Noether's razor jointly learns conservation laws and the symmetrised Hamiltonian on train data, without requiring validation data. By relying on differentiable model selection, we do not introduce additional regularizers that require tuning.š§µ7/16
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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. š§µ8/16
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 nĀ² dimensional unitary Lie group U(n) on N-harmonic oscillators.š§µ9/16
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). š§µ10/16
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. š§µ11/16
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. š§µ12/16
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.š§µ12/16
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?
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