Qi Tang: Stability and Accuracy of Neural Operators for Convection–Diffusion Equations

Abstract: Operator learning for convection–diffusion equations is challenging due to the interplay between transport and dissipation. Standard neural operators, such as the Fourier Neural Operator (FNO), can suffer from instability and poor generalization. We introduce a structure-preserving neural operator based on Strang splitting, which separates hyperbolic and parabolic dynamics, together with a learnable semi-Lagrangian scheme for convection to enhance stability. We provide numerical analyses of both FNO and the proposed method, demonstrating improved stability and accuracy. Experiments on variable-coefficient problems and the Vlasov–Poisson–Fokker–Planck system further confirm improved long-time rollout predictions.