Poisson-Algebraic Parallel Scan: A Fast Symplectic Framework for Neural Hamiltonians

Hamiltonian neural networks (HNNs) inherit physical inductive biases but remain constrained by sequential computation which limits their scalability. We introduce an algebraic framework that embeds the Hamiltonian representation learned by the network into a set of polynomial generators within a Poisson algebra, yielding a Lie group of flows with inherently associative composition. This structural property directly enables parallel scan (prefix-sum) algorithms, thereby reducing computational complexity from $\mathcal{O}(M)$ to $\mathcal{O}(\log M)$ while preserving symplectic consistency. Empirical results confirm significant runtime and memory improvements over sequential baselines, with the scaling benefit clearly observable over thousands of steps. Our approach highlights how Hamiltonian learning can be accelerated through parallel scan, supported by a Poisson algebraic structure. Consequently, this establishes a scalable foundation for extended timescale simulation.