What happens if you give the system not only the semi-mayor axis but also the semi-minor axis?
Have you tried with only the 6 planets Kepler know? (I don't expect this to change the result too much.)
Have you tired with noisy data?
to get a new paragraph here.
> b = a·sqrt(1−e²), so corr(a,b) = 1.00000 to 5 decimals
Isn't e different for each planet?
> Mercury is the only planet where they differ by more than 2%
I remember something about Mars been the planet with the most eccentric elipse
> *So fit quality degrades gracefully, symbolic recovery doesn't. Making that part noise-robust is pretty much the open frontier of the whole field, not just of my tool.
Nice. It's a hard problem. Which heuristic are you using to pick the "best" formula?
Isn't e different for each planet?
Yes, each planet got its own e (Mercury 0.206, Venus 0.007, Earth 0.017...). The correlation still comes out at 1.00000 because all eccentricities are small while a spans 0.39 to 30 AU — a ≤2% per-planet wobble is invisible to Pearson across two decades of range. That's exactly what makes it a nasty decoy: almost collinear, but not quite.
Mars been the planet with the most eccentric ellipse
Close — Mercury is actually the most eccentric (0.206), Mars is second (0.093). Funny enough, Mars is the famous one precisely because Kepler derived his laws fighting with Tycho's Mars data: its ellipse was just eccentric enough to kill every circular fit. If Tycho had handed him Venus data instead, we might have waited a while longer.
Which heuristic are you using to pick the "best" formula?
Hold-out validation during evolution, then the old CART-style one-standard-error rule: pick the smallest formula whose validation error is within ~1 SE of the best one. Constants get refit with Levenberg-Marquardt before that comparison, so small forms compete at their best. It also returns the full accuracy-vs-size Pareto front so you can override the choice. And to connect it to your noise question: under noise that 1-SE band is exactly where wrong-but-simpler formulas sneak in and tie the true one — the two problems are really the same problem.
sade_95•5d ago
GP_ELITE is a symbolic regression engine in pure Python: given (X, y) data, it searches for a readable mathematical formula linking them, instead of a black-box model.
To show what that means concretely: I gave it nothing but the 8 planets' distance from the Sun and orbital period — 8 data points — and asked for a formula. It returned:
T = a · sqrt(a) (i.e. a^1.5), R² = 1.000000
That's Kepler's Third Law (T² ∝ a³), which took Kepler ~10 years to find in 1618. GP_ELITE found it in ~3 seconds. Reproducible: examples/kepler_demo.py.
v0.2.0 (this week) added the parts that make it reliable: Levenberg-Marquardt constant fitting (constants come back at machine precision — Coulomb's q1·q2/(4πεr²) is recovered exactly), multi-restart with a merged candidate archive, a Pareto front output (the full complexity ↔ accuracy staircase, not just one champion), and a guarded forecasting mode for extrapolating trends beyond your data without the usual GP blow-ups.
Pure Python/NumPy — pip install gp-elite, no compiler, no Julia.
Target Audience
Anyone with small experimental datasets (≤10 variables, 100–5000 points) who wants to understand a relationship, not just predict it: lab engineers, scientists, students. One concrete use case that drove development: battery degradation (SOH) forecasting — the guarded mode gives you an honest bracket of scenarios (a Pareto front from a conservative straight line to richer bounded laws) instead of one overconfident curve. Production-usable for that niche (built-in hold-out validation, regression-tested); not aimed at large-scale ML.
Comparison
vs gplearn (the established pure-Python option): I ran both on the same frozen benchmark — 15 Feynman physics equations, identical data and splits, generous budget for gplearn. Exact symbolic recovery (machine precision): GP_ELITE 10/15 (67%) vs gplearn 6/15 (40%). gplearn recovers the constant-free formulas and stalls as soon as a ½ or a 4π appears (no real constant optimization); LM fitting is what closes that gap. Every number is reproducible: PYTHONHASHSEED=0 python benchmarks/feynman_bench.py 0 15 and benchmarks/duel.py in the repo.
vs PySR / Operon (the state of the art): they are stronger on speed and scale, and I'm not claiming otherwise — but they require a Julia or C++ toolchain. GP_ELITE's whole point is zero barrier: pip install and go.
vs neural nets / gradient boosting: those win on raw accuracy for large data, but give you a black box — GP_ELITE gives you the actual equation.
Honest limits: weak on chaotic targets (tested on Collatz), degrades past ~6 variables with decoy features, and pure Python costs wall-time on big data.
Code (MIT): https://github.com/ariel95500-create/gp-elite
srean•47m ago
There are infinite number of curves that agree on those 8 points and deviate from Kepler 's law everywhere else. On such 'trajectories' this algorithm would have performed badly.
sade_95•28m ago
aesthesia•5m ago