Deep Ritz method with Fourier feature mapping: A machine learning approach for solving variational models of microstructure

Principal Investigator:
Ensela Mema

Co-PIs:
Jaroslaw Knap, Ting Wang

Abstract:
Microstructures are structural features between the atomic and macroscopic scale that play a key role in the performance of engineering materials. They are spontaneously formed in nature to optimize performance (maximize strength at a given weight, minimize permeability, maximize energy), properties that can be useful in industrial and defense applications. We present a novel machine learning approach, Deep Ritz method with Fourier feature mapping, to approximate solutions to non-convex variational problems that arise in microstructure modeling applications. We focus our efforts in determining whether the algorithm is capable of approximating multi-scale, high frequency solutions, known to pose challenges to traditional numerical methods.

Description of Research:
This project uses a deep learning algorithm, called the Deep Ritz Method (DRM), to approximate solutions to non-convex variational problems that arise in microstructure modeling applications. We consider two benchmark problems: the first is a 1D problem made up of a double well potential, which generates an energy minimization problem that does not have a minimizer. Minimizing sequences oscillate and converge weakly, but not strongly, to zero. The second problem is a 2D scalar problem for twin branching: no minimizers exist here either, leading to minimizing sequences that develop rapid oscillations. These mathematical problems represent two examples of how minimization can lead to fine scale oscillations or microstructure formation.