A Learning-Based Ansatz Satisfying Boundary Conditions in Variational Problems

TL;DR

This paper introduces a neural network-based ansatz that inherently satisfies boundary conditions in variational problems, improving accuracy and reducing complexity over existing methods like the Deep Ritz Method.

Contribution

The proposed ansatz ensures boundary conditions are met inherently, eliminating the need for penalty terms and enhancing the reliability of neural network solutions in variational problems.

Findings

Eliminates misleading results caused by penalty terms

Reduces complexity of the neural network approach

Maintains high accuracy in solving variational problems

Abstract

Recently, innovative adaptations of the Ritz Method incorporating deep learning have been developed, known as the Deep Ritz Method. This approach employs a neural network as the test function for variational problems. However, the neural network does not inherently satisfy the boundary conditions of the variational problem. To resolve this issue, the Deep Ritz Method introduces a penalty term into the functional of the variational problem, which can lead to misleading results during the optimization process. In this work, an ansatz is proposed that inherently satisfies the boundary conditions of the variational problem. The results demonstrate that the proposed ansatz not only eliminates misleading outcomes but also reduces complexity while maintaining accuracy, showcasing its practical effectiveness in addressing variational problems.