Symmetry-Informed Governing Equation Discovery

TL;DR

This paper derives equivariance constraints from the time-independent symmetries of ODEs and develops a pipeline for incorporating symmetry constraints into various equation discovery algorithms, including sparse regression and genetic programming.

Abstract

Despite the advancements in learning governing differential equations from observations of dynamical systems, data-driven methods are often unaware of fundamental physical laws, such as frame invariance. As a result, these algorithms may search an unnecessarily large space and discover less accurate or overly complex equations. In this paper, we propose to leverage symmetry in automated equation discovery to compress the equation search space and improve the accuracy and simplicity of the learned equations. Specifically, we derive equivariance constraints from the time-independent symmetries of ODEs. Depending on the types of symmetries, we develop a pipeline for incorporating symmetry constraints into various equation discovery algorithms, including sparse regression and genetic programming. In experiments across diverse dynamical systems, our approach demonstrates better robustness against noise and recovers governing equations with significantly higher probability than baselines without symmetry. Our codebase is available at https://github.com/Rose-STL-Lab/symmetry-ode-discovery.

Figures (9)

• Figure 1: Pipeline for incorporating symmetries into equation discovery via solving linear symmetry constraint (Section \ref{['sec:esindy']}), regularization (Section \ref{['sec:symmreg']}) and symmetry discovery (Section \ref{['sec:symmdis']}). Given the trajectory data from the dynamical system, we first identify its symmetry based on prior knowledge or symmetry discovery techniques. We then enforce the symmetry by solving a set of constraints when possible and otherwise promote the symmetry through regularization.
• Figure 2: Solutions for equivariant constraints of (\ref{['eq:dosc,eq:growth']},\ref{['eq:growth']}). In \ref{['eq:dosc']}, the 2D parameter space is spanned by $Q_1$ and $Q_2$. In \ref{['eq:growth']}, the 3D parameter space is spanned by $Q_{1,2,3}$, each marked with a different color.
• Figure 3: Reaction-diffusion system and the prediction error in the high-dimensional input space using equations from different methods. The means and standard deviations (shaded area) of errors over 3 random runs are plotted.
• Figure 4: Long-term prediction error (MSE) against simulation time. The error curves are averaged over all discovered equations and random initial conditions in the test dataset. The shaded area indicates the standard deviation of prediction error at each timestep. Our EquivSINDy-c has the slowest error growth. Some algorithms, e.g. genetic programming and Weak SINDy, are not included in the plot when the simulation quickly diverges and the error grows to infinity.
• Figure 5: In the SEIR model, we search the equations of 4 variables with up to quadratic terms, leading to a 4$\times$15=60D parameter space. The symmetry $v=(S+E+I+R)\partial_R$ reduces it to 34D.

Theorems & Definitions (16)

• Definition 3.1
• Proposition 3.2
• Theorem 3.3
• Proposition 4.1
• proof
• Proposition 4.2
• Definition A.1: Def 2.23, olver1993applications
• Theorem A.2: The infinitesimal criterion. Thm 2.31, olver1993applications
• Proposition B.1
• proof