mlsanityData to 14 Oct 2026

A review on data-driven constitutive laws for solids

arXiv 2405.03658 ↗First posted 6 May 2024cs.CE cs.LG physics.app-phcited by 35

Not on the map. Why?

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.

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

89 of 423 references.

References · 89

  1. 2017Attention Is All You Needcited by 60,361
  2. 2015Deep Residual Learning for Image Recognitioncited by 52,096
  3. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,305
  4. 2013Auto-Encoding Variational Bayescited by 4,053
  5. 2017Non-local Neural Networkscited by 2,293
  6. 2016Density estimation using Real NVPcited by 2,042
  7. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  8. 2018Graph Neural Networks: A Review of Methods and Applicationscited by 1,576
  9. 2022Diffusion Models in Vision: A Surveycited by 1,015
  10. 2015Automatic differentiation in machine learning: a surveycited by 855
  11. 2018QT-Opt: Scalable Deep Reinforcement Learning for Vision-Based Robotic Manipulationcited by 743
  12. 2018GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Accelerationcited by 609
38 more
  1. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 583
  2. 2020What is the State of Neural Network Pruning?cited by 583
  3. 2019AI Feynman: a Physics-Inspired Method for Symbolic Regressioncited by 548
  4. 2017A Brief Survey of Deep Reinforcement Learningcited by 483
  5. 2019On Evaluating Adversarial Robustnesscited by 447
  6. 2018Fundamentals of Recurrent Neural Network (RNN) and Long Short-Term Memory (LSTM) Networkcited by 420
  7. 2016Benchmarking Deep Reinforcement Learning for Continuous Controlcited by 371
  8. 2017The Expressive Power of Neural Networks: A View from the Widthcited by 320
  9. 2019Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradientscited by 309
  10. 2016Safety Verification of Deep Neural Networkscited by 263
  11. 2021Contemporary Symbolic Regression Methods and their Relative Performancecited by 215
  12. 2018When Gaussian Process Meets Big Data: A Review of Scalable GPscited by 206
  13. 2018Disentangling Adversarial Robustness and Generalizationcited by 172
  14. 2017Sobolev Training for Neural Networkscited by 143
  15. 2019Adversarial Training Can Hurt Generalizationcited by 132
  16. 2017L2 Regularization versus Batch and Weight Normalizationcited by 124
  17. 2019Transfer learning enhanced physics informed neural network for phase-field modeling of fracturecited by 116
  18. 2021A physics-informed variational DeepONet for predicting the crack path in brittle materialscited by 98
  19. 2018Reachability Analysis of Deep Neural Networks with Provable Guaranteescited by 93
  20. 2019Deep Reinforcement Learning for Tradingcited by 73
  21. 2019Meta-Learning Acquisition Functions for Transfer Learning in Bayesian Optimizationcited by 70
  22. 2021Polyconvex anisotropic hyperelasticity with neural networkscited by 55
  23. 2020A Survey of Constrained Gaussian Process Regression: Approaches and Implementation Challengescited by 52
  24. 2020Unsupervised discovery of interpretable hyperelastic constitutive lawscited by 51
  25. 2020Geometric deep learning for computational mechanics Part I: Anisotropic Hyperelasticitycited by 49
  26. 2020Solving inverse problems using conditional invertible neural networkscited by 37
  27. 2019A Survey on GANs for Anomaly Detectioncited by 32
  28. 2018A deep material network for multiscale topology learning and accelerated nonlinear modeling of heterogeneous materialscited by 31
  29. 2022Automated discovery of generalized standard material models with EUCLIDcited by 31
  30. 2022NN-EUCLID: deep-learning hyperelasticity without stress datacited by 30
  31. 2018Model-Free Data-Driven Inelasticitycited by 29
  32. 2021Data-driven Tissue Mechanics with Polyconvex Neural Ordinary Differential Equationscited by 29
  33. 2022Discovering plasticity models without stress datacited by 28
  34. 2020A machine learning based plasticity model using proper orthogonal decompositioncited by 26
  35. 2018Verification for Machine Learning, Autonomy, and Neural Networks Surveycited by 24
  36. 2021On physics-informed data-driven isotropic and anisotropic constitutive models through probabilistic machine learning and space-filling samplingcited by 24
  37. 2023Denoising diffusion algorithm for inverse design of microstructures with fine-tuned nonlinear material propertiescited by 24
  38. 2022Bayesian-EUCLID: discovering hyperelastic material laws with uncertaintiescited by 23

50 most cited of 89 All 89 (JSON)

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 4; 2025: 15; 2026: 16.

35 papers.

Sort:

By area:

  1. 2025Can KAN CANs? Input-convex Kolmogorov-Arnold Networks (KANs) as hyperelastic constitutive artificial neural networks (CANs)cited by 13
  2. 2024Predicting Crack Nucleation and Propagation in Brittle Materials Using Deep Operator Networks with Diverse Trunk Architecturescited by 12
  3. 2024Accounting for plasticity: An extension of inelastic Constitutive Artificial Neural Networkscited by 6
  4. 2026Inelastic Constitutive Kolmogorov-Arnold Networks: A generalized framework for automated discovery of interpretable inelastic material modelscited by 6
  5. 2025Learning Memory and Material Dependent Constitutive Lawscited by 5
  6. 2025Polyconvex Physics-Augmented Neural Network Constitutive Models in Principal Stretchescited by 5
  7. 2025Stochastic and Non-local Closure Modeling for Nonlinear Dynamical Systems via Latent Score-based Generative Modelscited by 4
  8. 2025Monotone Peridynamic Neural Operator for Nonlinear Material Modeling with Conditionally Unique Solutionscited by 3
  9. 2025Automated Constitutive Model Discovery by Pairing Sparse Regression Algorithms with Model Selection Criteriacited by 3
  10. 2026Towards Rapid Constitutive Model Discovery from Multi-Modal Data: Physics Augmented Finite Element Model Updating (paFEMU)cited by 3
  11. 2024Differentiable Neural-Integrated Meshfree Method for Forward and Inverse Modeling of Finite Strain Hyperelasticitycited by 2
  12. 2025Input Specific Neural Networkscited by 2
23 more
  1. 2025COMMET: orders-of-magnitude speed-up in finite element method via batch-vectorized neural constitutive updatescited by 2
  2. 2025Beyond empirical models: Discovering new constitutive laws in solids with graph-based equation discoverycited by 2
  3. 2025Automating modeling in mechanics: LLMs as designers of physics-constrained neural networks for constitutive modeling of materialscited by 2
  4. 2025Discovering uncertainty: Gaussian constitutive neural networks with correlated weightscited by 1
  5. 2026Optimal Experimental Design for Reliable Learning of History-Dependent Constitutive Lawscited by 1
  6. 2024Learning Physics-Consistent Material Behavior from Dynamic Displacementscited by 0
  7. 2025The Illusion of Fit: Spatially Resolved Assessment of Constitutive Model Validity in Elastography and Physics-Based Inverse Problemscited by 0
  8. 2025A general physics-constrained method for the modelling of equation's closure terms with sparse datacited by 0
  9. 2025Crack Path Prediction with Operator Learning using Discrete Particle System data Generationcited by 0
  10. 2025A Comparison of Surrogate Constitutive Models for Viscoplastic Creep Simulation of HT-9 Steelcited by 0
  11. 2026Physics-Informed Gaussian Process Regression for the Constitutive Modeling of Concrete: A Data-Driven Improvement to Phenomenological Modelscited by 0
  12. 2026Systematic Performance Assessment of Deep Material Networks for Multiscale Material Modelingcited by 0
  13. 2026LLM-driven design of physics-constrained constitutive models: two agents are better than onecited by 0
  14. 2026Discovering Thermodynamically Admissible Dissipation Potentials via Grammar-Based Symbolic Regressioncited by 0
  15. 2026Automatic-Differentiation-Enabled Finite Element Model Updating: Learning Constitutive Neural Networks from Structural Deformation Datacited by 0
  16. 2026CANN-EUCLID: unsupervised constitutive artificial neural network model discovery from full-field datacited by 0
  17. 2026CodeJeNN: A simple C++ neural network generator for physics applicationscited by 0
  18. 2026Microstructure-Conditioned Surrogate Models for Graded Multiscale Optimization of Mycelium Compositescited by 0
  19. 2026Interval and fuzzy physics-augmented neural networks (iPANN and fPANN) for uncertainty quantification and propagation in constitutive modelingcited by 0
  20. 2026Benchmarking data-driven material models on the classic Treloar datasetcited by 0
  21. 2026Learning the Constitutive Behavior of Materials via Neural Operators and Causal Attention: Case Studies in Plasticity and Damagecited by 0
  22. 2026Data-Driven Discovery of Composition-Dependent Constitutive Models for Hyperelasticity and Viscoelasticity of Digital Materialscited by 0
  23. 2026Constitutive State-Space Modeling of Path-Dependent Plasticity: A Resolution-Consistent and Parallelizable Computational Frameworkcited by 0