An augmented Lagrangian deep learning method for variational problems with essential boundary conditions

TL;DR

This paper introduces a novel deep learning approach using augmented Lagrangian techniques to solve variational problems with essential boundary conditions, offering improved accuracy and robustness over traditional penalty methods.

Contribution

The paper develops a deep learning method based on augmented Lagrangian reformulation for variational problems with boundary conditions, enhancing stability and accuracy.

Findings

More accurate solutions at similar computational cost.

Flexible and robust penalty parameter selection.

Effective application to elliptic and eigenvalue problems.

Abstract

This paper is concerned with a novel deep learning method for variational problems with essential boundary conditions. To this end, we first reformulate the original problem into a minimax problem corresponding to a feasible augmented Lagrangian, which can be solved by the augmented Lagrangian method in an infinite dimensional setting. Based on this, by expressing the primal and dual variables with two individual deep neural network functions, we present an augmented Lagrangian deep learning method for which the parameters are trained by the stochastic optimization method together with a projection technique. Compared to the traditional penalty method, the new method admits two main advantages: i) the choice of the penalty parameter is flexible and robust, and ii) the numerical solution is more accurate in the same magnitude of computational cost. As typical applications, we apply the new approach to solve elliptic problems and (nonlinear) eigenvalue problems with essential boundary conditions, and numerical experiments are presented to show the effectiveness of the new method.