SympFNO: Structure-Preserving Fourier Neural Operators for Physical Surrogate Modeling

Neural operators can efficiently learn solution maps for partial differential equations, but standard architectures often violate fundamental physical structure during inference, leading to unstable long-horizon rollouts. We introduce the Symplectic Fourier Neural Operator (SympFNO), which is composed of three symplectic layers that embed structure preservation directly into Fourier-space and physical-space transformations, thereby preserving the geometric structure of Hamiltonian dynamics. Our theory provides a rigorous foundation for the architecture by showing that the structure-preserving design is mathematically consistent with Hamiltonian evolution and inherently more efficient than unconstrained operator parameterizations. Across five well-known Hamiltonian PDE problems, SympFNO achieves stable long-term rollouts, substantially lower trajectory error, more accurate conservation of physical invariants, and up to two orders of magnitude fewer parameters than state-of-the-art baselines, namely FNO and PINO.