Discovering Conservation Laws using Optimal Transport and Manifold Learning
This work addresses the challenge of discovering conservation laws for researchers modeling complex dynamical systems, offering a robust and interpretable method without requiring detailed system models or time information, though it appears incremental as it builds on existing manifold learning and optimal transport techniques.
The paper tackles the problem of identifying conservation laws in complex nonlinear dynamical systems, which are difficult to analyze without such laws, by proposing a non-parametric manifold learning approach using optimal transport, and demonstrates its ability to identify the number and values of conserved quantities across various physical systems.
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.