By learning the correct symmetries, the jointly learned Hamiltonians are more accurate, directly improving trajectory predictions at test time. We show this for n-harmonic oscillators, but also more complex N-body problems (see table below). 🧵10/16 - ThreadSky

tychovdo.bsky.social • 84 days ago

By learning the correct symmetries, the jointly learned Hamiltonians are more accurate, directly improving trajectory predictions at test time. We show this for n-harmonic oscillators, but also more complex N-body problems (see table below). 🧵10/16

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tychovdo.bsky.social•84 days ago

On more complex N-body problems, our method correctly discovers the correct 7 linear generators which correspond to the correct linear symmetries of rotation around center of mass, rotation around the origin, translations, and momentum-dependent translations. 🧵11/16

tychovdo.bsky.social•84 days ago

We use quadratic conserved quantities, which can represent any symmetry with an affine (linear + translation) action on the state space. This encompasses nearly all cases studied in geometric DL. 🧵12/16

tychovdo.bsky.social•84 days ago

Further, this does not limit the shapes of symmetry groups we can learn. For instance, we find the Euclidean group, which is itself a curved manifold with non-trivial topology.🧵12/16

tychovdo.bsky.social•84 days ago

If one wishes, we could extend to non-quadratic conserved quantities to model non-affine actions. More free-form conserved quantities would require ODE-solving instead of the matrix exponential and have more parameters, which could complicate the Bayesian model selection.🧵13/16

tychovdo.bsky.social•84 days ago

Our work demonstrates that approximate Bayesian model selection can be useful in neural nets, even in sophisticated use cases. We aim to further improve the efficiency and usability of neural model selection, making it a more integral part of training deep neural nets. 🧵14/16

tychovdo.bsky.social•84 days ago

Noether's razor allows symmetries to be parameterised in terms of conserved quantities and enables automatic symmetry at train time. This results in more accurate Hamiltonians that obey symmetry and trajectory predictions that remain more accurate over longer time spans.🧵15/16

andrescorrada.bsky.social•84 days ago

Very nice. Would you say that your work falls within the subject of Physics Influenced Machine Learning in the "meta" sense - your formalism discovers the physical laws the system is obeying?

tychovdo.bsky.social•84 days ago

Sure!

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