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Main discovery The authors present closed-form analytic equations that predict when a multi-planet system will undergo dynamical instability. These equations capture the neural network’s learned structure with a small number of variables that allow dynamical interpretation. (Dynamical instability means that planets’ orbits change until a close encounter or ejection occurs.)
How they found the equations The team used a two-stage discovery pipeline. First they trained a neural network that constructs an interpretable basis of learned features. Then they distilled the network’s predictor into algebraic formulas using symbolic regression. The network takes a short numerical integration of each system over 10^4 inner orbits and encodes 100 equally spaced samples of the system configuration to compute summary statistics. A sparse linear mapping W from a human-designed encoding of orbital elements produces N learned features; the mean and standard deviation of those features over the short integration form a 2N feature basis. During training W was sparsified with L1 regularization, pruned to keep the top k = 2 weights per feature, and fine-tuned. The predictor network outputs the mean (and variance) of a log-normal distribution for instability time. The symbolic regressor (PySR) then distilled the network’s predicted mean into closed-form equations by sampling 10,000 input–output pairs from the network. The distilled equations were learned from 113,543 N-body integrations of compact 3-planet systems.
Concrete results On a held-out resonant test set the reported root-mean-square error (RMSE, in dex) for predicting log10 instability time was: neural network 1.12 dex and distilled equations (“Ours”) 1.34 dex. For comparison, pure symbolic regression gave 1.41 dex and the human-derived analytic baseline Petit et al. (2020) gave 3.13 dex. (One dex corresponds to about a factor of ten error in predicted instability time.)
The distilled equations also extrapolate beyond training. When tested on equal-mass, equally spaced, 5-planet systems (trained on 3-planet systems) the equations achieved RMSE 1.52 dex versus 1.74 dex for the neural network on those inputs. The authors also tested on a period-ratio grid of 90,000 simulations covering adjacent planet period ratios 0.55–0.76 integrated to 10^9 orbits, and report that the equations extrapolate to systems with more bodies and to longer instability timescales than seen during training (including tests up to T_inst = 10^10 in other datasets).
Why this matters These equations are not black-box models. They reduce the prediction landscape to a few learned variables that can be interpreted in dynamical terms. They outperform prior human-derived analytic estimates in these tests. The two-stage approach—learning features then searching for equations—appears to aid extrapolation beyond the original training conditions.
Supported caveats and limitations The equations have a compressed dynamic range compared to the neural network. They show systematic bias: they tend to be biased upwards for unstable systems and downwards for stable systems, with a high-complexity term identified as a main source of that bias. The distilled formulas show a systematic limitation at low interplanetary separation Δ, where fine details of instability time are not fully captured. The symbolic distillation produced equations for the predicted mean only; the neural network’s predicted variance was not distilled because it would not yield calibrated uncertainty for the equations themselves. For 5-planet predictions the authors used either a constant standard-deviation placeholder or a fine-tuned network uncertainty. The discovered equations were not constrained to have correct physical units; variables such as mass and inclination are not unitless, so the equations do not carry correct units. The combined regression-plus-classification training objective led models to prefer higher false positive rates for instability (classifying some stable systems as unstable) as a way to reduce regression loss. Finally, restricting each learned sparse feature to use only k = 2 input variables can omit some variables from low-complexity equations; the authors identify this as a possible artifact of that restriction.
Summary Analytic, human-interpretable equations were distilled from a learned neural predictor. They predict instability times for compact planetary systems, match or beat some baselines in tests, and extrapolate in some settings. They also have clear limitations in range, bias, uncertainty calibration, and physical units that the authors document.