Physics Based Architectures for Deep Neural Networks

深度神经网络(DNN)是最近深度学习革命背后的引擎。这种网络在解决图像分类问题、图像分割问题和其他计算机视觉任务方面表现出了非凡的能力。这类任务在人脸识别、自动驾驶汽车和医学图像理解等应用中很常见。 近年来，已经提出了对这些网络的修改，特别是残差网络已显示出比现有技术更好的改进，并产生更好的结果。更具体地说，剩余网络更容易训练，已用于数百甚至数千层，并且可以提高内存效率。 受到这些结果的鼓舞，一些研究人员试图用常微分方程组和最优控制的理论，将这些网络作为离散动力系统进行分析，以解释这些网络的成功之处。 虽然初步结果很有趣，但当前的DNN训练和理解技术与其他使用类似方法的领域仍有很大差距，例如依赖时间的最优控制、反问题和基于偏微分方程组的优化。此外，DNN没有全面的连续模拟，这使得它们有点附加和问题/数据依赖。了解如何在不同比例之间移动网络、更改网络参数以及了解其属性都是困难的，在许多情况下，甚至是不可能的。 这项工作的目标是为理解DNN提供建议和开发新的想法。特别是，我们利用了这种解释 将神经网络作为动力系统，以便提出基于物理的或受物理激励的网络。这些网络表示物理过程，如非线性扩散和非线性波传播，类似于用于压缩传感的偏微分方程组。这种网络在整体特性方面可以具有良好的性质，并且可能更适合于使用这种结构的快速优化算法，例如多重网格法。 我将培训3名博士生和3-5名本科生研究助理，学习机器学习的基本原理及其在地学成像问题中的应用。总的来说，这个研究计划将增加机器学习计算的科学知识，并将 为加拿大越来越多致力于人工智能的公司做出贡献。我们预计，通过拟议的研究培训的学生将 在学术界或产业界追求有前途的职业生涯。

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.