<p>The Fourier neural operator (FNO) represents a state-of-the-art method in the field of artificial intelligence for science. It integrates theoretical principles and empirical data and applies the universal approximation theorem to learn mappings between different function spaces, thereby forging novel paths for intelligently solving complex nonlinear dynamic problems. This study aims to broaden the application scope of the FNO to develop a hydrodynamic surrogate model under dynamic open boundary conditions. However, natural river streamflow is a continuous-time stochastic process, presenting a challenge for the vanilla FNO in handling multistep prediction under open boundary conditions. To address this, we incorporate a time series forecasting method into the FNO architecture and propose a nonlinear autoregressive with exogenous inputs Fourier neural operator (NARX-FNO). NARX-FNO expands the FNO’s input dimension to encode the open boundary conditions of hydrodynamic simulation and adopts a NARX-based open/closed-loop architecture to achieve temporal evolution under dynamic boundary conditions. Taking the Yichang reach in the middle Yangtze River as a case study, we generate a dataset comprising water depth and flow velocity through numerical simulation to train the NARX-FNO model. The results show that the post-training NARX-FNO model effectively predicts unsteady flow, with Nash-Sutcliffe efficiency values of 0.99 and 0.97 during the flood and dry seasons, respectively. This study guarantees high-fidelity simulations of river hydrodynamics and enables real-time inference of actual river flow states and temporal synchronization of system parameters between physical and digital systems.</p>