Physics-Informed Neural Networks for Discovering Periodic Orbits in the Gravitational Three-Body Problem

Locating periodic solutions of chaotic dynamical systems normally requires an initial guess close enough to the target orbit for numerical continuation or gradient-based search to converge. We show that Physics-Informed Neural Networks (PINNs) trained on sparse, noisy observations without initial conditions recover periodic orbits of the gravitational three-body problem, including orbit families absent from the training data. The method rests on a second-order ODE formulation, fixed-frequency Fourier features, percentile-based adaptive refinement, and a trainable scaling parameter, each validated on forward problems. Across two 100-seed ensembles, 23--25% of runs converge to families not present in the training data. We then ask what determines which family emerges. Two chi2 tests give a consistent answer: changing the training data source significantly shifts the distribution of recovered families (p < 0.001, Cram'er's V = 0.339), whereas switching between the two initialization distributions tested does not (p = 0.620, V = 0.094). The random seed selects which family a given run recovers; the distribution the weights are drawn from does not shift the aggregate frequencies, but the training data does. The evidence is empirical: we do not characterize the loss landscape analytically, and PINNs remain slower than conventional integrators on well-posed initial-value problems. What the experiments establish is that the recovered orbits are verifiable rather than merely plausible: the identified ones refine to genuine periodic solutions, a network trained on Lagrange data recovers the figure-eight choreography (Li--Liao class I.A.1, matched to seven significant digits in T^*), and one trained on figure-eight data recovers a Broucke--Hadjidemetriou--H'enon orbit closing to δT < 10-9.