论文标题
神经PDE求解器的谎言点对称数据增强
Lie Point Symmetry Data Augmentation for Neural PDE Solvers
论文作者
论文摘要
神经网络越来越多地用于求解部分微分方程（PDE），以替换较慢的数值求解器。但是，一个关键的问题是，神经PDE求解器需要高质量的地面真相数据，这通常必须来自他们旨在替换的求解器。因此，向我们介绍了一个众所周知的鸡肉和鸡蛋问题。在本文中，我们提出了一种方法，可以通过改善神经PDE求解器样品复杂性（Lie Point对称数据增强（LPSDA））来部分缓解此问题。在PDE的上下文中，事实证明，我们能够根据所讨论的PDE的Lie Point对称组来定量得出详尽的数据转换列表，这在其他应用程序领域中是不可能的。我们提出了此框架，并证明了如何轻松部署它以通过数量级来改善神经PDE求解器样品的复杂性。
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.
