An Iterative Deep Ritz Method for Monotone Elliptic Problems

TL;DR

This paper introduces an iterative deep Ritz method (IDRM) for solving a broad class of elliptic problems involving monotone operators, improving accuracy and convergence without strict regularity assumptions.

Contribution

The paper proposes a novel IDRM that encodes geometric information and convex penalties, extending neural PDE solvers to more general elliptic problems with proven convergence rates.

Findings

Enhanced accuracy over existing neural PDE solvers

Proven convergence rate using Banach space geometry

Successful application to challenging elliptic problems

Abstract

In this work, we present a novel iterative deep Ritz method (IDRM) for solving a general class of elliptic problems. It is inspired by the iterative procedure for minimizing the loss during the training of the neural network, but at each step encodes the geometry of the underlying function space and incorporates a convex penalty to enhance the performance of the algorithm. The algorithm is applicable to elliptic problems involving a monotone operator (not necessarily of variational form) and does not impose any stringent regularity assumption on the solution. It improves several existing neural PDE solvers, e.g., physics informed neural network and deep Ritz method, in terms of the accuracy for the concerned class of elliptic problems. Further, we establish a convergence rate for the method using tools from geometry of Banach spaces and theory of monotone operators, and also analyze the learning error. To illustrate the effectiveness of the method, we present several challenging examples, including a comparative study with existing techniques.