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

TL;DR

This paper introduces a mesh-free deep learning method using a novel activation function to efficiently solve complex nonconvex energy minimization problems in martensitic phase transitions, capturing microstructures without mesh constraints.

Contribution

The paper presents a new deep learning framework with a specialized activation function for nonconvex energy minimization in phase transitions, enabling mesh-free approximation of microstructures.

Findings

Successfully approximates complex microstructures

Demonstrates effectiveness of the SmReLU activation

Achieves mesh-free computation of free boundaries

Abstract

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.