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

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.

Benchmarking data-driven material models on the classic Treloar dataset

Machine learning is rapidly reshaping constitutive modeling, offers new ways to learn material behavior directly from experimental data, and challenges long-established modeling paradigms. But with a growing number of machine-learning-based approaches available, how do they compare in practice? In this paper, we use the classic experimental data of Treloar to benchmark popular frameworks for hyperelasticity: (Generalized-Invariant) Constitutive Artificial Neural Networks, Physics-Augmented Neural Networks, (Adaptive) Material Fingerprinting, and Efficient Unsupervised Constitutive Law Identification & Discovery. We compare their fitting performance, computational cost, hyperparameter sensitivity, and ease of implementation. Furthermore, we discuss the trade-offs between predictive accuracy and model complexity. The latter is assessed by quantifying both the number of material parameters in the discovered models and the computational time required to evaluate the constitutive model and its derivatives. The results show that all methods can reproduce the benchmark data remarkably well. Rather than identifying a single winner, we highlight the strengths and limitations of each approach and provide practical guidance for their use. The source code for all six methods, including the training and comparison scripts, as well as all results and data used in this study, is publicly available via https://doi.org/10.5281/zenodo.21915635.

Interval and fuzzy physics-augmented neural networks (iPANN and fPANN) for uncertainty quantification and propagation in constitutive modeling

Constitutive modeling under uncertainty remains a central challenge for reliable mechanics simulations, particularly when the available stress-deformation data are sparse, noisy, or heterogeneous. We propose interval and fuzzy physics-augmented neural networks (iPANNs and fPANNs) for uncertainty-aware hyperelastic constitutive modeling. iPANNs learn sparse lower, mean, and upper free energy density branches whose stresses, obtained by automatic differentiation, ultimately enclose noisy stress observations. In contrast to this deterministic interval description, fPANNs embed the learned iPANN branches into a fuzzy-set representation through alpha-cut interpolation, yielding a nested family of admissible responses. iPANNs and fPANNs encode mechanistic constraints - preserving objectivity, consistency and promoting polyconvexity - and smoothed L0 regularization promotes interpretable energy representations. The bound models are trained through a two-stage transfer-learning procedure in which a sparse mean constitutive response is learned first and then fine-tuned into lower and upper energy branches. We evaluate the framework on synthetic isotropic hyperelastic data with heteroscedastic noise, varying random realizations, shifted noise means, and varying noise magnitudes. The results show that the learned bounds enclose noisy stress observations while generalizing to the test set. Further, we examine the propagation of uncertainty through the mean, upper and lower bound predictions of the learned iPANN models in a finite element setting. The proposed framework provides a compact, physics-consistent route for distribution-free aleatoric uncertainty quantification in hyperelastic constitutive modeling, and propagation in downstream finite element simulations.

CANN-EUCLID: unsupervised constitutive artificial neural network model discovery from full-field data

Constitutive artificial neural networks (CANNs) provide interpretable material model discovery, but have so far been used in stress-supervised settings based on apparent stress-strain data from homogeneous tests. Because each test samples only a narrow loading path and provides homogenized rather than local stress information, robust discovery typically requires multiple loading modes to constrain the multidimensional response. This is challenging for soft biological tissues, where repeated testing, damage, and sample variability limit reliable information from a single specimen. Here, we combine CANNs with the stress-unsupervised full-field discovery framework EUCLID to identify sparse hyperelastic laws directly from displacement fields and reaction forces in one heterogeneity-inducing loading case. CANN-EUCLID minimizes equilibrium imbalance with sparsity-promoting regularization selecting compact active terms, without local stress measurements or a prescribed law. We evaluate the approach on isotropic and anisotropic benchmarks with prescribed ground-truth laws. When the ground truth is representable by the chosen CANN basis, our method recovers the correct terms with near-exact accuracy, including exponential terms with embedded parameters. When it is not contained in the basis, the method retains shared terms and approximates missing contributions using available basis functions. Generalization depends strongly on sampled deformation states: exponential strain-stiffening terms can be recovered accurately when sufficiently probed, but can produce large extrapolation errors when the stiffening regime lies outside the sampled domain. Forward FE validation simulations show that the discovered behavior accurately replicates the ground truth. These results establish stress-unsupervised CANN discovery as a promising framework for interpretable full-field constitutive model identification.

GPT-Micro: A large language paradigm for accelerated, inexpensive, and thermodynamics-consistent discovery of constitutive models in manufacturing

Constitutive modeling of the relationship between process-imposed material states and fundamental material properties is critical to control of material microstructure in manufacturing processes. The limited accuracy resulting from the typical reliance on fallible human expertise and intuition for postulation and revision of the models functional form results in incremental and time consuming model discovery. Conventional Machine Learning (ML) incurs significant cost and time of data generation. Model discovery using Large Language Models (LLMs) suffers from the above issues and/or ignores the inviolability of fundamental thermodynamics laws. This work creates a novel GPT-Micro paradigm for autonomous, data sparse, and thermodynamics-compliant discovery of de-novo constitutive models. This framework seamlessly integrates semantic knowledge extraction from literature, enforcement of thermodynamics-based conservation laws, and sparse datasets, with LLM-driven generation and refinement of model hypotheses. Validation is performed for a long-intractable constitutive modeling problem in a printed electronics process testbed. This reveals significant and simultaneous advantages over the state-of-the-art including: (a) More than 70 percent reduction in data burden relative to ML-based modeling without loss in accuracy; (b) 400X reduction in discovery time after data generation, from months to hours, relative to human-driven modeling; (c) Discovery of models with novel functional forms without subjective human choice of a starting hypothesis; (d) Enhanced physics-rooted trustworthiness, human interpretability, and mechanistic insight via synthesis of compact, conservation-compliant, and physically complete analytical models. The potential of GPT-Micro to realize rapid, low-cost, physically trustworthy, and interpretable microstructure modeling across the manufacturing landscape is discussed.