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Explicit Discovery of Nonlinear Symmetries from Dynamic Data

Lexiang Hu, Yikang Li, Zhouchen Lin

TL;DR

Explicit Discovery of Nonlinear Symmetries from Dynamic Data introduces LieNLSD, a pipeline that identifies nonlinear infinitesimal generators and their explicit expressions directly in the observation space from dynamic data. It proves that the prolonged infinitesimal group action is linear in the coefficient matrix $W$, enabling a linear system based on the infinitesimal criterion that is solved via SVD, with a sparsification step (LADMAP) for interpretability. The method is validated on linear and nonlinear dynamics, including top quark tagging and Burgers', wave, and Schrödinger equations, and is shown to enhance data augmentation for neural PDE solvers by over 20% in long rollout accuracy. By decoupling neural fitting from symmetry discovery and offering a flexible function library, LieNLSD provides a practical, scalable approach to data-driven symmetry discovery with explicit, interpretable nonlinear generators.

Abstract

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.

Explicit Discovery of Nonlinear Symmetries from Dynamic Data

TL;DR

Explicit Discovery of Nonlinear Symmetries from Dynamic Data introduces LieNLSD, a pipeline that identifies nonlinear infinitesimal generators and their explicit expressions directly in the observation space from dynamic data. It proves that the prolonged infinitesimal group action is linear in the coefficient matrix , enabling a linear system based on the infinitesimal criterion that is solved via SVD, with a sparsification step (LADMAP) for interpretability. The method is validated on linear and nonlinear dynamics, including top quark tagging and Burgers', wave, and Schrödinger equations, and is shown to enhance data augmentation for neural PDE solvers by over 20% in long rollout accuracy. By decoupling neural fitting from symmetry discovery and offering a flexible function library, LieNLSD provides a practical, scalable approach to data-driven symmetry discovery with explicit, interpretable nonlinear generators.

Abstract

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.

Paper Structure

This paper contains 52 sections, 4 theorems, 69 equations, 8 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Group actions and infinitesimal group actions.
  4. Symmetries of differential equations.
  5. Prolongation.
  6. Related Work
  7. Symmetry discovery.
  8. Applications of symmetry.
  9. Method
  10. Prolongation Formula of Infinitesimal Group Actions
  11. Explicit Discovery of Infinitesimal Group Actions
  12. The Overall Algorithm
  13. Basis Sparsification
  14. Experiment
  15. Quantitative Metric: Grassmann Distance
  16. ...and 37 more sections

Key Result

Theorem 4.1

Let $G$ be a Lie group acting on $X \times U = \mathbb{R}^p \times \mathbb{R}^q$, with its corresponding Lie algebra $\mathfrak{g}$. Assume that the infinitesimal group action of $\mathbf{v} \in \mathfrak{g}$ takes the following form: where $W \in \mathbb{R}^{(p + q) \times r}$ and $\Theta(x, u) \in \mathbb{R}^{r \times 1}$. Then, the $n$-th prolongation of $\mathbf{v}$ is: where $J = (j_1, \dot

Figures (8)

  • Figure 1: Pipeline of LieNLSD.
  • Figure 2: Visualization result of symmetry discovery on top quark tagging by LieNLSD. The first subplot shows the last $8$ singular values. The other seven subplots display the infinitesimal generators corresponding to the nearly-zero singular values after sparsification.
  • Figure 3: Visualization result of symmetry discovery on Burgers' equation by LieNLSD. The first subplot shows the last $7$ singular values. The other six subplots display the infinitesimal generators corresponding to the nearly-zero singular values after sparsification.
  • Figure 4: Visualization result of symmetry discovery on the wave equation by LieNLSD. The first subplot shows the last $21$ singular values. The other twenty subplots display the infinitesimal generators corresponding to the nearly-zero singular values after sparsification.
  • Figure 5: Visualization result of symmetry discovery on the Schrödinger equation by LieNLSD. The first subplot shows the last $7$ singular values. The other six subplots display the infinitesimal generators corresponding to the nearly-zero singular values after sparsification.
  • ...and 3 more figures

Theorems & Definitions (18)

  • Theorem 4.1
  • Theorem 4.2
  • Definition 1.1
  • Proposition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • proof
  • ...and 8 more