Innovation of Numerical Methods for High-Dimensional Partial Differential Equations

高维偏微分方程数值方法的创新

• 批准号: 2309378
• 负责人: Jianfeng Lu
• 金额: $ 40万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2309378&HistoricalAwards=false
• 关键词: Innovation; Numerical; Methods; Dimensional; Partial

High dimensional partial differential equations (PDEs) arise ubiquitously from scientific and engineering problems with many degrees of freedom. Important examples include, but are not limited to, many-body quantum mechanics, dynamics of chemical systems, learning and control of complex systems, and spectral methods for high dimensional data. The numerical solution of high dimensional PDEs, such as the many-body Schrodinger equations, has been one of the greatest scientific challenges and remains a formidable task even with today's computational power and algorithmic advances. This project involves cross-fertilization of mathematical analysis and numerical algorithm development to address challenges in solving these nonlinear, high dimensional equations. The research results will advance our mathematical understanding and improve numerical algorithms for quantum many-body problems and other high dimensional PDE problems. The project involves new curriculum development and training of graduate students in applied mathematics and computational science.The research project aims to innovate numerical strategies for high dimensional PDEs by drawing from and further developing ideas and tools from recent advances in computational physics, quantum chemistry, and machine learning. In particular, modern techniques for nonlinear parametrization of high dimensional functions and sampling for high dimensional distributions. Specifically, the investigator will (1) design and analyze neural-network parametrization for high-dimensional functions with symmetry constraints, and (2) develop and analyze efficient adaptive sampling strategies for training neural-network solutions to high-dimensional PDEs. The research combines mathematical analysis and algorithm design to make progress in numerical methods for high dimensional PDEs.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.

高维偏微分方程（PDE）是由许多自由度的科学和工程问题无处不在的。重要的例子包括但不限于多体量子力学，化学系统动力学，复杂系统的学习和控制以及用于高维数据的光谱方法。 高维PDE的数值解决方案，例如多体Schrodinger方程，一直是最大的科学挑战之一，即使在当今的计算能力和算法进步中，也仍然是一项艰巨的任务。该项目涉及数学分析和数值算法开发的交叉利用，以解决解决这些非线性高维方程的挑战。研究结果将提高我们的数学理解，并改善量子多体问题和其他高维PDE问题的数值算法。 该项目涉及应用数学和计算科学研究生的新课程开发和培训。该研究项目旨在通过从最新的计算物理，量子化学和机器学习的进一步发展中提出进一步发展的思想和工具来创新高维PDE的数值策略。特别是，高维函数非线性参数化和高维分布的采样的现代技术。具体而言，研究者将（1）设计和分析具有对称性约束的高维函数的神经网络参数化，（2）开发和分析有效的自适应抽样策略，以训练高维PDE的神经网络解决方案。该研究结合了数学分析和算法设计，以在高维PDE的数值方法中取得进展。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估审查标准来通过评估来获得支持的。