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A model-free method for discovering symmetry in differential equations

Max Kreider, John Harlim, Daning Huang

TL;DR

This work tackles the problem of discovering Lie point symmetries directly from scattered data when the governing differential equations are unknown. It introduces a model-free pipeline that prolongs data to jet space using Generalized Moving Least Squares (GMLS) and enforces the infinitesimal invariance condition as a linear system whose null space yields infinitesimal generators. A rigorous convergence analysis based on perturbation theory (Davis–Kahan) provides guarantees for the recovered symmetries, and the method is demonstrated on both ODEs and PDEs, including first-order linear ODE, Stuart–Landau, transport, and heat equations. The approach offers a data-driven pathway to uncover symmetry structure and conserved quantities in dynamical systems without requiring explicit DE forms, with potential utility in physics, engineering, and data-driven modeling scenarios.

Abstract

Symmetry in differential equations reveals invariances and offers a powerful means to reduce model complexity. Lie group analysis characterizes these symmetries through infinitesimal generators, which provide a local, linear criterion for invariance. However, identifying Lie symmetries directly from scattered data, without explicit knowledge of the governing equations, remains a significant challenge. This work introduces a numerical scheme that approximates infinitesimal generators from data sampled on an unknown smooth manifold, enabling the recovery of continuous symmetries without requiring the analytical form of the differential equations. We employ a manifold learning technique, Generalized Moving Least Squares, to prolongate the data, from which a linear system is constructed whose null space encodes the infinitesimal generators representing the symmetries. Convergence bounds for the proposed approach are derived. Several numerical experiments, including ordinary and partial differential equations, demonstrate the method's accuracy, robustness, and convergence, highlighting its potential for data-driven discovery of symmetries in dynamical systems.

A model-free method for discovering symmetry in differential equations

TL;DR

This work tackles the problem of discovering Lie point symmetries directly from scattered data when the governing differential equations are unknown. It introduces a model-free pipeline that prolongs data to jet space using Generalized Moving Least Squares (GMLS) and enforces the infinitesimal invariance condition as a linear system whose null space yields infinitesimal generators. A rigorous convergence analysis based on perturbation theory (Davis–Kahan) provides guarantees for the recovered symmetries, and the method is demonstrated on both ODEs and PDEs, including first-order linear ODE, Stuart–Landau, transport, and heat equations. The approach offers a data-driven pathway to uncover symmetry structure and conserved quantities in dynamical systems without requiring explicit DE forms, with potential utility in physics, engineering, and data-driven modeling scenarios.

Abstract

Symmetry in differential equations reveals invariances and offers a powerful means to reduce model complexity. Lie group analysis characterizes these symmetries through infinitesimal generators, which provide a local, linear criterion for invariance. However, identifying Lie symmetries directly from scattered data, without explicit knowledge of the governing equations, remains a significant challenge. This work introduces a numerical scheme that approximates infinitesimal generators from data sampled on an unknown smooth manifold, enabling the recovery of continuous symmetries without requiring the analytical form of the differential equations. We employ a manifold learning technique, Generalized Moving Least Squares, to prolongate the data, from which a linear system is constructed whose null space encodes the infinitesimal generators representing the symmetries. Convergence bounds for the proposed approach are derived. Several numerical experiments, including ordinary and partial differential equations, demonstrate the method's accuracy, robustness, and convergence, highlighting its potential for data-driven discovery of symmetries in dynamical systems.

Paper Structure

This paper contains 21 sections, 7 theorems, 83 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Lie group theory for symmetry analysis
  3. Lie group theory
  4. Symmetry in differential equations
  5. Discovering symmetries in differential equations
  6. Numerical approximation
  7. Tangent vector approximation via SVD
  8. Tangent vector refinement via GMLS
  9. Approximation of prolongated data points
  10. Error analysis
  11. Numerical Experiments
  12. First order linear ODE
  13. Stuart-Landau oscillator
  14. Transport equation
  15. Heat equation
  16. ...and 6 more sections

Key Result

Theorem 2.1

Consider a system of differential equations of the form eqn_de with maximal rank defined over $M\subset \mathcal{X}\times \mathcal{U}$. If $G$ is a local group of transformations acting on $M$, and for every infinitesimal generator $X_g$ of $G$, then $G$ is a symmetry group of the system.

Figures (3)

  • Figure 1: Comparison of analytical and data-driven procedures of symmetry analysis.
  • Figure 2: Singular value decomposition of $\tilde{\mathbf P}$ for the linear ODE, shown for a family of solutions (a-c) and with $C=1$ fixed (d-f). (a) Semilog plot of singular values $\sigma_i$, showing two nearly vanishing modes. (b) and (c) Bar plots of the right singular vectors corresponding to $\sigma_5$ and $\sigma_6$, which span the numerical nullspace of $\tilde{\mathbf P}$. (d) Semilog plot of $\sigma_i$ for fixed $C$, revealing a single nearly trivial mode. (e) Bar plot of the corresponding right singular vector. (f) Comparison of numerical (black) and theoretical (red, rescaled) error $\|\sin(\bm\Theta)\|_2$, averaged over 100 trials for each $N$.
  • Figure 3: Convergence results for: (a) the SL oscillator with free integration constants, (b) the SL oscillator with fixed integration constants, (c) the transport equation, and (d) the heat equation. In each case, the theoretical error bound (red) is rescaled to match the initial value of the numerical error (black). The observed convergence rates consistently exceed the theoretical predictions, indicating that the bound is conservative.

Theorems & Definitions (21)

  • Example 2.1
  • Theorem 2.1: Theorem 2.31 in olver1993applications
  • Example 2.2
  • Remark 1
  • Example 2.3
  • Remark 2
  • Remark 3
  • Theorem 4.1: Davis and Kahan sine theorem, davis1970rotation
  • Proposition 4.1
  • Lemma 4.1
  • ...and 11 more