Our method discovers the correct symmetries from data. Learned Hamiltonians that obey the right symmetry generalise better as they remain more accurate in larger areas of the phase space, depicted here for a correctly learned SO(2) on a simple harmonic oscillator. 🧵8/16 - ThreadSky

tychovdo.bsky.social • 84 days ago

Our method discovers the correct symmetries from data. Learned Hamiltonians that obey the right symmetry generalise better as they remain more accurate in larger areas of the phase space, depicted here for a correctly learned SO(2) on a simple harmonic oscillator. 🧵8/16

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tychovdo.bsky.social•84 days ago

We verify the correct symmetries and group dimensionality are learned by inspecting parallelism, singular vectors, and transformations associated with learned generators. For instance, we correctly learn the n² dimensional unitary Lie group U(n) on N-harmonic oscillators.🧵9/16

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

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

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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