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arXiv · 2502.04495

Discovering Physics Laws of Dynamical Systems via Invariant Function Learning

Shurui Gui·Xiner Li·Shuiwang Ji
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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 $α^2 \sin{θ_t}$ by observing pendulum dynamics in different environments, such as the damped environment $α^2 \sin(θ_t) - ρω_t$ and powered environment $α^2 \sin(θ_t) + ρ\frac{ω_t}{\left|ω_t\right|}$. 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/}).

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Shurui Gui, Xiner Li, Shuiwang Ji. 2025-06-11. Discovering Physics Laws of Dynamical Systems via Invariant Function Learning. https://arxiv.org/abs/2502.04495

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