Solving Multiscale Problems and Data Classification with Subsampled Data by Integrating Partial Differential Equation Analysis with Data Science

在许多实际应用中，人们经常需要为大规模问题提供大量的解决方案，但只使用子采样数据和物理模型的部分信息。现有的计算解算器不能直接用于此目的。另一方面，在数据科学中已经开发了许多强大的技术，以极高的效率和低计算复杂性来表示和压缩数据以获得有用的信息。这些方法成功的一个关键因素是利用这些高维数据的一些特殊特征。该项目的目的是将物理模型与数据科学相结合，开发新一代的计算方法，仅使用次采样数据和物理模型的部分知识就可以解决大规模的物理或数据科学问题。数学分析将有助于揭示某些重要的解决方案结构，以便人们可以使用数据科学的技术为感兴趣的数量提供准确的近似值。如果不识别这些特殊的解决方案结构并使用物理模型作为约束，当前的数据科学技术就不能直接用于实现PI的目标。这个项目可以对计算科学和数据科学社区，对国家技术和社会产生重大影响。研究生的参与将会产生额外的影响。这项研究为数学分析、物理建模和数据科学提供了坚实的训练。他们在这个项目中接受的跨学科训练对他们未来在数学和科学领域的职业生涯非常重要。数据科学的最新进展为计算科学提供了巨大的机会。挖掘高维数据的一些特殊特征是数据科学取得成功的关键。传统的PDE求解器没有充分利用特殊的解结构。PDE分析和数据科学相辅相成。PDE分析可以识别一些重要的解结构，这些解结构可以帮助PI设计更有效的深度生成网络来解决物理问题。如果没有PDE分析的指导，当前机器学习算法在多尺度问题上的幼稚应用将会失败。非凸优化问题的解很容易陷入局部最小值，并可能收敛到错误的解。PI将确定使这种集成成功的一些关键因素，研究可以压缩哪种类型的偏微分方程，以及可以使用什么算法来近似使用一小部分子采样数据和部分物理模型知识的感兴趣的数量。该研究还将为解决多尺度问题的一些深度学习方法提供有价值的理论认识。PI将考虑逆问题和正问题。对于正演问题，他将开发一种新的基于下采样数据的多尺度方法，在保证精度的情况下重建解。对于反问题，PI将其作为贝叶斯反问题发布，并使用深度生成网络。该方法的一个重要组成部分是引入一种新的多尺度可逆流来近似运输图，这使得PI能够开发一种有效的采样算法来捕获后验分布中的多模态。该奖项反映了美国国家科学基金会的法定使命，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 many practical applications, one often needs to provide solutions to quantities of interest to a large-scale problem but with only subsampled data and partial information of the physical model. Existing computational solvers cannot be used directly for this purpose. On the other hand, many powerful techniques have been developed in data science to represent and compress data for useful information with extreme efficiency and low computational complexities. A crucial factor for the success of these methods is to exploit some special features in these high-dimensional data. The purpose of this project is to integrate physical models with data science to develop a new generation of computational methods that can solve large-scale physical or data science problems using only subsampled data and partial knowledge of the physical model. The mathematical analysis will help reveal certain important solution structures so that one can use techniques from data science to give accurate approximations for those quantities of interest. Without identifying these special solution structures and using the physical model as a constraint, the current techniques from data science cannot be used directly to achieve PI's goal. This project can have a substantial impact for the computational science and data science communities, for national technology and society. Additional impact will be the involvement of graduate students. This research provides a solid training in mathematical analysis, physical modeling, and data science. The interdisciplinary training they receive in this project will be very important for their future careers in mathematics and science. The recent advances in data science offer tremendous opportunities for computational sciences. A key to the success in data science is to exploit some special features in the high-dimensional data. Traditional PDE solvers have not taken full advantage of the special solution structures. PDE analysis and data science complement each other. PDE analysis can identify some important solution structures that can help the PI to design a more effective deep generative network to solve the physical problem. Without the guidance from the PDE analysis, naive application of current machine learning algorithms to multiscale problems would fail. The solution of the nonconvex optimization problem can easily get stuck in local minimum and may converge to the wrong solution. The PI will identify some key ingredients that would make such integration successful, investigate what type of PDEs can be compressed and what algorithms can be used to approximate quantities of interest with a small percentage of subsampled data and partial knowledge of the physical model. This research will also provide valuable theoretical understanding of some deep learning methods for solving multiscale problems. The PI will consider both inverse and forward problems. For the forward problem, he will develop a novel multiscale method based on subsampled data to reconstruct the solution with guaranteed accuracy. For the inverse problem, the PI will post it as a Bayesian inverse problem and use Deep Generative Networks. An essential ingredient in this approach is to introduce a novel multiscale invertible flow to approximate the transport map, which enables the PI to develop an efficient sampling algorithm to capture the multiple modes in the posterior distribution.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.