Geometry-physics confounding impairs PDE learning across varying domains

Yinghao Cheng; Gengxiang Chen; Xu Liu; Qinglu Meng; Yixin Jing; Xiangguo Tang; Wenping Mou; Lihui Wang

cs.LGSep 29, 2026

Geometry-physics confounding impairs PDE learning across varying domains

Authors: Yinghao Cheng, Gengxiang Chen, Xu Liu, Qinglu Meng, Yixin Jing, Xiangguo Tang, Wenping Mou, Lihui Wang, +1 more

Organizations: College of Mechanical & Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China · School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, Belfast, United Kingdom · School of Mechanical and Aerospace Engineering, Queen’s University Belfast, Belfast, United Kingdom · School of Mechanical and Power Engineering, Nanjing Tech University, Nanjing, China · Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hong Kong, China · Department of Production Engineering, KTH Royal Institute of Technology, Stockholm, Sweden

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.

Figures & tables

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 maps $F_{\omega}$ , 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 capacity $j_{g}$ and transformed conductivity $\widehat{\bm{\kappa}}_{g}$ yield identical pulled-back dynamics. Fitting a misspecified reference model yields identical apparent diffusion coefficients ( $\widehat{\kappa}_{\rm Ref}=0.010$ ) 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 ( $<10^{-14}$ ). Prediction curves and shaded bands show means and $\pm 1$ s.d. across five optimisation runs. For equation discovery (right), the apparent reference coefficient exhibits a 19.3% geometric drift and an equation residual of $0.223$ , whereas the geometry-complete candidate library recovers the generating coefficient with numerical-precision residual ( $<10^{-16}$ ).

Figure 4: Thermoelastic equation discovery and prediction on a held-out realization. a , Domain deformation $|d|$ and metric distortion $j_{d}=\det J_{d}$ 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 term $q$ and 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 at $t=2.0$ , 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 relative $L_{2}$ error under 0–5% additive state-field noise. g , Full-trajectory relative $L_{2}$ error across four held-out trajectories for temperature $T$ , displacement $\bm{d}$ , and von Mises stress $\sigma_{v}$ . The geometry closure is known.

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 area $A$ . b , Generating mapped-diffusion ( $0.012\mathcal{D}_{R}[c]$ ) and ALE-transport ( $\mathcal{A}_{R}[c]$ ) PDE contributions at $t=1.2$ . 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 nutrient $E_{c}$ , boundary radius $E_{R}$ , and domain area $E_{A}$ . e , Coefficient relative $L_{2}$ error under 0–5% state-field noise. f , Full-trajectory relative $L_{2}$ error across four held-out trajectories for nutrient $c$ , boundary $R$ , and area $A$ . The moving-boundary closure is known.

Term	Physical-domain form	Geometry-induced form	Short notation
Gradient	$\nabla_{x}u$	$\bm{J}_{g}^{-\mathsf{T}}\nabla_{\xi}\widehat{u}$	$\mathcal{G}_{g}[\widehat{u}]$
Advection	$\bm{b}\cdot\nabla_{x}u$	$(\bm{J}_{g}^{-1}\widehat{\bm{b}})\cdot\nabla_{\xi}\widehat{u}$	$\mathcal{C}_{g}[\widehat{u}]$
Diffusion	$\Delta_{x}u$	$j_{g}^{-1}\operatorname{Div}_{\xi}(\bm{K}_{g}\nabla_{\xi}\widehat{u})$	$\mathcal{D}_{g}[\widehat{u}]$
Time derivative	$\partial_{t}u\|_{x}$	$\partial_{t}\widehat{u}\|_{\xi}-\mathcal{A}_{g}[\widehat{u}]$	—
Reaction/source	$r(u)+s$	$r(\widehat{u})+s\circ\Phi_{g}$	—

Table 1: Common PDE terms and their geometry-induced forms.

Component	DIMON	DIMON-Geo
Geometry branch	$[20,100,100,100]$	Three $[121,32,32,100]$ branches
Boundary branch	$[68,150,150,150,100]$	$[68,150,150,150,100]$
Tensor-branch fusion	Not used	Elementwise product
Coordinate trunk	$[2,100,100,100]$	$[2,100,100,100]$

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

Component	DIMON	DIMON-Geo
Initial-condition branch	$[50,300,300,200]$	$[255,200,200,200]$
Geometry branch	$[20,200,200,200]$	Four $[2800,200,200,200]$ branches
Tensor-branch fusion	Not used	Affine $600\to 200$
Space–time trunk	$[3,200,200,200]$	$[3,200,200,200]$

Table S2: Reaction–diffusion network configurations.

Component	Transolver	Transolver-Geo
Input channels	2 physical coordinates	2 reference coordinates and 7 geometry channels
Input MLP	$[2,256,128]$	$[9,256,128]$
Attention blocks	8	8
Hidden width / heads	128 / 8	128 / 8
Slice tokens / MLP ratio	64 / 1	64 / 1
Output channels	1	1

Table S3: Pipe-flow network configurations.

Component	Transolver	Transolver-Geo
Input channels	2 physical coordinates	2 reference coordinates and 4 cofactor channels
Input MLP	$[2,256,128]$	$[6,256,128]$
Attention blocks	8	8
Hidden width / heads	128 / 8	128 / 8
Slice tokens / MLP ratio	64 / 1	64 / 1
Output channels	1	1

Table S4: Airfoil-flow network configurations. Both models use the structured-mesh backbone specified in S4.

Component	Transolver	Transolver-Geo
Input channels	2 physical coordinates	2 reference coordinates and 6 geometry channels
Input MLP	$[2,256,128]$	$[8,256,128]$
Attention blocks	8	8
Hidden width / heads	128 / 8	128 / 8
Slice tokens / MLP ratio	64 / 1	64 / 1
Output channels	1	1

Table S5: Elasticity network configurations. Both models use irregular-mesh physics attention.

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.

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 relative

L_{2}

error. The known elastic closure is used in both integrations.

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 relative $L_{2}$ error. The known moving-boundary closure is used in both integrations.

Extended Data Fig. 5: Sensitivity to state-field noise. Support

F_{1}

, coefficient relative

L_{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. Support

F_{1}

, coefficient relative

L_{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.

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