Adaptive importance sampling for Deep Ritz

TL;DR

This paper presents an adaptive importance sampling technique for the Deep Ritz method, utilizing deep generative models to improve PDE solutions, especially in high-dimensional and low-regularity cases.

Contribution

It introduces a novel adaptive sampling approach using deep generative models to enhance the Deep Ritz method for solving PDEs.

Findings

Improved accuracy in high-dimensional PDEs.

Effective sampling with bounded KRnet.

Enhanced solution quality for low-regularity problems.

Abstract

We introduce an adaptive sampling method for the Deep Ritz method aimed at solving partial differential equations (PDEs). Two deep neural networks are used. One network is employed to approximate the solution of PDEs, while the other one is a deep generative model used to generate new collocation points to refine the training set. The adaptive sampling procedure consists of two main steps. The first step is solving the PDEs using the Deep Ritz method by minimizing an associated variational loss discretized by the collocation points in the training set. The second step involves generating a new training set, which is then used in subsequent computations to further improve the accuracy of the current approximate solution. We treat the integrand in the variational loss as an unnormalized probability density function (PDF) and approximate it using a deep generative model called bounded KRnet. The new samples and their associated PDF values are obtained from the bounded KRnet. With these new samples and their associated PDF values, the variational loss can be approximated more accurately by importance sampling. Compared to the original Deep Ritz method, the proposed adaptive method improves accuracy, especially for problems characterized by low regularity and high dimensionality. We demonstrate the effectiveness of our new method through a series of numerical experiments.