Merging Memory and Space: A State Space Neural Operator

TL;DR

The paper introduces SS-NO, a compact neural operator architecture that effectively models time-dependent PDEs by combining state space models with adaptive damping and frequency modulation, achieving state-of-the-art results with fewer parameters.

Contribution

It extends structured state space models to joint spatiotemporal PDE modeling, introducing adaptive damping and learnable frequency modulation for improved efficiency and accuracy.

Findings

Achieves state-of-the-art performance on multiple PDE benchmarks.

Uses fewer parameters than competing methods.

Demonstrates scalable performance with a factorized variant.

Abstract

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.