S2NO: Coupling Aliasing-Free Convolutions with Fourier Neural Operators for Multiscale Modeling

Neural operators have emerged as a powerful paradigm for solving Partial Differential Equations. They learn mappings between infinite-dimensional function spaces and have been successfully applied in fields such as computational fluid dynamics and weather forecasting. However, existing methods face a fundamental dilemma. The Fourier Neural Operator is efficient at capturing global patterns but suffers from spectral leakage. Conversely, hybrid models combine spectral methods with standard convolutions but often introduce aliasing errors. This violates the critical property of resolution invariance. To address this, we propose the Spectral-Spatial Neural Operator. We introduce a dual-stream architecture that couples a spectral branch with an aliasing-free convolutional branch. This design allows the model to capture high-frequency residuals and sharp discontinuities without introducing grid-dependent artifacts. We conducted extensive experiments on the 1D Burgers, 2D Darcy Flow, and 2D Navier-Stokes equations. The results demonstrate that our method significantly outperforms state-of-the-art baselines in both prediction accuracy and zero-shot super-resolution stability.