Deep Learning for Inverse Problems

在过去的几年里，深度学习已经成为计算机视觉、图像处理、语音识别以及机器学习和数据科学中许多其他应用的主导方法。这种成功是几个因素的协同作用：（1）神经网络（NN）作为表示高维函数和映射的灵活框架，（2）简单的算法，如反向传播和随机梯度下降，用于调整模型参数，（3）高效的通用软件包，如Tensorflow和Pytorch，以及（4）前所未有的计算能力。然而，尽管取得了成功，但仍然存在几个关键挑战：（1）NN架构设计仍然是一门艺术，在许多情况下缺乏基本的数学原理，以及（2）NN训练通常需要大量的数据，这在许多应用中是不可用的。物理科学中的许多计算问题也面临着与数据科学相同的挑战：高维，复杂或未指定的模型，以及巨大的计算成本。一些著名的例子是多体量子系统，确定性和随机控制，分子动力学，不确定性量化和逆问题。在这些问题的研究中，利用NN的最新发展是很自然的。该项目的重点是逆问题，即从观测数据中恢复未知的问题参数。这是一个非常重要的领域，在物理学，化学，医学，地球科学和国防中有应用。然而，许多反问题是已知的计算上具有挑战性。该项目将使用深度学习作为更有效地解决这些逆问题的手段。PI计划开发一个新的应用线性代数课程，重点是机器学习，并在深度学习和物理科学的界面上组织研讨会。该项目将通过研究培养研究生和本科生，每年资助一名研究生。作为表示高维映射的工具，神经网络（NN）提供了一种灵活的方法来表示全逆映射，并通过训练学习数据分布先验。反问题背后丰富的数学和物理理论为设计紧凑而有效的NN架构提供了理论指导，因此可以避免对大量数据的需求。在理论部分，本计画计划辨识许多反问题中常用的数学运算子，并针对这些运算子设计新颖的神经网路模组。两个这样的例子是伪微分算子和傅立叶积分算子。利用偏微分方程和数值线性代数的分析结果，这两种类型的运营商可以表示为NN模块的复杂和准确。在应用部分，该项目考虑了五个逆问题：电阻抗层析成像、光学层析成像、逆声/电磁散射、地震成像和走时层析成像。然后使用理论部分的模块，沿着现有的原语（如卷积NN）组装逆映射的NN。然后，整个NN使用训练数据进行端到端的训练。研究成果，如出版物和软件，将在公共领域提供。该项目还将在若干领域产生更广泛的影响。首先，该项目主张基于数学理论，稀疏性考虑和不变性/等变性原则的NN架构设计的更有原则的方法。其次，该项目涵盖了五个不同主题的逆问题。通过与科学家和工程师合作，PI计划将开发的技术应用于实际应用。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

In the past few years, deep learning has become the dominant approach for computer vision, image processing, speech recognition, and many other applications in machine learning and data science. This success is a synergy of several ingredients: (1) neural networks (NNs) as a flexible framework for representing high-dimensional functions and maps, (2) simple algorithms such as back-propagation and stochastic gradient descent for tuning the model parameters, (3) highly effective general software packages such as Tensorflow and Pytorch, and (4) unprecedented computing power. Despite the successes however, several key challenges remain: (1) the NN architectural design is still an art and it lacks basic mathematical principles in many cases, and (2) the NN training often requires an enormous amount of data, which is not available in many applications. Many computational problems in physical sciences also face the same challenges as those in data science: high-dimensionality, complicated or unspecified models, and large computational cost. Some well-known examples are many-body quantum systems, deterministic and stochastic control, molecular dynamics, uncertainty quantification, and inverse problems. It is natural to leverage the recent developments of NNs in the study of these problems. This project focuses on inverse problems, that is, recovering unknown problem parameters from observation data. It is a field of enormous importance, with applications in physics, chemistry, medicine, earth sciences, and defense. However, many inverse problems are known to be computationally challenging. This project will use deep learning as a means for solving these inverse problems more efficiently. The PI plans to develop a new applied linear algebra course with a machine learning focus and to organize workshops at the interface of deep learning and physical sciences. The project will train graduate and undergraduate students through the research and support one graduate student per year.As a tool for representing high-dimensional maps, neural nets (NN) offer a flexible way for representing the full inverse maps and learn the data distribution prior via training. The rich mathematical and physical theories behind inverse problems provide theoretical guidance for designing compact yet effective NN architectures and hence it is possible to avoid the need for an enormous amount of data. In the theoretical part, this project plans to identify the commonly-used mathematical operators in many inverse problems and design novel NN modules for these operators. Two such examples are the pseudodifferential operators and the Fourier integral operators. Using analytical results from partial differential equations and numerical linear algebra, both types of operators can be represented compactly and accurately as NN modules. In the application part, this project considers five inverse problems: electric impedance tomography, optical tomography, inverse acoustic/electromagnetic scattering, seismic imaging, and travel-time tomography. The NN for the inverse map is then assembled using the modules from the theory part, along with existing primitives such as convolutional NN. The whole NN is then trained end-to-end with the training data. The research results such as publications and software will be made available in public domain. The project will also lead to broader impacts in several areas. First, this project advocates for a more principled approach for NN architecture design based on mathematical theories, sparsity considerations, and invariance/equivariance principles. Second, the project covers inverse problems from five different topics. By working with scientists and engineers, the PI plans to apply the technologies developed to practical applications.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.