Kernel Methods in the Deep Ritz framework: Theory and practice

TL;DR

This paper integrates kernel approximations into the Deep Ritz method to efficiently solve elliptic PDEs, providing theoretical error estimates and practical implementation insights validated by numerical experiments.

Contribution

It introduces a novel combination of kernel methods with the Deep Ritz framework, including rigorous error analysis and practical guidance for implementation.

Findings

The method achieves accurate approximations consistent with theoretical error bounds.

Numerical experiments confirm the effectiveness and practical viability of the approach.

The approach simplifies boundary condition enforcement in PDE solutions.

Abstract

In this contribution, kernel approximations are applied as ansatz functions within the Deep Ritz method. This allows to approximate weak solutions of elliptic partial differential equations with weak enforcement of boundary conditions using Nitsche's method. A priori error estimates are proven in different norms leveraging both standard results for weak solutions of elliptic equations and well-established convergence results for kernel methods. This availability of a priori error estimates renders the method useful for practical purposes. The procedure is described in detail, meanwhile providing practical hints and implementation details. By means of numerical examples, the performance of the proposed approach is evaluated numerically and the results agree with the theoretical findings.