Point-Calibrated Spectral Neural Operators

Two typical neural models have been extensively studied for operator learning, learning in spatial space via attention mechanism or learning in spectral space via spectral analysis technique such as Fourier Transform. Spatial learning enables point-level flexibility but lacks global continuity constraint, while spectral learning enforces spectral continuity prior but lacks point-wise adaptivity. This work innovatively combines the continuity prior and the point-level flexibility, with the introduced Point-Calibrated Spectral Transform. It achieves this by calibrating the preset spectral eigenfunctions with the predicted point-wise frequency preference via neural gate mechanism. Beyond this, we introduce Point-Calibrated Spectral Neural Operators, which learn operator mappings by approximating functions with the point-level adaptive spectral basis, thereby not only preserving the benefits of spectral prior but also boasting the superior adaptability comparable to the attention mechanism. Comprehensive experiments demonstrate its consistent performance enhancement in extensive PDE solving scenarios. Partial differential equations (PDEs) are widely used across a wide range of scientific and engineering tasks, such as airfoil design, plastic structure design, and blood flow simulation. However, traditional PDE solvers depend on high-precision meshes and substantial computational requirements, which significantly impedes efficiency in many engineering applications. To resolve these limitations, recent works (Li et al., 2020; Lu et al., 2019; Tripura & Chakraborty, 2022) introduce neural operators, a class of data-driven approaches that directly learn mappings between continuous function spaces for solving parametric partial differential equations. The most performed neural operators could be classified into two groups, i.e., attention-based neural operators (Cao, 2021; Hao et al., 2023; Wu et al., 2024) and spectral-based neural operators (Li et al., 2020; Tran et al., 2021; Gupta et al., 2021).