ScaleNet-NO: A Multi-Scale Neural Operator for Fluid Mechanics Problems and General PDE Solving

Abstract

Neural operators have shown strong potential for solving partial differential equations (PDEs) by learning mappings between high-dimensional function spaces. However, many existing models struggle to capture both high-frequency details and multi-scale structures, which are critical in turbulent flows, porous media heterogeneity, aerodynamic boundary layers, and Maxwell-type electromagnetic wave propagation. To overcome these limitations, this paper proposes ScaleNet-NO, a neural operator integrating two innovations: a Multi-Scale Parallel Large Convolution Kernel (MSPLCK) module for extended local context and fine-scale feature extraction, and an enhanced U-Net path for high-frequency representation. The MSPLCK module uses parallel large convolution kernels with effective receptive fields of 19, 13, and 7, enabling simultaneous modeling of meso-scale structures and localized features. Experiments on seven solid- and fluid-mechanics benchmarks show substantial gains, with relative error reductions of 59.0%, 34.6%, and 47.7% on Pipe, Airfoil, and Darcy flow compared with U-NO, and improvements over the state-of-the-art LSM model. ScaleNet-NO provides a general framework for complex PDE solving, with potential for fluid dynamics and electromagnetic-wave simulation.