Learning Modulated Spectral-Spatial State-Space Model for Solving Dynamic PDEs

Forecasting the evolution of systems governed by partial differential equations (PDEs) requires capturing interactions across scales and effectively conditioning on temporal history. We propose $S^4$-dyn, a spectral-spatial state-space neural operator that unifies cross-frequency modeling with history-conditioned spatial modulation. Its spectral-spatial backbone combines selective state-space modeling along bidirectional radial-angular scans of Fourier modes with local spatial processing, capturing complementary global and local interactions. A parallel modulation branch evolves historical observations in a latent spectral space and uses the resulting context to adaptively guide the forecasting backbone. Tri-plane factorization extends the selective state-space modeling in 2D Fourier space to 3D systems with reduced length of state-space sequence. We further establish an input-adaptive spatial kernel interpretation that characterizes the operator’s cross-frequency interactions. Across 15 diverse 2D and 3D physical systems, $S^4$-dyn demonstrates strong dynamic prediction performance, achieving the lowest error in most evaluations on The Well and on PDEArena-NS, while remaining comparable to the strongest baseline on FNO-NS. Ablation studies support the effectiveness of combining structured spectral modeling with adaptive temporal conditioning for PDE forecasting.