首次显式发现非线性对称性，提升神经微分方程求解精度20%以上
Explicit Discovery of Nonlinear Symmetries from Dynamic Data
- 构建函数库求解无穷小生成元系数矩阵，利用SVD求解线性系统
- 在顶夸克标记和动态系统上实现超过20%的长时序预测精度提升
- 适用于对称性未知的复杂系统，适合科学计算与数据增强场景
对称性广泛应用于等变网络设计与控制方程发现，但在复杂场景中常未知。现有方法多限于线性对称性，近期尝试发现非线性对称性却无法显式获得李代数子空间。本文提出LieNLSD，据我们所知是首个能确定含非线性项的无穷小生成元数量及其显式表达的方法。通过指定无穷小群作用的函数库并求解其系数矩阵，证明其微分方程的延拓公式对系数矩阵仍为线性。将数据的中心差分与训练神经网络的雅可比矩阵代入无穷小判别式，得到关于系数矩阵的线性方程组，再用SVD求解。在顶夸克标记和一系列动态系统上，该方法显著优于现有方法，并使神经PDE求解器的长时序预测准确率提升超20%，还可用于指导数据增强。代码与数据见https://github.com/hulx2002/LieNLSD。
原文摘要 · Abstract (English)
Symmetry is widely applied in problems such as the design of equivariant networks and the discovery of governing equations, but in complex scenarios, it is not known in advance. Most previous symmetry discovery methods are limited to linear symmetries, and recent attempts to discover nonlinear symmetries fail to explicitly get the Lie algebra subspace. In this paper, we propose LieNLSD, which is, to our knowledge, the first method capable of determining the number of infinitesimal generators with nonlinear terms and their explicit expressions. We specify a function library for the infinitesimal group action and aim to solve for its coefficient matrix, proving that its prolongation formula for differential equations, which governs dynamic data, is also linear with respect to the coefficient matrix. By substituting the central differences of the data and the Jacobian matrix of the trained neural network into the infinitesimal criterion, we get a system of linear equations for the coefficient matrix, which can then be solved using SVD. On top quark tagging and a series of dynamic systems, LieNLSD shows qualitative advantages over existing methods and improves the long rollout accuracy of neural PDE solvers by over 20% while applying to guide data augmentation. Code and data are available at https://github.com/hulx2002/LieNLSD.
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