Approximating Functions with Multi-Features on Sphere by Deep Convolutional Neural Networks

In recent years, deep convolutional neural networks (DCNNs) have demonstrated remarkable success in approximating functions with multiple features. However, several challenges remain unresolved, including the approximation of target functions in Sobolev spaces defined on the unit sphere, and the extension of the types for intrinsic functions. To address these issues, we propose a DCNNs architecture with multiple downsampling layers to approximate multi-feature functions in Sobolev spaces on the unit sphere. Our method facilitates automatic feature extraction without requiring prior knowledge of the underlying composite structure and alleviates the curse of dimensionality in function approximation by extracting general smooth and spherical polynomial features. Compared with previous approaches, the proposed DCNNs architecture is more effective in capturing a variety of features.