Abstract: Deep learning has achieved remarkable success in diverse applications; however, its use in scientific applications has emerged only recently. I have developed multifidelity neural networks to extract mechanical properties of solid materials (including 3-D printing materials) from instrumented indentation. I have improved the physics-informed neural networks and developed the library DeepXDE for solving forward and inverse problems for differential equations, including partial differential equations (PDEs), fractional PDEs, and stochastic PDEs. I have also developed the deep operator network (DeepONet) based on the universal approximation theorem of operators to learn nonlinear operators (e.g., dynamical systems) accurately and efficiently from a relatively small dataset. In addition, I will present my work on the deep learning theory of optimization and generalization.