Bringing Reservoir Computing to Coarse-Grained PDEs: An Open-Loop Two-Stage Design

Neural operators provide fast surrogates for spatiotemporal PDEs but require costly gradient-based training for each new system. Reservoir computing avoids this overhead by training only a linear read-out in closed form, but autoregressive feedback can amplify prediction errors over long horizons. We propose a two-stage reservoir architecture that confines feedback to a contractive linear backbone. A mode-wise propagator generates an autonomous base trajectory, while per-step circulant read-outs reconstruct the target without prediction feedback or gradient-based training. Experiments on Kuramoto--Sivashinsky and incompressible Navier--Stokes outperform closed-form autoregressive reservoirs and narrow the gap to neural operators, while Burgers reveals a failure regime. Overall, the approach substantially reduces training time and peak memory, highlighting a favorable accuracy--efficiency trade-off for coarse-grained PDE forecasting.