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Rediscovering orbital mechanics with machine learning

Pablo Lemos, Niall Jeffrey, Miles Cranmer, Shirley Ho, Peter Battaglia

arXiv:2202.02306v1astro-ph.EPastro-ph.IMcs.LG

TL;DR

The paper asks whether physical laws and hidden system properties can be recovered from observed trajectories. It trains a graph-network simulator, applies symbolic regression to its learned force law, and recovers Newtonian gravitation and accurate hidden-property estimates from solar-system data.

Abstract

from arXiv · show

We present an approach for using machine learning to automatically discover the governing equations and hidden properties of real physical systems from observations. We train a "graph neural network" to simulate the dynamics of our solar system's Sun, planets, and large moons from 30 years of trajectory data. We then use symbolic regression to discover an analytical expression for the force law implicitly learned by the neural network, which our results showed is equivalent to Newton's law of gravitation. The key assumptions that were required were translational and rotational equivariance, and Newton's second and third laws of motion. Our approach correctly discovered the form of the symbolic force law. Furthermore, our approach did not require any assumptions about the masses of planets and moons or physical constants. They, too, were accurately inferred through our methods. Though, of course, the classical law of gravitation has been known since Isaac Newton, our result serves as a validation that our method can discover unknown laws and hidden properties from observed data. More broadly this work represents a key step toward realizing the potential of machine learning for accelerating scientific discovery.

Model

The model represents the solar system as an interaction graph, predicts accelerations using Newtonian force and mass structure, and fits an analytic expression to the learned interaction function.

Model

  • Each body is a graph node with a trainable scalar, and directed edges encode possible pairwise interactions through relative displacements.Edges are instantiated between every pair because the interacting bodies are not known in advance.Source: Graph network-based learned simulator · p. 3

Graph network-based learned simulator.

  • For each body pair, the simulator computes one interaction vector and sets the reverse directed edge to its negative, imposing Newton's third law.The predicted per-body accelerations are formed by explicitly imposing Newton's second and third laws.Source: Graph network-based learned simulator · p. 3
  • The simulator enforces equal-and-opposite pair interactions, then sums incoming interactions and divides by each body's scalar to predict acceleration.The per-body scalar is interpreted as mass under Newton's second law.Source: Graph network-based learned simulator · p. 3
  • The simulator is trained by gradient descent to minimize discrepancies between predicted and observed accelerations, while its constant node scalars are fitted as mass-like properties.The observed states and accelerations are represented by the empirical training distribution.Source: Graph network-based learned simulator · p. 3

Symbolic regression of force function.

  • Symbolic regression searches for an analytic expression matching the learned interaction function while also favoring lower-complexity expressions.It fits the learned force function using sampled inputs of relative velocities and displacement.Source: Symbolic regression of force function · p. 3

Experimental methods

The experiments use long-span solar-system trajectory data, train a graph-network simulator with scaling and robustness choices, then use symbolic regression to search for compact expressions matching its interactions.

Experimental methods

  • The dataset covers 31 bodies from January 1980 to January 2013, with the first 30 years used for training and the final three for validation.Included moons exceed 10^18 kg; observations are sampled every 30 minutes in Cartesian coordinates relative to the solar-system barycenter.Source: Data · p. 4

Graph network implementation details.

  • The graph network uses three-layer MLPs with 128 hidden nodes per layer and transforms displacement and force magnitudes to handle their wide dynamic ranges.Displacements are converted to spherical coordinates with log10 magnitudes, while output magnitudes are transformed back through an exponential function.Source: Graph network implementation details · pp. 4–5
  • Training uses a relative mean weighted error to emphasize bodies equally, random 3D rotations, and Gaussian input noise to support robustness during rollouts.Training stops after 20 epochs without validation-loss improvement to prevent overfitting.Source: Graph network implementation details · p. 5

Symbolic regression implementation details.

  • PySR searches candidate equations that balance accuracy against expression complexity, but the authors do not claim it uniquely identifies a perfect equation.The candidate dataset contains 500 points not used to train the simulator; equation complexity is capped at 40.Source: Symbolic regression implementation… · pp. 5–6

Results

A graph-network simulator learned real solar-system dynamics, and symbolic regression recovered Newtonian gravity. Refitting hidden masses with the discovered law improved both mass estimates and trajectory predictions.

Learned Simulator Performance.

  • Predicted next-step relative accelerations had about 0.2% validation error, but rollout trajectories began to deviate after several months because training predicted only 30 minutes ahead.The graph-network simulator learned dynamics from real data, unlike prior work on simulated data.Source: Learned Simulator Performance · p. 6

Hidden Property Inference.

  • Learned scalars roughly matched body masses with 9.1% mean percent error, but masses of bodies with little gravitational influence were less accurate; influence was averaged over time.The error–influence correlation was −0.64 in log space.Source: Hidden Property Inference · p. 6, Figure 4 · p. 7

Symbolic Discovery.

  • Symbolic regression recovered Newtonian gravity among over 100 million tested equations; its rollout error was lowest, and its simpler expression predicted more accurately over time than the learned simulator.The learned gravitational constant was also very similar to the true value.Source: Symbolic Discovery · p. 7

Relearning the masses.

  • After replacing the learned edge function with the discovered law and refitting, mean mass percent error was approximately 1.6%; the influence-error correlation strengthened from −0.64 to −0.87.Fig. 4 shows the mass estimates and correlation before and after refitting.Source: Relearning the masses · pp. 7–8, Figure 4 · p. 7
  • Relearned masses with the discovered equation produced trajectories more accurate than the learned simulator and original-mass symbolic equation, matching Newtonian gravity with correct parameters.The rollout comparison is shown in Fig. 5.Source: Figure 5 · p. 8, Relearning the masses · p. 8

Discussion & Conclusions

The two-step approach recovered Newton’s gravitational force law and accurately estimated hidden system properties from real observations. The result depended on physical inductive biases and remains a tool to support, not replace, scientific expertise.

Discussion & Conclusions

  • The approach re-discovered Newton’s gravitational-force formula from real solar-system trajectories and made accurate estimates of hidden system properties.The authors describe the result as confirmation that their method can recover known laws and hidden properties from observations.Source: Discussion & Conclusions · p. 8
  • The discovery depended on inductive biases, particularly Newton’s second and third laws and spherical symmetry, as well as choices such as spherical coordinates and logarithmic units.The authors state that prior knowledge can facilitate machine-learning discovery of physical laws.Source: Discussion & Conclusions · p. 8
  • Mass-estimate variation across random seeds identifies local minima rather than providing a true error estimate; the current gradient-descent graph network produces point estimates, not Bayesian posteriors.The authors identify Bayesian neural networks as a possible future avenue.Source: Discussion & Conclusions · p. 8
  • The equation-quality scoring function, including its complexity-versus-accuracy trade-off, warrants further exploration and should express scientists’ preferences for a good equation.Candidate equations are intended as a narrower palette for further experimentation.Source: Discussion & Conclusions · p. 9
  • The work combines modern machine learning with automatic theory formation and demonstrates efficacy on complex real-world data, although the recovered gravitational law was already known.The stated purpose is to validate that the method can recover known laws and hidden properties.Source: Discussion & Conclusions · p. 9
  • The authors frame the approach as a step toward tools for automating data-driven scientific theory formation and evaluation, not as a replacement for scientists’ domain knowledge and intuition.They describe it as helping make parts of discovery more efficient and systematic.Source: Discussion & Conclusions · p. 9

Appendix

The appendix reports rollout errors for all bodies, calculated against real trajectories on data not used for training. Figure 6 shows the loss per step for bodies depicted in Figure 2.

Appendix

  • Figure 6 plots rollout errors for all bodies using data not used during training; the errors compare predicted trajectories with real data.The errors are defined as (x_truth − x_pred)^2 / x_truth^2.Source: Appendix · p. 11
  • Figure 6 shows loss per step, calculated using Eq. (3), when integrating trajectories for the bodies shown in Figure 2 with the learned simulator.Source: Figure 6 · p. 12
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