Discovering Physics Laws of Dynamical Systems via Invariant Function Learning

TL;DR

This paper introduces DIF, a causal-inference-based method for discovering invariant physical laws of dynamical systems from diverse environments, even when the underlying dynamics change in complex ways.

Contribution

It proposes a novel invariant function learning framework with a hypernetwork encoder-decoder architecture to disentangle environment-invariant laws from environment-specific dynamics.

Findings

Successfully discovers physical laws across different environments.

Outperforms meta-learning and invariant learning baselines.

Demonstrates effectiveness on three ODE systems.

Abstract

We consider learning underlying laws of dynamical systems governed by ordinary differential equations (ODE). A key challenge is how to discover intrinsic dynamics across multiple environments while circumventing environment-specific mechanisms. Unlike prior work, we tackle more complex environments where changes extend beyond function coefficients to entirely different function forms. For example, we demonstrate the discovery of ideal pendulum's natural motion $\alpha^2 \sin{\theta_t}$ by observing pendulum dynamics in different environments, such as the damped environment $\alpha^2 \sin(\theta_t) - \rho \omega_t$ and powered environment $\alpha^2 \sin(\theta_t) + \rho \frac{\omega_t}{\left|\omega_t\right|}$. Here, we formulate this problem as an \emph{invariant function learning} task and propose a new method, known as \textbf{D}isentanglement of \textbf{I}nvariant \textbf{F}unctions (DIF), that is grounded in causal analysis. We propose a causal graph and design an encoder-decoder hypernetwork that explicitly disentangles invariant functions from environment-specific dynamics. The discovery of invariant functions is guaranteed by our information-based principle that enforces the independence between extracted invariant functions and environments. Quantitative comparisons with meta-learning and invariant learning baselines on three ODE systems demonstrate the effectiveness and efficiency of our method. Furthermore, symbolic regression explanation results highlight the ability of our framework to uncover intrinsic laws. Our code has been released as part of the AIRS library (\href{https://github.com/divelab/AIRS/tree/main/OpenODE/DIF}{https://github.com/divelab/AIRS/}).