Fig. 3: Identifying the conserved energy for the simple pendulum.
From: Discovering conservation laws using optimal transport and manifold learning
a The simple pendulum has two degrees of freedom: angular position θ and angular velocity ω. b Sample trajectories show sample points plotted in the 2D phase space (θ, ω). 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, and d the extracted first component is monotonically related to the energy (rank correlation ρ = 0.9997). e, f With the addition of σ = 0.5 Gaussian noise to simulate measurement noise, g the heuristic score is still able to identify the first component as relevant, and (h) this first component corresponds well to the energy (ρ = 0.9978).
