Algorithmic and analytical development for solving and learning nonlocal models

非局部相互作用在自然系统中变得越来越突出，这导致了人们对非局部建模的兴趣日益浓厚。非局部模型在力学、材料科学、生命科学、数据科学和社会科学等学科中的应用，引起了应用数学家和计算数学家的广泛关注。本项目旨在研究一类重要的非局部模型，涉及有限范围的非局部相互作用。在各种现实世界的应用程序中，这些模型可以作为传统的基于PDE的模型的替代或补充，并帮助将PDE模型与离散模型连接起来。该项目与我们对信息和智能科学计算的愿景是一致的，这将广泛地促进计算科学和各种科学和工程研究领域的进步，特别是在世界日益联系（和非本地）的情况下。研究员将继续与不同研究机构的科学家建立密切的工作关系，以促进合作研究工作，并加强对年轻学生和初级研究人员的培训和指导。将努力确保将新的研究成果及时转化和整合到增强的应用建模和仿真能力中。培训至少一名研究生的主题提出的研究是预期的。非局部模型不同于由偏微分方程（PDEs）表示的更常见的局部模型，因为它们采用积分形式的非局部算子，可以显式地解释非局部相互作用并提供更一般的建模选择。非局部模型的数学理论和数值分析虽然取得了很大的进展，但仍处于起步阶段。在非局部建模、分析和计算的各个方面，许多关键问题仍未得到解答。在这个项目中，研究者旨在通过一个集成的解决方案和学习过程来推进具有有限范围相互作用的非局部模型的数学和算法发展。一方面，研究者将关注与非局部模型的制定、离散化和学习相关的几个关键问题，特别是涉及物理边界或界面。这表现在一些具体的任务中，这些任务涉及解决边界或边界附近的非均匀数据和/或异构非局部相互作用的非局部问题的分析和算法开发。众所周知，这些问题具有挑战性，它们令人满意的解决方案将对应用程序产生重大影响。同时，该项目将帮助我们建立集成的建模和学习过程。将开发新的策略和算法，以帮助为非局部模型的制定和学习以及相关的数值方法带来潜在的变革，包括它们的开发，分析和应用。通过与本地偏微分方程、分数和离散图模型的密切联系，该项目还将揭示许多其他相关学科的数学和计算研究。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智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Nonlocal interactions have become increasingly prominent in natural systems, which has led to the growing interest in nonlocal modeling. The study of nonlocal models has attracted much attention from applied and computational mathematicians, motivated by applications in disciplines like mechanics, materials science, life science, data science, and social science. This project aims to study an important class of nonlocal models involving nonlocal interactions of a finite range. In a variety of real world applications, these models can serve as an alternative or complement to traditional PDE-based models and help connect PDE models with discrete models. The project is consistent with our vision of informative and intelligent scientific computing, which will broadly contribute to the advance of computation science and various scientific and engineering research fields, particularly as the world becomes increasingly connected (and nonlocal). The investigator will continue to build a close working relationship with research scientists at different institutions to facilitate the collaborative research effort and to strengthen the training and mentoring of young students and junior researchers. Effort will be made to ensure the timely translation and integration of new research findings into enhanced modeling and simulation capabilities for applications. Training at least one graduate student on the topics of the proposed research is expected.Nonlocal models differ from the more common local models represented by partial differential equations (PDEs) in that they employ nonlocal operators in integral forms, which can account for nonlocal interactions explicitly and provide more general modeling choices. Despite much progress, the mathematical theory and numerical analysis of nonlocal models are still at a nascent stage. Many key questions remain unanswered in all aspects of nonlocal modeling, analysis, and computation. In this project, the investigator aims to advance the mathematical and algorithmic development of nonlocal models with a finite range of interactions through an integrated solution and learning process. On one hand, the investigator will focus on a few key questions related to the formulation, discretization, and learning of nonlocal models, particularly involving physical boundaries or interfaces. This is represented by some specific tasks concerning the analytical and algorithmic development for solving nonlocal problems with inhomogeneous data at or near the boundary and/or with heterogeneous nonlocal interactions. These problems are notoriously challenging and their satisfactory solutions will have a significant impact on applications. Meanwhile, the project will help us build integrated modeling and learning processes. New strategies and algorithms will be developed to help bring potentially transformative changes to how nonlocal models are formulated and learned and to the related numerical methods, including their development, analysis, and application. By drawing close connections to local PDEs, fractional, and discrete graph models, the project will also shed light on the mathematical and computational studies of many other related subjectsThis 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.