Physics Based Architectures for Deep Neural Networks

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Deep Neural Networks (DNNs) have been the engine behind the recent deep learning revolution. Such networks have demonstrated remarkable capabilities in solving image classification problems, image segmentation problems and other computer vision tasks. Such tasks are common in applications such as face recognition, autonomous cars and medical image understanding. In recent years, modifications to these networks have been proposed, and in particular, residual networks have shown to improve over existing techniques, and yield even better results. More specifically, residual networks are easier to train, have been used with hundreds and even thousands of layers and can be made memory efficient.  Encouraged from these results, a few researchers have attempted to explain the success of such networks by analyzing them as discrete dynamical systems, using theory from ordinary differential equations and optimal control. While initial results are interesting, there is still a large gap between the current technology of training and understanding DNNs, to other fields that use similar methodologies, such as time dependent optimal control, inverse problems and partial differential equations based optimization. Furthermore,  DNNs do not have an over-arching continuous analog which makes them somewhat add-hoc and problem/data dependent. Understanding how to move networks between scales, changing the network parameters and understanding its properties are all difficult and in many cases, impossible. The goal of this work is to suggest and develop new ideas for the understanding of DNNs. In particular, we exploit the interpretation of neural networks as dynamical systems in order to propose physics based, or physically motivated networks. These networks represent physical processes such as nonlinear diffusion and nonlinear wave propagation similar to the partial differential equations that have been used for compressed sensing. Such networks can have favorable properties in term of their overall character and may be more amendable to fast optimization algorithms that utilize this structure such as multigrid methods. I will be training 3 Ph.D. students as well as 3-5 undergraduate research assistants in the fundamentals of machine learning and its application to problems in geoscience imaging.Overall, this research program will increase scientific knowledge in the computing of machine learning and will contribute to the growing number of companies working on AI in Canada.  We expect students trained via the proposed research will pursue promising professional careers in either academia or industry.