Discovering Physics Laws of Dynamical Systems via Invariant Function Learning

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 *invariant function learning* task and propose a new method, known as **D**isentanglement of **I**nvariant **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.

Many natural systems, like swinging pendulums or spreading diseases, follow rules that stay the same even when the environment changes. Our goal is to uncover these hidden rules, even when different situations make the system behave differently. For example, a pendulum swings differently in air, water, or when powered by a motor, but the basic motion rule stays the same. We created a new method called DIF that helps find these shared rules by separating what’s common from what’s specific to each environment. Our method learns from different examples and uses ideas from cause-and-effect reasoning to make sure it finds only the shared physical laws. We tested it on several systems and showed that it works better than existing methods, even revealing clear formulas that describe how the systems behave.