CAREER: Learning Kernels in Operators from Data: Learning Theory, Scalable Algorithms and Applications

从数据中学习算子中的核/函数是连接计算数学、反问题、统计推理和机器学习的一个新的前沿。这样的算子学习问题出现在物理、生物、工程、经济学和水文学等学科的各种应用中。这些核/函数代表粒子之间相互作用的内在物理规律或运算符的内在结构。因此，它们就像物理学中的物理引力定律一样基本，在应用中粉碎了对模型的洞察。因此，当数据规模增大时，构造稳健的收敛估计以揭示对数据不敏感的内在规律是至关重要的。这个项目将开发一种统一的计算方法，用于从高维或无限维数据中学习算子中的核/函数，并且产品是可扩展的算法，在统一的变分框架下具有性能保证。它将研究四组应用，其中目标是恢复PDE算子、非马尔科夫过程、状态空间模型和加权卷积算子中的核/函数。值得注意的是，涵盖可辨识性和正则化的系统学习理论将解决来自非局部依赖和高维数据的挑战。此外，这一理论为一般框架奠定了数学基础，使算法和理论适用于反问题，远远超出了本提案中研究的范围。最后，这项研究将被整合到多学科的本科生和研究生教学中，促进计算数学在统计/机器学习中的基础作用，并为研究生提供许多研究机会。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。

Learning kernels/functions in operators from data is a new frontier that bridges computational mathematics, inverse problems, statistical inference, and machine learning. Such operator learning problems arise in various applications in disciplines such as physics, biology, engineering, economics, and hydrography. These kernels/functions represent the intrinsic physical laws of interactions between particles or the inherent structures of the operators. Thus, they are as fundamental as the physical laws of gravity in physics, shredding insight into models in their applications. Therefore, it is paramount to construct robust convergent estimators when the data size increases to reveal the intrinsic laws that are not sensitive to the data. This project will develop a unified computational approach for the nonparametric learning of kernels/functions in operators from high- or infinite dimensional data, and the products are scalable algorithms with performance guarantees in a unifying variational framework. It will study four groups of applications where the goal is to recover the kernels/functions in PDE operators, non-Markovian processes, state-space models, and weighted convolution operators. Notably, a systematic learning theory, covering identifiability and regularization, will address the challenges from nonlocal dependence and high-dimensional data. Furthermore, this theory builds the mathematical foundations for a general framework, making the algorithms and theory applicable to inverse problems far beyond those studied in this proposal. Finally, the research will be integrated into undergraduate and graduate teaching in multiple disciplines, promote the fundamental role of computational math in statistical/machine learning, and provide many research opportunities for graduate students.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.