Neural Networks as Physics-Consistent Surrogates: An \textit{Explainable AI} Validation Framework for Learning Constitutive Relations

TL;DR

This work tackles the interpretability challenge of neural constitutive models by introducing a Physics-XAI framework that validates learned material behavior against classical mechanics across hyperelasticity, elastoplasticity, and viscoelasticity. It combines synthetic data generation, physics-informed surrogate models, and a suite of explainability tools (Grad×Input, Integrated Gradients, SHAP, PCA, and wavelet analysis) to verify that networks learn stiffness, history-dependence, and fractional damping rather than mere curve fitting. A Physics-Aware Distillation step maps neural predictions to interpretable constants such as $\hat{\mu},\hat{C}_1,\hat{C}_2$ for Mooney–Rivlin and related parameters for Chaboche, strengthening trust and providing a bridge to traditional constitutive models. The framework demonstrates that neural surrogates can serve as physically meaningful, verifiable components in solid-mechanics simulations, enabling potential discovery of new constitutive insights while guiding robust deployment in engineering contexts.

Abstract

This paper presents a Physics-\textit{Explainable AI} (XAI) framework to validate and interpret neural networks for the constitutive modeling of solid materials. The study bridges the gap between data-driven models and continuum mechanics by applying a suite of explainability methods to neural networks trained on three distinct material behaviors: hyperelasticity (\textit{Mooney-Rivlin}), elastoplasticity (\textit{Chaboche}), and viscoelasticity (\textit{Fractional Zener}). First, high-fidelity surrogate models, including dense feed-forward networks, LSTMs, and GRUs, are trained on synthetically generated data to accurately capture complex material responses. The core of the work then employs XAI techniques to "open the black box" and confirm that the networks learn physically meaningful principles. For hyperelasticity, gradient-based attributions (\textit{Grad Input} (GI)) successfully match the analytical tangent modulus, proving the network learned material stiffness. For elastoplasticity, \textit{SHapley Additive exPlanations} (SHAP) and \textit{Principal Component Analysis} (PCA) demonstrate the \textit{Recurrent Neural Network} (RNN) internalizes path-dependent memory, with SHAP identifying \textit{plastic strain} as the dominant feature governing the stress prediction. For viscoelasticity, latent-space and wavelet analyses of the \textit{Gated Recurrent Unit. } GRU layers reveal a clear temporal hierarchy, with different layers encoding instantaneous elastic response, intermediate relaxation, and long-term fractional memory. Ultimately, the study demonstrates that the XAI framework can verify that the neural networks are not merely curve-fitting but are, in fact, learning the underlying physical mechanisms of stiffness, history-dependence, and temporal damping.

Figures (17)

• Figure 1: Comparison of analytical and neural model stress--stretch curves for the Mooney--Rivlin model.
• Figure 2: Training and validation loss for the Chaboche RNN. The model exhibits smooth convergence and negligible overfitting.
• Figure 3: Comparison between analytical and RNN-predicted stress--strain response. The network accurately reproduces hysteresis and hardening behavior.
• Figure 4: Representative stress--strain--time response of the Fractional Zener model, showing instantaneous elastic deformation followed by delayed viscoelastic relaxation.
• Figure 5: Training and validation loss history for the Fractional Zener GRU model. The log-scale plot illustrates rapid early convergence followed by stable low-error behavior.