Fig. 1: Proposed non-parametric method for discovering conservation laws illustrated using a simple pendulum example.

a First, we collect and normalize the trajectory data from the dynamical system. Two example trajectories are highlighted in red and blue. b Then, we use the Wasserstein metric from optimal transport to compute the distance between each pair of trajectories and construct a distance matrix. For the two example trajectories, the optimal transport plan is shown as lines connecting pairs of points. The constructed distance matrix is plotted with color representing the computed Wasserstein distance between each pair of trajectories. The computed distance between the two example trajectories is marked (black dots) on the distance matrix plot. c An embedding of the shape space manifold \({{{{{{{\mathcal{C}}}}}}}}\) is extracted from the distance matrix using diffusion maps. The embedding plot is colored by the conserved energy of the pendulum E. The points corresponding to the two example trajectories are marked in red and blue. d Finally, a heuristic score (Supplementary Note 1) is used to select relevant components. In this case, only component 1 is relevant, corresponding to a single conserved quantity—the energy E. Again, the embedding plot is colored by E, and the two example trajectories are marked in red and blue.