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

TL;DR

This work introduces IS-FNO, an inverse-scattering-inspired Fourier neural operator designed to learn stable, long-horizon time-advancement operators for nonlinear PDEs. By enforcing a near-invertible lifting–projection pair and modeling latent evolution with exponential Fourier layers, IS-FNO integrates reversibility and spectral dynamics into neural operators. Across eight benchmark PDEs spanning chaotic, stiff, and integrable regimes (MS/KS/KdV/KP in 1D and 2D), IS-FNO, particularly with nonlinear latent evolution, achieves superior short-term accuracy and long-horizon stability compared to baseline FNO and Koopman-inspired variants, with specialized KdV-inspired variants offering competitive performance with reduced capacity. The results highlight the value of embedding physical structure into neural operators to enhance robustness, interpretability, and generalization for nonlinear PDE dynamics.

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.

Figures (10)

• Figure 1: Schematic comparison of inverse scattering based and Koopman inspired Fourier Neural Operator architectures.
• Figure 2: Short-term relative $L_2$ errors for training (wide bars, dotted outlines) and validation (thin bars, solid outlines) across eight datasets: five 1d cases (KS and MS at $\beta=10$ and $\beta=40$, and KdV) and three 2d cases (KS and MS at $\beta=15$, and KDV/KP). Numerical values are listed in Table \ref{['Table1']} in the Appendix.
• Figure 3: Long-horizon error evolution \ref{['eq:long_horizon_error']} for seven models evaluated on four 1d datasets (KS and MS at $\beta = 10$ and $\beta = 40$). For each model and dataset, errors are averaged over 20 rollout sequences by uniform random initialization, Eq. \ref{['eq:MS_KS_init']}.
• Figure 4: Long-horizon error evolution for nine models trained on the 1d-KdV dataset. Error is averaged over 20 rollout sequences initialized by low-wavenumber randomization (Eq. \ref{['eq:init_lowwavenumber']}, left ) and random soliton superposition (Eq. \ref{['eq:init_1dsol']}, right).
• Figure 5: Long-horizon error evolution \ref{['eq:long_horizon_error']} for three models trained on three 2d datasets. Errors are averaged over five rollout sequences initialized by low-wavenumber randomization.