Symplectic Parallel Scan: A Neural Hamiltonian Framework for Accelerated Scientific Simulation

Neural Hamiltonian models provide a principled approach to learning dynamical systems by embedding the symplectic structure of physical evolution into the model class. However, this geometric prior alone has not made Hamiltonian neural networks efficient long-horizon simulators: an M-step trajectory is still typically generated by M sequential updates, and common training objectives based on force matching or one-step prediction provide only local supervision. We introduce Symplectic Parallel Scan (SPS), a neural Hamiltonian framework that algebraizes the learned dynamics through a finite-dimensional Poisson algebra of Hamiltonian generators. SPS constructs a Neural--Poisson Composition DAG closed under projected products and Poisson brackets, whose elements induce symplectic Hamiltonian drivers organized as a Lie-group walk. The associativity of this flow composition enables a parallel prefix-scan over Hamiltonian drivers, reducing the sequential depth of trajectory propagation from $\mathcal O(M)$ to $\mathcal O(\log M)$ while keeping each prefix inside the symplectic flow family. The resulting model is trained through a global trajectory-matching objective rather than only local force labels. Across quantum spin and molecular dynamics benchmarks, SPS achieves substantial wall-clock acceleration over sequential HNN baselines, improves long-horizon prediction accuracy, and reduces energy drift and symplectic violation under extrapolation.