Merging Memory and Space: A State Space Neural Operator

We propose the *State Space Neural Operator* (SS-NO), a compact architecture for learning solution operators of time-dependent partial differential equations (PDEs). Our formulation extends structured state space models (SSMs) to joint spatiotemporal modeling, introducing two key mechanisms: *adaptive damping*, which stabilizes learning by localizing receptive fields, and *learnable frequency modulation*, which enables data-driven spectral selection. These components provide a unified framework for capturing long-range dependencies with parameter efficiency. Theoretically, we establish connections between SSMs and neural operators, proving a universality theorem for convolutional architectures with full field-of-view. Empirically, SS-NO achieves state-of-the-art performance across diverse PDE benchmarks-including 1D Burgers' and Kuramoto-Sivashinsky equations, and 2D Navier-Stokes and compressible Euler flows-while using significantly fewer parameters than competing approaches. A factorized variant of SS-NO further demonstrates scalable performance on challenging 2D problems. Our results highlight the effectiveness of damping and frequency learning in operator modeling, while showing that lightweight factorization provides a complementary path toward efficient large-scale PDE learning.