Learning convection–diffusion dynamics with neural operators is difficult because transport and dissipation act on different scales, and standard neural operators often lose stability across regimes. We propose a Structure-Preserving Neural Operator that captures this transport–dissipation interplay. The method uses Strang splitting to evolve hyperbolic and parabolic dynamics in substeps. Convection is handled by a learnable semi-Lagrangian approach that follows characteristics and embeds flow structure directly into the architecture, while diffusion is treated through a residual correction neural operator. Experiments on variable-coefficient problems and the Vlasov–Poisson–Fokker–Planck system show improved stability, accuracy, and long-time performance with large time steps.