# Exploring energy minimization to model strain localization as a strong discontinuity using Physics Informed Neural Networks

arXiv: https://arxiv.org/abs/2409.13241 · HTML: https://mlsanity.com/paper/2409.13241/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 35 references that are in the map.
- First posted on arXiv 20 Sep 2024.

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

- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [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)
- [Transfer learning enhanced physics informed neural network for phase-field modeling of fracture](https://mlsanity.com/paper/1907.02531/) (2019)
- [A Neural Network-enhanced Reproducing Kernel Particle Method for Modeling Strain Localization](https://mlsanity.com/paper/2204.13821/) (2022)
- [An introduction to programming Physics-Informed Neural Network-based computational solid mechanics](https://mlsanity.com/paper/2210.09060/) (2022)
- [Phase-field modeling of fracture for quasi-brittle materials](https://mlsanity.com/paper/1902.10759/) (2019)

## Built on by (most cited first, 1 of 1)

- [Physics-based machine learning for fatigue lifetime prediction under non-uniform loading scenarios](https://mlsanity.com/paper/2503.05419/) (2025)

## Abstract (arXiv)

We explore the possibilities of using energy minimization for the numerical modeling of strain localization in solids as a sharp discontinuity in the displacement field. For this purpose, we consider (regularized) strong discontinuity kinematics in elastoplastic solids. The corresponding mathematical model is discretized using Artificial Neural Networks (ANNs), aiming to predict both the magnitude and location of the displacement jump from energy minimization, textiti.e., within a variational setting. The architecture takes care of the kinematics, while the loss function takes care of the variational statement of the boundary value problem. The main idea behind this approach is to solve both the equilibrium problem and the location of the localization band by means of trainable parameters in the ANN. As a proof of concept, we show through both 1D and 2D numerical examples that the computational modeling of strain localization for elastoplastic solids using energy minimization is feasible.