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

Area of Science:

• Scientific computing
• Machine learning for physics

Background:

• Neural operators like GNNs and FNNs excel at solving PDEs.
• GNNs offer interpretability by modeling physical fields as graphs.
• Current GNNs struggle with deep node feature extraction, limiting accuracy.

Purpose of the Study:

• To enhance the accuracy of GNNs for solving PDEs.
• To address limitations in mining deep-level graph node features.
• To develop a GNN that performs well with reduced data requirements.

Main Methods:

• Proposed a novel Node Feature Optimization GNN (NFO-GNO).
• Introduced a multiscale graph building module to capture PDE information at various scales.
• Employed a node feature optimization network (NFON) with space-frequency cross-attention (CA) for feature extraction and fusion.

Main Results:

• NFO-GNO demonstrated superior performance over baseline methods on four benchmarks.
• The approach covers both solid mechanics and fluid dynamics simulations.
• Achieved robust performance with limited training samples and low-resolution data.

Conclusions:

• NFO-GNO effectively extracts and optimizes deep-level graph node features.
• The method significantly improves the accuracy of solving PDEs.
• NFO-GNO is adaptable to data-scarce environments, reducing data dependency.