A growing body of literature has been leveraging techniques of machine learning (ML) to build novel approaches to approximating the solutions to partial differential equations. Noticeably absent from the literature is a systematic exploration of the stability of the solutions generated by these ML approaches. Here, a recurrent network is introduced that matches precisely the evaluation of a multistep method paired with a collocation method for approximating spatial derivatives in the advection diffusion equation. This allows for two things: 1) the use of traditional tools for analyzing the stability of a numerical method for solving PDEs and 2) bringing to bear efficient techniques of ML for the training of approximations for the action of (spatial) linear operators. Observations on impacts of varying the large number of parameters in even this simple linear problem are presented. Further, it is demonstrated that stable solutions can be found even where traditional numerical methods may fail.

翻译：近年来，越来越多的研究利用机器学习技术构建偏微分方程求解的新颖近似方法。值得注意的是，现有文献中缺乏对这些机器学习方法所生成解稳定性的系统性探讨。本文提出一种循环网络，其计算过程与多步法完全匹配，并结合配置法来近似对流-扩散方程中的空间导数。这实现了两个目标：1）能够运用传统工具分析偏微分方程数值求解方法的稳定性；2）可借助高效的机器学习技术来训练（空间）线性算子作用的近似表示。研究展示了即使在此简单线性问题中，调整大量参数所产生的影响。此外，实验证明即使在传统数值方法可能失效的情况下，仍能找到稳定的解。