Merging Memory and Space: A Spatiotemporal State Space Neural Operator

Abstract: We propose the Spatiotemporal State Space Neural Operator (ST-SSM), a compact architecture for learning solution operators of time-dependent partial differential equations (PDEs). ST-SSM introduces a novel factorization of the spatial and temporal dimensions, using structured state-space models to independently model temporal evolution and spatial interactions. This design enables parameter efficiency and flexible modeling of long-range spatiotemporal dynamics. A theoretical connection is established between SSMs and neural operators, and a unified universality theorem is proved for the resulting class of architectures. Empirically, we demonstrate that our factorized formulation outperforms alternative schemes such as zigzag scanning and parallel independent processing on several PDE benchmarks, including 1D Burgers' equation, 1D Kuramoto-Sivashinsky equation, and 2D Navier-Stokes equations under varying physical conditions. Our model performs competitively with existing baselines while using significantly fewer parameters. In addition, our results reinforce previous findings on the benefits of temporal memory by showing improved performance under partial observability. Our results highlight the advantages of dimensionally factorized operator learning for efficient and generalizable PDE modeling, and put this approach on a firm theoretical footing.