Discovering conservation laws using optimal transport and manifold learning

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. Conservation laws are crucial for analyzing and modeling nonlinear dynamical systems; however, identification of conserved quantities is often quite challenging. The authors propose here a geometric approach to discovering conservation laws directly from trajectory data that does not require an explicit dynamical model of the system or detailed time information.

Automated summary

Conservation laws are essential for understanding and modeling nonlinear dynamical systems. However, identifying conserved quantities is challenging. This paper proposes a non-parametric approach based on manifold learning to discover conservation laws directly from trajectory data, without the need for an explicit model or accurate time information.