Curse of dimensionality has been a notorious difficulty in scientific computing. Recent advances in machine learning, especially in deep learning, have ushered some new hope in addressing this difficulty. In this dissertation, we extend the power of deep learning to two new domains that suffer from this difficulty: (1) large-scale molecular dynamics and (2) high-dimensional stochastic control and parabolic partial differential equations. Accordingly, the main body of the dissertation consists of two independent parts.

In the first part, we present a simple yet general scheme for molecular simulations, the Deep Potential Molecular Dynamics (DPMD). It is based on a many-body potential and interatomic forces generated by a carefully crafted deep neural network trained with ab initio data. The model preserves all the natural symmetries of the system and is "first principle-based" in the sense that there are no ad hoc components aside from the network specification. The proposed scheme provides an efficient and accurate protocol in a variety of systems, including bulk materials and molecules. In all these cases, DPMD gives results that are essentially indistinguishable from the original data, at a cost that scales linearly with system size.

In the second part, we first develop a deep learning approach that directly solves high-dimensional stochastic control problems in the finite horizon. The time-dependent controls are approximated by neural networks, which are then stacked together through model dynamics. Numerical results in examples from the areas of optimal trading and energy storage suggest that the proposed algorithm achieves satisfactory accuracy with great applicability to high-dimensional problems. Furthermore, we introduce a deep learning approach, called deep BSDE method, which can solve general high-dimensional parabolic partial differential equations. To this end, the partial differential equations are reformulated using backward stochastic differential equations and the gradient of the unknown solution is approximated by neural networks. Numerical results illustrate the efficiency and accuracy of the deep BSDE method for a wide variety of problems. This opens up new possibilities in many disciplines by considering all participating units together at the same time, instead of making ad hoc assumptions on their inter-relationships.