# A new family of Constitutive Artificial Neural Networks towards automated model discovery

arXiv: https://arxiv.org/abs/2210.02202 · HTML: https://mlsanity.com/paper/2210.02202/

## Key facts

- Cited by 49 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 62 references that are in the map.
- First posted on arXiv 15 Sep 2022.

## Builds on (most cited first, 6 of 6)

- [Polyconvex anisotropic hyperelasticity with neural networks](https://mlsanity.com/paper/2106.14623/) (2021)
- [Bayesian Physics-Informed Neural Networks for real-world nonlinear dynamical systems](https://mlsanity.com/paper/2205.08304/) (2022)
- [NN-EUCLID: deep-learning hyperelasticity without stress data](https://mlsanity.com/paper/2205.06664/) (2022)
- [Data-driven Tissue Mechanics with Polyconvex Neural Ordinary Differential Equations](https://mlsanity.com/paper/2110.03774/) (2021)
- [Multiscale modeling of inelastic materials with Thermodynamics-based Artificial Neural Networks (TANN)](https://mlsanity.com/paper/2108.13137/) (2021)
- [Model-free Data-Driven Computational Mechanics Enhanced by Tensor Voting](https://mlsanity.com/paper/2004.02503/) (2020)

## Built on by (most cited first, 30 of 49)

- [Discovering a reaction-diffusion model for Alzheimer's disease by combining PINNs with symbolic regression](https://mlsanity.com/paper/2307.08107/) (2023)
- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [Peridynamic Neural Operators: A Data-Driven Nonlocal Constitutive Model for Complex Material Responses](https://mlsanity.com/paper/2401.06070/) (2024)
- [Data-driven anisotropic finite viscoelasticity using neural ordinary differential equations](https://mlsanity.com/paper/2302.03598/) (2023)
- [Extreme sparsification of physics-augmented neural networks for interpretable model discovery in mechanics](https://mlsanity.com/paper/2310.03652/) (2023)
- [On sparse regression, Lp-regularization, and automated model discovery](https://mlsanity.com/paper/2310.06872/) (2023)
- [Viscoelastic Constitutive Artificial Neural Networks (vCANNs) - a framework for data-driven anisotropic nonlinear finite viscoelasticity](https://mlsanity.com/paper/2303.12164/) (2023)
- [Can KAN CANs? Input-convex Kolmogorov-Arnold Networks (KANs) as hyperelastic constitutive artificial neural networks (CANs)](https://mlsanity.com/paper/2503.05617/) (2025)
- [Theory and implementation of inelastic Constitutive Artificial Neural Networks](https://mlsanity.com/paper/2311.06380/) (2023)
- [Discovering interpretable elastoplasticity models via the neural polynomial method enabled symbolic regressions](https://mlsanity.com/paper/2307.13149/) (2023)
- [Stress representations for tensor basis neural networks: alternative formulations to Finger-Rivlin-Ericksen](https://mlsanity.com/paper/2308.11080/) (2023)
- [A Generative Modeling Framework for Inferring Families of Biomechanical Constitutive Laws in Data-Sparse Regimes](https://mlsanity.com/paper/2305.03184/) (2023)
- [Generative Hyperelasticity with Physics-Informed Probabilistic Diffusion Fields](https://mlsanity.com/paper/2310.03745/) (2023)
- [Accounting for plasticity: An extension of inelastic Constitutive Artificial Neural Networks](https://mlsanity.com/paper/2407.19326/) (2024)
- [Automated Model Discovery for Tensional Homeostasis: Constitutive Machine Learning in Growth and Remodeling](https://mlsanity.com/paper/2410.13645/) (2024)
- [Inelastic Constitutive Kolmogorov-Arnold Networks: A generalized framework for automated discovery of interpretable inelastic material models](https://mlsanity.com/paper/2602.17750/) (2026)
- [Bayesian neural networks for predicting uncertainty in full-field material response](https://mlsanity.com/paper/2406.14838/) (2024)
- [Latent Dynamics Networks (LDNets): learning the intrinsic dynamics of spatio-temporal processes](https://mlsanity.com/paper/2305.00094/) (2023)
- [A physics-encoded Fourier neural operator approach for surrogate modeling of divergence-free stress fields in solids](https://mlsanity.com/paper/2408.15408/) (2024)
- [A Complement to Neural Networks for Anisotropic Inelasticity at Finite Strains](https://mlsanity.com/paper/2510.04187/) (2025)
- [From Perception to Autonomous Computational Modeling: A Multi-Agent Approach](https://mlsanity.com/paper/2604.06788/) (2026)
- [Input convex neural networks: universal approximation theorem and implementation for isotropic polyconvex hyperelastic energies](https://mlsanity.com/paper/2502.08534/) (2025)
- [Generalized invariants meet constitutive neural networks: A novel framework for hyperelastic materials](https://mlsanity.com/paper/2508.12063/) (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)
- [Similarity Equivariant Graph Neural Networks for Homogenization of Metamaterials](https://mlsanity.com/paper/2404.17365/) (2024)
- [COMMET: orders-of-magnitude speed-up in finite element method via batch-vectorized neural constitutive updates](https://mlsanity.com/paper/2510.00884/) (2025)
- [Unsupervised full-field Bayesian inference of orthotropic hyperelasticity from a single biaxial test: a myocardial case study](https://mlsanity.com/paper/2510.09498/) (2025)
- [Hyperelastic constitutive model discovery with differentiable finite elements and structure-preserving neural networks](https://mlsanity.com/paper/2603.26517/) (2026)
- [Predicting and explaining nonlinear material response using deep Physically Guided Neural Networks with Internal Variables](https://mlsanity.com/paper/2308.03915/) (2023)

## Abstract (arXiv)

For more than 100 years, chemical, physical, and material scientists have proposed competing constitutive models to best characterize the behavior of natural and man-made materials in response to mechanical loading. Now, computer science offers a universal solution: Neural Networks. Neural Networks are powerful function approximators that can learn constitutive relations from large data without any knowledge of the underlying physics. However, classical Neural Networks ignore a century of research in constitutive modeling, violate thermodynamic considerations, and fail to predict the behavior outside the training regime. Here we design a new family of Constitutive Artificial Neural Networks that inherently satisfy common kinematic, thermodynamic, and physic constraints and, at the same time, constrain the design space of admissible functions to create robust approximators, even in the presence of sparse data. We revisit the non-linear field theories of mechanics and reverse-engineer the network input to account for material objectivity, symmetry, and incompressibility; the network output to enforce thermodynamic consistency; the activation functions to implement physically reasonable restrictions; and the network architecture to ensure polyconvexity. We demonstrate that this new class of models is a generalization of the classical neo Hooke, Blatz Ko, Mooney Rivlin, Yeoh, and Demiray models and that the network weights have a clear physical interpretation. When trained with classical benchmark data for rubber, our network autonomously selects the best constitutive model and learns its parameters. Our findings suggests that Constitutive Artificial Neural Networks have the potential to induce a paradigm shift in constitutive modeling, from user-defined model selection to automated model discovery. Our source code, data, and examples are available at https://github.com/LivingMatterLab/CANN.