Fig. 2: Identifying the conserved energy for the simple harmonic oscillator (SHO).
From: Discovering conservation laws using optimal transport and manifold learning
a The SHO has two degrees of freedom: position q and momentum p. b Sample trajectories from the SHO dataset show sample points plotted in the 2D phase space (q, p). c The heuristic score (with cutoff 0.6) correctly identifies that the first embedding component extracted by the diffusion map is the only relevant component. d The extracted first component closely matches the analytically predicted first component (Eq. (5)) for the SHO (R2 = 0.9995). e Next, consider the SHO dataset with a partially observed phase space containing position only. f For each sample trajectory, the sample points are shown as a histogram. g The heuristic score is still able to identify the first component as relevant, and h this first component matches the analytic prediction (R2 = 0.9961).
