用微分不变量缩小方程搜索空间，更准更快发现物理规律
Governing Equation Discovery from Data Based on Differential Invariants
- 基于对称性构造微分不变量，自动筛选方程候选项
- 在多个偏微分方程上，准确率和成功率均优于现有方法
- 适合需要从数据中挖掘物理模型的研究者
显式控制方程是描述物理定律最简洁直观的形式。然而，直接从数据中发现偏微分方程（PDE）面临巨大挑战，主要在于如何从庞大的候选项空间中确定相关项。对称性作为科学领域的关键先验知识，已被广泛用于设计等变网络和引导神经PDE求解器。本文提出一种基于微分不变量的控制方程发现流程，可无损压缩现有方法的搜索空间，同时严格遵守对称性约束。具体而言，我们计算对称群无穷小生成元对应的微分不变量集合，并将其作为方程发现的候选项。以基于微分不变量的SINDy（DI-SINDy）为例，我们在一系列偏微分方程上验证了其发现成功率和精度均超越其他对称性引导的方法。
原文摘要 · Abstract (English)
The explicit governing equation is one of the simplest and most intuitive forms for characterizing physical laws. However, directly discovering partial differential equations (PDEs) from data poses significant challenges, primarily in determining relevant terms from a vast search space. Symmetry, as a crucial prior knowledge in scientific fields, has been widely applied in tasks such as designing equivariant networks and guiding neural PDE solvers. In this paper, we propose a pipeline for governing equation discovery based on differential invariants, which can losslessly reduce the search space of existing equation discovery methods while strictly adhering to symmetry. Specifically, we compute the set of differential invariants corresponding to the infinitesimal generators of the symmetry group and select them as the relevant terms for equation discovery. Taking DI-SINDy (SINDy based on Differential Invariants) as an example, we demonstrate that its success rate and accuracy in PDE discovery surpass those of other symmetry-informed governing equation discovery methods across a series of PDEs.
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