Gaussian Variational Schemes on Bounded and Unbounded Domains
<|reference_start|>Gaussian Variational Schemes on Bounded and Unbounded Domains: 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.<|reference_end|>
@article{actor2024gaussian,
    title={Gaussian Variational Schemes on Bounded and Unbounded Domains},
    author={Jonas A. Actor, Anthony Gruber, Eric C. Cyr, Nathaniel Trask},
    journal={arXiv preprint arXiv:2410.06219},
    year={2024},
    archivePrefix={arXiv},
    eprint={2410.06219},
    primaryClass={math.NA cs.LG cs.NA}
}