Towards Rapid Constitutive Model Discovery from Multi-Modal Data: Physics Augmented Finite Element Model Updating (paFEMU)

Abstract: Recent progress in AI-enabled constitutive modeling has concentrated on moving from a purely data-driven paradigm to the enforcement of physical constraints and mechanistic principles, a concept referred to as physics augmentation. Classical phenomenological approaches rely on selecting a pre-defined model and calibrating its parameters, while machine learning methods often focus on discovery of the model itself. Sparse regression approaches lie in between, where large libraries of pre-defined models are probed during calibration. Sparsification in the aforementioned paradigm, but also in the context of neural network architecture, has been shown to enable interpretability, uncertainty quantification, but also heterogeneous software integration due to the low-dimensional nature of the resulting models. Most works in AI-enabled constitutive modeling have also focused on data from a single source, but in reality, materials modeling workflows can contain data from many different sources (multi-modal data), and also from testing other materials within the same materials class (multi-fidelity data). In this work, we introduce physics augmented finite element model updating (paFEMU), as a transfer learning approach that combines AI-enabled constitutive modeling, sparsification for interpretable model discovery, and finite element-based adjoint optimization utilizing multi-modal data. This is achieved by combining simple mechanical testing data, potentially from a distinct material, with digital image correlation-type full-field data acquisition to ultimately enable rapid constitutive modeling discovery. The simplicity of the sparse representation enables easy integration of neural constitutive models in existing finite element workflows, and also enables low-dimensional updating during transfer learning.

• The paper introduces paFEMU for rapid model discovery using sparse, physics-augmented neural networks integrated with differentiable finite element methods.
• It demonstrates extreme model sparsity with as few as 9–13 parameters while maintaining robust performance and adherence to physical constraints.
• The approach enables efficient transfer learning from low-fidelity tests to high-fidelity full-field data, ensuring reliable simulations even under extrapolated conditions.

Physics-Augmented Transfer Learning for Constitutive Model Discovery: The paFEMU Framework

Introduction

"Towards Rapid Constitutive Model Discovery from Multi-Modal Data: Physics Augmented Finite Element Model Updating (paFEMU)" (2604.07746) addresses the challenge of rapid, interpretable constitutive model identification in solid mechanics, leveraging limited, heterogeneous datasets typical in experimental workflows. The research advances the state-of-the-art in AI-driven constitutive modeling by introducing a transfer learning paradigm, grounded in differentiable finite element frameworks, that unifies sparse neural architecture discovery, physics augmentation, and multi-fidelity data integration. The emphasis is on interpretable neural models that respect thermodynamic and polyconvexity constraints, enabling efficient integration with classical simulation environments and robust extrapolation from small data.

Context and Core Advances

Prevailing approaches in AI-enabled constitutive modeling span phenomenological paradigms (fixed model forms with data-fitting), sparse regression (library-based model selection), and black-box neural surrogates. While ML models deliver functional flexibility, their deployment in mechanics is hampered by poor interpretability, limited generalization, and frequent violation of physical laws. The paFEMU framework circumvents these pitfalls through several innovations:

• Sparse Physics-Augmented Neural Networks (PANNs): Neural constitutive models are aggressively sparsified via $L_0$-based stochastic regularization, yielding low-dimensional, interpretable representations that generalize better and are more easily incorporated into FE pipelines.
• Differentiable FEM-based Adjoint Optimization: Integration of discovered models into an FEM/adjoint setting enables analytical sensitivity computation and PDE-constrained optimization against full-field, multi-modal observables.
• Multi-stage Transfer Learning with Multi-modal Data: Constitutive priors are learned from simple (potentially cross-material) tests and then effectively transferred and fine-tuned on high-fidelity, limited, full-field data, reflecting realistic materials development workflows.

Framework and Methodology

Constitutive Model Construction

The material law is represented as a neural energy potential $\Psi(\mathbf{F})$ formulated on invariants $(I_1, I_2, J)$ for isotropic hyperelasticity, with energy-derived stresses ensuring mechanical consistency. Critical to the approach is embedding physics both in architecture (e.g., Input Convex NNs, monotonic constraints, polyconvexity guarantees) and in training (objectivity, normalization at reference state, analytic differentiation). At the same time, physical expressivity is maximized by partially relaxing polyconvexity constraints, and monitoring them via differentiable polyconvexity indicators.

Extreme Model Parsimony

Model interpretability and computational integration stem from extreme sparsification during pre-training. Here, the high-dimensional neural model (O(10⁴) parameters) is collapsed via $L_0$ regularization (implemented with differentiable hard-concrete stochastic gates) to as few as 9–13 nonzero parameters, ensuring the discovered law is both compact and directly embeddable in differentiable FE solvers.

Transfer Learning via paFEMU

The model discovery proceeds in two phases:

1. Pre-training: The neural model is trained and sparsified on synthetic or simple experiment data (e.g., homogeneous stress/strain states from uniaxial, biaxial tests), possibly from reference materials. All relevant physical constraints (objectivity, normalization, polyconvexity either as hard or soft constraint) are enforced.
2. FE-based Model Updating: The low-dimensional neural law is deployed as the constitutive kernel in a differentiable FEM environment. High-fidelity, full-field data (e.g., DIC displacements from complex tests) is used as target, and the neural parameters are optimized via adjoint-based PDE-constrained minimization of the observation/model discrepancy. The paFEMU pipeline thus enables highly data-efficient, physically trustworthy parameter and (to some extent) model-form adaptation.

Numerical Results and Strong Claims

Rigorous computational studies demonstrate:

• Extreme Sparsity without Loss of Expressivity: $L_0$ regularized training yields neural potentials with O(10) parameters but with predictive performance competitive with highly overparameterized networks. Compact models generalize better in data-scarce regimes and are robust in deployment.
• Transferability with Minimal Data: Pre-trained neural laws (e.g., Gent-type) can be efficiently adapted to new, more complex target materials (e.g., Neo-Hookean, generalized Ogden) using as little as a handful of full-field DIC datasets. Convergence occurs in few tens of iterations.
• Polyconvexity/Trustworthiness-Expressivity Trade-off: While polyconvexity-by-construction ensures absolute stability (no nonphysical model output), empirically it can compromise expressivity and fit for real, complex materials. Relaxed-constraint networks using soft polyconvexity penalties achieve a balance, matching unconstrained performance in most cases and outperforming over-restricted counterparts especially in generalization and deployment.
• Robust Deployment in Unseen Scenarios: Final transferred models, once integrated into FE simulations, remain numerically stable and predictive even under large, out-of-training-distribution deformations (e.g., 3D torsion up to 458°), with relative stress errors well under 10% despite the model operating far outside its training regime.

Practical and Theoretical Implications

The paFEMU architecture demonstrates how interpretable, sparse, and physically augmented neural models can close the loop between model discovery and practical simulation/deployment in solid mechanics. This has multiple implications:

• Enabling Transfer Across Materials/Conditions: The physics-augmented transfer learning paradigm allows leveraging existing materials data across a class, reducing experimental load and accelerating characterization for novel composites, polymers, or biological tissues, especially in data-limited regimes.
• Efficient Simulation Pipeline Integration: Sparse, interpretable neural material laws discovered via paFEMU are immediately usable in established FE codes, in contrast to high-parameter, black-box surrogates.
• Certifiable Physical Admissibility: The combination of architecture-level physics augmentation, empirical polyconvexity checks, and PDE-constrained optimization ensures that deployed models remain within physical and mechanical admissibility bounds.

Prospects and Future Directions

Future developments include extending paFEMU to history/path-dependent materials (plasticity, viscoelasticity), integrating active experimental design for optimal data collection, scaling to heterogeneous/anisotropic behavior, and enabling real-time, closed-loop experimental/model update cycles. The paradigm is also relevant for coupled multi-physics discovery in computational science.

Conclusion

paFEMU—in combining model sparsification, differentiable adjoint optimization, and physics-aware transfer learning on multi-modal data—provides an efficient, interpretable, and certifiable route to rapid constitutive model discovery. This framework represents a significant methodological unification, enabling practical and generalizable integration of neural constitutive models in computational mechanics.