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Automated Discovery of Conservation Laws via Hybrid Neural ODE-Transformers

Vivan Doshi

TL;DR

The paper addresses the challenge of discovering conservation laws from noisy observational data by proposing a hybrid, three-stage pipeline: (1) learn a continuous vector field with a Neural ODE, (2) generate symbolic candidate invariants with a Transformer conditioned on the learned dynamics, and (3) verify candidates through a symbolic-numeric certificate. The key contribution is a decoupled learn-then-search framework that yields robust invariant discovery despite data imperfections, outperforming end-to-end baselines on canonical physical systems. The approach combines adjoint-sensitivity-based dynamics learning, a PPO-tuned symbolic generator with a physics-informed reward, and a grid-based numerical certificate to ensure invariants hold for the learned model, not just the observed trajectories. This work demonstrates a principled path to extracting mathematical laws from imperfect data and suggests directions for scaling and formal verification in more complex settings.

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.

Automated Discovery of Conservation Laws via Hybrid Neural ODE-Transformers

TL;DR

The paper addresses the challenge of discovering conservation laws from noisy observational data by proposing a hybrid, three-stage pipeline: (1) learn a continuous vector field with a Neural ODE, (2) generate symbolic candidate invariants with a Transformer conditioned on the learned dynamics, and (3) verify candidates through a symbolic-numeric certificate. The key contribution is a decoupled learn-then-search framework that yields robust invariant discovery despite data imperfections, outperforming end-to-end baselines on canonical physical systems. The approach combines adjoint-sensitivity-based dynamics learning, a PPO-tuned symbolic generator with a physics-informed reward, and a grid-based numerical certificate to ensure invariants hold for the learned model, not just the observed trajectories. This work demonstrates a principled path to extracting mathematical laws from imperfect data and suggests directions for scaling and formal verification in more complex settings.

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.

Paper Structure

This paper contains 9 sections, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. The Proposed Framework
  4. Module 1: Learning System Dynamics with Neural ODEs
  5. Module 2: Symbolic Candidate Generation with a Transformer
  6. Module 3: Symbolic-Numeric Verification
  7. Experiments and Results
  8. Limitations and Future Work
  9. Conclusion

Figures (3)

  • Figure 1: Our Hybrid Architecture. The process flows from data to a learned continuous model, then to symbolic candidates, and finally to a rigorous numerical verification stage.
  • Figure 2: Learned pendulum phase portrait. Contours are level sets of the discovered energy; dashed lines are ground-truth trajectories, confirming the high fidelity of the learned dynamics model.
  • Figure 3: Sample efficiency on the harmonic oscillator (2% noise). Shaded regions are 95% CIs. Our method learns faster and more robustly from fewer trajectories.