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

arXiv: https://arxiv.org/abs/2607.23501 · HTML: https://mlsanity.com/paper/2607.23501/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 29 references that are in the map.
- First posted on arXiv 26 Jul 2026.

## Builds on (most cited first, 8 of 8)

- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Comparing Physics-Informed and Neural ODE Approaches for Modeling Nonlinear Biological Systems: A Case Study Based on the Morris-Lecar Model](https://mlsanity.com/paper/2603.26921/) (2026)
- [A hybrid approach for solving the gravitational N-body problem with Artificial Neural Networks](https://mlsanity.com/paper/2310.20398/) (2023)
- [Advancing Solutions for the Three-Body Problem Through Physics-Informed Neural Networks](https://mlsanity.com/paper/2503.04585/) (2025)

## Abstract (arXiv)

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.