Noether's theorem states that symmetries in a dynamical system have an associated conserved quantity, i.e. an observable that remains invariant over a trajectory. We use this result to parameterise symmetries in Hamiltonian neural networks in terms of conserved quantities. 🧵2/16 - ThreadSky

tychovdo.bsky.social • 84 days ago

Noether's theorem states that symmetries in a dynamical system have an associated conserved quantity, i.e. an observable that remains invariant over a trajectory. We use this result to parameterise symmetries in Hamiltonian neural networks in terms of conserved quantities. 🧵2/16

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tychovdo.bsky.social•84 days ago

We let our Hamiltonian neural network be a flexible deep neural network that is symmetrized by integrating it over the flow induced by a set of learnable conserved quantities. This construction allows us to define a flexible function with learnable symmetry. 🧵3/16

tychovdo.bsky.social•84 days ago

We jointly learn the Hamiltonian and conserved quantities with Bayesian model selection by optimising marginal likelihood estimates. This yields a Noether's Razor effect favouring simpler symmetric solutions if this describe data well, avoiding collapse into no symmetry. 🧵4/16

tychovdo.bsky.social•84 days ago

We scale to deep neural networks using Variational Inference (VI). To obtain a sufficiently tight bound for model selection, we avoid mean-field and instead use efficient matrix normal posteriors and closed-form updates of prior and output variance (see tricks in App. D). 🧵5/16

tychovdo.bsky.social•84 days ago

As far as we know, this is the first case in which Bayesian model selection with VI is successfully scaled to deep neural networks, an achievement in its own right, whereas most works so far have relied on Laplace approximations. 🧵6/16

tychovdo.bsky.social•84 days ago

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.🧵7/16

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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