Nonlocality in Continuum Mechanics, Population Dynamics, and Neural Networks

非局部系统已被用来预测动态断裂中的裂纹和损伤，对图像进行去噪，并在二元流体和相分离中提供更准确的模型。该项目涉及用于对此类应用中遇到的不连续、奇异或不规则行为进行建模的理论。目前，它们被应用于研究抗损伤的盖子玻璃，例如用于手机、复合材料飞机、风力涡轮机叶片、石油和天然气开采的玻璃，以及用于核能行业的抗损伤盖子玻璃，如在极端情况下混凝土的破裂和乏核燃料的储存。该项目还旨在开发数学工具来促进对非局部模型的理解，目的是回答以下问题：特定的力会对材料造成多大的破坏，在给定的增长、衰退和扩散规则下，种群的组成是什么，以及如何提高神经网络的效率和有效性。该项目为研究生和本科生的研究性培训提供了机会。研究人员将对板材动态断裂、图像处理和相变中出现的非局部模型进行数学和计算研究。他们将通过考虑隐藏状态中的层的累积效应，创建涉及非局部性的连续深度的神经网络模型。为了在应用数学领域内为非局部模型的研究提供理论和方法论基础，研究人员将把现有的技术从局部理论调整到非局部框架，并开发新的方法。来自学术界和研究实验室的应用数学家将提供建模和计算方面的专业知识。该项目由DMS应用数学计划和既定的激励竞争研究计划(EPSCoR)联合资助。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。

Nonlocal systems have been used to predict cracks and damage in dynamic fracture, to denoise images, and to provide more accurate models in binary fluids and in phase separation. This project concerns theories employed to model discontinuous, singular, or irregular behavior encountered in such applications. Currently, they are applied to study damage-resistant cover glass such as the one produced for cellphones, composite-based aircrafts, blades for wind turbines, oil and gas extraction, and in the nuclear energy industry, as in fracturing of concrete in extreme situations and storage of spent nuclear fuel. The project also aims to develop mathematical tools to advance the understanding of nonlocal models, with the goal of answering questions such as how much damage a specified force will induce in a material, what is the make-up of a population under given growth, decay, and diffusion rules, and how can efficiency and efficacy of neural networks performance be improved. The project provides opportunities for research training of graduate and undergraduate students. The investigators will conduct a mathematical and computational investigation of nonlocal models that arise in dynamic fracture in plates, image processing, and phase transitions. They will create neural networks models with continuous depth, which involve nonlocality, by considering a cumulative effect of layers in the hidden states. To provide a theoretical and methodological foundation for the study of nonlocal models within the realm of applied mathematics, the researchers will adapt existing techniques from the local theory to the nonlocal framework, as well as develop new methods. Applied mathematicians from academia and research laboratories will provide expertise in modeling and computational aspects.This project is jointly funded by the DMS Applied Mathematics Program and the Established Program to Stimulate Competitive Research (EPSCoR).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.