Applied Machine Learning with Kernel Features for Variational Partial Differential Equations

Abstract

This thesis develops mesh-free approximation methods for linear elliptic and free-boundary partial differential equations within variational formulations. It consists of two main contributions.

First, we construct radial basis function approximations for linear elliptic partial differential equations and free-boundary value problems. Classical problems, including the Poisson and reaction–diffusion equations, are reformulated as minimization tasks, with boundary conditions imposed through penalty methods. For obstacle problems, we consider a variational formulation with a non-smooth ℓ¹-regularized objective. To address the ill-conditioning of the discrete linear systems, a truncated singular value decomposition is employed as a stabilization mechanism. Numerical experiments demonstrate that the proposed method achieves high accuracy and efficiency, and that it outperforms competing approaches, including finite-difference methods, Galerkin methods, and neural network-based solvers.

The second contribution extends this variational framework to high-dimensional problems by means of random feature approximations. The resulting formulation yields scalable convex optimization problems that are well suited for high-dimensional settings. Numerical results indicate that the proposed approach surpasses other mesh-free methodologies, particularly physics-informed neural networks and the Deep Ritz method, while preserving high accuracy.

In general, these contributions provide effective variational mesh-free algorithms for linear elliptic, obstacle-type, and high-dimensional partial differential equations. The findings highlight the potential of radial basis function networks and random feature models as accurate, stable, and scalable methods for the numerical approximation of variational partial differential equations.