# Physically recurrent neural networks for path-dependent heterogeneous materials: embedding constitutive models in a data-driven surrogate

arXiv: https://arxiv.org/abs/2209.07320 · HTML: https://mlsanity.com/paper/2209.07320/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Sep 2022.

## Built on by (most cited first, 13 of 13)

- [A Microstructure-based Graph Neural Network for Accelerating Multiscale Simulations](https://mlsanity.com/paper/2402.13101/) (2024)
- [Learning solution of nonlinear constitutive material models using physics-informed neural networks: COMM-PINN](https://mlsanity.com/paper/2304.06044/) (2023)
- [History-Aware Neural Operator: Robust Data-Driven Constitutive Modeling of Path-Dependent Materials](https://mlsanity.com/paper/2506.10352/) (2025)
- [Physically recurrent neural network for rate and path-dependent heterogeneous materials in a finite strain framework](https://mlsanity.com/paper/2404.17583/) (2024)
- [Recurrent neural networks and transfer learning for elasto-plasticity in woven composites](https://mlsanity.com/paper/2311.13434/) (2023)
- [Towards scientific machine learning for granular material simulations -- challenges and opportunities](https://mlsanity.com/paper/2504.08766/) (2025)
- [Embedding an ANN-Based Crystal Plasticity Model into the Finite Element Framework using an ABAQUS User-Material Subroutine](https://mlsanity.com/paper/2410.08214/) (2024)
- [Systematic Performance Assessment of Deep Material Networks for Multiscale Material Modeling](https://mlsanity.com/paper/2602.07192/) (2026)
- [Non-linear mechanical field reconstruction coupling recurrent neural networks with physics-informed graph neural networks](https://mlsanity.com/paper/2606.10909/) (2026)
- [One-Step Flow Matching for Generative Modeling of Path-Dependent Physical Fields](https://mlsanity.com/paper/2606.22752/) (2026)
- [Microstructure-Conditioned Surrogate Models for Graded Multiscale Optimization of Mycelium Composites](https://mlsanity.com/paper/2607.13688/) (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)
- [On architectural choices for interpretability and thermodynamic consistency in Physically Recurrent Neural Networks in the low-data regime](https://mlsanity.com/paper/2610.04067/) (2026)

## Abstract (arXiv)

Driven by the need to accelerate numerical simulations, the use of machine learning techniques is rapidly growing in the field of computational solid mechanics. Their application is especially advantageous in concurrent multiscale finite element analysis (FE2) due to the exceedingly high computational costs often associated with it and the high number of similar micromechanical analyses involved. To tackle the issue, using surrogate models to approximate the microscopic behavior and accelerate the simulations is a promising and increasingly popular strategy. However, several challenges related to their data-driven nature compromise the reliability of surrogate models in material modeling. The alternative explored in this work is to reintroduce some of the physics-based knowledge of classical constitutive modeling into a neural network by employing the actual material models used in the full-order micromodel to introduce non-linearity. Thus, path-dependency arises naturally since every material model in the layer keeps track of its own internal variables. For the numerical examples, a composite Representative Volume Element with elastic fibers and elasto-plastic matrix material is used as the microscopic model. The network is tested in a series of challenging scenarios and its performance is compared to that of a state-of-the-art Recurrent Neural Network (RNN). A remarkable outcome of the novel framework is the ability to naturally predict unloading/reloading behavior without ever seeing it during training, a stark contrast with popular but data-hungry models such as RNNs. Finally, the proposed network is applied to FE2 examples to assess its robustness for application in nonlinear finite element analysis.