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Nonlocality in Continuum Mechanics, Population Dynamics, and Neural Networks

连续体力学、群体动力学和神经网络中的非定域性

基本信息

批准号：
2109149
负责人：
Petronela Radu
金额：
$ 34.31万
依托单位：
University of Nebraska-Lincoln
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-08-15 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2109149&HistoricalAwards=false
关键词：
Nonlocality Continuum Mechanics Population Dynamics

项目摘要

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.

非局部系统已被用于预测动态断裂中的裂纹和损伤，对图像进行降噪，并在二元流体和相分离中提供更精确的模型。该项目涉及用于模拟此类应用中遇到的不连续、单一或不规则行为的理论。目前，它们被应用于研究抗损伤盖板玻璃，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，以及核能工业中，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，以促进对非局部模型的理解，其目标是回答诸如特定力会在材料中引起多大的损伤，在给定的生长、衰减和扩散规则下种群的构成是什么，以及如何提高神经网络性能的效率和功效等问题。该项目为研究生和本科生提供了研究训练的机会。研究人员将对在板的动态断裂、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。他们将创建具有连续深度的神经网络模型，该模型涉及非局域性，通过考虑隐藏状态中的层的累积效应。为了为应用数学领域的非局部模型研究提供理论和方法基础，研究人员将把现有的局部理论技术应用于非局部框架，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应用数学项目和促进竞争性研究的既定项目（EPSCoR）共同资助。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。