Coupling PDE-Based Computational Inversion and Learning Via Weighted Optimization

通过加权优化耦合基于偏微分方程的计算反演和学习

• 批准号: 2309802
• 负责人: Kui Ren
• 金额: $ 40.95万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2309802&HistoricalAwards=false
• 关键词: Coupling; PDE; Based; Computational; Inversion

This project aims at developing a mathematically rigorous computational framework to improve classical image reconstruction algorithms in several application areas, such as medical imaging and seismic imaging. Through intelligent utilization of the recent advances in deep learning techniques, the investigator plans to implement and analyze novel computational inversion methods that have the potential to be significantly more efficient than existing image reconstruction methods. This project also involves an integrated educational component that aims at training students at different levels. The investigator intends to organize a reading group for junior researchers to work on simplified problems from this project to enrich their research experiences. Research internships on the topics of this project will be offered to undergraduate students from the investigator's institution as well as some high school students in its surrounding area.The main technical novelty of the project lies in an efficient computational coupling strategy that integrates offline and online reconstruction algorithms to take their complementary advantages. The coupling is implemented with a weighted optimization method, a general framework to approximate novel objective functions such as those based on the Wasserstein metrics induced in optimal transport theory. In a nutshell, the proposed weighted optimization methods make it possible to reliably train neural networks to learn the dominant components (when represented in an appropriate basis) of an inverse operator. This learned approximate inverse is then utilized to improve model-based iterative inversion schemes. Specifically, the primary proposed research efforts in this project include three main components: (i) to develop a systematic understanding of the stability property of different components of inverse operators and design corresponding weighted optimization schemes to focus on such components in the learning process; (ii) to develop a general computational framework to couple the complementary benefits of learning and classical iterative reconstruction methods; and (iii) to characterize the impact of learning uncertainty on the reconstruction results from new data.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.

该项目旨在开发一个数学上严格的计算框架，以改进几个应用领域的经典图像重建算法，如医学成像和地震成像。通过智能地利用深度学习技术中的最新进展，研究人员计划实施和分析新的计算反演方法，这些方法有可能比现有的图像重建方法更有效。该项目还包括一个旨在培训不同级别学生的综合教育部分。研究人员打算为初级研究人员组织一个阅读小组，以解决这个项目中的简化问题，以丰富他们的研究经验。该项目将为研究机构的本科生以及周边地区的一些高中生提供研究实习机会。该项目的主要技术创新在于一种高效的计算耦合策略，该策略整合了离线和在线重建算法，以发挥它们的互补优势。这种耦合是通过加权最优化方法实现的，这是一种逼近新目标函数的通用框架，例如基于最优传输理论中引入的沃瑟斯坦度量的目标函数。简而言之，所提出的加权优化方法使得可靠地训练神经网络来学习逆算子的主要分量(当以适当的基来表示时)成为可能。这种学习的近似逆然后被用来改进基于模型的迭代反演方案。具体地说，该项目的主要拟议研究工作包括三个主要部分：(I)系统地了解逆算子不同组件的稳定性属性，并设计相应的加权优化方案，以关注学习过程中的这些组件；(Ii)开发一个通用计算框架，将学习的互补好处与经典迭代重建方法结合起来；以及(Iii)表征学习不确定性对新数据重建结果的影响。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。