Symplectic-inspired neural operators for discretization-invariant light propagation with exact power conservation

Computational optics faces a dilemma between the prohibitive cost of split-step Fourier methods and the numerical dissipation of deep-learning surrogates. Neither paradigm simultaneously achieves physical fidelity, computational efficiency, and end-to-end differentiability. We introduce the Symplectic-inspired Neural Operator (SympNO), a physics-enforced architecture that embeds the symplectic Strang splitting structure into the network. A hard unitary constraint ensures optical power preservation to machine precision, eliminating soft penalty terms. By learning data-driven, effective corrections to splitting errors over macro-scale steps, SympNO achieves a Pareto-optimal balance between accuracy and latency. It outperforms the finest beam-propagation configuration in both NMSE and inference time, and remains the only compared method that is simultaneously differentiable, mesh-generalizable, and physically consistent. The Iterative SympNO Solver (ISS) extends this to high-fidelity multi-step propagation, achieving cross-wavelength generalization and consistent accuracy across spatial scales (64² to 4096²) without retraining. In strong turbulence (σ _R ²≈1), the ISS reproduces reference statistics with NMSE <10^-4. For inverse design in thick random media, the differentiable surrogate enables gradient-based optimization with diffraction efficiency within 2% of the physics-informed BPM upper bound while reducing forward steps by 15 ×. These results demonstrate that embedding geometric conservation into neural operators unifies computational efficiency with physical accuracy in Hamiltonian wave systems, offering a robust simulation tool for real-time atmospheric optics and nanophotonic device design.