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Learning the Constitutive Behavior of Materials via Neural Operators and Causal Attention: Case Studies in Plasticity and Damage

Rishabh Arora, Lisa Scheunemann, Tim Brepols, Shahed Rezaei

arXiv:2609.02194v1cs.LGcs.CE

TL;DR

Path-dependent constitutive modeling often relies on internal variables that are difficult to measure or prescribe without domain knowledge. This paper learns a material operator mapping full strain histories to stress trajectories with causal attention, spectral convolutions, and sinusoidal layers. Across rate-independent plasticity and damage benchmarks, the framework reports low error and minimal sensitivity to sampling resolution, while remaining limited to rate-independent materials.

  • Problem

    Constitutive responses for path-dependent inelastic materials must often be inferred without measurable internal variables or fully specified evolution laws.

  • Method

    The framework treats the entire strain trajectory as one function input and predicts the full stress trajectory in parallel using causal attention, FNO spectral convolution, and SIREN layers.

  • Results

    Across three benchmarks, the proposed model achieved low prediction error and the smallest temporal-discretization sensitivity, with essentially flat error curves from N = 50 to N = 1000.

  • Takeaways & Limitations

    The architecture provides resolution-invariant predictions for rate-independent path-dependent constitutive behavior while preserving parallel inference.

  • Takeaways & Limitations

    The present study is limited to rate-independent materials; viscoelastic and viscoplastic extensions require physical-time frequency parameterization.

Abstract

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Classical constitutive modeling of path-dependent inelastic materials relies on internal state variables whose evolution equations must be postulated based on domain knowledge and calibrated against experimental data. However, in many practical settings, the relevant internal variables are typically not measurable in experiments, and the constitutive response must be inferred entirely from measured strain-stress data without any prior knowledge of the material's internal state. We propose a data-driven constitutive modeling framework based on the concept of a material operator, which treats a deforming material as a functional mapping from its entire strain history to the corresponding stress response. In contrast to traditional autoregressive or recurrent formulations, the model is trained directly on full loading paths as function-to-function mappings, predicting complete stress trajectories in a single parallel forward pass. Temporal path dependence is enforced through a causally masked attention mechanism embedded within the operator, which restricts the model's attention to past material states while preserving computational parallelizability. Spectral convolutions provide discretization-invariant representations in the frequency domain, while causal attention captures highly adaptive, non-local history dependence. Furthermore, sinusoidal activation functions are used to resolve the strong nonlinear transitions inherent in inelastic regimes. The framework is evaluated across multidimensional, rate-independent material models exhibiting complex phenomena, with an emphasis on nonlinear plasticity and ductile damage accumulation. The results demonstrate accurate and robust predictions of irreversible deformation mechanisms while simultaneously achieving resolution invariance and excellent parallel efficiency.

Nomenclature

The paper uses abbreviations for neural-network architectures, mechanics methods, physical concepts, and evaluation metrics.

  • ANN denotes Artificial Neural Network, while FFNN, RNN, GRU, LSTM, MLP, FNO, and SIREN identify network architectures or components.
  • CANN, TANN, and PINN refer to constitutive, thermodynamics-based, and physics-informed neural-network approaches.
  • CP, FEM, FFT, GP, PDE, RVE, and ISV denote crystal plasticity, finite element method, fast Fourier transform, Gaussian process, partial differential equation, representative volume element, and internal state variable.
  • KKT and MSE denote Karush–Kuhn–Tucker conditions and mean squared error.

1. Introduction

The introduction motivates learning path-dependent constitutive behavior without prescribing unmeasurable internal variables and presents a causal sequence-to-sequence neural operator. Its FNO-SIREN-attention design targets resolution-invariant, accurate predictions across plasticity and damage benchmarks.

  • Motivation: Path-dependent inelastic constitutive laws remain difficult because stress depends on memory, including coupled plasticity and damage evolution.
  • Existing approaches: Physics-structured models improve consistency and interpretability but require domain assumptions such as free-energy forms, invariants, or internal-variable families.
  • Existing approaches: Earlier FFNN and recurrent approaches encode history autoregressively or sequentially, with fixed windows limiting memory and temporal-sampling transfer.
  • Proposed framework: The proposed operator maps an entire discretely sampled strain trajectory to its full stress trajectory in one parallel forward pass.
  • Proposed framework: Causal attention restricts each output to past and present strains, while spectral convolution mixes these features globally over the temporal axis.
  • Proposed framework: FNO-SIREN combines resolution-invariant spectral filters with sinusoidal activations suited to sharp yield, switching, and damage transitions.
  • Evaluation: The framework demonstrates path-family generalization from Gaussian-Process paths to unseen zig-zag and sinusoidal paths across three benchmark problems.

2. Thermodynamics-based material modeling

The paper reviews thermodynamics-based constitutive modeling through internal variables and specifies benchmark plasticity, damage-plasticity, and plane-strain J2 models. These formulations define stress, evolution, admissibility, and coupled updates for the evaluation problems.

  • Thermodynamic formulation: Rate-independent inelastic constitutive models conventionally use internal state variables z to represent plastic strain, hardening, and damage memory.
  • Thermodynamic formulation: The constitutive response is path-dependent because stress depends on prior strain history through the evolution of internal variables.
  • One-dimensional elastoplasticity: The one-dimensional elastoplastic benchmark uses additive elastic-plastic strain decomposition, linear elasticity, nonlinear isotropic hardening, and KKT plasticity conditions.
  • Coupled damage-plasticity: The coupled one-dimensional model includes local isotropic damage, degraded stiffness, plastic hardening, damage hardening, and independent KKT conditions.
  • Coupled damage-plasticity: When plastic and damage surfaces are simultaneously active, the stress update solves two consistency equations using Newton iteration on a 2 × 2 residual system.
  • Two-dimensional plane strain: The two-dimensional plane-strain benchmark predicts three stress components from three strain components and learns normal-shear coupling under J2 plasticity.
  • Two-dimensional plane strain: The J2 formulation uses deviatoric stress invariants, additive elastic-plastic strain decomposition, nonlinear isotropic hardening, and plane-strain kinematics.

3. Data-driven constitutive modeling of path-dependent materials

The framework learns a causal operator mapping complete strain trajectories to complete stress trajectories in one parallel pass. It combines spectral convolutions, causal attention, and SIREN layers to represent resolution-invariant global history dependence and sharp inelastic transitions.

  • Material operator formulation: The proposed operator maps an entire discretely sampled strain path to its entire stress path in one parallel forward pass.Path dependence is enforced without sliding-window iteration or hidden-state recurrence.
  • Fourier Neural Operator backbone: Fourier convolutions provide global temporal mixing and discretization invariance by learning filters indexed by physical wavenumbers rather than grid points.The same trained filter can apply across coarse and fine temporal discretizations when the signal is adequately resolved.
  • SIREN representation: SIREN layers are used for lifting, Fourier-block nonlinearities, and projection, targeting sharp yield, elastic-to-plastic, and damage transitions.Their sinusoidal activations help represent features that standard non-periodic activations tend to smooth.
  • Causal self-attention: Causality restricts each stress prediction to present and past strain values, while causal attention assigns non-uniform relevance to history positions.The mask makes each output a weighted combination of value vectors from positions j ≤ i only.
  • FNO-SIREN architecture: Each operator block combines causal attention, spectral convolution, and SIREN transformations with residual connections.The architecture stacks these blocks between SIREN-based lifting and stress projection operators.

3.2. Autoregressive Feed Forward Neural Networks with explicit history windows

Autoregressive FFNN surrogates approximate constitutive behavior using finite explicit windows of recent strain–stress observations. Their sequential rollout creates error accumulation and a trade-off between memory capacity, input size, and overfitting.

  • Windowed formulation: An FFNN predicts the next stress state from a finite recent window of strain–stress observations and the upcoming strain increment.The reported window sizes are W ∈ {1, 5, 10}.
  • Autoregressive inference: The FFNN is evaluated n−W+1 times sequentially to produce a trajectory of length n, feeding each prediction back into the next input window.Training uses overlapping historical windows and fully populated history windows beginning at time step W.
  • Autoregressive inference: Autoregressive rollout accumulates errors because each prediction conditions subsequent steps.The model’s own predicted stresses replace unobserved true stresses during deployment.
  • Window-length trade-off: Increasing W enlarges the input dimension and computational burden, whereas a window that is too small cannot capture relevant material memory.Window size is therefore the primary structural hyperparameter controlling temporal memory capacity.

3.3. Recurrent neural networks with hidden-state memory

RNN-GRU surrogates encode loading memory in a persistent hidden state updated sequentially from instantaneous strain. This avoids explicit history windows but makes predictions dependent on the temporal discretization through recurrent state evolution.

  • Hidden-state memory: RNN and GRU surrogates represent loading memory in an evolving hidden state rather than an explicit input window.The hidden state acts as a learned counterpart to physical internal state variables.
  • Sequential inference: A trajectory of length n requires N sequential network evaluations, so the RNN processes loading paths sequentially.Unlike the FFNN, it does not maintain a sliding window or feed prior stress predictions back into the input stream.
  • Discretization dependence: RNN hidden-state evolution depends on the training resolution, causing different stress predictions when the same continuous strain history is resampled.This discretization dependence conflicts with the expected self-consistency of rate-independent constitutive laws.
  • Sequential inference: At each timestep, the GRU updates its persistent hidden state from the instantaneous strain and maps that state to the predicted stress.The hidden state is not reset between timesteps.

4. Results

The evaluation compares the proposed operator with windowed FFNN and RNN-GRU baselines across three rate-independent material models and multiple loading-path distributions. Testing includes in-distribution, zig-zag, sinusoidal, and resolution-varying trajectories.

  • Evaluation scope: Three material models are evaluated: one-dimensional elastoplasticity, one-dimensional coupled damage-plasticity, and two-dimensional elastoplasticity.All models are rate-independent and include nonlinear isotropic hardening where specified.
  • Training data: 9000 Gaussian-Process paths are used for training and 1000 for validation, with all paths uniformly sampled at N = 50 steps.The paths exercise repeated yielding, unloading, and reloading.
  • Test distributions: Generalization is tested on fresh Gaussian-Process, zig-zag, and sinusoidal path families, each containing 100 trajectories at N = 50.The sinusoidal paths contain multi-cycle oscillatory loading–unloading sequences.
  • Discretization study: The discretization study resamples identical loading paths from N = 50 to N = 1000 and compares them against a high-resolution reference at N = 5000.This tests whether predictions remain consistent across temporal resolutions.

sPN

The operator models maintain low stress-prediction error at the training resolution and preserve accuracy as loading paths are refined, unlike autoregressive baselines whose errors increase with resolution.

  • All evaluated models achieve relative error below 1% against the reference solution at the reported baseline resolution.
  • 100 canonical zig-zag paths are resampled from N = 50 through N = 1000 while models remain trained only at N = 50.The underlying physical paths remain identical across resolutions, differing only in temporal discretization.
  • FFNN and RNN-GRU mean relative L2 error and standard deviation grow with sampling resolution.Their stepwise mappings are tied to training-time sampling, and autoregressive evaluations accumulate error as paths are refined.
  • FNO-SIREN errors remain essentially flat across resolutions, while causal self-attention further reduces error and standard deviation at every resolution.Spectral filters are indexed by physical wavenumbers rather than sample indices, supporting consistent application to fine discretizations.

4.2.3. Discretization Study

The coupled damage-plasticity and two-dimensional evaluations show that operator-based models retain discretization robustness under more complex constitutive behavior and multicomponent loading.

  • 4.2.3. Discretization Study: The coupled damage-plasticity study evaluates six architectures on 100 zig-zag paths at N = 50 through N = 1000 after training exclusively at N = 50.The test uses the same temporal resolutions as the earlier discretization-invariance evaluation.
  • 4.2.3. Discretization Study: FFNN accuracy degrades sharply as resolution increases, with the window-size-1 exception plateauing only after its error exceeds 80%.
  • 4.2.3. Discretization Study: Both operator models remain essentially flat across the full resolution range, and FNO-SIREN+Attention has the lowest error at every resolution.The result extends discretization invariance from elastoplasticity to coupled damage and plasticity.
  • Multidimensional evaluation: The two-dimensional plane-strain model predicts three stress components from three strain components while learning normal–shear coupling.The input is therefore a three-channel strain tensor rather than a single-channel input.
  • 4.2.3. Discretization Study: FNO-SIREN and FNO-SIREN+Attention maintain nearly constant mean error across uniaxial, biaxial, and multiaxial loading resolutions.The narrow standard-deviation band indicates that the invariance holds path-by-path rather than only after averaging trajectories.

5. Accuracy and computational efficiency of surrogate models

Across accuracy and efficiency comparisons, FNO-SIREN+Attention offers the strongest resolution generalization and fastest inference, while requiring substantially longer training; its energy predictions also show the smallest accumulated error at high resolution.

  • RNN-GRU has the best training-resolution accuracy at 0.32%, but its error rises to 12.41% at N = 1000.FFNN variants similarly degrade to 9.87–13.83% at N = 1000.
  • FNO-SIREN+Attention achieves 0.75% error at training resolution and 2.01% at N = 1000, the best overall accuracy balance.The result is reported as substantially better resolution generalization than FFNN and RNN-GRU.
  • 103.45 min training time accompanies FNO-SIREN+Attention, roughly 4–9× longer than the other models.The added cost reflects combining spectral convolutions with attention.
  • 2.617 msec/signal inference for FNO-SIREN+Attention is faster than the 12.8–18.6 msec/signal of FFNN and RNN-GRU.
  • Thermodynamic consistency: The thermodynamic consistency assessment uses implied cumulative mechanical work because the reference satisfies E(t) ≥ 0 by construction.The study measures surrogate departure from this reference-consistent behavior.
  • Thermodynamic consistency: At higher resolution, FNO-SIREN+Attention has the smallest accumulated mechanical-work error, followed by RNN-GRU and then FFNN.This indicates that its stress predictions track the reference energy accurately under high-resolution, complex paths.

6. Potential extension to FE solver

The operator can be queried on progressively longer strain histories, supporting an incremental finite-element workflow without explicit internal-state tracking. Independent horizon evaluations avoid accumulating earlier prediction errors.

  • The proposed deployment strategy requires no explicit internal state to be tracked between solver steps.The passage frames this as a practical coupling strategy for an operator that does not predict or calculate classical internal variables.
  • The operator predicts accurate stress responses across progressively longer loading horizons without error growth.Across ten windows, relative error remains between 0.33% and 1.06%.
  • Each horizon is recomputed independently from the complete strain history available at that point.This differs from extending an earlier prediction and prevents small early inaccuracies from compounding as the horizon grows.
  • An FE solver could supply the accumulated strain history at each global step and retain only the current stress.The operator returns the full stress trajectory, while the solver discards the rest after using the current-time value for equilibrium iteration.

7. Conclusion and Outlook

The work addresses temporal-discretization dependence in data-driven models by mapping complete strain trajectories to complete stress trajectories with causal attention and spectral components. Across three constitutive benchmarks, the framework maintains low error and weak sensitivity to temporal resolution, while its current scope is limited to rate-independent materials.

  • Conclusion and Outlook: Step-wise and recurrent surrogates can degrade when test loading paths use a different temporal resolution than training data.For rate-independent laws, dependence on sampling rather than the loading path is identified as a mechanical inconsistency.
  • Conclusion and Outlook: The proposed sequence-to-sequence operator maps an entire strain trajectory to an entire stress trajectory in one parallel pass.A strict causal mask preserves path dependence, while FNO, SIREN, and causal attention provide complementary architectural components.
  • Conclusion and Outlook: Across three constitutive benchmarks, the model achieves low prediction error and the smallest sensitivity to temporal discretization.The evaluated cases progress from one-dimensional elastoplasticity to coupled damage-plasticity and two-dimensional plane-strain J2 elastoplasticity.
  • Conclusion and Outlook: The architecture produces discretization-invariant behavior without an additional constraint or penalty during training.The interpretation given is that the model learns the continuous strain-to-stress mapping rather than a representation tied to one step size.
  • Conclusion and Outlook: The present study is limited to rate-independent materials.Extending it to viscoelastic and viscoplastic materials requires physical-time information and spectral filters parameterized by true frequency.
  • Conclusion and Outlook: Future extensions include thermodynamic admissibility and conditioning the operator on microstructural descriptors.These directions aim toward physics-consistent, microstructure-aware operators integrated with finite-element frameworks.

Appendix A. Discretization study on sinusoidal loading paths

The sinusoidal-loading discretization study evaluates models trained at N = 50 across temporal resolutions from N = 50 to 1000. The reported comparisons focus on how relative L2 error changes with resolution for elastoplastic and coupled damage-plasticity cases.

  • Discretization study on sinusoidal loading paths: Models trained at N = 50 are evaluated across temporal resolutions from N = 50 to N = 1000.The study uses 100 canonical sinusoidal test paths and reports mean relative L2 error versus temporal resolution.
  • Discretization study on sinusoidal loading paths: The study reports discretization results for one-dimensional elastoplastic and coupled damage-plasticity material models.Figures 21–23 cover sinusoidal loading paths for these constitutive settings.
  • Discretization study on sinusoidal loading paths: Mean relative L2 error is plotted as a function of temporal resolution.The comparison is performed across six models and multiple discretizations.
  • Discretization study on sinusoidal loading paths: The FFNN(W = 1), FFNN(W = 5), FFNN(W = 10), and RNN-GRU models degrade steadily as resolution increases.This reported trend is part of the overall discretization study across the canonical sinusoidal test paths.
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