To extend SConvCNPs to higher-dimensional input spaces like images or time series data while preserving their ability to capture long-range dependencies, several modifications can be implemented. One approach is to incorporate convolutional layers with higher dimensions, such as 2D or 3D convolutions, to handle spatial or spatiotemporal relationships in the data. By adapting the Fourier Neural Operators (FNOs) used in SConvCNPs to operate in multiple dimensions, the model can effectively capture complex patterns across different axes.
Additionally, leveraging hierarchical structures like hierarchical Fourier Neural Operators can help in processing multi-scale features present in high-dimensional data. By cascading multiple layers of FNOs with varying receptive fields, the model can learn representations at different levels of abstraction, enabling it to capture dependencies at various scales within the input space.
Moreover, incorporating attention mechanisms, similar to those used in Transformer models, can enhance the model's capability to focus on relevant regions in the input space, especially in scenarios where long-range dependencies are crucial. By combining convolutional operations with attention mechanisms, the model can effectively capture both local and global dependencies in the data, making it suitable for handling higher-dimensional inputs while maintaining its advantages in capturing long-range dependencies.