Discovering Conservation Laws via Manifold Learning*

Conservation laws are key theoretical and practical tools for understanding, characterizing, and modeling nonlinear dynamical systems. However, for many complex dynamical systems, the corresponding conserved quantities are difficult to identify, making it hard to analyze their dynamics and build efficient, stable predictive models. Many current approaches for discovering conservation laws rely on fine-grained time measurements and dynamical information. We instead reformulate this task as a manifold learning problem and propose a non-parametric approach, combining the Wasserstein metric from optimal transport with diffusion maps, to determine all the conserved quantities that vary across trajectories sampled from a dynamical system. We test this new approach on a variety of physical systems and demonstrate that our manifold learning method is able to both identify the number of conserved quantities and extract their values.