Discovering Conservation Laws using Optimal Transport and Manifold Learning

TL;DR

This paper introduces a non-parametric, geometry-based method using optimal transport and manifold learning to discover conservation laws in complex dynamical systems, avoiding reliance on detailed models or time data.

Contribution

It reformulates conservation law discovery as a manifold learning problem, providing a robust, interpretable approach that identifies conserved quantities without explicit system models.

Findings

Successfully identifies the number of conserved quantities

Accurately extracts conserved values across physical systems

Operates without detailed dynamical information or explicit models

Abstract

Conservation laws are key theoretical and practical tools for understanding, characterizing, and modeling nonlinear dynamical systems. However, for many complex systems, the corresponding conserved quantities are difficult to identify, making it hard to analyze their dynamics and build stable predictive models. Current approaches for discovering conservation laws often depend on detailed dynamical information or rely on black box parametric deep learning methods. We instead reformulate this task as a manifold learning problem and propose a non-parametric approach for discovering conserved quantities. We test this new approach on a variety of physical systems and demonstrate that our method is able to both identify the number of conserved quantities and extract their values. Using tools from optimal transport theory and manifold learning, our proposed method provides a direct geometric approach to identifying conservation laws that is both robust and interpretable without requiring an explicit model of the system nor accurate time information.