Open-Loop, Closed-Form: Solver-Driven Reservoir Computing for Coarse-Grid PDE Forecasting

Hybrid numerical–learning models can improve coarse-grid PDE forecasts, but training corrections through coupled rollouts is computationally expensive. We propose an open-loop solver–reservoir framework with closed-form readout training. A numerical solver initialized solely from the coarse initial state generates a base trajectory. A fixed reservoir transforms this trajectory into history-dependent features, and a linear Fourier readout, shared across forecast times, predicts the downsampled high-fidelity states. Training reduces to ridge regression independently across Fourier modes, without differentiation through the solver or reservoir. Learned predictions never enter either state update, so readout errors cannot alter subsequent features. We analyze the guarantees and limitations of this design: feature consistency for a fixed construction under a common train–test trajectory law, a prediction-risk decomposition, and an expected-risk bound using independent trajectories as the sampling units. Across five configurations of the Kuramoto–Sivashinsky, Burgers, and Navier–Stokes equations, our method reduces test nRMSE by 31–88% relative to the uncorrected solver and achieves the lowest error among the compared methods in three configurations. Measured training times, including validation-based model selection, correspond to speedups of 4.0–38.9$\times$ over solver-in-the-loop hybrids with FFNO and S4FFNO correctors under the evaluated protocols, although the best such hybrid remains more accurate in the other two configurations. Our method also outperforms the evaluated closed-loop hybrid reservoir baseline in all four shared configurations.