Gaussian Variational Schemes on Bounded and Unbounded 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.
Cite
@article{arxiv.2410.06219,
title = {Gaussian Variational Schemes on Bounded and Unbounded Domains},
author = {Jonas A. Actor and Anthony Gruber and Eric C. Cyr and Nathaniel Trask},
journal= {arXiv preprint arXiv:2410.06219},
year = {2024}
}