Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Control

• URL: http://arxiv.org/abs/2105.14094v1
• Date: Fri, 28 May 2021 20:25:40 GMT
• Title: Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Control
• Authors: Mark Ainsworth and Justin Dong
• Abstract summary: We present a new approach to using neural networks to approximate the solutions of variational equations. We use a sequence of finite-dimensional subspaces whose basis functions are realizations of a sequence of neural networks.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: We present a new approach to using neural networks to approximate the solutions of variational equations, based on the adaptive construction of a sequence of finite-dimensional subspaces whose basis functions are realizations of a sequence of neural networks. The finite-dimensional subspaces are then used to define a standard Galerkin approximation of the variational equation. This approach enjoys a number of advantages, including: the sequential nature of the algorithm offers a systematic approach to enhancing the accuracy of a given approximation; the sequential enhancements provide a useful indicator for the error that can be used as a criterion for terminating the sequential updates; the basic approach is largely oblivious to the nature of the partial differential equation under consideration; and, some basic theoretical results are presented regarding the convergence (or otherwise) of the method which are used to formulate basic guidelines for applying the method.

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