Gaussian Variational Schemes on Bounded and Unbounded Domains
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Abstract (arXiv)
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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References · 10
- 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
- 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 634
- 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 608
- 2023An Expert's Guide to Training Physics-informed Neural Networkscited by 126
- 2024Challenges in Training PINNs: A Loss Landscape Perspectivecited by 113
- 2023Tackling the Curse of Dimensionality with Physics-Informed Neural Networkscited by 79
- 2019Robust Training and Initialization of Deep Neural Networks: An Adaptive Basis Viewpointcited by 42
- 2022Critical Investigation of Failure Modes in Physics-informed Neural Networkscited by 15
- 2022Tensor Moments of Gaussian Mixture Models: Theory and Applicationscited by 7
- 2024RBF-PINN: Non-Fourier Positional Embedding in Physics-Informed Neural Networkscited by 5
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