Neural networks are increasingly being used to solve partial differential
equations (PDEs), replacing slower numerical solvers. However, a critical issue
is that neural PDE solvers require high-quality ground truth data, which
usually must come from the very solvers they are designed to replace. Thus, we
are presented with a proverbial chicken-and-egg problem. In this paper, we
present a method, which can partially alleviate this problem, by improving
neural PDE solver sample complexity -- Lie point symmetry data augmentation
(LPSDA). In the context of PDEs, it turns out that we are able to
quantitatively derive an exhaustive list of data transformations, based on the
Lie point symmetry group of the PDEs in question, something not possible in
other application areas. We present this framework and demonstrate how it can
easily be deployed to improve neural PDE solver sample complexity by an order
of magnitude.

本文介绍了一种基于 Lie 对称数据增强的方法，能够解决神经偏微分方程求解器中需要高质量数据的问题，并能够轻松地将其部署到神经偏微分方程求解器中以提高样本复杂度约一个数量级。