Discovering Physics Laws of Dynamical Systems via Invariant Function Learning

2025-02-06

AI Summary

This work addresses automatic discovery of intrinsic ODE dynamics under varying environmental conditions, mitigating interference from environment-specific effects—such as drastic changes in functional forms—that hinder identification of fundamental physical laws. We propose the Dynamical Invariant Function (DIF) method: the first framework to formulate dynamics discovery as invariant function learning. DIF integrates causal graph modeling to characterize environmental confounding mechanisms, employs an encoder-decoder hypernetwork to learn environment-agnostic invariant dynamical representations, and incorporates information-theoretic constraints to explicitly disentangle environmental variables from invariant functions. We provide theoretical guarantees that the learned functions are provably environment-invariant. Empirical evaluation on three canonical ODE systems—including damped and driven pendula—demonstrates significant improvements over meta-learning and invariant learning baselines. Notably, DIF accurately recovers the ideal pendulum’s $sin heta$ dynamics from complex, confounded environments, enabling interpretable and generalizable physical law discovery.

Technology Category

• Search and Optimization: Learning to Search
• Machine Learning: Causal Learning
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 $alpha^2 sin{ heta_t}$ by observing pendulum dynamics in different environments, such as the damped environment $alpha^2 sin( heta_t) - ho omega_t$ and powered environment $alpha^2 sin( heta_t) + ho frac{omega_t}{left|omega_t ight|}$. Here, we formulate this problem as an emph{invariant function learning} task and propose a new method, known as extbf{D}isentanglement of extbf{I}nvariant extbf{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.

Problem

• Discovering invariant laws in dynamical systems
• Disentangling invariant functions across environments
• Ensuring independence of invariant functions from environments

Innovation

• Causal graph for invariant functions
• Encoder-decoder hypernetwork design
• Information-based independence principle