🚀 AI for Science: Machine Learning « re-discovers » the Law of Eccentricity (e) — Without Knowing Physics!

Hey r/LearningMachineLearning!
I just had a wild experience I HAVE to share. My ML workflow, using only geometric features (no physical laws!), managed to « rediscover » the core formula for the eccentricity of an ellipse from pure Kepler orbit data.

The Law That Emerged

e = 0.5 × r_range (when a=1)
or, in general,
e = (r_max – r_min) / (r_max + r_min)

I didn’t hardcode physics at all.
The model just found this from patterns in |ΛF| and Q_Λ — the « structural » changes along the orbit.

1. Data Generation: Just Kepler’s Law

200 orbits generated with random eccentricities, all a=1 for simplicity. Extracted pure structural features:
|ΛF| (« transactional structure change » per step) Q_Λ (« topological charge », cumulative log-derivative) No physics! No energy, no velocity, no Newton.

2. ML Pattern Mining

Correlated features like LF_std, Q_range, r_range, etc., with eccentricity e. Model « noticed » that r_range is the key: correlation r=1.000. It derived the formula:
e = 0.5 * r_range (with a=1) Generalizes to e = (r_max – r_min) / (r_max + r_min).

3. Here’s the Actual Python Code (core part):

« `python import numpy as np

… [code for generating orbit, extracting features, fitting, etc.] …

TL;DR — data only, model only, no physics assumptions.

« `

4. Results (see figure!):

AI directly predicts e from r_range with R² = 1.000 Other structural parameters (LF_std, Q_range) also map almost perfectly. The model « discovered » the underlying law, the same as in textbooks — but it had NO prior knowledge of orbits!

5. Why is This Important?

Shows that ML can « discover » physical laws from structure alone. No energy, force, or velocity needed — just patterns! Next step: try with orbits where a ≠ 1, noise, real data… Can the model generalize to other domains?

🔗 I’d love your feedback, thoughts, or if you want the full notebook, let me know!

This, to me, is « AI for Science » in its purest, most beautiful form.

Github:https://github.com/miosync-masa/LambdaOrbitalFinder

Note: I’m Japanese and not a native English speaker — so I used an AI language model to help translate and write this post! If anything is unclear, please let me know, and I really appreciate your understanding and advice. (日本人なのでAI翻訳サポート入りです)