A Few Thoughts on Deep Learning-Based PDE Solvers

The remarkable success of deep learning in computer science has evinced potentially great applications of deep learning in computational and applied mathematics. Understanding the mathematical principles of deep learning is crucial to validating and advancing deep learning-based PDE solvers. We present a few thoughts on the theoretical foundation of this topic for high-dimensional partial differential equations including approximation, optimization, and generalization. Though our analysis is not a complete story and there are many missing pieces to make it well-justified, it may still be helpful to provide some insights into deep learning.