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

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

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

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

🌟Noether's razor⭐️ Our NeurIPS 2024 paper connects ML symmetries to conserved quantities through a seminal result in mathematical physics: Noether's theorem. We can learn neural network symmetries from data by learning associated conservation laws. Learn more👇. 1/16🧵