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A variational physics-informed graph neural network for heterogeneous solid mechanics

Aashay Rajan Yadav, Amiya Prakash Das, Ratna Kumar Annabattula

arXiv:2609.10983v1math.NAcs.LG

TL;DR

Heterogeneous interfaces challenge coordinate-based PINNs because smooth trial fields struggle to represent stiffness-induced stress jumps. The PI-GNN instead embeds heterogeneity in a conforming mesh graph and minimizes an FE-aligned variational objective, achieving robust accuracy across contrasts while incurring substantially higher training cost than a single FE solve.

  • Problem

    Coordinate-based PINNs have representational bias at bimaterial interfaces because a single smooth trial field cannot reproduce stress jumps and requires transition widths or interface penalties.

  • Method

    The PI-GNN assigns constitutive behavior per mesh element and minimizes a discrete total potential energy equal to the FE Ritz functional on P1 elements, with Dirichlet conditions enforced by construction.

  • Results

    Across E_inc/E_mat ∈[10^-2, 10^2], PI-GNN von Mises error remains below 3.58% versus 5.58% for a strong-form PINN, while message passing lowers fixed-budget error by 19-43%.

  • Takeaways & Limitations

    The construction provides a variationally consistent, penalty-free representation of heterogeneous interfaces for parametric surrogates and inverse identification.

  • Takeaways & Limitations

    Training time exceeds a single FE solve by 55 times, so the method is not positioned as a replacement for one-off FE analysis.

Abstract

from arXiv · show

Stress localization in heterogeneous solids is governed by the bimaterial interface, where the displacement field remains $C^0$-continuous, while in-plane stresses jump due to the stiffness mismatch. Coordinate-based physics-informed neural networks (PINNs) represent this jump via a prescribed regularization width or a weighted interface penalty, making their accuracy sensitive to how phase-contrast changes are handled. This work presents a variational, label-free physics-informed graph neural network (PI-GNN) in which the heterogeneity is carried by the discretization rather than by the trial field. The solver operates on a conforming adaptive mesh graph, assigns constitutive behavior per element, and minimizes the discrete total potential energy as a single unweighted objective in which only first derivatives appear. The discrete energy on piecewise-linear elements coincides with the finite element (FE) Ritz functional. Dirichlet conditions are enforced by construction, with no penalty term, no interface weight, and no prescribed transition width. Using one fixed architecture, optimizer, and loss across small-strain elasticity and finite-strain Neo-Hookean hyperelasticity in two and three dimensions, the von Mises error remains below $3.58\%$ across a stiffness-contrast sweep spanning $(E_{\mathrm{inc}}/E_{\mathrm{mat}}\in[10^{-2},10^{2}])$, where a strong-form PINN degrades to $5.58\%$, and its displacement error reaches $7.66\%$ against $0.49\%$ for the PI-GNN. A trained network halves the ($σ_{xx}$) error of an energy-based PINN ($5.01\%$ versus $10.94\%$). Training cost exceeds a single FE solve by more than an order of magnitude, so the construction is a variationally consistent, penalty-free interface representation for parametric surrogates and inverse identification rather than a replacement for a one-off FE analysis.

1. Introduction

The PI-GNN addresses interface-induced stress jumps by carrying material heterogeneity in a conforming mesh graph rather than a smooth coordinate-based trial field. Across fixed settings, it reports robust accuracy and faster optimization from message passing, while retaining FEM-aligned variational structure.

  • The PI-GNN assigns constitutive behavior per element and minimizes a discrete total-potential-energy functional that coincides with the FE Ritz functional on P1 elements.This makes the conforming mesh carry the bimaterial interface and aligns the training objective with the FEM benchmark.
  • Across E_inc/E_mat ∈[10^-2, 10^2], von Mises error stays below 3.58% for PI-GNN versus 5.58% for a strong-form PINN with fixed transition width.The comparison uses frozen settings across the stiffness-contrast sweep.
  • At a fixed training budget, message passing lowers error by 19-43% across a mesh sweep, with the advantage widening as elements coarsen.At convergence, both variants reach a similar loss, indicating that message passing accelerates optimization toward the shared Ritz minimizer.
  • On an unseen discretization, PI-GNN roughly halves σxx error relative to an energy-based PINN: 5.01% versus 10.94%, and 7.57% versus 15.08%.These errors measure transfer from training to an unseen mesh.
  • Using one fixed architecture, hyperparameter set, and loss, the framework achieves displacement L2 errors below 1.5% across 2D and 3D verification and application examples.The examples include smooth and re-entrant inclusions, combined loading, and 3D torsion.

2. Preliminaries: Kinematics and PI-GNN Formulation

The PI-GNN formulates heterogeneous solid mechanics as a variational graph problem on an adaptive mesh, combining constitutive models, hard boundary enforcement, and discrete potential-energy minimization. Its mesh and graph construction refine interfaces, encode material information, and yield an objective identical to the FE Ritz functional.

  • 2.1. Governing equations and material models: The framework combines governing kinematics, small-strain linear elasticity, and finite-strain Neo-Hookean hyperelasticity within a minimum-potential-energy formulation.The equilibrium displacement minimizes the total potential energy over admissible fields satisfying prescribed displacement conditions.
  • 2.2. PI-GNN solver formulation: The PI-GNN workflow maps an adaptive mesh and material-labeled graph to nodal displacements, then assembles element strains, constitutive responses, and discrete potential energy.The graph-based pipeline requires no labeled training data and supports heterogeneous boundary-value problems.
  • 2.2.1. Adaptive mesh generation and graph construction: Mesh construction samples boundaries and interiors adaptively, triangulates the points with Delaunay connectivity, and assigns per-element material labels to the resulting graph.Nodes become graph vertices, shared triangle edges become directed connections, and geometric edge attributes encode relative position and length.
  • 2.2.1. Adaptive mesh generation and graph construction: Adaptive spacing places the finest resolution at interfaces and geometric features, then grades toward coarser far-field spacing without an abrupt fixed-width refinement band.The size field uses feature distance, minimum spacing, grading, and far-field spacing to control resolution.
  • 2.2.2. Graph neural network architecture: The graph processor propagates information through four message-passing layers using nodal embeddings, neighbor differences, and edge geometry.The state-difference feature emphasizes local contrast near interfaces, while mesh-derived edge features support evaluation on unseen discretizations.
  • 2.2.3. Discrete potential energy assembly: The training objective coincides identically with the FE Ritz functional, so the converged FE displacement field is the exact minimizer of the discrete loss.The network minimizes this single potential-energy objective without PDE-residual or boundary-penalty terms.

3. Numerical Verification and Reference Solver

Across hole, heterogeneous-inclusion, and stiffness-contrast benchmarks, the PI-GNN closely matches FE and analytical references while generally outperforming the energy-based or strong-form PINN comparisons. Its fixed configuration remains accurate across geometries and contrasts, although training is substantially more expensive than a single FE solve and refinement can strain a fixed optimization budget.

  • 3.2.1. Single and multiple holes: The PI-GNN reproduces analytical and FE stress fields for single-hole plates and resolves approximately 5T peak stress in the ligament of the asymmetric three-hole case.For the single hole, it recovers the Kirsch concentration of 3T; for three holes, the peak occurs above the largest hole.
  • 3.2.2. Two-phase composite: In the heterogeneous-inclusion benchmark, both methods achieve displacement L2 error below 1%, while PI-GNN errors remain centered near zero and improve near-interface field agreement.The conforming mesh represents the stress discontinuity directly through adjacent element materials, unlike the PINN’s diffuse transition band.
  • 3.2.2. Two-phase composite: Across stiffness contrasts, PI-GNN von Mises error stays below 3.58%, whereas PINN error reaches 5.58% at contrast 100; displacement error reaches 0.49% versus 7.66% for the soft-inclusion case.The methods are comparable below approximately contrast 2, with separation emerging beyond approximately 3.
  • 3. Numerical Verification and Reference Solver: Training is more costly than a single FE solve, and fixed optimization budgets can make finer meshes appear less accurate because they contain more nodal unknowns.The authors therefore position the method as a fixed-configuration neural surrogate rather than a replacement for one-off FE analysis.
  • 3.2.3. Isolating the effect of message passing: Message passing contributes larger error reductions on coarser meshes and away from unit stiffness contrast, where neighborhood exchange carries more interface information.The reported error drop grows from 19% at h = 0.01 to 43% at h = 0.05, while message passing costs 1.2 to 2.2 times the ablated model’s training time.

4. Numerical applications

The PI-GNN accurately resolves heterogeneous interfaces, stress concentrations, and torsional responses across two- and three-dimensional applications, including irregular and re-entrant inclusions. Agreement with FE references remains strong for displacement, dominant stresses, and von Mises stress, while localized errors grow in the most complex multiaxial setting.

  • 4.1. Material discontinuities: Approximately 100 Pa stress lobes and steep notch gradients were reproduced around the stiff inclusion, while the surrounding matrix relaxed toward 40 Pa.The PI-GNN matched the position and magnitude of every lobe along the loading direction.
  • 4.1. Re-entrant inclusion: R2 ≥0.99 for displacements and R2 ≥0.98 for dominant σxx and von Mises stress in the re-entrant inclusion benchmark.The result holds globally and near the interface, with displacement-component signed median errors of −0.7161% and +0.6681%.
  • 4.2. Composite cube under torsion: R2 > 0.99 was achieved for displacement components under inclined loading, while dominant σxx relative error increased from 2.8% to 6.8%.The more complex tensile-compressive state retained R2 ≥0.99 for σxx and R2 >0.97 for all stress components.
  • 4.2. Composite cube under torsion: 2.2% to 4.9%: von Mises relative error increased under multiaxial loading, despite high-fidelity recovery of the more complex stress field.The increase was reported alongside the transition from predominantly tensile to combined tensile and compressive regions.
  • 4.3. Composite rod under torsion: R2 ≥0.99 was obtained for displacement and load-carrying shear fields in the three-dimensional torsion problem, with signed median stress errors below 0.5%.The axial displacement and the shear components σxz and σyz, together with σvM, all met this accuracy level.
  • 4.3. Composite rod under torsion: 0.72% relative energy difference separated the converged PI-GNN from the FE reference, while training took approximately 55 times longer than the FE solve and inference took 0.02 times as long.The reported values were Π = −5.978 × 10−2 kPa mm3 for PI-GNN and −6.022 × 10−2 kPa mm3 for FE; a forward pass took 12.42 s versus 672.35 s for FE.

5. Summary and Outlook

The PI-GNN provides a variational, label-free solver for heterogeneous solids by carrying material heterogeneity in a conforming mesh graph and minimizing the discrete total potential energy. Across elasticity and hyperelasticity benchmarks, it improves accuracy and convergence relative to PINN baselines, while remaining more expensive than a one-off FE solve.

  • The PI-GNN minimizes a single discrete total-potential-energy objective on a conforming mesh graph, with element-level constitutive behavior and no penalty term or prescribed transition width.Its P1 discrete energy coincides with the FE Ritz functional, making the converged FE field the exact training-loss minimizer.
  • Across stiffness contrasts Einc/Emat ∈[10^-2, 10^2], PI-GNN von Mises error stays below 3.58%, versus 5.58% for a strong-form PINN.The PI-GNN error depends on contrast magnitude but not its sign, whereas PINN accuracy depends on both.
  • At fixed reduced training budgets, message passing lowers error by 19–43% across a mesh sweep, while both architectures converge to the shared Ritz minimizer.The advantage is faster convergence rather than a lower final minimum, with a persistent benefit especially for near-interface displacement fields.
  • The framework remains a per-instance solver requiring separate training for each new boundary-value problem, and its accuracy is bounded by the P1 Ritz minimizer.The message-passing advantage narrows under strong mesh refinement and lenient training budgets.
  • Training exceeds a single FE solve by 55 times, although inference produces the full field in one forward pass and is reported as 50 times faster than a FEniCSx Newton–Raphson solve.These timings are case-specific and depend mainly on graph size.
  • Future work targets parametric surrogates over geometry, loading, and material variations, along with elastoplasticity, anisotropic hyperelasticity, and inverse problems.

Data availability

The study states that its data and source code will be openly available in a repository containing the implementation, input files, and documentation needed to reproduce the reported experiments and figures.

  • The repository will provide the implementation, input files, and documentation required to reproduce the numerical experiments and figures.

Appendix A. Mesh Convergence study for Kirsch verification

The mesh-convergence study verifies that the FEM reference for the single-hole problem is sufficiently resolved near the hole boundary. Fine meshes reach the expected stress-concentration plateau and satisfy the prescribed acceptance band.

  • For the 20 × 20 plane-stress plate, refining hole-boundary element size raises the SCF from 2.88 to approximately 3.04, consistent with analytical finite- and infinite-plate estimates.
  • Every mesh with hhole ≤ 0.09 lies within the 3.0 ± 2% acceptance band, whereas coarser meshes are under-resolved and underestimate the SCF.
  • The PI-GNN training mesh with hhole ≈ 0.025 and 3577 nodes produces SCF 3.04 within the acceptance band, confirming a sufficiently mesh-converged FEM benchmark.

Appendix B. Architecture Ablation Study

The architecture ablation varies message-passing depth and hidden width while holding training conditions fixed, evaluating the trade-off between stress-field accuracy and model complexity. The resulting Pareto front identifies architectures that cannot reduce error without increasing parameter count.

  • The ablation varies message-passing layers and hidden width on a stiff re-entrant inclusion while keeping the learning rate, epochs, materials, dataset, and random seed fixed.
  • The Pareto front evaluates area-weighted von Mises L2 error against trainable-parameter count and marks architectures where lower error requires greater complexity.

Appendix C. Stiffness-contrast sweep

The stiffness-contrast sweep spans heterogeneous inclusion-to-matrix ratios across four orders of magnitude using fixed solver configurations and a common held-out mesh. Both methods are evaluated with consistent area weighting on that mesh.

  • Twenty-five logarithmically spaced contrasts cover Einc/Emat ∈[10^-2, 10^2], including twelve contrasts on each side of the homogeneous limit.
  • The comparison fixes the PINN at four 64-neuron tanh layers and the PI-GNN at L4H64 across the sweep.
  • Both solvers are evaluated on a common 3034-node, 5944-element mesh also used for the FE reference solution.
  • Area weighting is necessary because element areas vary by a factor of 168, with the smallest elements concentrated near the material interface.

Appendix C.1. The unit contrast limit

At unit stiffness contrast, the PINN achieves an exceptionally low error because its prescribed background field already matches the exact uniform-stress solution. The PI-GNN instead continues optimizing the discrete energy, and its residual error is linked to incomplete convergence.

  • At Einc/Emat = 1, the PINN error is 0.002% versus 0.598% for the PI-GNN, but this isolated result is not representative of heterogeneous cases.
  • The PINN’s unit-contrast advantage follows from its prescribed background field coinciding with the exact uniform-stress solution.
  • The PI-GNN residual error reflects incomplete optimization: all stored solutions have higher discrete energy than FE, and their error fields are weakly correlated.
  • Reducing the learning rate to 10^-4 and selecting the lowest-energy checkpoint lowers the PI-GNN error from 0.598% to 0.368%.

Appendix C.2. Comparison of converged solutions

After sufficient training, the PI-GNN and mesh-based PINN approach the FE energy minimum, while the PI-GNN retains faster convergence and lower displacement and interface-band errors. The converged von Mises advantage largely disappears.

  • The total-potential curve descends faster for PI-GNN during the first ∼3000 epochs, reaching an energy level the mesh-based PINN attains near 30,000 epochs.
  • The PI-GNN reaches Π = −5.885 × 10^-7 at epoch 29,997 versus Π = −5.869×10^-7 for the mesh-based PINN at epoch 29,818, near ΠFEM = −5.889×10^-7.
  • After convergence, von Mises error is 4.60% versus 4.64%, so the earlier advantage closes.
  • Converged displacement error is 0.239% versus 0.709%, while interface-band error is 0.250% versus 0.837%.
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