Distilling neural networks into equations to predict planetary instability
Abstract
The problem of predicting when a multi-planet system will become dynamically unstable has defied analytical progress over the past three hundred years. Recent machine learning advances achieve high predictive performance of instability dynamics, but as black-box systems cannot be interpreted to connect to existing scientific understanding. Here we show analytic equations that predict when a multi-planet system will undergo instability. We discover these equations by first training a neural network on a dataset of N-body simulations, and then distilling its latent knowledge into closed-form algebraic expressions. Our two-stage pipeline first learns an interpretable variable basis, then searches for equations using symbolic regression. The resulting equations outperform existing analytic estimates of instability and apply across wide ranges of mass ratios, spacing, and orbital eccentricities. They capture structure in the prediction landscape with a small number of variables amenable to dynamical interpretation. These results demonstrate how symbolic learning can be combined with neural networks to extract scientific insight from complex dynamical systems.
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BibTeX
@misc{alford2026distilling,
title = {{Distilling neural networks into equations to predict planetary instability}},
author = {Simon Alford and Edward Gu and Omar Costilla-Reyes and Dan Tamayo and Miles Cranmer and Kevin Ellis},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35617},
eprint = {2610.35617},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35617},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Distilling neural networks into equations to predict planetary instability
AU - Simon Alford
AU - Edward Gu
AU - Omar Costilla-Reyes
AU - Dan Tamayo
AU - Miles Cranmer
AU - Kevin Ellis
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35617
UR - https://arxiv.org/abs/2610.35617
AB - The problem of predicting when a multi-planet system will become dynamically unstable has defied analytical progress over the past three hundred years. Recent machine learning advances achieve high predictive performance of instability dynamics, but as black-box systems cannot be interpreted to connect to existing scientific understanding. Here we show analytic equations that predict when a multi-planet system will undergo instability. We discover these equations by first training a neural network on a dataset of N-body simulations, and then distilling its latent knowledge into closed-form algebraic expressions. Our two-stage pipeline first learns an interpretable variable basis, then searches for equations using symbolic regression. The resulting equations outperform existing analytic estimates of instability and apply across wide ranges of mass ratios, spacing, and orbital eccentricities. They capture structure in the prediction landscape with a small number of variables amenable to dynamical interpretation. These results demonstrate how symbolic learning can be combined with neural networks to extract scientific insight from complex dynamical systems.
ER -