An Inverse Scattering Inspired Fourier Neural Operator for Time-Dependent PDE Learning

Abstract: Learning accurate and stable time-advancement operators for nonlinear partial differential equations (PDEs) remains challenging, particularly for chaotic, stiff, and long-horizon dynamical systems. While neural operator methods such as the Fourier Neural Operator (FNO) and Koopman-inspired extensions achieve good short-term accuracy, their long-term stability is often limited by unconstrained latent representations and cumulative rollout errors. In this work, we introduce an inverse scattering inspired Fourier Neural Operator(IS-FNO), motivated by the reversibility and spectral evolution structure underlying the classical inverse scattering transform. The proposed architecture enforces a near-reversible pairing between lifting and projection maps through an explicitly invertible neural transformation, and models latent temporal evolution using exponential Fourier layers that naturally encode linear and nonlinear spectral dynamics. We systematically evaluate IS-FNO against baseline FNO and Koopman-based models on a range of benchmark PDEs, including the Michelson-Sivashinsky and Kuramoto-Sivashinsky equations (in one and two dimensions), as well as the integrable Korteweg-de Vries and Kadomtsev-Petviashvili equations. The results demonstrate that IS-FNO achieves lower short-term errors and substantially improved long-horizon stability in non-stiff regimes. For integrable systems, reduced IS-FNO variants that embed analytical scattering structure retain competitive long-term accuracy despite limited model capacity. Overall, this work shows that incorporating physical structure -- particularly reversibility and spectral evolution -- into neural operator design significantly enhances robustness and long-term predictive fidelity for nonlinear PDE dynamics.