SHNO: Symplectic Operator Learning with Size-Independent Parameter Counts

Learning Hamiltonian dynamics across system sizes requires reusable parameterizations that preserve symplectic structure. We introduce the Shared Hénon Neural Operator (SHNO), which combines a tied local encoder–decoder pair shared across canonical patches, a shared HénonNet for latent evolution, and optional symplectic edge exchange between patches. The assembled operator preserves the symplectic form on the represented manifold. For fixed local architecture and propagation depth, its parameter count is independent of patch count, while computation scales linearly with the numbers of patches and edges. For compatible local target compositions, we establish a size-uniform normalized approximation bound under uniform block-approximation and Lipschitz assumptions. Experiments on Klein–Gordon, Fermi–Pasta–Ulam–Tsingou, and Toda lattices evaluate prediction accuracy, symplectic structure, and parameter reuse. With separate training at each size, the 5,574-parameter SHNO without edge exchange achieves lower Klein–Gordon rollout errors than Global Hénon ROM, FNO, and a shared residual baseline. Matched unshared models use 8–32 times as many parameters, with accuracy differences depending on the system and size. Fixed-checkpoint tests evaluate deployment from 64 to 128 and 256 canonical pairs without target-size training. On Toda, edge exchange reduces geometric mean rollout error by 62.8% relative to identity coupling with 3.6% more parameters. Code is available anonymously at https://anonymous.4open.science/r/SHNO-6547.