The objective of the proposed research is to use tools from partial differential equations (PDEs) to understand and construct deep learning algorithms. This research includes the theory-inspired design of optimization algorithms for deep learning, theoretical analysis of deep network training, and applications in computer vision. We introduce a recently developed framework (PDE acceleration), which is a variational approach to accelerated optimization with partial differential equations (PDE), in the context of optimization of deep networks, which leads to a novel and simple extension of stochastic gradient descent (SGD) with momentum. We empirically validate the theory and evaluate our new algorithm on image classification showing empirical improvement over SGD. To further enhance the performance of deep learning algorithms, we need a better understanding of the stability and convergence properties. We discovered restrained numerical instabilities in current training practices of deep networks. To explain this phenomenon, we present a theoretical framework using numerical analysis of PDE and analyzing the gradient descent PDE of a simplified convolutional neural network (CNN). Further, the special potential of “geometric PDEs in particular to advance deep learning applications remains to be explored. We plan to complete our research program by unifying two deep learning methods, surface rendering and volume rendering, under the stereoscopic segmentation framework, which is a geometrically driven PDE approach for 3D reconstruction. In summary, we believe the tools we’ve introduced could improve deep learning practice.