Gaussian Variational Schemes on Bounded and Unbounded Domains

TL;DR

This paper introduces a Gaussian radial basis function variational scheme for approximating linear problems on various domains, leveraging exact quadrature and a flexible Galerkin framework for improved surrogate modeling.

Contribution

It develops a novel, machine-learnable variational method using GRBFs that exploits integral relationships for exact quadrature, applicable to both bounded and unbounded domains.

Findings

Error rates are derived for the proposed scheme.

The method demonstrates utility as a surrogate modeling technique.

The scheme is conforming in infinite domains and adaptable to bounded domains.

Abstract

A machine-learnable variational scheme using Gaussian radial basis functions (GRBFs) is presented and used to approximate linear problems on bounded and unbounded domains. In contrast to standard mesh-free methods, which use GRBFs to discretize strong-form differential equations, this work exploits the relationship between integrals of GRBFs, their derivatives, and polynomial moments to produce exact quadrature formulae which enable weak-form expressions. Combined with trainable GRBF means and covariances, this leads to a flexible, generalized Galerkin variational framework which is applied in the infinite-domain setting where the scheme is conforming, as well as the bounded-domain setting where it is not. Error rates for the proposed GRBF scheme are derived in each case, and examples are presented demonstrating utility of this approach as a surrogate modeling technique.