# A review on data-driven constitutive laws for solids

arXiv: https://arxiv.org/abs/2405.03658 · HTML: https://mlsanity.com/paper/2405.03658/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- Builds on 89 of its 423 references that are in the map.
- First posted on arXiv 6 May 2024.

## Builds on (most cited first, 30 of 89)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Non-local Neural Networks](https://mlsanity.com/paper/1711.07971/) (2017)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/1812.08434/) (2018)
- [Diffusion Models in Vision: A Survey](https://mlsanity.com/paper/2209.04747/) (2022)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [QT-Opt: Scalable Deep Reinforcement Learning for Vision-Based Robotic Manipulation](https://mlsanity.com/paper/1806.10293/) (2018)
- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [What is the State of Neural Network Pruning?](https://mlsanity.com/paper/2003.03033/) (2020)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [A Brief Survey of Deep Reinforcement Learning](https://mlsanity.com/paper/1708.05866/) (2017)
- [On Evaluating Adversarial Robustness](https://mlsanity.com/paper/1902.06705/) (2019)
- [Fundamentals of Recurrent Neural Network (RNN) and Long Short-Term Memory (LSTM) Network](https://mlsanity.com/paper/1808.03314/) (2018)
- [Benchmarking Deep Reinforcement Learning for Continuous Control](https://mlsanity.com/paper/1604.06778/) (2016)
- [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [Safety Verification of Deep Neural Networks](https://mlsanity.com/paper/1610.06940/) (2016)
- [Contemporary Symbolic Regression Methods and their Relative Performance](https://mlsanity.com/paper/2107.14351/) (2021)
- [When Gaussian Process Meets Big Data: A Review of Scalable GPs](https://mlsanity.com/paper/1807.01065/) (2018)
- [Disentangling Adversarial Robustness and Generalization](https://mlsanity.com/paper/1812.00740/) (2018)
- [Sobolev Training for Neural Networks](https://mlsanity.com/paper/1706.04859/) (2017)
- [Adversarial Training Can Hurt Generalization](https://mlsanity.com/paper/1906.06032/) (2019)
- [L2 Regularization versus Batch and Weight Normalization](https://mlsanity.com/paper/1706.05350/) (2017)
- [Transfer learning enhanced physics informed neural network for phase-field modeling of fracture](https://mlsanity.com/paper/1907.02531/) (2019)
- [A physics-informed variational DeepONet for predicting the crack path in brittle materials](https://mlsanity.com/paper/2108.06905/) (2021)

## Built on by (most cited first, 30 of 35)

- [Can KAN CANs? Input-convex Kolmogorov-Arnold Networks (KANs) as hyperelastic constitutive artificial neural networks (CANs)](https://mlsanity.com/paper/2503.05617/) (2025)
- [Predicting Crack Nucleation and Propagation in Brittle Materials Using Deep Operator Networks with Diverse Trunk Architectures](https://mlsanity.com/paper/2501.00016/) (2024)
- [Accounting for plasticity: An extension of inelastic Constitutive Artificial Neural Networks](https://mlsanity.com/paper/2407.19326/) (2024)
- [Inelastic Constitutive Kolmogorov-Arnold Networks: A generalized framework for automated discovery of interpretable inelastic material models](https://mlsanity.com/paper/2602.17750/) (2026)
- [Learning Memory and Material Dependent Constitutive Laws](https://mlsanity.com/paper/2502.05463/) (2025)
- [Polyconvex Physics-Augmented Neural Network Constitutive Models in Principal Stretches](https://mlsanity.com/paper/2503.00575/) (2025)
- [Stochastic and Non-local Closure Modeling for Nonlinear Dynamical Systems via Latent Score-based Generative Models](https://mlsanity.com/paper/2506.20771/) (2025)
- [Monotone Peridynamic Neural Operator for Nonlinear Material Modeling with Conditionally Unique Solutions](https://mlsanity.com/paper/2505.01060/) (2025)
- [Automated Constitutive Model Discovery by Pairing Sparse Regression Algorithms with Model Selection Criteria](https://mlsanity.com/paper/2509.16040/) (2025)
- [Towards Rapid Constitutive Model Discovery from Multi-Modal Data: Physics Augmented Finite Element Model Updating (paFEMU)](https://mlsanity.com/paper/2604.07746/) (2026)
- [Differentiable Neural-Integrated Meshfree Method for Forward and Inverse Modeling of Finite Strain Hyperelasticity](https://mlsanity.com/paper/2407.11183/) (2024)
- [Input Specific Neural Networks](https://mlsanity.com/paper/2503.00268/) (2025)
- [COMMET: orders-of-magnitude speed-up in finite element method via batch-vectorized neural constitutive updates](https://mlsanity.com/paper/2510.00884/) (2025)
- [Beyond empirical models: Discovering new constitutive laws in solids with graph-based equation discovery](https://mlsanity.com/paper/2511.09906/) (2025)
- [Automating modeling in mechanics: LLMs as designers of physics-constrained neural networks for constitutive modeling of materials](https://mlsanity.com/paper/2512.01735/) (2025)
- [Discovering uncertainty: Gaussian constitutive neural networks with correlated weights](https://mlsanity.com/paper/2503.12679/) (2025)
- [Optimal Experimental Design for Reliable Learning of History-Dependent Constitutive Laws](https://mlsanity.com/paper/2603.12365/) (2026)
- [Learning Physics-Consistent Material Behavior from Dynamic Displacements](https://mlsanity.com/paper/2407.20273/) (2024)
- [The Illusion of Fit: Spatially Resolved Assessment of Constitutive Model Validity in Elastography and Physics-Based Inverse Problems](https://mlsanity.com/paper/2502.07415/) (2025)
- [A general physics-constrained method for the modelling of equation's closure terms with sparse data](https://mlsanity.com/paper/2505.03783/) (2025)
- [Crack Path Prediction with Operator Learning using Discrete Particle System data Generation](https://mlsanity.com/paper/2506.01976/) (2025)
- [A Comparison of Surrogate Constitutive Models for Viscoplastic Creep Simulation of HT-9 Steel](https://mlsanity.com/paper/2509.22667/) (2025)
- [Physics-Informed Gaussian Process Regression for the Constitutive Modeling of Concrete: A Data-Driven Improvement to Phenomenological Models](https://mlsanity.com/paper/2601.03367/) (2026)
- [Systematic Performance Assessment of Deep Material Networks for Multiscale Material Modeling](https://mlsanity.com/paper/2602.07192/) (2026)
- [LLM-driven design of physics-constrained constitutive models: two agents are better than one](https://mlsanity.com/paper/2605.23754/) (2026)
- [Discovering Thermodynamically Admissible Dissipation Potentials via Grammar-Based Symbolic Regression](https://mlsanity.com/paper/2605.31532/) (2026)
- [Automatic-Differentiation-Enabled Finite Element Model Updating: Learning Constitutive Neural Networks from Structural Deformation Data](https://mlsanity.com/paper/2606.05199/) (2026)
- [CANN-EUCLID: unsupervised constitutive artificial neural network model discovery from full-field data](https://mlsanity.com/paper/2606.14565/) (2026)
- [CodeJeNN: A simple C++ neural network generator for physics applications](https://mlsanity.com/paper/2607.02746/) (2026)
- [Microstructure-Conditioned Surrogate Models for Graded Multiscale Optimization of Mycelium Composites](https://mlsanity.com/paper/2607.13688/) (2026)

## Abstract (arXiv)

This review article highlights state-of-the-art data-driven techniques to discover, encode, surrogate, or emulate constitutive laws that describe the path-independent and path-dependent response of solids. Our objective is to provide an organized taxonomy to a large spectrum of methodologies developed in the past decades and to discuss the benefits and drawbacks of the various techniques for interpreting and forecasting mechanics behavior across different scales. Distinguishing between machine-learning-based and model-free methods, we further categorize approaches based on their interpretability and on their learning process/type of required data, while discussing the key problems of generalization and trustworthiness. We attempt to provide a road map of how these can be reconciled in a data-availability-aware context. We also touch upon relevant aspects such as data sampling techniques, design of experiments, verification, and validation.