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Nonlinear Partial Differential Equations and Applications

非线性偏微分方程及其应用

基本信息

批准号：
2153822
负责人：
Panagiotis Souganidis
金额：
$ 29.28万
依托单位：
University of Chicago
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-08-01 至 2025-07-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2153822&HistoricalAwards=false
关键词：
Nonlinear Partial Differential Equations Applications

项目摘要

Fluctuations are ubiquitous both in real world contexts and in key technological challenges like, among others, thermal fluctuations in physical systems, algorithmic stochasticity in machine learning and modeling of weather patterns in climate dynamics. At the same time, such complex systems are subject to an abundance of influences, and depend on a large variety of parameters and interactions. In addition, for many complex phenomena most of the available information is very often "statistical" (random) and not "exact" (deterministic). A systematic understanding of the interplay of stochasticity and complex dynamical behavior aims at unveiling universal properties, irrespectively of the many details of the concrete systems at hand. Its development relies on the derivation and analysis of universal concepts for their scaling limits, capturing not only their average behavior, but also their fluctuations. Stochastic partial differential equations are the natural mathematical object to study and understand the role of fluctuations. Another very current and important issue arising in science and technology is the analysis, both theoretically and computationally, of problems that involve several disparate length-scales at once. For example, understanding, modeling and accurately predicting the behavior of materials at the macroscopic scale necessitates to consider their microscopic structure. Instead of considering materials in one scale and neglecting the finer scales, modern materials science increasingly explicitly and concurrently deals with models of a given material at many different scales. Homogenization and multiscale approaches are the two, respectively theoretical and computational, facets of the mathematical theory to study such problems. As a matter of fact, it is necessary to consider homogenization in random environments since periodicity is rather restrictive for the modeling of real materials. Growth models are natural mathematical models used in probability and mathematical physics to study stochastic partial differential equations exhibiting a universal scaling limit. Mean-field games are the ideal mathematical structures to study the quintessential problems in the social-economic sciences, which differ from physical settings because of the forward looking behavior on the part of individual agents. In this context, an agent aims to optimize certain criteria which, together with her/his dynamics, depend on the other agents and their actions. Agents react, anticipate and strategize instead of simply reacting instantaneously. Examples of applications include the modeling of the macro-economy and conflicts in the modern era. In both cases, a large number of agents interact strategically in a stochastically evolving environment, all responding to partly common and partly idiosyncratic incentives, and all trying to simultaneously forecast the dynamic decisions of others. Some mean-field games models in, for example, telecommunications are naturally set on networks (graphs). This raises the need of the development of the mathematical tools to study equations on graphs and to understand their behavior across nodes. The project provides research training opportunities for graduate students.The project is a continuation of the PI's program to develop novel methodologies and techniques for the qualitative and quantitative study of nonlinear first- and second-order deterministic and stochastic partial differential equations (PDEs and SPDEs, respectively) arising in natural and social sciences and engineering. The emphasis is on (i) PDEs with multiplicative ``rough'' path dependence; (ii) homogenization in random media; (iii) well-posedness in domains with singularities; (iv) mean-field games; and (v) convergence of growth models. Nonlinear, first- and second-order partial differential equations with rough and, in particular, stochastic time dependence arise naturally in the study of fluctuations. The further development of the theory of pathwise solutions is important, for it allows to study of new classes of nonlinear SPDEs, and is expected to play a crucial role in applied areas by providing the tools to analyze previously intractable models. Studying qualitatively and quantitatively stochastic homogenization problems requires the development of novel arguments and methodologies to address the loss of compactness when going from periodic to stationary ergodic media. The analysis of the behavior of scaled growth models gives rise to new equations. Their well-posedness requires the refinement of some of the by now classical tools from the theory of viscosity solutions. It is by now well understood that a very important element of the mean-field game theory is the so-called master equation, an infinite-dimensional partial differential equation which subsumes in a single equation both the individual and collective behaviors of agents. In spite of considerable progress in the study of the properties of the smooth solution of the master equation in the presence of both idiosyncratic and common noises, less is known in the absence of the former in which case smooth solutions do not exist in general. Another important question in this context is the development of a notion of weak solution in the relevant for application nonmonotone setting. Two of the most important questions in the study of equations of graphs is the well-posedness across vertices and the identification of the correct coupling condition.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.

波动在真实的世界背景和关键技术挑战中无处不在，例如物理系统中的热波动，机器学习中的算法随机性和气候动力学中的天气模式建模。同时，这种复杂的系统受到大量的影响，并取决于各种各样的参数和相互作用。此外，对于许多复杂的现象，大多数可用的信息往往是“统计的”（随机的），而不是“精确的”（确定的）。系统地理解随机性和复杂动力学行为的相互作用，旨在揭示普遍的性质，而不管手头的具体系统的许多细节。它的发展依赖于对标度极限的普遍概念的推导和分析，不仅捕捉它们的平均行为，而且捕捉它们的波动。随机偏微分方程是研究和理解波动作用的自然数学对象。另一个在科学和技术中出现的非常当前和重要的问题是，从理论上和计算上分析同时涉及几个不同长度尺度的问题。例如，在宏观尺度上理解、建模和准确预测材料的行为需要考虑它们的微观结构。现代材料科学不再以一个尺度来考虑材料而忽略更精细的尺度，而是越来越明确地同时处理给定材料在许多不同尺度上的模型。均匀化和多尺度方法是研究这类问题的数学理论的两个方面，分别是理论和计算。事实上，由于周期性对真实的材料的建模有相当大的限制，因此在随机环境中考虑均匀化是必要的。增长模型是概率论和数学物理学中用来研究具有普遍尺度极限的随机偏微分方程的自然数学模型。平均场博弈是研究社会经济科学中典型问题的理想数学结构，它不同于物理环境，因为个体代理人的前瞻性行为。在这种情况下，一个代理的目的是优化某些标准，连同她/他的动态，取决于其他代理和他们的行动。智能体做出反应、预测和制定战略，而不是简单地立即做出反应。应用的例子包括宏观经济和现代冲突的建模。在这两种情况下，大量的代理商在随机演变的环境中进行战略性互动，所有这些代理商都对部分共同和部分特殊的激励做出反应，并且所有代理商都试图同时预测其他代理商的动态决策。例如，电信领域的一些平均场博弈模型自然是建立在网络（图）上的。这就需要开发数学工具来研究图上的方程，并了解它们在节点上的行为。该项目为研究生提供了研究培训机会。该项目是PI计划的延续，旨在为自然科学和社会科学以及工程领域中出现的非线性一阶和二阶确定性和随机偏微分方程（PDE和SPDE）的定性和定量研究开发新的方法和技术。重点是（一）与乘法“粗糙”路径依赖偏微分方程;（二）均匀化随机介质;（三）在域奇异性适定性;（四）平均场游戏;和（五）收敛的增长模型。非线性，一阶和二阶偏微分方程的粗糙，特别是随机时间依赖性自然出现在波动的研究。路径解理论的进一步发展是重要的，因为它允许研究新的非线性SPDE类，并预计将在应用领域发挥关键作用，提供工具来分析以前棘手的模型。定性和定量研究随机均匀化问题需要发展新的论点和方法，以解决从周期性到平稳遍历介质时的紧凑性损失。对尺度增长模型的行为分析产生了新的方程。他们的适定性需要改进的一些经典的工具，从理论的粘度解决方案。到目前为止，人们已经很好地理解了平均场博弈论的一个非常重要的元素是所谓的主方程，这是一个无限维的偏微分方程，它将个体和个体的集体行为都包含在一个方程中。尽管有相当大的进展，在研究的性质的光滑的解决方案的主方程的存在下的特质和共同的噪音，少有人知，在没有前者的情况下，在这种情况下，光滑的解决方案一般不存在。在这方面的另一个重要问题是弱解的概念在相关的应用程序非单调设置的发展。图方程研究中最重要的两个问题是顶点间的适定性和正确耦合条件的识别。该奖项反映了NSF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。