\_\{g\}, as does the temporal term\.

Pipe flow\.Let𝒗\\bm\{v\}andppdenote velocity and pressure per unit density\. The steady incompressible equations are\(𝒗⋅∇x\)​𝒗−ν​Δx​𝒗\+∇xp=0\(\\bm\{v\}\\cdot\\nabla\_\{x\}\)\\bm\{v\}\-\\nu\\Delta\_\{x\}\\bm\{v\}\+\\nabla\_\{x\}p=0and∇x⋅𝒗=0\\nabla\_\{x\}\\cdot\\bm\{v\}=0, with constant kinematic viscosityν\\nu\. For prescribed inlet velocity, no\-slip walls and the homogeneous natural outlet condition\(ν​∇x𝒗−p​𝑰\)​𝒏x=0\(\\nu\\nabla\_\{x\}\\bm\{v\}\-p\\bm\{I\}\)\\bm\{n\}\_\{x\}=0, their equivalent weak form is

∫Ω^\[\(∇ξ𝒗^\)​Bg​𝒗^\]⋅𝑽^​𝑑ξ\+ν​∫Ω^\[\(∇ξ𝒗^\)​𝑲g\]:∇ξ𝑽^​𝑑ξ\\displaystyle\\int\_\{\\widehat\{\\Omega\}\}\\bigl\[\(\\nabla\_\{\\xi\}\\widehat\{\\bm\{v\}\}\)B\_\{g\}\\widehat\{\\bm\{v\}\}\\bigr\]\\cdot\\widehat\{\\bm\{V\}\}\\,d\\xi\+\\nu\\int\_\{\\widehat\{\\Omega\}\}\\bigl\[\(\\nabla\_\{\\xi\}\\widehat\{\\bm\{v\}\}\)\\bm\{K\}\_\{g\}\\bigr\]:\\nabla\_\{\\xi\}\\widehat\{\\bm\{V\}\}\\,d\\xi\(20\)−∫Ω^p^Cg:∇ξ𝑽^dξ=0,\\displaystyle\-\\int\_\{\\widehat\{\\Omega\}\}\\widehat\{p\}\\,C\_\{g\}:\\nabla\_\{\\xi\}\\widehat\{\\bm\{V\}\}\\,d\\xi=0,∫Ω^ψ^​Cg:∇ξ𝒗^​dξ=0,\\displaystyle\\int\_\{\\widehat\{\\Omega\}\}\\widehat\{\\psi\}\\,C\_\{g\}:\\nabla\_\{\\xi\}\\widehat\{\\bm\{v\}\}\\,d\\xi=0,for all vector tests𝑽^\\widehat\{\\bm\{V\}\}vanishing on the inlet and walls and scalar pressure testsψ^\\widehat\{\\psi\}\. The flux factors areBg=jg​𝑱g−1B\_\{g\}=j\_\{g\}\\bm\{J\}\_\{g\}^\{\-1\}andCg=jg​𝑱g−𝖳=Bg𝖳=cof⁡𝑱gC\_\{g\}=j\_\{g\}\\bm\{J\}\_\{g\}^\{\-\\mathsf\{T\}\}=B\_\{g\}^\{\\mathsf\{T\}\}=\\operatorname\{cof\}\\bm\{J\}\_\{g\}\. Thus convection usesBgB\_\{g\}, viscous diffusion uses𝑲g\\bm\{K\}\_\{g\}, and pressure and incompressibility useCg=Bg𝖳C\_\{g\}=B\_\{g\}^\{\\mathsf\{T\}\}\. The four entries ofBgB\_\{g\}and the three independent entries of𝑲g\\bm\{K\}\_\{g\}give the seven geometric feature channels, without a separate set of channels forCgC\_\{g\}\. The prediction target is the horizontal Cartesian velocity componentvx=𝒗⋅𝒆xv\_\{x\}=\\bm\{v\}\\cdot\\bm\{e\}\_\{x\}\. Pressure enters the governing momentum balance used to derive these fields, but is not the learned output in this benchmark\.

Airfoil flow\.For the steady compressible Euler system∇x⋅𝑭⁡\(𝑼\)=0\\nabla\_\{x\}\\cdot\\bm\{F\}\(\\bm\{U\}\)=0, let𝑼\\bm\{U\}be the conservative state and let the rows of𝑭⁡\(𝑼\)\\bm\{F\}\(\\bm\{U\}\)contain its mass, momentum and energy flux vectors\. Integration by parts and pull\-back give the equivalent weak balance

∫Ω^\[𝑭⁡\(𝑼^\)​Cg\]:∇ξ𝑽^​𝑑ξ=∫∂Ω^\[𝑭⁡\(𝑼^\)​Cg​𝒏ξ\]⋅𝑽^​d​Sξ\\int\_\{\\widehat\{\\Omega\}\}\\bigl\[\\bm\{F\}\(\\widehat\{\\bm\{U\}\}\)C\_\{g\}\\bigr\]:\\nabla\_\{\\xi\}\\widehat\{\\bm\{V\}\}\\,d\\xi=\\int\_\{\\partial\\widehat\{\\Omega\}\}\\bigl\[\\bm\{F\}\(\\widehat\{\\bm\{U\}\}\)C\_\{g\}\\bm\{n\}\_\{\\xi\}\\bigr\]\\cdot\\widehat\{\\bm\{V\}\}\\,dS\_\{\\xi\}\(21\)for every admissible vector test field𝑽^\\widehat\{\\bm\{V\}\}, with the wall and far\-field conditions imposed in the boundary flux\. Here𝒏ξ\\bm\{n\}\_\{\\xi\}is the outward unit normal, and the transformed surface vector satisfies𝒏x​d​Sx=Cg​𝒏ξ​d​Sξ\\bm\{n\}\_\{x\}\\,dS\_\{x\}=C\_\{g\}\\bm\{n\}\_\{\\xi\}\\,dS\_\{\\xi\}\. Hence the same cofactor field acts on the volume and boundary fluxes\. Its four entries are the geometric feature channels for this steady problem\. No temporal volume\-weight term is present\.

Elasticity\.For static hyperelasticity, let𝒖\\bm\{u\}be the mechanical displacement on the physical undeformed domainΩg\\Omega\_\{g\}, and let𝚺\\bm\{\\Sigma\}denote the first Piola–Kirchhoff stress\. With no body force, equilibrium is∇x⋅𝚺=0\\nabla\_\{x\}\\cdot\\bm\{\\Sigma\}=0\. If𝒕N=𝚺​𝒏x\\bm\{t\}\_\{N\}=\\bm\{\\Sigma\}\\bm\{n\}\_\{x\}is the prescribed traction onΓN\\Gamma\_\{N\}, the equivalent weak problem is

∫Ω^\(𝚺^​Cg\):∇ξ𝑽^​𝑑ξ=∫Γ^N‖Cg​𝒏ξ‖​𝒕^N⋅𝑽^​d​Sξ,\\int\_\{\\widehat\{\\Omega\}\}\(\\widehat\{\\bm\{\\Sigma\}\}C\_\{g\}\):\\nabla\_\{\\xi\}\\widehat\{\\bm\{V\}\}\\,d\\xi=\\int\_\{\\widehat\{\\Gamma\}\_\{N\}\}\\\|C\_\{g\}\\bm\{n\}\_\{\\xi\}\\\|\\,\\widehat\{\\bm\{t\}\}\_\{N\}\\cdot\\widehat\{\\bm\{V\}\}\\,dS\_\{\\xi\},\(22\)for all vector tests𝑽^\\widehat\{\\bm\{V\}\}vanishing on the prescribed\-displacement boundary, whereΓ^N=Φg−1​\(ΓN\)\\widehat\{\\Gamma\}\_\{N\}=\\Phi\_\{g\}^\{\-1\}\(\\Gamma\_\{N\}\)\. The surface factor‖Cg​𝒏ξ‖\\\|C\_\{g\}\\bm\{n\}\_\{\\xi\}\\\|converts traction per unit physical boundary measure\. For strain\-energy densityWW, the pulled\-back constitutive relation is

𝑭^m=𝑰\+\(∇ξ𝒖^\)​𝑱g−1,𝚺^=∂W∂𝑭m​\(𝑭^m\),\\widehat\{\\bm\{F\}\}\_\{m\}=\\bm\{I\}\+\(\\nabla\_\{\\xi\}\\widehat\{\\bm\{u\}\}\)\\bm\{J\}\_\{g\}^\{\-1\},\\qquad\\widehat\{\\bm\{\\Sigma\}\}=\\frac\{\\partial W\}\{\\partial\\bm\{F\}\_\{m\}\}\(\\widehat\{\\bm\{F\}\}\_\{m\}\),\(23\)where𝑭m=𝑰\+∇x𝒖\\bm\{F\}\_\{m\}=\\bm\{I\}\+\\nabla\_\{x\}\\bm\{u\}is the material deformation gradient, distinct from the domain\-map Jacobian𝑱g\\bm\{J\}\_\{g\}\. The weak form identifiesCgC\_\{g\}as the stress\-flux factor, while the stress itself also depends on the unknown displacement through the transformed constitutive law\. ThusCgC\_\{g\}is not, by itself, the assembled elasticity operator\. The implemented input pairs its four entries with the two\-component domain\-map displacementdg=x−Xd\_\{g\}=x\-X, whereXXdenotes the corresponding reference point\. This additional field retains the global shape and boundary position\. It is supplied geometric information, not the mechanical response being predicted\.

#### Geometry\-induced operator terms for equation discovery\.

For equation discovery, geometry\-dependent coefficient fields are combined with state derivatives to form complete reference\-domain candidate terms\. We use pointwise strong\-form pull\-backs rather than the weak integrals above\. The weak\-integration volume weight is therefore not appended to each term, although factors such asjgj\_\{g\}remain within the transformed operators\.

For a physical vector flux𝒒\\bm\{q\}, the Piola identity reads\(∇x⋅𝒒\)∘Φg=jg−1​∇ξ⋅\(Bg​𝒒^\)\(\\nabla\_\{x\}\\cdot\\bm\{q\}\)\\circ\\Phi\_\{g\}=j\_\{g\}^\{\-1\}\\nabla\_\{\\xi\}\\cdot\(B\_\{g\}\\widehat\{\\bm\{q\}\}\), where𝒒^=𝒒∘Φg\\widehat\{\\bm\{q\}\}=\\bm\{q\}\\circ\\Phi\_\{g\}\. Together with the gradient transformation above, this identity yields the mapped diffusion term\. Table[1](https://arxiv.org/html/2609.38623#Sx4.T1)summarises common transformation rules, rather than a complete regression library\. Here𝒃^=𝒃∘Φg\\widehat\{\\bm\{b\}\}=\\bm\{b\}\\circ\\Phi\_\{g\}is a prescribed advective velocity pulled back to the reference domain, andssis a physical\-domain source\.

Table 1:Common PDE terms and their geometry\-induced forms\.A pointwise reaction retains its functional form, while a prescribed spatial source is composed with the map\. A supplied source field may enter the candidate library with an unknown multiplier\. For a moving map, Eq\. \([13](https://arxiv.org/html/2609.38623#Sx1.E13)\) gives the Eulerian time\-derivative transformation\. Writing𝒘g=∂tΦg\\bm\{w\}\_\{g\}=\\partial\_\{t\}\\Phi\_\{g\}, the induced arbitrary Lagrangian–Eulerian \(ALE\) term is𝒜g​\[u^\]=\(𝑱g−1​𝒘g\)⋅∇ξu^\\mathcal\{A\}\_\{g\}\[\\widehat\{u\}\]=\(\\bm\{J\}\_\{g\}^\{\-1\}\\bm\{w\}\_\{g\}\)\\cdot\\nabla\_\{\\xi\}\\widehat\{u\}\. It enters with a positive sign on the right\-hand side when the reference\-coordinate time derivative is isolated on the left\. If the map follows material motion and the balance uses the corresponding material\-time derivative, this explicit contribution is absorbed\.

Each assembled candidate term is assigned one coefficient\. Its coordinate\-expanded components are not fitted independently, even when it contains several reference derivatives and geometry\-dependent factors\. State\-modulated candidates, such asu^​𝒟g​\[u^\]\\widehat\{u\}\\mathcal\{D\}\_\{g\}\[\\widehat\{u\}\], are formed after assembly and assigned their own coefficients\. In the experiments below, mapped diffusion, ALE transport and their state\-modulated counterparts augment the retained reference\-coordinate library\. System\-specific libraries and generating balances are detailed in the equation\-discovery protocol below\. Hats are suppressed on pulled\-back state fields in those protocols\.

### Diffusion example of geometry–physics confounding

#### Exact equivalence and scalar attribution\.

The reference domain was the unit square\. A scalar deformation levelssdefined the orientation\-preserving analytic map

x⁡\(ξ,η\)\\displaystyle x\(\\xi,\\eta\)=ξ\+s⁡\[0\.11​\(2​ξ−1\)​sin⁡\(π​η\)\+0\.16​η\],\\displaystyle=\\xi\+s\\left\[0\.11\(2\\xi\-1\)\\sin\(\\pi\\eta\)\+0\.16\\eta\\right\],\(24\)y⁡\(ξ,η\)\\displaystyle y\(\\xi,\\eta\)=η\+0\.09​s​\(2​η−1\)​sin⁡\(π​ξ\)\.\\displaystyle=\\eta\+0\.09s\(2\\eta\-1\)\\sin\(\\pi\\xi\)\.\(25\)The system visualised in Fig\.[1](https://arxiv.org/html/2609.38623#Sx1.F1)b used a101×81101\\times 81grid,s=1s=1andκ=0\.01\\kappa=0\.01\. A two\-peak initial condition with homogeneous Dirichlet boundaries was advanced tot=0\.55t=0\.55by the second\-order Heun method with time step2×10−42\\times 10^\{\-4\}\. After pull\-back, the deformed\-domain and fixed\-domain descriptions yield the same discrete equation and therefore share one integrated trajectory\. Their common stored solution gives the exact zero observation mismatch\.

For the fixed\-deformation identification experiment,s∈\{0,0\.1,…,1\.2\}s\\in\\\{0,0\.1,\\ldots,1\.2\\\}and nine intrinsic diffusivities from 0\.006 to 0\.014 were varied independently\. Eight fixed smooth probe fields were evaluated on the101×81101\\times 81grid, with boundary\-adjacent points excluded\. The misspecified coefficient was the least\-squares projection

κ^Ref=arg⁡min⁡∑ma⁡‖κ​ℒg​\[um\]−a​Δξ​um‖22,ℒg​\[u\]=jg−1​∇ξ⋅\(𝑲g​∇ξu\),\\widehat\{\\kappa\}\_\{\\rm Ref\}=\\arg\\min\_\{a\}\\sum\_\{m\}\\left\\\|\\kappa\\mathcal\{L\}\_\{g\}\[u\_\{m\}\]\-a\\Delta\_\{\\xi\}u\_\{m\}\\right\\\|\_\{2\}^\{2\},\\qquad\\mathcal\{L\}\_\{g\}\[u\]=j\_\{g\}^\{\-1\}\\nabla\_\{\\xi\}\\\!\\cdot\(\\bm\{K\}\_\{g\}\\nabla\_\{\\xi\}u\),\(26\)whereas the geometry\-complete fit replacedΔξ​um\\Delta\_\{\\xi\}u\_\{m\}byℒg​\[um\]\\mathcal\{L\}\_\{g\}\[u\_\{m\}\]\. Residuals were normalised by the norm of the generated response\. The second pair on theκ^Ref=0\.01\\widehat\{\\kappa\}\_\{\\rm Ref\}=0\.01contour was obtained from the unit\-diffusivity projection ats=1\.2s=1\.2using the fitted scalar attribution\.

#### Five\-term operator expansion\.

The prediction experiments used the same map on a41×4141\\times 41reference grid\. Writing𝑲g=\(Ka​b\)a,b=12\\bm\{K\}\_\{g\}=\(K\_\{ab\}\)\_\{a,b=1\}^\{2\}and expandingjg−1​∇ξ⋅\(𝑲g​∇ξu\)j\_\{g\}^\{\-1\}\\nabla\_\{\\xi\}\\cdot\(\\bm\{K\}\_\{g\}\\nabla\_\{\\xi\}u\)gives

ℒg​\[u\]=∑i=15qg,i​Di​\[u\],𝑫⁡\[u\]=\[uξuηuξ​ξuξ​ηuη​η\]𝖳,\\mathcal\{L\}\_\{g\}\[u\]=\\sum\_\{i=1\}^\{5\}q\_\{g,i\}D\_\{i\}\[u\],\\quad\\bm\{D\}\[u\]=\\begin\{bmatrix\}u\_\{\\xi\}&u\_\{\\eta\}&u\_\{\\xi\\xi\}&u\_\{\\xi\\eta\}&u\_\{\\eta\\eta\}\\end\{bmatrix\}^\{\\mathsf\{T\}\},\(27\)with

𝒒g=\[jg−1​\(∂ξK11\+∂ηK12\)jg−1​\(∂ξK12\+∂ηK22\)jg−1​K112​jg−1​K12jg−1​K22\]\.\\bm\{q\}\_\{g\}=\\begin\{bmatrix\}j\_\{g\}^\{\-1\}\(\\partial\_\{\\xi\}K\_\{11\}\+\\partial\_\{\\eta\}K\_\{12\}\)\\\\ j\_\{g\}^\{\-1\}\(\\partial\_\{\\xi\}K\_\{12\}\+\\partial\_\{\\eta\}K\_\{22\}\)\\\\ j\_\{g\}^\{\-1\}K\_\{11\}\\\\ 2j\_\{g\}^\{\-1\}K\_\{12\}\\\\ j\_\{g\}^\{\-1\}K\_\{22\}\\end\{bmatrix\}\.\(28\)The geometry map and𝑲g\\bm\{K\}\_\{g\}were evaluated analytically\. Derivatives of the tensor components in Eq\. \([28](https://arxiv.org/html/2609.38623#Sx4.E28)\) were evaluated by complex\-step differentiation\. Ats=0s=0,𝒒=\(0,0,1,0,1\)𝖳\\bm\{q\}=\(0,0,1,0,1\)^\{\\mathsf\{T\}\}, and the expansion reduces exactly to the reference Laplacian\. The prediction target for every representation was the same noiseless field,ut=κ​∑iqg,i​Di​\[u\]u\_\{t\}=\\kappa\\sum\_\{i\}q\_\{g,i\}D\_\{i\}\[u\], withκ=0\.01\\kappa=0\.01\.

Smooth states were random combinations of 12 low\-frequency sine modes with independent Gaussian coefficients scaled by inverse wavenumber and normalised to unit root\-mean\-square amplitude\. All five derivatives were evaluated analytically from the same modal expansion\. These exact derivatives provide an exact\-derivative control for state\-differentiation error\. Prediction was evaluated using neural representations based on state and coordinates, state derivatives and geometry\-induced coefficient fields\. The final representation supplied the five geometry\-induced PDE termsqg,i​Di​\[u\]q\_\{g,i\}D\_\{i\}\[u\]to a least\-squares calculation\. The complete mode set and implementation details are provided in Section S1\.

#### Training and evaluation\.

Eight training geometries were uniformly spaced overs∈\[0,1\]s\\in\[0,1\], and prediction was evaluated at the held\-out geometrys=0\.75s=0\.75\. The pointwise experiment used balanced, nested training sets ranging from 800 to 819,200 samples, whereas the field experiment used balanced, nested training sets ranging from 16 to 8,192 complete fields\. Active inputs and targets were standardised using only the selected training samples at each training size\.

For each task and sample count, the three neural representations began from the same weights and saw the same shuffled samples\. The learning curves in Fig\.[1](https://arxiv.org/html/2609.38623#Sx1.F1)report the mean and standard deviation across five runs\. Training schedules, exact seeds and the analytical reconstruction check are reported in Section S1\.

### Operator\-learning benchmark design and evaluation

#### Benchmark design and model comparisons\.

We compared baseline and geometry\-conditioned models on the Laplace and nonlinear reaction–diffusion problems from DIMON and the pipe\-flow, airfoil and elasticity benchmarks distributed with Geo\-FNO and evaluated with Transolver\[[17](https://arxiv.org/html/2609.38623#bib.bib4),[2](https://arxiv.org/html/2609.38623#bib.bib2),[3](https://arxiv.org/html/2609.38623#bib.bib7)\]\. DIMON and DIMON\-Geo used a multi\-input branch–trunk formulation, with the latter supplying geometry\-induced fields through dedicated branches\. Transolver\-Geo retained the case\-specific attention backbone and replaced physical\-coordinate inputs with reference coordinates and geometric channels\. The field choices are motivated by the weak forms above\. All models were trained on solution observations\.

Within each baseline–Geo pair, we matched model architectures and training settings as closely as possible, allowing for the different input representations and their associated input\-processing and architectural adaptations\. Both variants used the same labelled training and test samples, prediction targets, optimisation schedule and number of epochs or outer optimisation steps\. Training sets comprised the firstnnsamples of the case\-specific training pool and were nested across training sizes, with five paired random\-seed values at each size\. Data partitions, network configurations, and training procedures, etc\., are specified separately for each case in Supplementary Sections S2–S6\.

#### Geometry preprocessing\.

Geometric features were constructed from corresponding reference and physical mesh points\. Laplace and reaction–diffusion used the supplied DIMON reference coordinates and domain correspondences\. For pipe flow, airfoil flow and elasticity, the fixed reference mesh was the pointwise mean of the full 1,000\-case training\-pool meshes\. This reference was reused at every labelled training size, without using test geometries or additional solution labels\.

For each domain correspondence, writing𝒅i=xi−ξi\\bm\{d\}\_\{i\}=x\_\{i\}\-\\xi\_\{i\}, we estimated the local displacement gradient by ridge\-regularised least squares and recovered the domain\-map Jacobian as

𝑨i=arg⁡min⁡∑j∈𝒩⁡\(i\)𝑨⁡‖\(𝒅j−𝒅i\)−𝑨⁡\(ξj−ξi\)‖22\+λ​‖𝑨‖F2,𝑱i=𝑰\+𝑨i\.\\bm\{A\}\_\{i\}=\\arg\\min\_\{\\bm\{A\}\}\\sum\_\{j\\in\\mathcal\{N\}\(i\)\}\\left\\\|\(\\bm\{d\}\_\{j\}\-\\bm\{d\}\_\{i\}\)\-\\bm\{A\}\(\\xi\_\{j\}\-\\xi\_\{i\}\)\\right\\\|\_\{2\}^\{2\}\+\\lambda\\\|\\bm\{A\}\\\|\_\{F\}^\{2\},\\qquad\\bm\{J\}\_\{i\}=\\bm\{I\}\+\\bm\{A\}\_\{i\}\.\(29\)Here𝒩⁡\(i\)\\mathcal\{N\}\(i\)is a neighbourhood in the reference mesh andλ\\lambdais the ridge parameter\. The reconstructed Jacobians supplied the local geometric information for the feature channels\. Neighbourhood definitions, regularisation, mapping\-quality diagnostics and case\-retention rules are given in Sections S2–S6\. Where feature or target standardisation was used, its statistics were estimated from the selected training samples and applied unchanged to the test data\. This fitting subset is distinct from the full geometry pool used to construct the Transolver reference meshes\.

#### Data\-efficiency analysis\.

Training sizenncounts complete geometry–solution samples\. For each trained model, the unweighted discrete relativeL2L\_\{2\}error was computed for each test sample and then averaged over the test set\. At each measured sizenin\_\{i\}, the five run\-level averages were summarised by their meanE¯i\\overline\{E\}\_\{i\}and sample standard deviationsis\_\{i\}\. The following procedure was applied to both the DIMON and Transolver comparisons in Figs\.[2](https://arxiv.org/html/2609.38623#Sx1.F2)c,e and[3](https://arxiv.org/html/2609.38623#Sx2.F3)b,d,f\.

Required sample counts were estimated by inverting each model’s mean learning curve using piecewise log–log interpolation\. For a target errorE∗E\_\{\*\}between the mean errors at two adjacent measured sizesnin\_\{i\}andni\+1n\_\{i\+1\},

n⁡\(E∗\)=exp⁡\[\(1−θ\)​log⁡ni\+θ​log⁡ni\+1\],θ=log⁡E∗−log⁡E¯ilog⁡E¯i\+1−log⁡E¯i\.n\(E\_\{\*\}\)=\\exp\\\!\\left\[\(1\-\\theta\)\\log n\_\{i\}\+\\theta\\log n\_\{i\+1\}\\right\],\\qquad\\theta=\\frac\{\\log E\_\{\*\}\-\\log\\overline\{E\}\_\{i\}\}\{\\log\\overline\{E\}\_\{i\+1\}\-\\log\\overline\{E\}\_\{i\}\}\.\(30\)Only target errors within the common measured error range of the two models were compared, without extrapolation\. The percentage reduction in required labelled samples was

100​\[1−nGeo​\(E∗\)nbase​\(E∗\)\],100\\left\[1\-\\frac\{n\_\{\\rm Geo\}\(E\_\{\*\}\)\}\{n\_\{\\rm base\}\(E\_\{\*\}\)\}\\right\],\(31\)where the baseline is DIMON or Transolver, as appropriate\. These estimates invert the across\-seed mean curves, rather than averaging separately inverted seed\-level curves\. They describe interpolated labelled\-sample requirements, not additional training experiments or savings in geometric information or computational cost\.

For visualisation, the mean error was interpolated in log–log space and the sample standard deviation was interpolated linearly againstlog⁡n\\log n, retaining its original error units\. Shaded bands show the interpolated mean plus or minus this standard deviation\. They represent between\-run variation and its visual interpolation, not confidence intervals or uncertainty estimates for the inferred sample requirements\.

### Geometry\-complete equation discovery

#### Identification setting and candidate libraries\.

Equation discovery was evaluated on a deforming thermoelastic solid and a nutrient\-limited tissue undergoing free\-boundary growth\. In both systems, the stateu⁡\(ξ,t\)u\(\\xi,t\)and evolving geometryx=Φg​\(ξ,t\)x=\\Phi\_\{g\}\(\\xi,t\)were observed, the geometry\-evolution closure was known and the prescribed inputs were collected insknowns^\{\\rm known\}, excluding all candidate coefficients to be inferred\. Known input fields, such as a source profile, can enter the library with unknown multipliers\. The transport PDE was identified in the reference\-coordinate regression

ut=Θ​𝒄,u\_\{t\}=\\Theta\\,\\bm\{c\},\(32\)whereΘ\\Thetadenotes the candidate\-library matrix and𝒄\\bm\{c\}contains one coefficient for every column, with zero entries for inactive candidates\. A single vector is shared across all training realizations\. These are coefficients of the chosen equation representation, not a physical\-parameter vector\. Individual entries may coincide with physical parameters only for particular candidate definitions and scalings\.

The geometry\-complete library retained the reference\-coordinate terms and added the mapped diffusion and ALE transport operators in Table[1](https://arxiv.org/html/2609.38623#Sx4.T1)\. Both systems used the candidate block\[𝒟g​\[u\],u​𝒟g​\[u\],𝒜g​\[u\],u​𝒜g​\[u\]\]\[\\mathcal\{D\}\_\{g\}\[u\],\\,u\\mathcal\{D\}\_\{g\}\[u\],\\,\\mathcal\{A\}\_\{g\}\[u\],\\,u\\mathcal\{A\}\_\{g\}\[u\]\]\. Here, “geometry\-complete” refers to this prespecified transformed\-operator block\. Both libraries used the same aggregation, normalisation and model\-selection protocol\.

#### Thermoelastic solid system\.

On a reference annular domain0\.5≤r≤10\.5\\leq r\\leq 1,T⁡\(ξ,t\)T\(\\xi,t\)denotes temperature,q⁡\(ξ,t\)q\(\\xi,t\)the pulled\-back heat source, and𝒅⁡\(ξ,t\)\\bm\{d\}\(\\xi,t\)the mechanical displacement field definingx=ξ\+𝒅⁡\(ξ,t\)x=\\xi\+\\bm\{d\}\(\\xi,t\)\. The known quasi\-static elastostatic closure was governed by

μsΔ𝒅\+\(λs\+μs\)∇\(∇⋅𝒅\)=βT∇T,\\mu\_\{s\}\\Delta\\bm\{d\}\+\(\\lambda\_\{s\}\+\\mu\_\{s\}\)\\nabla\(\\nabla\\\!\\cdot\\bm\{d\}\)=\\beta\_\{T\}\\nabla T,\(33\)whereλs\\lambda\_\{s\}andμs\\mu\_\{s\}are Lamé elastic moduli andβT\\beta\_\{T\}governs isotropic thermal expansion\. The transported temperature obeyedTt=κ​𝒟d​\[T\]\+qT\_\{t\}=\\kappa\\mathcal\{D\}\_\{d\}\[T\]\+qwith intrinsic thermal diffusivityκ=0\.01\\kappa=0\.01and unit source gain\. The reference grid comprised2525radial and7272angular points, with8181temporal snapshots recorded over0≤t≤20\\leq t\\leq 2\. The geometry\-induced candidate block was\[𝒟d​\[T\],T​𝒟d​\[T\],𝒜d​\[T\],T​𝒜d​\[T\]\]\[\\mathcal\{D\}\_\{d\}\[T\],\\,T\\mathcal\{D\}\_\{d\}\[T\],\\,\\mathcal\{A\}\_\{d\}\[T\],\\,T\\mathcal\{A\}\_\{d\}\[T\]\], where𝒜d​\[T\]=\(𝑱d−1​dt\)⋅∇ξT\\mathcal\{A\}\_\{d\}\[T\]=\(\\bm\{J\}\_\{d\}^\{\-1\}d\_\{t\}\)\\cdot\\nabla\_\{\\xi\}Twas evaluated from the observed displacement velocity\. The temperature balance uses the material\-time derivative\. Consequently, the two ALE candidates have zero generating coefficients\. Downstream displacement fields, Jacobian distributions and von Mises stresses were evaluated via the elastostatic closure as validation targets\.

#### Free\-boundary tumour\-growth system\.

Letc⁡\(ρ,θ,t\)c\(\\rho,\\theta,t\)denote biochemical nutrient concentration,R⁡\(θ,t\)R\(\\theta,t\)the local boundary radius,SSthe continuous nutrient supply, andϵ\\epsilonthe metabolic uptake field\. The physical domain was parameterised in polar coordinates byx=ρ​R​\(θ,t\)​𝒆rx=\\rho R\(\\theta,t\)\\bm\{e\}\_\{r\}\. On this common coordinate template, nutrient transport was governed by

ct=D​𝒟R​\[c\]\+ρ​RtR​cρ\+S−λ​ϵ​c,c\_\{t\}=D\\,\\mathcal\{D\}\_\{R\}\[c\]\+\\rho\\frac\{R\_\{t\}\}\{R\}c\_\{\\rho\}\+S\-\\lambda\\epsilon c,\(34\)with intrinsic diffusivityD=0\.012D=0\.012, unit coefficients for boundary\-motion ALE transport and nutrient supply, and metabolic consumption rateλ=0\.72\\lambda=0\.72\. The moving boundary was generated by the known kinematic closure

Rt=μ​ϵb​cb1\+α​cb−δ​mb,R\_\{t\}=\\mu\\frac\{\\epsilon\_\{b\}c\_\{b\}\}\{1\+\\alpha c\_\{b\}\}\-\\delta m\_\{b\},\(35\)where subscriptbbindicates boundary values, capturing nutrient\-driven mitotic expansion against boundary cell loss\. The computational mesh utilized2929radial and9696angular points, storing101101snapshots over0≤t≤2\.40\\leq t\\leq 2\.4\. The geometry\-induced candidate block was\[𝒟R​\[c\],c​𝒟R​\[c\],𝒜R​\[c\],c​𝒜R​\[c\]\]\[\\mathcal\{D\}\_\{R\}\[c\],\\,c\\mathcal\{D\}\_\{R\}\[c\],\\,\\mathcal\{A\}\_\{R\}\[c\],\\,c\\mathcal\{A\}\_\{R\}\[c\]\]\. Evolving boundary radius, total domain area, and nutrient profiles served as downstream closed\-loop validation targets\.

#### Trajectory partitioning and sparse regression protocol\.

For each benchmark, four evolving\-domain realizations generated under distinct forcing configurations and initial geometries were used for identification, while four geometrically shifted realizations were held out for predictive validation\. Temporal derivatives and candidate fields were locally aggregated before sparse regression, following a formulation inspired by weak\-form methods\[[22](https://arxiv.org/html/2609.38623#bib.bib20)\]\.

Candidate library columns were normalised to unit Euclidean norm\. Sparse selection used sequential threshold ridge regression \(STRidge\), with ridge and threshold parameters selected by leave\-one\-trajectory\-out validation, followed by an ordinary least\-squares refit on the selected support\[[18](https://arxiv.org/html/2609.38623#bib.bib18),[19](https://arxiv.org/html/2609.38623#bib.bib19)\]\. Candidate lists, aggregation settings and hyperparameter grids are provided in Section S7\.

Noise sensitivity was evaluated at0%0\\%,0\.5%0\.5\\%,1%1\\%,2%2\\%and5%5\\%state\-field noise across five realizations per level\. The trajectory\-diversity analysis evaluated every non\-empty subset of the four training trajectories using the main\-experiment hyperparameters\. Smoothing and subset protocols are given in Section S8\.

### Evaluation and statistical reporting

For a predicted fieldy^\\widehat\{y\}and targetyyevaluated at the same points, relative error is

EL2=‖y^−y‖2‖y‖2\.E\_\{L\_\{2\}\}=\\frac\{\\\|\\widehat\{y\}\-y\\\|\_\{2\}\}\{\\\|y\\\|\_\{2\}\}\.\(36\)Operator\-learning errors are computed per test sample and then averaged over the corresponding test set\. In the pointwise diffusion experiment, the relative norm is evaluated across the eight held\-out states, whereas in the field task it is averaged over 16 held\-out state fields\. Parameter\-sweep residuals are computed at each geometry–diffusivity pair, and PDE\-identification prediction errors are averaged over four held\-out trajectories\. Stress, displacement, boundary and area are generated by coupling the identified state equation to the known geometry closure during closed\-loop validation\.

For equation residuals,yydenotes the sampled time derivative andΘ​𝒄^\\Theta\\widehat\{\\bm\{c\}\}the derivative reconstructed from the fitted candidate coefficients, giving‖Θ​𝒄^−y‖2/‖y‖2\\\|\\Theta\\widehat\{\\bm\{c\}\}\-y\\\|\_\{2\}/\\\|y\\\|\_\{2\}\. Fold reductions are calculated from unrounded results\. Discovery support and coefficient errors are evaluated on the ordered union of terms in the geometry\-complete candidate library\. In this evaluation,𝒄⋆\\bm\{c\}^\{\\star\}and𝒄^\\widehat\{\\bm\{c\}\}denote the generating and fitted candidate\-coefficient vectors on that full ordered union, with zero entries for inactive terms\. An unselected or unavailable candidate has fitted coefficient zero\. A generating term absent from the reference\-coordinate library therefore counts as a false negative\. SupportF1F\_\{1\}is the harmonic mean of precision and recall, and coefficient relative error is‖𝒄^−𝒄⋆‖2/‖𝒄⋆‖2\\\|\\widehat\{\\bm\{c\}\}\-\\bm\{c\}^\{\\star\}\\\|\_\{2\}/\\\|\\bm\{c\}^\{\\star\}\\\|\_\{2\}\.

The exact\-equivalence, analytical\-reconstruction and clean equation\-identification calculations are deterministic\. Diffusion\-example neural curves and operator\-learning benchmarks report the mean and standard deviation across five runs\. Noise\-study lines show medians across five realizations per level, with individual realizations displayed as pale points\. Trajectory\-diversity plots enumerate every subset\.

## Competing interests

The authors declare no competing interests\.

## Additional information

Supplementary information accompanies this manuscript\.

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Supplementary Information

Geometry–physics confounding impairs PDE learning across varying domains

Yinghao Cheng1,2, Gengxiang Chen1,3, Xu Liu4, Qinglu Meng1, Yixin Jing1, Xiangguo Tang1, Wenping Mou1, Lihui Wang5,6, and Yingguang Li1,∗

1College of Mechanical & Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China 2School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, Belfast, United Kingdom 3School of Mechanical and Aerospace Engineering, Queen’s University Belfast, Belfast, United Kingdom 4School of Mechanical and Power Engineering, Nanjing Tech University, Nanjing, China 5Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hong Kong, China 6Department of Production Engineering, KTH Royal Institute of Technology, Stockholm, Sweden ∗Correspondence: liyingguang@nuaa\.edu\.cn\.

## S1\. Diffusion implementation details

The prediction experiments in Fig\. 1d compare neural inputs comprising state and coordinates, exact state derivatives, or state derivatives together with the five geometry\-induced coefficient fieldsqiq\_\{i\}defined in Methods\. The final representation supplies the five geometry\-induced PDE termsqi​Di​\[u\]q\_\{i\}D\_\{i\}\[u\]directly\.

The state\-derivative representation uses exact first\- and second\-order state derivatives to remove differentiation error while the geometry\-induced coefficient fields remain implicit\. The first three representations are matched within each task\. Pointwise inputs are padded to 37 dimensions and use the same 37–64–64–1 GELU MLP\. Field inputs are padded to 13 channels and use the same13→32→32→113\\to 32\\to 32\\to 1three\-layer3×33\\times 3GELU CNN\. Padded channels remain exactly zero\. At each count the three neural representations share initialization, shuffled examples, optimizer, learning\-rate schedule, epoch budget and target\. Input and target normalisation is recomputed from the selected training samples at each training size and then applied to the test data\. The final representation uses a double\-precision, five\-column least\-squares problem with five independently fitted weights\.

Prediction uses a41×4141\\times 41reference grid\. Twelve sine products with mode pairs from\(1,1\)\(1,1\)to\(4,4\)\(4,4\)generate smooth states\. Gaussian modal weights are attenuated by\(m2\+n2\)−1/2\(m^\{2\}\+n^\{2\}\)^\{\-1/2\}, and every state is normalised to unit root\-mean\-square magnitude\. The five derivatives in the expanded operator are differentiated analytically from the modal series\. Theqiq\_\{i\}are evaluated from the analytical map, with complex\-step derivatives for the first derivatives of𝑲\\bm\{K\}\. These analytical derivatives remove finite\-difference error from the comparison between the state\-derivative and geometry\-induced\-coefficient\-field representations\.

Eight training geometries are uniformly spaced overs∈\[0,1\]s\\in\[0,1\], and prediction is evaluated at the held\-out geometrys=0\.75s=0\.75\.

The pointwise pool contains 100 states, eight geometries and 1,024 nested interior points per state–geometry pair, or 819,200 samples\. The same point ordering is used for a state on all eight geometries\. Nested training sets have sizes

\{800,1,600,3,200,6,400,12,800,25,600,51,200,102,400,204,800,409,600,819,200\},\\begin\{split\}\\\{&800,1\{,\}600,3\{,\}200,6\{,\}400,12\{,\}800,25\{,\}600,\\\\ &51\{,\}200,102\{,\}400,204\{,\}800,409\{,\}600,819\{,\}200\\\},\\end\{split\}\(S1\)with each set taking the same number of leading points from all 800 state–geometry blocks\. Evaluation uses all39×3939\\times 39eligible centres from eight disjoint held\-out states\.

The field pool contains 1,024 states paired with all eight geometries, or 8,192 complete fields\. Nested training sets comprise the first16,32,64,128,256,512,1,024,2,048,4,09616,32,64,128,256,512,1,024,2,048,4,096and8,1928,192fields in the stored order\. Because fields are stored in state\-major groups of eight, every training set is exactly balanced across geometry\. Sixteen independent states are held out\. The CNN sees the full grid, while training loss and relative error are computed over the central35×3535\\times 35region to avoid convolution\-padding effects at the boundary\.

All three neural representations are trained for 1,000 epochs with Adam and cosine learning\-rate decay from10−310^\{\-3\}to10−510^\{\-5\}\. Maximum batch sizes are 2,048 point samples and 32 fields\. The extended count curves and error bands report the mean and standard deviation across five independent optimisation runs\.

## Shared settings and reporting conventions for operator\-learning benchmarks

Sections S2–S6 specify the data, network architectures and optimisation protocols for the five operator\-learning comparisons\. Each training size uses five runs with seeds0,7,42,1234,20260,7,42,1234,2026, paired between the baseline and its geometry\-induced counterpart\. A training size denotes the number of complete geometry–solution samples, not the number of spatial query points\. Errors are computed as the unweighted discrete relativeL2L\_\{2\}norm for each test sample, averaged over test samples and then summarised by the mean and sample standard deviation across seeds\. Network\-width lists below include input and output dimensions\.

## S2\. Laplace benchmark

#### Data and geometric representation\.

The Laplace problem uses the DIMON dataset of harmonic solutions on deformed planar domains with varying Dirichlet boundary data\[[17](https://arxiv.org/html/2609.38623#bib.bib4)\]\. Each solution is represented at 2,601 corresponding points on a51×5151\\times 51reference grid\. The original ordered partition contains 3,303 training cases and 197 test cases\. Local Jacobians are reconstructed from the reference\-to\-physical displacements using 25 nearest reference points, including the centre, and ridge parameter10−810^\{\-8\}in Eq\. \([29](https://arxiv.org/html/2609.38623#Sx4.E29)\)\. Retaining cases withminξ⁡jg\>0\.5\\min\_\{\\xi\}j\_\{g\}\>0\.5within each partition gives 3,233 training cases and 193 test cases\. Their original order is preserved\. Both models use the firstn∈\{300,500,1000,2000,3000\}n\\in\\\{300,500,1000,2000,3000\\\}retained training cases and the same 193 test cases\. The reference coordinates, physical meshes, boundary values and solution targets in the two processed datasets are identical\.

The baseline geometry input contains 20 principal\-component coefficients, with ten modes for each Cartesian component of the mapping displacement\. The two principal\-component bases and their centring means are fitted anew on the selected training samples and applied unchanged to the test set\. DIMON\-Geo instead receives the three independent symmetric geometry\-tensor channels stored in the processed dataset\. Each channel is sampled every fifth point along both reference\-grid axes, giving an11×1111\\times 11array and 121 values per branch\. This subsampling applies only to the geometric branch inputs\. Both networks predict at all 2,601 query points and receive the same 68 sampled boundary values\. Apart from the baseline PCA centring, no input or target standardisation is applied\.

#### Network comparison\.

Both models use fully connected branch–trunk networks with a latent width of 100\. A hyperbolic\-tangent activation follows every affine layer in each branch and trunk, including the final layer\. The baseline multiplies the geometry and boundary branch outputs elementwise and takes their inner product with the coordinate\-trunk output\. DIMON\-Geo multiplies the outputs of its three tensor branches and boundary branch before the same trunk contraction\. No additional scalar output bias is used\. Table[S1](https://arxiv.org/html/2609.38623#Sx9.T1)lists the dimensions\. The boundary branch and trunk are unchanged, while the geometry branches differ in both input representation and internal widths\.

Table S1:Laplace network configurations\.Each tensor component in DIMON\-Geo has a separate branch with the listed dimensions\.
#### Training and evaluation\.

Both networks are trained using full\-batch mean\-squared error over the selected training samples and all spatial points\. Training comprises 10,000 Adam steps at a learning rate of10−310^\{\-3\}and 39,000 steps at10−410^\{\-4\}, with Adam reinitialised between these stages, followed by 1,000 L\-BFGS outer steps\. The final trained weights are used for evaluation, with relativeL2L\_\{2\}errors computed over all 2,601 spatial points\.

## S3\. Reaction–diffusion benchmark

#### Data and geometric representation\.

The nonlinear reaction–diffusion benchmark uses the DIMON data on affinely deformed domains, with varying initial conditions and the no\-flux boundary condition described in Methods\[[17](https://arxiv.org/html/2609.38623#bib.bib4)\]\. Each sample contains 2,800 spatial points and 26 solution snapshots att=0,0\.08,…,2\.0t=0,0\.08,\\ldots,2\.0, giving 72,800 space–time queries\. The first 2,700 cases form the training pool and the last 300 form the fixed test set\. Training uses the firstn∈\{500,1000,1500,2000,2700\}n\\in\\\{500,1000,1500,2000,2700\\\}cases in the pool, giving nested training sets\. The geometry preprocessing uses 25 nearest reference points, including the centre, with ridge parameter10−810^\{\-8\}\. Its retention criterion isminξ⁡jg\>0\.4\\min\_\{\\xi\}j\_\{g\}\>0\.4\. The processed dataset contains 3,000 cases and has a minimum determinant of approximately 0\.428\.

The baseline reads the supplied 50 initial\-condition coefficients and 20 geometry coefficients\. These precomputed encodings are not refitted when the selected training samples change\. DIMON\-Geo uses 255 sampled initial\-condition values instead of the 50 coefficients, together with four full\-resolution geometric inputs: the determinant fieldjgj\_\{g\}and the three independent symmetric geometry\-tensor channels in the processed dataset\. Each geometric branch therefore receives 2,800 values\. Both trunks query the same three coordinates\(t,ξ1,ξ2\)\(t,\\xi\_\{1\},\\xi\_\{2\}\)\. No input or target standardisation is applied\.

#### Network comparison\.

The branch and trunk dimensions are listed in Table[S2](https://arxiv.org/html/2609.38623#Sx10.T2)\. Each subnetwork uses a hyperbolic\-tangent activation after every affine layer, including its final layer\. In the baseline, the initial\-condition and geometry branch outputs are multiplied elementwise and contracted with the 200\-component trunk output\. In DIMON\-Geo, the three tensor\-branch outputs are concatenated into a 600\-component vector and passed through an affine600→200600\\to 200fusion layer without a further activation\. The fused vector is multiplied elementwise by the determinant\-branch and initial\-condition\-branch outputs before contraction with the trunk\. The comparison thus changes the initial\-condition encoding and branch dimensions as well as the geometric representation\.

Table S2:Reaction–diffusion network configurations\.
#### Training and evaluation\.

Both networks are trained using full\-batch mean\-squared error over the selected training samples and all spatial points and times\. Training comprises 100,000 Adam steps with learning rates2×10−42\\times 10^\{\-4\}for the first 50,000 steps,10−410^\{\-4\}for the next 40,000 and10−510^\{\-5\}for the final 10,000, with Adam reinitialised at each transition\. The final trained weights are used for evaluation, with each sample’s relativeL2L\_\{2\}error computed jointly over all 26 snapshots and 2,800 spatial points\.

## S4\. Pipe\-flow benchmark

#### Data and geometric representation\.

The pipe\-flow comparison uses the geometry\-varying flow benchmark distributed with Geo\-FNO and Transolver\[[2](https://arxiv.org/html/2609.38623#bib.bib2),[3](https://arxiv.org/html/2609.38623#bib.bib7)\]\. The raw dataset contains 2,310 cases on129×129129\\times 129grids\. We use the first 1,200 cases in source order, with the first 1,000 forming the training pool and the next 200 forming the test set\. Sampling every point along the first grid axis and every seventh point along the second gives a129×19129\\times 19grid with 2,451 points\. The scalar target is horizontal velocityvxv\_\{x\}\. Both models use identical sampled target arrays\.

The reference mesh is the pointwise mean of the 1,000 training\-pool meshes\. It remains fixed for all training sizes, so small\-data runs use geometry from the full training pool when constructing this reference, but no additional solution labels or test geometries\. Local Jacobians are recovered using radius\-2 structured index patches, containing at most 25 points including the centre, with truncated patches at the boundary and ridge parameter10−810^\{\-8\}\. No cases are filtered or sorted by Jacobian quality\. All 1,200 selected cases havejg\>0\.2j\_\{g\}\>0\.2, with a global minimum of 0\.785 and maximum Jacobian condition number below 4\.120\.

The baseline receives the two sample\-dependent physical coordinates\. Transolver\-Geo receives the two reference coordinates and seven additional channels in the order\(Kg,11,Kg,12,Kg,22,Bg,11,Bg,12,Bg,21,Bg,22\)\(K\_\{g,11\},K\_\{g,12\},K\_\{g,22\},B\_\{g,11\},B\_\{g,12\},B\_\{g,21\},B\_\{g,22\}\)\. The matrices𝑲g\\bm\{K\}\_\{g\}andBgB\_\{g\}are defined in Methods\. Coordinates and geometric channels are standardised channelwise using means and standard deviations over the selected training cases and their mesh points\. A scalar target normaliser is fitted over the same training cases\. Test data use these training statistics, with10−810^\{\-8\}added to each standard deviation\.

#### Network comparison\.

Both models use the structured\-mesh Transolver backbone in Table[S3](https://arxiv.org/html/2609.38623#Sx11.T3)\. The input projection is a two\-layer MLP with widths\[din,256,128\]\[d\_\{\\rm in\},256,128\]and an intermediate GELU activation\. Each of the eight blocks contains pre\-normalised physics attention and a pre\-normalised\[128,128,128\]\[128,128,128\]GELU MLP, both with residual connections\. Physics attention uses eight heads, 64 slice tokens and two3×33\\times 3feature projections on the structured grid\. A final layer normalisation and affine128→1128\\to 1map produce the scalar prediction\. Dropout is zero\. Neither time embeddings nor the optional distance\-to\-reference\-grid positional encoding are used\. The coordinate\-only implementation adds a learned 128\-component vector after the input projection\. The feature\-conditioned structured\-mesh path does not add this vector\. The attention backbone is otherwise unchanged\.

Table S3:Pipe\-flow network configurations\.
#### Training and evaluation\.

Both models use the firstn∈\{200,400,600,800,1000\}n\\in\\\{200,400,600,800,1000\\\}cases of the fixed training pool\. Training uses 500 epochs of AdamW with shuffled batches of four, weight decay10−510^\{\-5\}, gradient\-norm clipping at 0\.1 and a batch\-wise OneCycleLR schedule with maximum learning rate10−310^\{\-3\}\. The loss sums per\-sample relativeL2L\_\{2\}errors over each batch\. Losses and reported errors are evaluated on the physical velocity scale over all 2,451 spatial points, and the final trained weights are used for evaluation\.

## S5\. Airfoil\-flow benchmark

#### Data and geometric representation\.

The airfoil comparison uses the compressible\-flow benchmark distributed with Geo\-FNO and Transolver\[[2](https://arxiv.org/html/2609.38623#bib.bib2),[3](https://arxiv.org/html/2609.38623#bib.bib7)\]\. Although the raw dataset contains 2,490 cases, the experiments use only the first 1,200 in source order\. The first 1,000 form the training pool and the following 200 form the fixed test set\. Each case retains all221×51=11,271221\\times 51=11\{,\}271mesh points, without spatial subsampling\. The scalar prediction target is Mach number, taken from channel 4, using zero\-based indexing, of the flow array\.

The reference mesh is the pointwise mean of all 1,000 training\-pool meshes and is fixed across training sizes, as in S4\. Jacobian reconstruction uses radius\-2 structured patches with ridge parameter10−810^\{\-8\}\. The source\-order split is unchanged by the Jacobian diagnostics\. No case filtering, sorting or determinant clipping is applied\. The minimum determinant over the 1,200 selected cases is approximately 0\.423\. The baseline receives physical coordinates, whereas Transolver\-Geo receives reference coordinates and the four cofactor channels\(Cg,11,Cg,12,Cg,21,Cg,22\)\(C\_\{g,11\},C\_\{g,12\},C\_\{g,21\},C\_\{g,22\}\)\. Coordinate, geometry\-channel and target normalisation use only the selected training samples and follow the procedure in S4\.

#### Network comparison\.

Both models use the same structured\-mesh attention blocks, activation, normalisation, output head and positional\-encoding settings as the pipe models\. Their grid dimensions are221×51221\\times 51\. The geometry\-conditioned input projection has six input channels rather than the baseline’s two\. The learned\-vector convention is also the same as in S4\. The resulting configurations are given in Table[S4](https://arxiv.org/html/2609.38623#Sx12.T4)\.

Table S4:Airfoil\-flow network configurations\.Both models use the structured\-mesh backbone specified in S4\.
#### Training and evaluation\.

Both models use the firstn∈\{200,400,600,800,1000\}n\\in\\\{200,400,600,800,1000\\\}cases of the fixed training pool\. Training uses 500 epochs, shuffled batches of four and the AdamW, gradient\-clipping and batch\-wise OneCycleLR settings specified in S4\. The loss sums per\-sample relativeL2L\_\{2\}errors over each batch\. Losses and reported errors are computed from decoded Mach numbers over all 11,271 spatial points\. The final trained weights are used for evaluation\.

## S6\. Elasticity benchmark

#### Data and geometric representation\.

The elasticity comparison uses the varying\-unit\-cell benchmark distributed with Geo\-FNO and Transolver\[[2](https://arxiv.org/html/2609.38623#bib.bib2),[3](https://arxiv.org/html/2609.38623#bib.bib7)\]\. The source dataset contains 2,000 cases, each represented by a 972\-point unstructured mesh\. The first 1,000 cases form the training pool, and the final 200 cases, with zero\-based source indices 1,800–1,999, form the test set\. The intervening 800 cases are unused\. The target is the scalar stress field, and both models predict this same field at all 972 points\.

The fixed reference point cloud is the pointwise mean of the 1,000 training\-pool meshes\. Jacobians are reconstructed from 24 nearest reference points, including the centre, with ridge parameter10−810^\{\-8\}\. The reference mesh and neighbour stencils do not depend on the test geometries and are reused at every training size\. The 1,200 selected cases are retained in their prescribed order without Jacobian\-based filtering\. Their minimum determinant is approximately 0\.405, so all exceed 0\.2\. This value is a diagnostic of the selected dataset, not a criterion used to change its membership\.

The baseline receives unnormalised physical coordinates\. Transolver\-Geo receives the two reference coordinates and six channels ordered as\(Cg,11,Cg,12,Cg,21,Cg,22,dg,1,dg,2\)\(C\_\{g,11\},C\_\{g,12\},C\_\{g,21\},C\_\{g,22\},d\_\{g,1\},d\_\{g,2\}\), where𝒅g=x−ξ\\bm\{d\}\_\{g\}=x\-\\xidescribes the benchmark domain correspondence, not the predicted mechanical response\. The reference coordinates and six geometric channels are standardised using the selected training samples\. Both models use a scalar stress normaliser fitted to those samples\.

#### Network comparison\.

Both models use the irregular\-mesh Transolver variant, with eight blocks, hidden width 128, eight heads, 64 slice tokens, MLP ratio one, GELU activation and zero dropout\. The block structure and scalar output head follow S4, but the two feature projections in physics attention are pointwise affine maps rather than structured\-grid convolutions\. No time or optional distance\-based positional encoding is used\. In this irregular\-mesh implementation, both input paths add the learned 128\-component vector after their input MLP\. The attention backbone is held fixed and the input widths are listed in Table[S5](https://arxiv.org/html/2609.38623#Sx13.T5)\.

Table S5:Elasticity network configurations\.Both models use irregular\-mesh physics attention\.
#### Training and evaluation\.

Both models use the firstn∈\{200,400,600,800,1000\}n\\in\\\{200,400,600,800,1000\\\}cases of the fixed training pool\. Training uses 500 epochs, shuffled batches of one and the AdamW, gradient\-clipping and batch\-wise OneCycleLR settings specified in S4\. The relativeL2L\_\{2\}training loss and reported errors are evaluated on the physical stress scale over all 972 spatial points\. The final trained weights are used for evaluation\.

## S7\. Equation\-discovery protocols and identified equations

#### Shared identification protocol\.

The thermoelastic and tumour\-growth cases use the same sparse\-identification protocol\. For each system, four trajectories are used for identification and four geometrically shifted trajectories are held out for predictive validation\. The transported\-state equation is identified while the geometry\-evolution closure remains known\.

Both systems use a five\-point fourth\-order centred finite difference for time derivatives\. Candidate columns and target derivatives are averaged over local3×53\\times 5spatial boxes before equidistant subsampling, and candidate columns are normalised to unit Euclidean norm\. Sequential threshold ridge regression \(STRidge\)\[[18](https://arxiv.org/html/2609.38623#bib.bib18),[19](https://arxiv.org/html/2609.38623#bib.bib19)\]is evaluated over ridge penalties\{10−10,10−8,10−6,10−5\}\\\{10^\{\-10\},10^\{\-8\},10^\{\-6\},10^\{\-5\}\\\}and relative thresholds\{0\.002,0\.005,0\.01,0\.02,0\.04,0\.06,0\.08,0\.14\}\\\{0\.002,0\.005,0\.01,0\.02,0\.04,0\.06,0\.08,0\.14\\\}\. Leave\-one\-trajectory\-out validation selects the most parsimonious model whose mean residual is within20%20\\%of the minimum, with ties resolved by residual magnitude\. The selected support is then refitted by ordinary least squares across all four training trajectories\.

Support and coefficient errors follow the definitions in Methods\. Coefficients are indexed over the full evaluation library, with zeros for inactive terms, and refer to the chosen candidate representation rather than a physical\-parameter vector\. In both cases, the geometry\-complete library retains the reference\-coordinate candidates and adds mapped diffusion, ALE transport and their state\-modulated counterparts\. The case\-specific libraries and fitted equations are given below\.

#### Thermoelastic system\.

Identification targets the temperature law in Fig\.[4](https://arxiv.org/html/2609.38623#Sx2.F4), with the mechanical response supplied by the known elastic closure\. The prescribed heat\-source fieldqqbelongs tosknowns^\{\\rm known\}, whereas its multiplier is an unknown candidate coefficient\.

The reference\-coordinate candidate library is

\[1,T,T2,T3,q,Tξ1,Tξ2,Δξ​T,T​Δξ​T,\|∇ξT\|2\]\.\[1,T,T^\{2\},T^\{3\},q,T\_\{\\xi\_\{1\}\},T\_\{\\xi\_\{2\}\},\\Delta\_\{\\xi\}T,T\\Delta\_\{\\xi\}T,\|\\nabla\_\{\\xi\}T\|^\{2\}\]\.The geometry\-complete candidate library adds

\[𝒟d​\[T\],T​𝒟d​\[T\],𝒜d​\[T\],T​𝒜d​\[T\]\],\[\\mathcal\{D\}\_\{d\}\[T\],T\\mathcal\{D\}\_\{d\}\[T\],\\mathcal\{A\}\_\{d\}\[T\],T\\mathcal\{A\}\_\{d\}\[T\]\],where𝒜d​\[T\]=\(𝑱d−1​dt\)⋅∇ξT\\mathcal\{A\}\_\{d\}\[T\]=\(\\bm\{J\}\_\{d\}^\{\-1\}d\_\{t\}\)\\cdot\\nabla\_\{\\xi\}Tis evaluated from the observed displacement velocity\. The material\-time formulation assigns zero generating coefficients to the two thermal ALE candidates, which remain available to the sparse selector\.

The fitted reference\-coordinate equation was

Tt=0\.999061​q\+0\.014685​Δξ​T−0\.049062​T​Δξ​T,T\_\{t\}=0\.999061q\+0\.014685\\Delta\_\{\\xi\}T\-0\.049062\\,T\\Delta\_\{\\xi\}T,\(S2\)and the geometry\-complete equation was

Tt=0\.999980​q\+0\.00999978​𝒟d​\[T\]\.T\_\{t\}=0\.999980q\+0\.00999978\\,\\mathcal\{D\}\_\{d\}\[T\]\.\(S3\)The reference\-coordinate and geometry\-complete fits have supportF1F\_\{1\}scores of 0\.4 and 1 and held\-out residuals of 0\.171 and3\.02×10−53\.02\\times 10^\{\-5\}, respectively\. The geometry\-complete coefficient relativeL2L\_\{2\}error is2\.03×10−52\.03\\times 10^\{\-5\}\. Both fitted temperature equations were integrated with the same elastic closure\. Fig\.[4](https://arxiv.org/html/2609.38623#Sx2.F4)and Extended Data Fig\. 3 report the resulting held\-out predictions\.

#### Tumour\-growth system\.

The nutrient law in Fig\.[5](https://arxiv.org/html/2609.38623#Sx2.F5)is identified while the boundary\-growth closure remains known\. The reference\-coordinate candidate library is

\[1,c,c2,c3,S,−ϵ​c,cρ,ρ−1​cθ,Δref​c,c​Δref​c,\|∇refc\|2\]\.\[1,c,c^\{2\},c^\{3\},S,\-\\epsilon c,c\_\{\\rho\},\\rho^\{\-1\}c\_\{\\theta\},\\Delta\_\{\\rm ref\}c,c\\Delta\_\{\\rm ref\}c,\|\\nabla\_\{\\rm ref\}c\|^\{2\}\]\.The geometry\-complete candidate library adds

\[𝒟R​\[c\],c​𝒟R​\[c\],𝒜R​\[c\],c​𝒜R​\[c\]\]\.\[\\mathcal\{D\}\_\{R\}\[c\],c\\mathcal\{D\}\_\{R\}\[c\],\\mathcal\{A\}\_\{R\}\[c\],c\\mathcal\{A\}\_\{R\}\[c\]\]\.The prescribed fieldsSSandϵ\\epsilonbelong tosknowns^\{\\rm known\}, while the multipliers ofSSand−ϵ​c\-\\epsilon care candidate coefficients to be inferred\. These terms remain in the physical block because they acquire no new differential geometry field in the pointwise strong form\.

The fitted reference\-coordinate equation was

ct=0\.109157​c2−0\.128395​c3\+0\.987610​S\+0\.732135​\(−ϵ​c\)\+0\.062263​cρ\+0\.008024​Δref​c,c\_\{t\}=0\.109157c^\{2\}\-0\.128395c^\{3\}\+0\.987610S\+0\.732135\(\-\\epsilon c\)\+0\.062263c\_\{\\rho\}\+0\.008024\\Delta\_\{\\rm ref\}c,\(S4\)and the geometry\-complete equation was

ct=1\.000001​S\+0\.719997​\(−ϵ​c\)\+0\.0120013​𝒟R​\[c\]\+1\.000067​𝒜R​\[c\]\.c\_\{t\}=1\.000001S\+0\.719997\(\-\\epsilon c\)\+0\.0120013\\,\\mathcal\{D\}\_\{R\}\[c\]\+1\.000067\\,\\mathcal\{A\}\_\{R\}\[c\]\.\(S5\)The reference\-coordinate and geometry\-complete fits have supportF1F\_\{1\}scores of 0\.4 and 1, coefficient relativeL2L\_\{2\}errors of 0\.640 and4\.25×10−54\.25\\times 10^\{\-5\}, and held\-out residuals of 0\.180 and6\.93×10−46\.93\\times 10^\{\-4\}, respectively\. Both fitted nutrient equations were integrated with the same boundary\-growth closure\. Fig\.[5](https://arxiv.org/html/2609.38623#Sx2.F5)and Extended Data Fig\. 4 report the resulting held\-out predictions\.

## S8\. Sensitivity to noise and training\-set size in equation discovery cases

Gaussian noise at 0, 0\.5%, 1%, 2% and 5% of the state\-field standard deviation is applied with five seeds per level\. Geometry remains noise\-free\. Non\-zero\-noise fields are smoothed sequentially in time, radius and angle with third\-order Savitzky–Golay filters using windows 7, 5 and 7 before locally aggregated STRidge\[[22](https://arxiv.org/html/2609.38623#bib.bib20)\]\.

Median thermoelastic support F1 remains 1 through 5% noise, and neither inactive thermal ALE candidate is selected in any realization\. Tumour support remains exact through 1% and decreases at 2% and 5%\. A separate clean\-data analysis enumerates all subsets of one to four training realizations using fixed main\-experiment hyperparameters\. One trajectory recovers exact support in both evolving\-domain systems\.

Extended Data Fig\. 5 reports the three identification metrics at each noise level\. Thermoelastic reference\-library residuals remained about an order of magnitude larger throughout\. Tumour reference residuals were larger through 2% noise\. At 5%, both libraries selected the same degraded sparse model and had the same median residual\. Extended Data Fig\. 6 reports all clean\-data training subsets\.

Extended Data Fig\. 1:Candidate\-library availability and selection\.Term\-by\-term comparison of candidate construction and sparse selection\. The geometry\-complete library for each system contains the same four\-term template: mapped diffusion, state\-modulated mapped diffusion, ALE transport and state\-modulated ALE transport\. For each system, the first column marks the generating support\. Ref and Geo denote the reference\-coordinate and geometry\-complete libraries, each shown through library\-membership and selected\-support columns\. Open squares mark available candidates and grey dashes mark terms absent from a library\. Purple and blue circles mark terms selected by Ref and Geo, respectively\. Purple triangles mark additional Ref selections, and red crosses mark missed generating terms\.![Refer to caption](https://arxiv.org/html/2609.38623v1/Extended_Data_Fig02_geometry_gallery.png)Extended Data Fig\. 2:Training and held\-out evolving\-domain realizations\.Final\-state gallery for the four training and four held\-out trajectories in each system\. The held\-out set changes source placement and geometric response beyond the combinations used for sparse identification\.Extended Data Fig\. 3:Thermoelastic transfer across all held\-out realizations\.Final\-state temperature truth, reference\-coordinate prediction and absolute error, and geometry\-complete prediction and absolute error for all four held\-out trajectories\. Prediction rows share a physical\-value colour scale and error rows share an absolute\-error scale\. Labels report final\-state relativeL2L\_\{2\}error\. The known elastic closure is used in both integrations\.![Refer to caption](https://arxiv.org/html/2609.38623v1/Extended_Data_Fig04_tumor_transfer.png)Extended Data Fig\. 4:Tumour transfer across all held\-out realizations\.Final\-state nutrient truth, reference\-coordinate prediction and absolute error, and geometry\-complete prediction and absolute error for all four held\-out trajectories\. Prediction rows share a physical\-value colour scale and error rows share an absolute\-error scale\. Labels report final\-state relativeL2L\_\{2\}error\. The known moving\-boundary closure is used in both integrations\.Extended Data Fig\. 5:Sensitivity to state\-field noise\.SupportF1F\_\{1\}, coefficient relativeL2L\_\{2\}error and clean held\-out PDE residual after identification from state fields with 0–5% additive noise relative to the field standard deviation\. Large symbols show medians across five noise realizations and pale points show individual realizations\. Geometry variables remain noise\-free\.Extended Data Fig\. 6:Training\-set size and subset stability\.SupportF1F\_\{1\}, coefficient relativeL2L\_\{2\}error and held\-out PDE residual versus the number of clean training realizations for the geometry\-complete candidate library\. All subsets of one to four trajectories are enumerated\. Large symbols show the subset median and pale points show individual subsets\.