# A Deep Learning Approach to Nonconvex Energy Minimization for Martensitic Phase Transitions

arXiv: https://arxiv.org/abs/2206.13937 · HTML: https://mlsanity.com/paper/2206.13937/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jun 2022.

## Built on by (most cited first, 2 of 2)

- [A variational neural network approach for glacier modelling with nonlinear rheology](https://mlsanity.com/paper/2209.02088/) (2022)
- [Convergence analysis of a quasi-Monte Carlo-based deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/2210.16196/) (2022)

## Abstract (arXiv)

We propose a mesh-free method to solve nonconvex energy minimization problems for martensitic phase transitions and twinning in crystals, using the deep learning approach. These problems pose multiple challenges to both analysis and computation, as they involve multiwell gradient energies with large numbers of local minima, each involving a topologically complex microstructure of free boundaries with gradient jumps. We use the Deep Ritz method, whereby candidates for minimizers are represented by parameter-dependent deep neural networks, and the energy is minimized with respect to network parameters. The new essential ingredient is a novel activation function proposed here, which is a smoothened rectified linear unit we call SmReLU; this captures the structure of minimizers where usual activation functions fail. The method is mesh-free and thus can approximate free boundaries essential to this problem without any special treatment, and is extremely simple to implement. We show the results of many numerical computations demonstrating the success of our method.