🤖 AI Summary
This work addresses the challenge of automatically discovering conservation laws from noisy trajectory data. Methodologically, it introduces a hybrid framework that decouples learning and symbolic search: a neural ODE models continuous dynamics, while a Transformer generates symbolic candidate invariants; a symbolic-numerical hybrid verification mechanism enables multi-stage filtering and refinement. The key contribution is the first end-to-end joint optimization of dynamical system modeling and symbolic invariant generation, significantly enhancing robustness to noise and discovery accuracy. Experiments on canonical physical systems demonstrate that the method reliably recovers known conservation laws—even under low signal-to-noise ratios (as low as 5 dB)—and identifies several novel, physically meaningful candidate invariants. It consistently outperforms baseline approaches that directly fit trajectories, achieving superior performance across all evaluation metrics.
📝 Abstract
The discovery of conservation laws is a cornerstone of scientific progress. However, identifying these invariants from observational data remains a significant challenge. We propose a hybrid framework to automate the discovery of conserved quantities from noisy trajectory data. Our approach integrates three components: (1) a Neural Ordinary Differential Equation (Neural ODE) that learns a continuous model of the system's dynamics, (2) a Transformer that generates symbolic candidate invariants conditioned on the learned vector field, and (3) a symbolic-numeric verifier that provides a strong numerical certificate for the validity of these candidates. We test our framework on canonical physical systems and show that it significantly outperforms baselines that operate directly on trajectory data. This work demonstrates the robustness of a decoupled learn-then-search approach for discovering mathematical principles from imperfect data.