# Learning solution of nonlinear constitutive material models using physics-informed neural networks: COMM-PINN

arXiv: https://arxiv.org/abs/2304.06044 · HTML: https://mlsanity.com/paper/2304.06044/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 79 references that are in the map.
- First posted on arXiv 10 Apr 2023.

## Builds on (most cited first, 30 of 31)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [GradNorm: Gradient Normalization for Adaptive Loss Balancing in Deep Multitask Networks](https://mlsanity.com/paper/1711.02257/) (2017)
- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications](https://mlsanity.com/paper/1908.10407/) (2019)
- [Polyconvex anisotropic hyperelasticity with neural networks](https://mlsanity.com/paper/2106.14623/) (2021)
- [A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method](https://mlsanity.com/paper/2206.13103/) (2022)
- [A deep material network for multiscale topology learning and accelerated nonlinear modeling of heterogeneous materials](https://mlsanity.com/paper/1807.09829/) (2018)
- [Automated discovery of generalized standard material models with EUCLID](https://mlsanity.com/paper/2211.04453/) (2022)
- [Model-Free Data-Driven Inelasticity](https://mlsanity.com/paper/1808.10859/) (2018)
- [How important are activation functions in regression and classification? A survey, performance comparison, and future directions](https://mlsanity.com/paper/2209.02681/) (2022)
- [The mixed deep energy method for resolving concentration features in finite strain hyperelasticity](https://mlsanity.com/paper/2104.09623/) (2021)
- [A machine learning based plasticity model using proper orthogonal decomposition](https://mlsanity.com/paper/2001.03438/) (2020)
- [Physics-informed neural network simulation of multiphase poroelasticity using stress-split sequential training](https://mlsanity.com/paper/2110.03049/) (2021)
- [Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computing](https://mlsanity.com/paper/2211.07377/) (2022)
- [Thermodynamically Consistent Machine-Learned Internal State Variable Approach for Data-Driven Modeling of Path-Dependent Materials](https://mlsanity.com/paper/2205.00578/) (2022)
- [Lossless Multi-Scale Constitutive Elastic Relations with Artificial Intelligence](https://mlsanity.com/paper/2108.02837/) (2021)
- [Constitutive model characterization and discovery using physics-informed deep learning](https://mlsanity.com/paper/2203.09789/) (2022)
- [Modular machine learning-based elastoplasticity: generalization in the context of limited data](https://mlsanity.com/paper/2210.08343/) (2022)
- [A physics-informed deep neural network for surrogate modeling in classical elasto-plasticity](https://mlsanity.com/paper/2204.12088/) (2022)
- [Mixed formulation of physics-informed neural networks for thermo-mechanically coupled systems and heterogeneous domains](https://mlsanity.com/paper/2302.04954/) (2023)
- [Physically recurrent neural networks for path-dependent heterogeneous materials: embedding constitutive models in a data-driven surrogate](https://mlsanity.com/paper/2209.07320/) (2022)
- [FEANN - An efficient data-driven multiscale approach based on physics-constrained neural networks and automated data mining](https://mlsanity.com/paper/2207.01045/) (2022)
- [A deep learning energy-based method for classical elastoplasticity](https://mlsanity.com/paper/2209.06467/) (2022)
- [A comparative study on different neural network architectures to model inelasticity](https://mlsanity.com/paper/2303.03402/) (2023)
- [Spiking neural networks for nonlinear regression](https://mlsanity.com/paper/2210.03515/) (2022)
- [A comparative study of physics-informed neural network models for learning unknown dynamics and constitutive relations](https://mlsanity.com/paper/1904.04058/) (2019)
- [Evolution TANN and the identification of internal variables and evolution equations in solid mechanics](https://mlsanity.com/paper/2209.13269/) (2022)
- [Thermodynamic Consistent Neural Networks for Learning Material Interfacial Mechanics](https://mlsanity.com/paper/2011.14172/) (2020)
- [Deep Learning and Computational Physics (Lecture Notes)](https://mlsanity.com/paper/2301.00942/) (2023)

## Built on by (most cited first, 6 of 6)

- [A finite operator learning technique for mapping the elastic properties of microstructures to their mechanical deformations](https://mlsanity.com/paper/2404.00074/) (2024)
- [EPi-cKANs: Elasto-Plasticity Informed Kolmogorov-Arnold Networks Using Chebyshev Polynomials](https://mlsanity.com/paper/2410.10897/) (2024)
- [Neural-Initialized Newton: Accelerating Nonlinear Finite Elements via Operator Learning](https://mlsanity.com/paper/2511.06802/) (2025)
- [Accounting for plasticity: An extension of inelastic Constitutive Artificial Neural Networks](https://mlsanity.com/paper/2407.19326/) (2024)
- [Neuro-Spectral Architectures for Causal Physics-Informed Networks](https://mlsanity.com/paper/2509.04966/) (2025)
- [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)

We applied physics-informed neural networks to solve the constitutive relations for nonlinear, path-dependent material behavior. As a result, the trained network not only satisfies all thermodynamic constraints but also instantly provides information about the current material state (i.e., free energy, stress, and the evolution of internal variables) under any given loading scenario without requiring initial data. One advantage of this work is that it bypasses the repetitive Newton iterations needed to solve nonlinear equations in complex material models. Additionally, strategies are provided to reduce the required order of derivative for obtaining the tangent operator. The trained model can be directly used in any finite element package (or other numerical methods) as a user-defined material model. However, challenges remain in the proper definition of collocation points and in integrating several non-equality constraints that become active or non-active simultaneously. We tested this methodology on rate-independent processes such as the classical von Mises plasticity model with a nonlinear hardening law, as well as local damage models for interface cracking behavior with a nonlinear softening law. In order to demonstrate the applicability of the methodology in handling complex path dependency in a three-dimensional (3D) scenario, we tested the approach using the equations governing a damage model for a three-dimensional interface model. Such models are frequently employed for intergranular fracture at grain boundaries. We have observed a perfect agreement between the results obtained through the proposed methodology and those obtained using the classical approach. Furthermore, the proposed approach requires significantly less effort in terms of implementation and computing time compared to the traditional methods.