SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2474-A2501, January 2021.
We present a new approach to using neural networks to approximate 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 can be used to define a standard Galerkin approximation of the variational equation. This approach enjoys advantages including the following: 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 to some extent 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.

中文翻译：

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Galerkin 神经网络：使用误差控制逼近变分方程的框架

SIAM 科学计算杂志，第 43 卷，第 4 期，第 A2474-A2501 页，2021 年 1 月。
我们提出了一种使用神经网络逼近变分方程的新方法，基于有限维子空间序列的自适应构造，其基函数是神经网络序列的实现。有限维子空间可用于定义变分方程的标准伽辽金近似。这种方法具有以下优点：算法的顺序特性提供了一种系统方法来提高给定近似值的准确性；顺序增强为错误提供了一个有用的指标，可以用作终止顺序更新的标准；基本方法在某种程度上忽略了所考虑的偏微分方程的性质；