Space-Frequency Cross-Attention Node Feature Optimization Graph Neural Operator for Partial Differential Equations.

Abstract

Neural operators, such as graph neural operators (GNOs) and Fourier neural operators (FNOs), directly learn the mapping from any functional parametric dependence to the solution and have achieved remarkable progress in solving partial differential equations (PDEs). GNOs exhibit excellent interpretability, as they construct graph models of physical fields to mine the mutual relationships between different nodes. However, existing methods are unable to mine the deep-level graph node features, which leads to insufficient information from neighboring nodes during the graph information aggregation process, ultimately resulting in a decline in solution accuracy. This article proposes a space-frequency cross-attention (CA) node feature optimization GNO (NFO-GNO) to address this issue. Considering the multiscale nature of the PDEs, we first construct a multiscale graph building module to obtain the PDEs information at different scales by processing graphs of different scales. After obtaining the multiscale graph models, we use a node feature optimization network (NFON) to extract and optimize the node features of the graphs in the spatial and frequency domains, and utilize CA to fuse them, thereby obtaining deep-level graph node features. Finally, we use a GNO to solve the optimized node features. NFO-GNO achieves superior solving performance compared to the baselines on four standard benchmarks covering both solid mechanics and fluid dynamics simulations. Notably, NFO-GNO maintains better performance with limited training samples and low-resolution training data, reducing data requirements and making it more adaptable to scenarios where high-quality datasets are difficult to obtain.