Sparse RBF Networks for PDEs and nonlocal equations: function space theory, operator calculus, and training algorithms

TL;DR

This paper advances the theoretical understanding and computational methods of sparse radial basis function networks for solving nonlinear PDEs, including function space characterization, operator evaluation, and training strategies.

Contribution

It provides a unified function space description, operator evaluation techniques, and a comprehensive analysis of training algorithms for SparseRBFnet in PDE solving.

Findings

Solution space characterized as a Besov space, kernel-independent.

Explicit kernel structure allows quasi-analytical operator evaluation.

Numerical experiments show insensitivity to kernel choice and trade-offs in accuracy and efficiency.

Abstract

This work presents a systematic analysis and extension of the sparse radial basis function network (SparseRBFnet) previously introduced for solving nonlinear partial differential equations (PDEs). Based on its adaptive-width shallow kernel network formulation, we further investigate its function-space characterization, operator evaluation, and computational algorithm. We provide a unified description of the solution space for a broad class of radial basis functions (RBFs). Under mild assumptions, this space admits a characterization as a Besov space, independent of the specific kernel choice. We further demonstrate how the explicit kernel-based structure enables quasi-analytical evaluation of both differential and nonlocal operators, including fractional Laplacians. On the computational end, we study the adaptive-width network and related three-phase training strategy through a comparison with variants concerning the modeling and algorithmic details. In particular, we assess the roles of second-order optimization, inner-weight training, network adaptivity, and anisotropic kernel parameterizations. Numerical experiments on high-order, fractional, and anisotropic PDE benchmarks illustrate the empirical insensitivity to kernel choice, as well as the resulting trade-offs between accuracy, sparsity, and computational cost. Collectively, these results consolidate and generalize the theoretical and computational framework of SparseRBFnet, supporting accurate sparse representations with efficient operator evaluation and offering theory-grounded guidance for algorithmic and modeling choices.