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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, Yingguang Li

arXiv:2609.38623v1cs.LG

TL;DR

Geometric transformations can make physically distinct systems produce the same observed dynamics, complicating PDE learning across domains. The paper supplies geometry-induced representations for operator learning and geometry-complete libraries for equation discovery. These representations improve operator learning across architectures and support recovery of governing equations across fixed and evolving domains.

  • Problem

    Distinct physical systems can yield identical pulled-back trajectories when the domain map is not used separately, making physical attribution ambiguous.

    Source: Geometry–physics confounding and… · p. 5

  • Method

    The framework makes geometry-to-operator transformations explicit through geometry-induced fields for operator learning and transformed, geometry-complete candidate libraries for equation discovery.

    Source: Problem setup · p. 5, Discussion · pp. 14–15

  • Results

    Geometry-induced fields improve operator learning across DIMON and Transolver, while geometry-complete libraries recover shared coefficients and governing equations across fixed-deformation and evolving-domain cases.

    Source: Discussion · pp. 14–15

  • Takeaways & Limitations

    The results identify implicit geometry as a shared representation burden in operator learning and a source of candidate-space misspecification in equation discovery.

    Source: Discussion · p. 14

    The cases primarily use relatively ideal geometric maps, and sample-savings estimates are interpolated rather than additional training experiments or savings in geometric information or computational cost.

    Source: Discussion · p. 15, Operator-learning benchmark design and… · p. 20

Abstract

from arXiv · show

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.

Geometry–physics confounding and de-confounding methods

Across varying and evolving domains, geometric transformations can make distinct physical systems observationally indistinguishable and complicate both operator learning and equation discovery. The paper makes those transformations explicit through task-matched geometry-induced representations.

Problem setup

  • The framework treats operator learning as learning geometry-conditioned solution maps and equation discovery as selecting active terms from a prespecified operator library across related geometries.Both tasks use a shared coefficient vector across sampled geometries, while seeking different objects.Source: Problem setup · p. 3

Geometry–physics confounding and failure mechanism

  • Distinct physical descriptions—a homogeneous material on a deformed domain and a heterogeneous anisotropic material on a fixed domain—can produce identical pulled-back trajectories when attribution omits the domain map.A single apparent diffusion coefficient cannot determine how much observed change comes from geometric distortion versus material properties.Source: Geometry–physics confounding and… · p. 5
  • At the largest training sets, geometry-induced coefficient fields reduce held-out rate-prediction errors by factors of 5.4 for pointwise prediction and 5.3 for complete-field prediction; geometry-induced PDE terms reduce both errors to numerical precision.The coefficient-field and derivative-input models share the same neural architecture and differ in whether they expose geometry-to-operator action.Source: Geometry–physics confounding and… · p. 6
  • A geometry-complete candidate library recovers the generating diffusion coefficient and reduces the equation residual to numerical precision, whereas a reference Laplacian incurs 19.3% coefficient bias and a 0.223 residual.The apparent coefficient drift reflects geometric action absorbed by the misspecified term; the remaining residual reflects the operator the library cannot represent.Source: Geometry–physics confounding and… · p. 6

De-confounding through geometry-induced representations

  • For operator learning, an equivalent weak-form pull-back transforms trial and test fields and problem data to reference spaces; integration by parts can avoid higher-order state and map derivatives.The transformed specification includes applicable initial and boundary conditions, forcing, known coefficients, controls, and fields generated by the change of variables.Source: De-confounding through… · pp. 6–7
  • For equation discovery, a strong-form pull-back transforms each physical candidate into a structured reference-domain operator and includes an ALE transport candidate when domain motion contributes.Under a time-independent map the ALE term is zero; under the prescribed Eulerian description its form and sign are determined by the map.Source: De-confounding through… · p. 7

Results

Geometry-conditioned representations lowered prediction errors across five operator-learning benchmarks, while geometry-complete libraries recovered generating equations and substantially reduced held-out errors in both evolving-domain systems.

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

  • For DIMON, Laplace error fell 54.6% at n = 300 and 2%-target sample need fell 36.7%; reaction–diffusion sample savings reached 70.0% at a 10% target error.The respective comparisons were DIMON-Geo against DIMON.Source: Geometry-induced representations… · p. 9
  • Transolver-Geo cut errors by 19.8–21.8% across pipe-flow training sizes, 36.5% at n = 200 for airfoil flow, and 18.4% at n = 200 for elasticity.Figure 3 compares errors and estimated sample requirements for these three benchmarks.Source: Geometry-induced representations… · pp. 9, 11, Figure 3 · p. 10
  • Across all five benchmarks, geometry-conditioned variants achieved lower mean prediction errors at every tested training size under the case-specific protocols.The reported evidence spans the tested model families, equations and discretizations.Source: Geometry-induced representations… · p. 11

Geometry-complete libraries enable equation discovery on evolving domains

  • 3.02 × 10−5 versus 0.171 held-out PDE residuals accompanied geometry-complete thermoelastic discovery, which reduced full-trajectory errors by more than four orders of magnitude.The geometry-complete library recovered the generating terms and rejected inactive ALE candidates.Source: Geometry-complete libraries enable… · pp. 11–13, Figure 4 · p. 12
  • With noise-free geometry, thermoelastic support remained exact through 5% state-field noise, while tumour-growth support was exact only through 1% and its median F1 score fell to 0.444 at 5%.These tests distinguish candidate-library misspecification from state-field noise and trajectory selection in the tested systems.Source: Geometry-complete libraries enable… · p. 13
  • 6.93 × 10−4 versus 0.180 equation residuals accompanied recovery of all four tumour-growth generating terms; held-out trajectory errors fell by three to four orders of magnitude.The comparison used geometry-complete and reference-coordinate candidate libraries.Source: Geometry-complete libraries enable… · p. 13, Figure 5 · p. 14

Discussion

Geometry–physics confounding connects the representation burden in operator learning with candidate-space misspecification in equation discovery. Geometry-complete representations address these issues, but their benefits depend on the supplied fields and the quality of state derivatives and geometric maps.

Discussion

  • Geometry-induced coefficient fields reduce operator-learning representation burden across DIMON and Transolver, but their effect depends on whether the fields fully describe the transformed PDE.Elasticity requires both a local cofactor field and a displacement field to represent local deformation and global shape.Source: Discussion · pp. 14–15
  • Reference-coordinate libraries absorb omitted geometric action into biased coefficients and extra terms, while geometry-complete libraries recover shared candidate coefficients across training trajectories.The thermoelastic system rejects inactive ALE candidates under its material-time description, whereas the tumour system retains active domain-motion transport.Source: Discussion · p. 15
  • Correct operators remove library misspecification but do not eliminate sensitivity to noisy state derivatives; the cases primarily use relatively ideal geometric maps.The authors identify sensitivity to mapping and representation errors as a subject for future work.Source: Discussion · p. 15

Methods

The methods make geometry-to-operator action explicit, then evaluate geometry-aware operator inputs and candidate libraries in diffusion, benchmark, and evolving-domain settings. Shared evaluation procedures quantify prediction errors, equation residuals, and identification quality.

Domain maps and geometry-induced representation

  • Geometry-induced coefficient fields represent the pulled-back operators, while equation discovery assembles transformed differential terms into candidate libraries, including diffusion and ALE transport when appropriate.ALE transport depends on domain motion and the adopted physical description; state-modulated candidates are assembled as separate terms.Source: Domain maps and geometry-induced… · pp. 15, 17

Diffusion example of geometry–physics confounding

  • The diffusion example compares deformed- and fixed-domain descriptions that yield the same discrete equation and integrated trajectory, producing zero observation mismatch.The geometry-complete fit replaces the reference Laplacian with the mapped operator; prediction also tests representations using geometry-induced PDE terms.Source: Diffusion example of geometry–physics… · p. 18

Operator-learning benchmark design and evaluation

  • Five operator-learning benchmarks compare matched baseline and geometry-conditioned models using the same labelled samples, targets, and training schedules.Required sample counts are interpolated from mean learning curves, and the estimates describe labelled-sample requirements rather than savings in geometry information or computation.Source: Operator-learning benchmark design and… · pp. 19–20

Geometry-complete equation discovery

  • Equation discovery uses observed states and evolving geometry, a prespecified geometry-complete block of mapped diffusion and ALE terms, and sparse regression validated across trajectories.Four realizations are used for identification and four geometrically shifted realizations for predictive validation; noise and trajectory-diversity analyses are also conducted.Source: Geometry-complete equation discovery · pp. 20–21

Evaluation and statistical reporting

  • Evaluation reports operator errors averaged over test samples and equation residuals against sampled time derivatives, alongside support F1 and coefficient relative error.Clean equation-identification calculations are deterministic; neural and operator-learning curves summarize five runs, while noise-study lines show medians across five realizations per level.Source: Evaluation and statistical reporting · p. 21

S1. Diffusion implementation details

The diffusion implementation controls state-derivative error analytically and compares geometry-aware and geometry-implicit representations on nested pointwise and field datasets.

S1. Diffusion implementation details

  • Analytical derivatives of smooth sine-mode states and geometry-induced coefficients remove finite-difference error from comparisons of derivative-based and geometry-aware representations.The pointwise models share a 37–64–64–1 MLP, while the field models share a three-layer 3 × 3 GELU CNN.Source: S1. Diffusion implementation details · p. 25
  • Prediction uses eight training geometries and evaluates the held-out geometry s = 0.75, with nested pointwise training sets drawn from 800 state–geometry blocks.The pointwise pool contains 100 states, eight geometries, and 1,024 interior points per pair; evaluation uses eight disjoint held-out states.Source: S1. Diffusion implementation details · pp. 25–26
  • The field pool contains 1,024 states paired with all eight geometries, and every nested training set is balanced across geometry.Sixteen states are held out, and training loss and relative error use the central 35 × 35 region to avoid convolution-padding effects.Source: S1. Diffusion implementation details · p. 26

Shared settings and reporting conventions for operator-learning benchmarks

Operator-learning comparisons use paired runs, nested training sizes, and a shared relative-error convention across test samples.

Shared settings and reporting conventions for operator-learning benchmarks

  • Each training size counts complete geometry–solution samples, and five paired seeds are used for baseline and geometry-induced models.Errors are computed per test sample, averaged over the test set, then summarized by the mean and sample standard deviation across seeds.Source: Shared settings and reporting… · p. 26
  • Benchmark errors use the unweighted discrete relative L2 norm, averaged across test samples before summarizing runs.The reported run summaries are the mean and sample standard deviation across seeds.Source: Shared settings and reporting… · p. 26
  • Training sizes are nested, with each size defined by the number of complete geometry–solution samples rather than spatial query points.Network-width lists include input and output dimensions.Source: Shared settings and reporting… · p. 26

S2. Laplace benchmark

The Laplace benchmark compares baseline and geometry-conditioned models on identical solution, boundary, query-point, and test data. DIMON-Geo replaces PCA geometry coefficients with sampled symmetric geometry-tensor channels while retaining the branch–trunk formulation.

S2. Laplace benchmark

  • Both models use the same 193 test cases and predict at all 2,601 spatial points, with identical boundary values and solution targets.The retained dataset has 3,233 training cases; runs use the first 300, 500, 1,000, 2,000, or 3,000 cases.Source: S2. Laplace benchmark · p. 26
  • DIMON-Geo receives three geometry-tensor channels, each sampled to 121 values, while the baseline receives 20 geometry PCA coefficients.Both models also receive the same 68 sampled boundary values.Source: S2. Laplace benchmark · p. 26
  • Both models use fully connected branch–trunk networks; the baseline combines geometry and boundary branches, while DIMON-Geo combines three tensor branches and the boundary branch.The boundary branch and coordinate trunk are unchanged; geometry branches differ in input representation and internal widths.Source: S2. Laplace benchmark · pp. 26–27
  • Training uses full-batch mean-squared error, 50,000 Adam steps across two learning-rate stages, then 1,000 L-BFGS outer steps.Evaluation uses relative L2 error over all 2,601 spatial points.Source: S2. Laplace benchmark · p. 27

S3. Reaction–diffusion benchmark

The nonlinear reaction–diffusion benchmark compares models on nested training sets and a fixed test set. DIMON-Geo uses sampled initial-condition values and geometry-induced fields, with a modified branch-fusion design.

S3. Reaction–diffusion benchmark

  • The processed benchmark contains 3,000 cases, with 2,700 training-pool cases and 300 fixed test cases; each sample has 2,800 spatial points and 26 snapshots.Training uses nested sets of 500, 1,000, 1,500, 2,000, or 2,700 cases.Source: S3. Reaction–diffusion benchmark · p. 27
  • DIMON-Geo uses 255 sampled initial-condition values and four full-resolution geometry inputs, while the baseline uses 50 initial-condition and 20 geometry coefficients.The four geometry inputs are the determinant field and three independent symmetric geometry-tensor channels.Source: S3. Reaction–diffusion benchmark · p. 27
  • DIMON-Geo concatenates three tensor-branch outputs, fuses them to 200 components, then combines them with determinant and initial-condition branches before trunk contraction.The baseline instead multiplies its initial-condition and geometry branch outputs elementwise; the comparison also changes encoding and branch dimensions.Source: S3. Reaction–diffusion benchmark · p. 27
  • Both networks train with full-batch mean-squared error over samples, spatial points, and times, using 100,000 Adam steps across three learning-rate stages.Evaluation computes each sample’s relative L2 error jointly over 26 snapshots and 2,800 spatial points.Source: S3. Reaction–diffusion benchmark · pp. 27–28

S4. Pipe-flow benchmark

The pipe-flow comparison uses a fixed source-order subset of the public benchmark and identical sampled velocity targets. Transolver-Geo replaces physical-coordinate inputs with reference coordinates and geometry channels while retaining the structured-mesh attention backbone.

S4. Pipe-flow benchmark

  • The comparison uses 1,200 cases from a 2,310-case dataset, with 1,000 cases in the training pool and 200 in the test set.Sampling yields a 129 × 19 grid of 2,451 points, and the target is horizontal velocity vx.Source: S4. Pipe-flow benchmark · p. 28
  • The reference mesh is the mean of the 1,000 training-pool meshes and stays fixed across training sizes; no cases are filtered or sorted.All selected cases have jg > 0.2; the global minimum is 0.785 and maximum Jacobian condition number is below 4.120.Source: S4. Pipe-flow benchmark · p. 28
  • The baseline receives physical coordinates, whereas Transolver-Geo receives reference coordinates and seven channels from Kg and Bg.Coordinates and geometry channels are standardised using statistics from selected training cases; the target is also normalised.Source: S4. Pipe-flow benchmark · p. 28
  • Both models use the same structured-mesh Transolver attention backbone; only the input-path convention differs, including a learned vector for the coordinate-only model.The backbone has eight attention blocks, eight heads, and 64 slice tokens; neither time embeddings nor optional distance encoding are used.Source: S4. Pipe-flow benchmark · p. 28
  • Training uses 500 AdamW epochs on nested sets of 200, 400, 600, 800, or 1,000 cases, with shuffled batches of four.Reported relative L2 errors are evaluated on physical velocity values over all 2,451 points.Source: S4. Pipe-flow benchmark · pp. 28–29

S5. Airfoil-flow benchmark

The airfoil comparison uses a fixed source-order subset of the compressible-flow benchmark and predicts Mach number at every mesh point. Transolver-Geo uses reference coordinates and cofactor channels with the shared structured-mesh attention backbone.

S5. Airfoil-flow benchmark

  • Experiments use the first 1,200 of 2,490 cases, with the first 1,000 forming the training pool and the next 200 the fixed test set.Each case retains all 11,271 points on its 221 × 51 mesh; the target is Mach number from channel 4.Source: S5. Airfoil-flow benchmark · p. 29
  • The baseline receives physical coordinates, while Transolver-Geo receives reference coordinates and four cofactor channels; no cases are filtered, sorted, or determinant-clipped.The minimum determinant across selected cases is approximately 0.423; normalisation uses selected training samples.Source: S5. Airfoil-flow benchmark · p. 29
  • Both models use the same structured-mesh attention blocks and settings, but the geometry-conditioned input projection has six channels rather than two.The grid is 221 × 51, and the learned-vector convention matches the pipe comparison.Source: S5. Airfoil-flow benchmark · p. 29
  • Training uses 500 epochs of AdamW with shuffled batches of four, and evaluation computes decoded Mach-number relative L2 error over all 11,271 points.Runs use nested training sets of 200, 400, 600, 800, or 1,000 cases.Source: S5. Airfoil-flow benchmark · p. 30

S6. Elasticity benchmark

The elasticity benchmark compares a baseline with Transolver-Geo on a shared varying-unit-cell dataset, testing whether geometric input improves stress-field prediction as training data increase.

S6. Elasticity benchmark

  • The benchmark uses 1,000 training-pool cases from a 2,000-case dataset, with 200 final-index cases held out for testing and 972 mesh points per case.The target is the scalar stress field predicted at all 972 points.Source: S6. Elasticity benchmark · p. 30
  • Transolver-Geo receives reference coordinates and six geometry channels, while the baseline receives unnormalised physical coordinates.The channels encode the components of Cg and the domain correspondence dg = x − ξ, not predicted mechanical response.Source: S6. Elasticity benchmark · p. 30
  • Both models use irregular-mesh physics attention and a shared eight-block backbone; training uses up to 1,000 cases, and errors are evaluated on physical-scale stress.Training sizes are 200, 400, 600, 800, and 1,000 cases; the final trained weights are evaluated over all 972 spatial points.Source: S6. Elasticity benchmark · pp. 30–31, Table S5 · p. 30

S7. Equation-discovery protocols and identified equations

The equation-discovery protocols identify transported-state laws from four trajectories while keeping each system’s geometry-evolution closure known. Geometry-complete libraries improve support recovery and held-out residuals in both evolving-domain systems.

S7. Equation-discovery protocols and identified equations

  • Each system uses four identification trajectories and four geometrically shifted held-out trajectories; the transported-state equation is identified while its geometry-evolution closure remains known.STRidge selection uses leave-one-trajectory-out validation, then refits the selected support by ordinary least squares across all four training trajectories.Source: S7. Equation-discovery protocols and… · p. 31
  • The geometry-complete library retains reference-coordinate candidates and adds mapped diffusion, ALE transport, and their state-modulated counterparts.The thermoelastic and tumour systems instantiate these additions as [Dd[T], TDd[T], Ad[T], TAd[T]] and [DR[c], cDR[c], AR[c], cAR[c]], respectively.Source: S7. Equation-discovery protocols and… · pp. 31–32
  • Thermoelastic identification targets the temperature law with a known elastic closure; the prescribed heat-source field q is known, but its multiplier is inferred.Thermal ALE candidates remain available to selection although their generating coefficients are zero in the material-time formulation.Source: S7. Equation-discovery protocols and… · p. 31
  • Thermoelastic support F1 rises from 0.4 (reference-coordinate) to 1 (geometry-complete), while held-out residual falls from 0.171 to 3.02 × 10−5.The geometry-complete coefficient relative L2 error is 2.03 × 10−5.Source: S7. Equation-discovery protocols and… · p. 31
  • Tumour identification targets the nutrient law with a known boundary-growth closure; prescribed fields S and ϵ remain known while their multipliers are inferred.These source and sink terms stay in the physical block because they acquire no new differential geometry field in the pointwise strong form.Source: S7. Equation-discovery protocols and… · p. 32
  • Tumour support F1 rises from 0.4 (reference-coordinate) to 1 (geometry-complete), coefficient relative L2 error falls from 0.640 to 4.25 × 10−5, and held-out residual from 0.180 to 6.93 × 10−4.Source: S7. Equation-discovery protocols and… · p. 32

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

Equation-discovery sensitivity tests vary state-field noise and clean training-set size while keeping geometry noise-free. Thermoelastic support remains exact through the tested noise range, while tumour support degrades at higher noise; one clean trajectory suffices for exact support in both systems.

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

  • Thermoelastic median support F1 remains 1 through 5% state noise, while tumour support remains exact through 1% and decreases at 2% and 5%.The noise is additive relative to state-field standard deviation; geometry remains noise-free.Source: S8. Sensitivity to noise and… · p. 33, Figure 5 · p. 38
  • One clean training trajectory recovers exact support in both evolving-domain systems.The clean-data analysis enumerates all subsets of one to four trajectories using fixed main-experiment hyperparameters.Source: S8. Sensitivity to noise and… · p. 33, Figure 6 · p. 39
  • At 5% noise, both libraries select the same degraded sparse model and have the same median residual; reference-library residuals are larger in thermoelastic tests throughout.Tumour reference residuals are larger through 2% noise.Source: S8. Sensitivity to noise and… · p. 33
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