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Gaussian Variational Schemes on Bounded and Unbounded Domains

Jonas A. Actor, Anthony Gruber, Eric C. Cyr, Nathaniel Trask

TL;DR

A machine-learnable variational scheme using Gaussian radial basis functions (GRBFs) is presented and used to approximate linear problems on bounded and unbounded domains to produce exact quadrature formulae which enable weak-form expressions.

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.

Gaussian Variational Schemes on Bounded and Unbounded Domains

TL;DR

A machine-learnable variational scheme using Gaussian radial basis functions (GRBFs) is presented and used to approximate linear problems on bounded and unbounded domains to produce exact quadrature formulae which enable weak-form expressions.

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.
Paper Structure (24 sections, 23 theorems, 121 equations, 3 figures, 4 tables, 1 algorithm)

This paper contains 24 sections, 23 theorems, 121 equations, 3 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The method at a glance
  3. Approximation with Radial Basis Functions
  4. The GRBF approach as a generalized Galerkin method
  5. Numerical Scheme
  6. An Illustration: Poisson's Equation
  7. Examples
  8. Poisson Problem on Infinite Domain
  9. Poisson Problem on Bounded Domain
  10. Trained Solutions to High-Dimensional PDEs
  11. Whitney Forms
  12. Discussion and Conclusions
  13. Appendix A: Facts and Omitted Proofs
  14. Smooth Extensions to Unbounded Domains
  15. Product of Gaussian PDFs
  16. ...and 9 more sections

Key Result

Theorem 3.1

Let $f \in \mathcal{N}_{\phi,\Omega}$ and let $P_f$ be its GRBF interpolant on $X$. For any $x \in \Omega$, denote $S_r(~x) = \Omega \cap B_{r}(~x)$. Then, for any $k \in \mathbb{N}$ there exist constants $C, r$ independent of $X$, $f$, and $\phi$, such that for any $x \in \Omega$, where $h$ is the fill distance and

Figures (3)

  • Figure 1: Log-log convergence plots for our GRBF method on Problem 1 (unbounded, turquoise) and Problem 2 (bounded, magenta), both with and without training. Note that the error rates of Section \ref{['sec:fem']} are observed, although poor conditioning causes a slight climb in the Problem 1 error for large basis sizes. \newlabelfig:convergence0
  • Figure 1: Solutions to Problem 1 for $N=8$ using our novel GRBF method (top), a standard PINN (middle), and an RBFNet PINN (bottom). From left to right: plot of true (blue) and predicted (orange) solutions; pointwise absolute error; scaled Gaussian basis found during training. \newlabelfig:prob1-train8-accuracy0
  • Figure 2: Solutions to Problem 1 for $N=16$ using our novel GRBF method (top), a standard PINN (middle), and an RBFNet PINN (bottom). From left to right: plot of true (blue) and predicted (orange) solutions; pointwise absolute error; scaled Gaussian basis found during training. \newlabelfig:prob1-train16-accuracy0

Theorems & Definitions (44)

  • Remark 2.1
  • Theorem 3.1
  • Corollary 3.2
  • Remark 3.3
  • Theorem 4.1
  • Lemma 4.2
  • Proof 1
  • Lemma 5.1
  • Definition 5.2
  • Theorem 5.3
  • ...and 34 more