Neural Hamiltonian Diffusions for Modeling Structured Geometric Dynamics

We introduce Neural Hamiltonian Diffusion, a unified framework for learning stochastic Hamiltonian dynamics on differentiable manifolds. While Hamiltonian Neural Networks (HNNs) model conservative systems in flat Euclidean space, they fail to account for geometric structure and intrinsic stochasticity. Conversely, diffusion models on Riemannian manifolds offer geometry-aware stochastic modeling but lack physical inductive biases. Our method parameterizes a Hamiltonian with a neural network and defines its dynamics as a stochastic differential equation on a (pseudo-)Riemannian manifold equipped with a Poisson structure. This enables physically consistent modeling of dynamics on curved, periodic, or causally structured spaces. We demonstrate that the proposed geometric dynamics generalizes existing approaches and applies to systems ranging from molecular dynamics to relativistic n-body problems.