Spherical mapping of cortical surface meshes provides a more convenient and accurate space for cortical surface registration and analysis and thus has been widely adopted in neuroimaging field. Conventional approaches typically first inflate and project the original cortical surface mesh onto a sphere to generate an initial spherical mesh which contains large distortions. Then they iteratively reshape the spherical mesh to minimize the metric (distance), area or angle distortions. However, these methods suffer from two major issues: 1) the iterative optimization process is computationally expensive, making them not suitable for large-scale data processing; 2) when metric distortion cannot be further minimized, either area or angle distortion is minimized at the expense of the other, which is not flexible to generate application-specific meshes based on both of them. To address these issues, for the first time, we propose a deep learning-based algorithm to learn the mapping between the original cortical surface and spherical surface meshes. Specifically, we take advantage of the Spherical U-Net model to learn the spherical diffeomorphic deformation field for minimizing the distortions between the icosahedron-reparameterized original surface and spherical surface meshes. The end-to-end unsupervised learning scheme is very flexible to incorporate various optimization objectives. We further integrate it into a coarse-to-fine multi-resolution framework for better correcting fine-scaled distortions. We have validated our method on 800+ cortical surfaces, demonstrating reduced distortions than FreeSurfer (the most popularly used tool), while speeding up the process from 20 min to 5 s.