Long-Horizon SPDE Forecasting with Stochastic Latent Operators

Stochastic partial differential equations (SPDEs) describe spatial systems whose evolution is shaped by both resolved dynamics and intrinsic random forcing. Learning their long-horizon evolution is particularly challenging, with field error, spectral attenuation, recurrent drift, and forcing-response error all compounding under autoregressive rollout. To address these limitations, we introduce the Stochastic Latent Operator (SLO), a recurrent latent neural operator with an encoder-decoder architecture that separates learned evolution in latent space from the physical forcing action in field space. Under temporal extrapolation, SLO delivers its largest gains, reducing field error relative to the next-best baseline by 42\% on Burgers and by 61\% on two-dimensional Navier-Stokes. Under interpolation, it achieves the lowest error on cylindrical KdV and remains competitive on reaction-diffusion and Q-Wiener KdV. Across these benchmarks, SLO preserves spectral fidelity, yielding the best or near-best log-spectrum errors on the extrapolation and KdV benchmarks, with improvements of up to 51\% relative to DeepOMamba.