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Governing Equation Discovery from Data Based on Differential Invariants

Lexiang Hu, Yikang Li, Zhouchen Lin

TL;DR

This work tackles data-driven PDE discovery by enforcing symmetry through differential invariants derived from Lie group generators. By proving that a symmetry can be represented solely with its differential invariants, the authors provide a lossless, plug-and-play replacement for the equation skeleton in existing discovery methods (e.g., SINDy), yielding improved accuracy and stability. The proposed DI-SINDy workflow automatically obtains invariants from data (or symmetry discovery methods), constructs the invariant-based skeleton, and performs sparse regression to recover the explicit PDE; it outperforms baseline methods across KdV, KS, Burgers, and nKdV, achieving near-perfect success rates and reduced long-term errors. This approach enhances interpretability and reliability of data-driven PDE discovery and can be combined with symmetry discovery when symmetries are not known a priori, making it broadly applicable to physics-informed model discovery and beyond.

Abstract

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.

Governing Equation Discovery from Data Based on Differential Invariants

TL;DR

This work tackles data-driven PDE discovery by enforcing symmetry through differential invariants derived from Lie group generators. By proving that a symmetry can be represented solely with its differential invariants, the authors provide a lossless, plug-and-play replacement for the equation skeleton in existing discovery methods (e.g., SINDy), yielding improved accuracy and stability. The proposed DI-SINDy workflow automatically obtains invariants from data (or symmetry discovery methods), constructs the invariant-based skeleton, and performs sparse regression to recover the explicit PDE; it outperforms baseline methods across KdV, KS, Burgers, and nKdV, achieving near-perfect success rates and reduced long-term errors. This approach enhances interpretability and reliability of data-driven PDE discovery and can be combined with symmetry discovery when symmetries are not known a priori, making it broadly applicable to physics-informed model discovery and beyond.

Abstract

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.

Paper Structure

This paper contains 24 sections, 1 theorem, 33 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Symmetry discovery.
  4. Governing equation discovery.
  5. Applications of symmetry.
  6. Preliminary
  7. Partial differential equations.
  8. Lie point symmetries.
  9. Infinitesimal criteria.
  10. Method
  11. Calculation of differential invariants
  12. Governing equation discovery based on differential invariants
  13. Example algorithm: DI-SINDy
  14. Experiment
  15. Experimental setup
  16. ...and 9 more sections

Key Result

Proposition 4.1

Let $G$ be a Lie group acting on $X \times U$, and $\eta^1(x, u^{(n)}), \dots, \eta^k(x, u^{(n)})$ be a complete set of functionally independent $n$-th order differential invariants. An $n$-th order differential equation $F(x, u^{(n)}) = 0$ admits $G$ as a symmetry group if and only if there is an e involving only the differential invariants of $G$.

Figures (3)

  • Figure 1: Comparison between the existing equation discovery method and our differential invariant-based equation discovery method for the nKdV equation $e^{-\frac{t}{t_0}} u_t + u u_x + u_{xxx} = 0$. The former struggles with selecting relevant terms, whereas our relevant terms are directly determined by the symmetry group.
  • Figure 2: Pipeline of our differential invariant-based equation discovery method.
  • Figure 3: Long-term prediction errors of different equation discovery methods for the KdV, KS, Burgers, and nKdV equations. The MSE at each time step is averaged over $4$ initial conditions and $50$ runs, with the shaded area representing the standard deviation.

Theorems & Definitions (2)

  • Definition 4.1
  • Proposition 4.1