用可学习的高斯基函数构建变分框架，解决有界与无界域上的线性问题。
Gaussian Variational Schemes on Bounded and Unbounded Domains
- 基于高斯基函数积分与多项式矩的关系，构造精确积分公式。
- 在无界域上误差率可分析，有界域上具灵活逼近能力。
- 适合做代理模型，尤其适用于需可学习参数的物理建模场景。
本文提出一种基于高斯径向基函数（GRBFs）的可机器学习变分方法，用于近似求解有界域与无界域上的线性问题。与传统无网格方法不同，该方法利用GRBFs及其导数的积分与多项式矩之间的关系，推导出精确的数值积分公式，从而实现弱形式表达。结合可训练的GRBF均值与协方差，构建了一个灵活的广义伽辽金变分框架：在无限域情形下方案保持协调性，在有界域情形下则不协调。针对两种情形分别推导了误差率，并通过实例展示了该方法作为代理建模技术的实用性。
原文摘要 · Abstract (English)
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.
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