Autoencoder discovers periodic orbits in the restricted three-body problem

In a development that could reshape how mission designers chart the pathways of future spacecraft, a team of researchers from the University of Strathclyde and the Polytechnic University of Madrid has shown that a generative artificial intelligence model can discover entirely new families of periodic orbits in the three-body problem of celestial mechanics. The study, published in the journal Celestial Mechanics and Dynamical Astronomy, demonstrates that a variational autoencoder trained on thousands of known orbits can learn the deep structure of orbital motion well enough to invent trajectories no human has ever catalogued, then refine them into physically valid solutions that spacecraft could actually follow. The work, carried out within the European Space Agency-supported OrbitGPT project, represents one of the most concrete demonstrations yet that generative AI can move beyond producing plausible-looking images and text to producing mathematically rigorous solutions of classical physics problems.

The stage for the research is the circular restricted three-body problem, or CR3BP, a foundational model of astrodynamics that describes the motion of a massless object, such as a spacecraft, under the gravitational influence of two massive bodies moving in circular orbits about their common centre of mass. In the Earth-Moon system, this simplified setting captures the essential dynamics that govern the behaviour of satellites near the famous Lagrange points, the five equilibrium locations where gravitational and centrifugal effects balance. Since Henry Poincaré proved in 1892 that the three-body problem admits infinitely many periodic solutions arranged in continuous families, generations of celestial mechanicians have mapped these orbits by hand and by computer, using continuation methods that step gradually from one known orbit to the next. The new work injects a fundamentally different tool into this centuries-old enterprise: a neural network that compresses the essence of an orbit into a handful of numbers and then recombines those numbers to generate orbits that never existed in its training data.

The technical heart of the approach is a variational autoencoder, a deep generative architecture first introduced by Diederik Kingma and Max Welling in 2013. Like a conventional autoencoder, a VAE consists of an encoder that compresses input data into a low-dimensional latent representation and a decoder that reconstructs the input from that representation. What distinguishes the variational version is its probabilistic character: the encoder does not produce a single point in the latent space but rather the parameters of a Gaussian distribution, from which samples are drawn during decoding. This seemingly small modification enforces a smooth, regular latent landscape governed by a known prior distribution, and it is precisely this regularity that makes generation possible. During training, the network minimises a loss function composed of a reconstruction term, which measures how faithfully the decoded trajectories match the originals, and a Kullback-Leibler divergence term, weighted by a parameter beta, which pulls the learned distribution toward the standard normal prior. Because sampling from a distribution is not directly differentiable, the team employed the well-known reparametrisation trick, expressing each latent sample as a deterministic mean shifted by a noise vector scaled by the learned standard deviation.

The researchers built their model from one-dimensional convolutional neural networks, a choice motivated by the need for a simple and robust architecture capable of processing orbital trajectories across many different families. Each orbit in the training data was represented as a time series of 300 equally spaced nodes containing the full six-dimensional state vector, position and velocity, along with the corresponding time stamp. The convolutional filters sweep along the temporal dimension with progressively increasing receptive fields, achieved through an exponentially increasing dilation schedule, allowing the network to capture both fine local structure and broad global trends in the trajectories. The team conducted an extensive trade-off analysis of hyperparameters, including kernel sizes, network depth, dropout regularisation and the beta weighting of the KL term, ultimately settling on a configuration whose reconstruction loss on the test set was an order of magnitude better than that of a more sophisticated InceptionTime-based alternative, which suffered from overfitting and training instability on this comparatively uniform dataset.

The training data itself was drawn from an extensive NASA catalogue of periodic orbits in the Earth-Moon CR3BP, comprising 44,112 initial states grouped into 40 orbital families, including the well-known Lyapunov, Halo, Vertical and Axial orbits around the libration points as well as exotic resonant families such as Butterfly and Dragonfly orbits. The authors selected 33 families with well-conditioned dynamics for their working dataset, sampling 250 orbits per family and splitting them into training, validation and test subsets. Each orbit was integrated over one full period using an explicit Vern7 solver in Julia with tight tolerances of 10 to the minus 13. To automatically verify which orbital families the model’s outputs belonged to, the researchers also built a random forest classifier trained on physically meaningful features: the orbital period, the Jacobi constant, which is the conserved energy-like integral of the CR3BP, and the coefficients of a Fast Fourier Transform of the state time series. This classifier achieved perfect classification precision on nearly all families in the held-out test set, providing a reliable yardstick for assessing the novelty of generated orbits.

Perhaps the most striking discovery came when the researchers examined the structure of the learned latent space. Although the model was trained entirely without labels, orbits belonging to the same family spontaneously clustered together, forming what the authors call family manifolds: smooth, low-dimensional structures within the latent space corresponding to distinct families of periodic motion. Moreover, certain latent dimensions turned out to encode physical quantities in an interpretable way. The second latent dimension correlated strongly with the orbital period, yielding a coefficient of determination of 0.789, meaning that sliding along that axis of the latent space produced orbits whose periods changed smoothly and predictably. When the team increased the dimensionality of the latent space to 12, however, most of the extra dimensions proved unstructured, suggesting that the effective information content of the orbit catalogue is captured by only a handful of meaningful coordinates.

Generation proceeds through two complementary strategies. In grid sampling, the latent space is covered with a uniform mesh of points, each decoded into a candidate trajectory. In latent-space continuation, the team identifies the direction along which a family manifold develops, using principal component analysis of the family’s encodings, and then samples along that direction or perpendicular to it. Because the decoded trajectories from the VAE only approximately satisfy the equations of motion, they are fed into a multiple-shooting refinement algorithm, a damped Newton-Raphson scheme that enforces both the dynamical constraints and the periodicity condition until the residuals fall below a tolerance of one part in a million. The damped version of the algorithm proved essential for handling dynamically unstable configurations such as Halo orbits, where tiny perturbations can cause trajectories to diverge wildly. Out of 22,500 trajectories generated in a dense grid search of the latent space, 10,420 converged to genuine periodic orbits, revealing well-defined basins of attraction surrounding each family manifold.

The novel orbits themselves emerged in a particularly elegant way: at the boundaries of the convergence regions and in the transition zones between neighbouring family manifolds. A quantitative novelty metric, combining Mahalanobis distances to the global orbit distribution and to individual families with the uncertainty of the random forest classification, identified the hundred most unusual trajectories, and these clustered precisely at the edges of the latent landscape. In one documented case, a single generated guess converged to a completely new three-dimensional orbital family when refined on a coarse 11-node discretisation, while the same guess refined on 100 nodes produced a conventional resonant orbit. Continuation analysis by Jacobi constant established that the new family was dynamically connected to the known 1:3 resonant family, demonstrating that even the apparent surprises of the generative model possess coherent physical provenance rather than arising from random noise. Sampling in the immediate neighbourhood of a novel orbit systematically produced trajectories belonging to the same new family, confirming that the latent space preserves local structure even where it departs from the training data.

The implications extend well beyond the Earth-Moon system. Periodic orbits in multibody environments underpin mission designs ranging from the James Webb Space Telescope’s station at Sun-Earth L2 to proposed cislunar surveillance constellations and lunar gateway infrastructure. Traditional continuation methods demand careful step-size control and can only inch outward from known solutions, whereas the latent-space approach can generate candidate orbits at arbitrary distances from the starting family in a single decode, then rely on the classical refinement machinery to guarantee physical validity. The authors also demonstrated a rediscovery experiment, in which the model, trained with entire families removed from its dataset, successfully regenerated those families by exploring its latent space, indicating that the generative pipeline captures and recombines underlying dynamical features rather than merely memorising training samples. A companion analysis noted that the KL regularisation, while essential for stable training, prevents a strict one-to-one mapping between latent coordinates and physical orbit elements, a limitation the team plans to address with hierarchical and disentangled architectures, physics-informed constraints embedded directly in the loss function, and conditional generative models capable of producing trajectories on demand with prescribed periods, energies or stability properties. The model and code have been released openly, inviting the astrodynamics community to explore a new continent of orbital possibilities that, until now, no one knew how to ask the mathematics to reveal.