DS4D-FFNO: Hidden-State Retrieval for PDEs with Fixed-Delay Boundary Feedback

Partial differential equations (PDEs) with fixed-delay boundary feedback arise in fields such as wave propagation, structural dynamics, electromagnetics, and flow control, where boundary feedback depends on past states. In recent years, neural operators have been widely used to solve PDEs, but most lack a dedicated mechanism for modeling fixed-delay boundary feedback. We propose DS4D-FFNO, combining factorized Fourier neural operator (FFNO) spatial mixing with DS4D, an S4D extension that models delayed dynamics by retrieving interpolated past hidden states via a delayed-feedback branch. DS4D-FFNO consists of an encoder followed by two FFNO layers, one DS4D layer, two further FFNO layers, and an output head. Compared with seven other models, DS4D-FFNO achieves the lowest error in solving each of six such PDEs, which describe wave propagation, structural vibrations, thermoelastic coupling, boundary dynamics, and reaction–diffusion processes. To assess its applicability beyond delayed systems, we compare DS4D-FFNO with S4-FFNO on six classical PDEs without physical delay and find comparable predictive accuracy. To test whether its learned delay indices contribute to the gains on delayed PDEs, we permute or randomly replace them, and both interventions increase error, indicating that learned delayed access is essential for maintaining the trained model's predictive accuracy.