Figure 1: Dynamical inference of the law of gravity.

A particle is released with velocity v₀ perpendicular to the line connecting it to the sun, with varying initial distance r₀ from the sun. (a) With only N=150 examples (each consisting of just a single noisy observation of r at a random time t after the release; Supplementary Note 2), we infer a single dynamical model in the S-systems class that reproduces the data. With no supervision, adaptive dynamical inference produces bifurcations that lead to qualitatively different behaviour: in this case, a single model produces both oscillations (elliptical orbits) and monotonic growth (hyperbolic trajectories). Inferred trajectories are shown with solid coloured lines, and the corresponding true trajectories are shown with dashed lines. (b) Similar to the true model (left), the inferred model (right) contains a single hidden variable X₂ and works using a similar phase space structure. Specifically, the location of nullclines (green lines) and a single fixed point (green circle) as a function of r₀ are recovered well by the fit. Note that the hidden variable is defined up to a power (Supplementary Note 2), and we choose to plot here.