3 ms·

This is not surprising at all and depends on the inductive bias hardcoded in the search. There are infinite number of curves that agree on those 8 points and d

by srean 4mo ago

This is not surprising at all and depends on the inductive bias hardcoded in the search. 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 4mo ago
You're right, and I'd go further: the inductive bias isn't incidental, it's the whole product. Short trees over {+, *, sqrt, exp, ...} plus a parsimony penalty is basically Occam's razor made executable. A bias-free learner can't generalize at all (no free lunch), so the honest question is whether this particular bias matches the domain. For physics it has an unreasonably good track record though that's the mystery of physics, not of my library. Two nuances though. The evidence here isn't "a curve fits 8 points" infinitely many do, as you say. It's that a 3-node formula fits them to machine precision (1−R² ≈ 1e-15). Under an MDL view that's not nothing: the probability that such a short description nails 8 independent points exactly, if the truth were some unrelated wiggly curve, is astronomically small. The shortness is the evidence. Second: your adversarial curve would fool Kepler too, and any finite-data method ever. The practical mitigation is the boring one held-out validation, and in the planetary case, extrapolation: the law found on 8 planets keeps working on moons, asteroids and exoplanets. On the tool side I try to keep the failure mode visible rather than hidden: it returns the full accuracy-vs-complexity Pareto front, and the docs say plainly that noise breaks symbolic recovery long before it breaks fit quality. So yes: it finds simple laws when simple laws exist. When they don't, it fails ideally loudly.
- aesthesia 4mo ago
  Thank you, Claude.
  - deleted 4mo ago
    [deleted]
- srean 4mo ago
  I would argue that Kepler's success influenced the choice of the inductive bias. In that case the claim that Kepler took years what this can do in seconds is not an unbiased position to take. I do see a great value in conjecturing possible solutions that needs to be verified with domain specific knowledge. Recall that Ptolemaic epicycles were a great fit, in fact a better fit than Copernicus's heliocentric model. This makes me wary of deep NNs in Physics.
  - sade_95 4mo ago
    You are right, and I concede the point. I chose the operators and the parsimony rule knowing already that laws like Kepler exist. So "Kepler took 10 years" is a bit of theater, the honest claim is only: if a short law exists in this basis, the search finds it fast. Kepler had to invent the idea that such law exists at all, and fight Tycho's data with no computer. Not the same job. For the epicycles, for me this is exactly the argument for parsimony pressure. Epicycles fit better because you can always add one more circle, same as adding parameters in a NN. They lose on description length, and that is the only defense I have too. And it is not perfect: on noisy data my tool produces its own epicycles, formulas that fit very well and mean nothing. One battery model it gave me predicted the battery heals itself after cycle 264. Great fit, zero physics. So yes, a conjecture machine that a domain expert must verify. No more than that.
    - srean 3mo ago
      I agree and I vehemently share your concern about profusion of tweakable parameters in the model. There is some misconception in the wild about epicycles models that need not be shared by you specifically. There weren't that many epicycles per orbiting body, but every orbiting body had a few that had to be 'trained' specifically for them. My fear, and I suspect yours too is that good curve fits done one at a time with mathematical models that are universal approximators (*) rarely, if at all lead to causally explanatory models. In Physics it's the latter that we seek. (*) Epicycloidal models are a form of Fourier analysis and are a class of universal approximators for periodic trajectories.
      - 3mo ago

by srean 4mo ago

sade_95 4mo ago

You're right, and I'd go further: the inductive bias isn't incidental, it's the whole product. Short trees over {+, *, sqrt, exp, ...} plus a parsimony penalty is basically Occam's razor made executable. A bias-free learner can't generalize at all (no free lunch), so the honest question is whether this particular bias matches the domain. For physics it has an unreasonably good track record though that's the mystery of physics, not of my library. Two nuances though. The evidence here isn't "a curve fits 8 points" infinitely many do, as you say. It's that a 3-node formula fits them to machine precision (1−R² ≈ 1e-15). Under an MDL view that's not nothing: the probability that such a short description nails 8 independent points exactly, if the truth were some unrelated wiggly curve, is astronomically small. The shortness is the evidence. Second: your adversarial curve would fool Kepler too, and any finite-data method ever. The practical mitigation is the boring one held-out validation, and in the planetary case, extrapolation: the law found on 8 planets keeps working on moons, asteroids and exoplanets. On the tool side I try to keep the failure mode visible rather than hidden: it returns the full accuracy-vs-complexity Pareto front, and the docs say plainly that noise breaks symbolic recovery long before it breaks fit quality. So yes: it finds simple laws when simple laws exist. When they don't, it fails ideally loudly.

aesthesia 4mo ago
Thank you, Claude.
- deleted 4mo ago
  [deleted]
srean 4mo ago
I would argue that Kepler's success influenced the choice of the inductive bias. In that case the claim that Kepler took years what this can do in seconds is not an unbiased position to take. I do see a great value in conjecturing possible solutions that needs to be verified with domain specific knowledge. Recall that Ptolemaic epicycles were a great fit, in fact a better fit than Copernicus's heliocentric model. This makes me wary of deep NNs in Physics.
- sade_95 4mo ago
  You are right, and I concede the point. I chose the operators and the parsimony rule knowing already that laws like Kepler exist. So "Kepler took 10 years" is a bit of theater, the honest claim is only: if a short law exists in this basis, the search finds it fast. Kepler had to invent the idea that such law exists at all, and fight Tycho's data with no computer. Not the same job. For the epicycles, for me this is exactly the argument for parsimony pressure. Epicycles fit better because you can always add one more circle, same as adding parameters in a NN. They lose on description length, and that is the only defense I have too. And it is not perfect: on noisy data my tool produces its own epicycles, formulas that fit very well and mean nothing. One battery model it gave me predicted the battery heals itself after cycle 264. Great fit, zero physics. So yes, a conjecture machine that a domain expert must verify. No more than that.
  - srean 3mo ago
    I agree and I vehemently share your concern about profusion of tweakable parameters in the model. There is some misconception in the wild about epicycles models that need not be shared by you specifically. There weren't that many epicycles per orbiting body, but every orbiting body had a few that had to be 'trained' specifically for them. My fear, and I suspect yours too is that good curve fits done one at a time with mathematical models that are universal approximators (*) rarely, if at all lead to causally explanatory models. In Physics it's the latter that we seek. (*) Epicycloidal models are a form of Fourier analysis and are a class of universal approximators for periodic trajectories.
    - 3mo ago