Learning Partial Differential Equation (PDE) and Beyond

学习偏微分方程 (PDE) 及其他内容

• 批准号: 2309551
• 负责人: Hong-Kai Zhao
• 金额: $ 25万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2309551&HistoricalAwards=false
• 关键词: Learning; Partial; Differential; Equation; PDE

Partial differential equations (PDE) are powerful tools in modeling physical, biological and social worlds. They are used to understand, predict, simulate and infer complex phenomena and quantities across disciplines of science and engineering, from heat transfer, wave propagation, and biological systems to weather forecast, climate change, and economic and social behavior. In the past, most of the equations are derived from basic physical laws and assumptions. With the advancement of technologies, abundant data are available from measurements and observations in many complex situations where the underlying model is not yet available or not accurate enough. Whether one can learn a PDE model directly from observed data becomes a both interesting and important question. In this project, a new perspective of PDE learning based on observed solution data, from theory and methodology to efficient algorithms and applications, will be developed. Mathematical theories and computational tools developed in this project will be useful for researchers and practitioners in many disciplines that have to deal with computational modeling of systems using differential equations. The developed mathematical tools and computational algorithms will be disseminated broadly for advancing scientific and technological progress. Integration with education at different levels will be designed to provide training for young computational mathematicians. A data driven approach to PDE learning can provide a useful tool to discover new or more accurate quantitative models for complex systems and dynamics not only in traditional physical sciences but also in biological, social and other disciplines where basic principles and laws are not available but data are abundant. In return, the learned PDE model will provide more understandings and insights of the underlying problem as well as computational tools. For any data driven, data intensive and data-enabled approach, it is a fundamental task to study and understand the complexity of the underlying problem, model, or data and using this understanding to design effective representation, dimension reduction and fast algorithm that are problem specific. This project is aimed to address a few important questions and challenges which include (1) understanding and characterization of the data, e.g., how much data are available and how much data are needed, (2) data selection and data processing to deal with numerical errors and noise, and (3) development of efficient and accurate algorithms based on different formulations. Systematic experiments will be designed to verify the developed theories and test proposed methodologies and algorithms. The insights obtained in this project will also provide new perspectives on mathematics and physics based learning.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.

偏微分方程（PDE）是建模物理、生物和社会世界的有力工具。它们用于理解，预测，模拟和推断科学和工程学科的复杂现象和数量，从传热，波传播和生物系统到天气预报，气候变化以及经济和社会行为。在过去，大多数方程都是从基本的物理定律和假设推导出来的。随着技术的进步，在许多基础模型尚不具备或不够准确的复杂情况下，可以从测量和观测中获得丰富的数据。人们是否可以直接从观测数据中学习偏微分方程模型成为一个既有趣又重要的问题。在这个项目中，一个新的角度偏微分方程学习的基础上观察到的解决方案的数据，从理论和方法，以有效的算法和应用程序，将开发。在这个项目中开发的数学理论和计算工具将是有用的研究人员和从业人员在许多学科，必须处理使用微分方程的系统的计算建模。开发的数学工具和计算算法将广泛传播，以推动科学和技术进步。将与不同级别的教育相结合，为年轻的计算数学家提供培训。数据驱动的偏微分方程学习方法不仅可以为传统的物理科学，而且还可以为生物，社交及其他学科提供一个有用的工具，以发现新的或更精确的复杂系统和动力学的定量模型，这些学科的基本原理和定律不可用，但数据丰富。作为回报，学习的PDE模型将提供对潜在问题的更多理解和见解以及计算工具。对于任何数据驱动、数据密集和数据启用的方法，研究和理解底层问题、模型或数据的复杂性，并使用这种理解来设计针对特定问题的有效表示、降维和快速算法是一项基本任务。该项目旨在解决一些重要的问题和挑战，其中包括（1）数据的理解和表征，例如，有多少数据可用以及需要多少数据，（2）数据选择和数据处理以处理数值误差和噪声，以及（3）开发基于不同公式的有效和准确的算法。系统的实验将被设计来验证所开发的理论和测试所提出的方法和算法。该奖项反映了NSF的法定使命，并被认为值得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来支持。