To extend the PDEformer architecture to handle higher-dimensional PDEs, several modifications and enhancements can be implemented:
Increased Dimensionality in Graph Transformer: The Graph Transformer component of PDEformer can be adapted to handle higher-dimensional input data. This would involve adjusting the embedding dimensions, attention mechanisms, and network architecture to accommodate the additional dimensions.
Expansion of INR Decoder: The Implicit Neural Representation (INR) decoder in PDEformer can be modified to accept multi-dimensional input coordinates. By incorporating additional dimensions into the coordinate input, the model can effectively handle higher-dimensional PDEs.
Integration of Spatial and Temporal Features: For spatio-temporal PDEs, the model can be designed to capture both spatial and temporal dependencies across multiple dimensions. This would involve incorporating multi-dimensional convolutions or recurrent layers to process the input data effectively.
Enhanced Hypernet Architecture: In the Poly-INR component, the hypernets can be extended to generate scale and shift modulations for higher-dimensional hidden layers. This adaptation would enable the model to learn complex patterns and relationships in multi-dimensional data.
Data Preprocessing and Augmentation: Preprocessing techniques such as data normalization, dimensionality reduction, and data augmentation can be tailored to suit higher-dimensional datasets. By optimizing the data preprocessing pipeline, the model can effectively handle the increased complexity of higher-dimensional PDEs.
By implementing these enhancements and adjustments, the PDEformer architecture can be extended to effectively tackle higher-dimensional PDEs across various scientific and engineering domains.