Approximating Young Measures With Deep Neural Networks

TL;DR

This paper introduces a deep neural network method to approximate Young measures, transforming the problem into learning push-forward maps of Gaussian measures, enabling applications in complex microstructure modeling and game theory.

Contribution

We propose a novel neural network framework to approximate Young measures by representing them as push-forwards of Gaussian measures, reformulating the variational problem.

Findings

Successfully approximated Young measures in numerical examples

Demonstrated applicability to microstructure modeling

Provided a new computational pathway for measure approximation

Abstract

Parametrized measures (or Young measures) enable to reformulate non-convex variational problems as convex problems at the cost of enlarging the search space from space of functions to space of measures. To benefit from such machinery, we need powerful tools for approximating measures. We develop a deep neural network approximation of Young measures in this paper. The key idea is to write the Young measure as push-forward of Gaussian measures, and reformulate the problem of finding Young measures to finding the corresponding push-forward. We approximate the push-forward map using deep neural networks by encoding the reformulated variational problem in the loss function. After developing the framework, we demonstrate the approach in several numerical examples. We hope this framework and our illustrative computational experiments provide a pathway for approximating Young measures in their wide range of applications from modeling complex microstructure in materials to non-cooperative games.