# Constitutive model characterization and discovery using physics-informed deep learning

arXiv: https://arxiv.org/abs/2203.09789 · HTML: https://mlsanity.com/paper/2203.09789/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 74 references that are in the map.
- First posted on arXiv 18 Mar 2022.

## Builds on (most cited first, 12 of 12)

- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Speech Recognition with Deep Recurrent Neural Networks](https://mlsanity.com/paper/1303.5778/) (2013)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Physics informed deep learning for computational elastodynamics without labeled data](https://mlsanity.com/paper/2006.08472/) (2020)
- [Unsupervised discovery of interpretable hyperelastic constitutive laws](https://mlsanity.com/paper/2010.13496/) (2020)
- [Physics-Informed Neural Network for Modelling the Thermochemical Curing Process of Composite-Tool Systems During Manufacture](https://mlsanity.com/paper/2011.13511/) (2020)
- [Physics-informed neural network simulation of multiphase poroelasticity using stress-split sequential training](https://mlsanity.com/paper/2110.03049/) (2021)
- [A nonlocal physics-informed deep learning framework using the peridynamic differential operator](https://mlsanity.com/paper/2006.00446/) (2020)
- [Energy-based error bound of physics-informed neural network solutions in elasticity](https://mlsanity.com/paper/2010.09088/) (2020)
- [Deep Learning for One-dimensional Consolidation](https://mlsanity.com/paper/2004.11689/) (2020)
- [Learning Unknown Physics of non-Newtonian Fluids](https://mlsanity.com/paper/2009.01658/) (2020)
- [Data-driven discovery of interpretable causal relations for deep learning material laws with uncertainty propagation](https://mlsanity.com/paper/2105.09980/) (2021)

## Built on by (most cited first, 17 of 17)

- [A review on data-driven constitutive laws for solids](https://mlsanity.com/paper/2405.03658/) (2024)
- [Solving Forward and Inverse Problems of Contact Mechanics using Physics-Informed Neural Networks](https://mlsanity.com/paper/2308.12716/) (2023)
- [Theory and implementation of inelastic Constitutive Artificial Neural Networks](https://mlsanity.com/paper/2311.06380/) (2023)
- [Evolutionary Optimization of Physics-Informed Neural Networks: Evo-PINN Frontiers and Opportunities](https://mlsanity.com/paper/2501.06572/) (2025)
- [A finite operator learning technique for mapping the elastic properties of microstructures to their mechanical deformations](https://mlsanity.com/paper/2404.00074/) (2024)
- [Transfer learning for improved generalizability in causal physics-informed neural networks for beam simulations](https://mlsanity.com/paper/2311.00578/) (2023)
- [Learning solution of nonlinear constitutive material models using physics-informed neural networks: COMM-PINN](https://mlsanity.com/paper/2304.06044/) (2023)
- [Learning Homogenization for Elliptic Operators](https://mlsanity.com/paper/2306.12006/) (2023)
- [Learning Memory and Material Dependent Constitutive Laws](https://mlsanity.com/paper/2502.05463/) (2025)
- [History-Aware Neural Operator: Robust Data-Driven Constitutive Modeling of Path-Dependent Materials](https://mlsanity.com/paper/2506.10352/) (2025)
- [Monotone Peridynamic Neural Operator for Nonlinear Material Modeling with Conditionally Unique Solutions](https://mlsanity.com/paper/2505.01060/) (2025)
- [Beyond empirical models: Discovering new constitutive laws in solids with graph-based equation discovery](https://mlsanity.com/paper/2511.09906/) (2025)
- [Predicting and explaining nonlinear material response using deep Physically Guided Neural Networks with Internal Variables](https://mlsanity.com/paper/2308.03915/) (2023)
- [Finding the Underlying Viscoelastic Constitutive Equation via Universal Differential Equations and Differentiable Physics](https://mlsanity.com/paper/2501.00556/) (2024)
- [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)
- [Finite basis physics-informed neural networks with hard constraints for viscous fluid flow in highly perforated domains](https://mlsanity.com/paper/2608.08114/) (2026)
- [Learning the Constitutive Behavior of Materials via Neural Operators and Causal Attention: Case Studies in Plasticity and Damage](https://mlsanity.com/paper/2609.02194/) (2026)

## Abstract (arXiv)

Classically, the mechanical response of materials is described through constitutive models, often in the form of constrained ordinary differential equations. These models have a very limited number of parameters, yet, they are extremely efficient in reproducing complex responses observed in experiments. Additionally, in their discretized form, they are computationally very efficient, often resulting in a simple algebraic relation, and therefore they have been extensively used within large-scale explicit and implicit finite element models. However, it is very challenging to formulate new constitutive models, particularly for materials with complex microstructures such as composites. A recent trend in constitutive modeling leverages complex neural network architectures to construct complex material responses where a constitutive model does not yet exist. Whilst very accurate, they suffer from two deficiencies. First, they are interpolation models and often do poorly in extrapolation. Second, due to their complex architecture and numerous parameters, they are inefficient to be used as a constitutive model within a large-scale finite element model. In this study, we propose a novel approach based on the physics-informed learning machines for the characterization and discovery of constitutive models. Unlike data-driven constitutive models, we leverage foundations of elastoplasticity theory as regularization terms in the total loss function to find parametric constitutive models that are also theoretically sound. We demonstrate that our proposed framework can efficiently identify the underlying constitutive model describing different datasets from the von Mises family.