Innovation of Numerical Methods for High-Dimensional Partial Differential Equations

高维偏微分方程(PDE)是科学和工程问题中普遍存在的具有多个自由度的问题。重要的例子包括但不限于，多体量子力学、化学系统的动力学、复杂系统的学习和控制，以及高维数据的光谱方法。高维偏微分方程组的数值解，如多体薛定谔方程，一直是最大的科学挑战之一，即使在今天的计算能力和算法进步的情况下，仍然是一项艰巨的任务。这个项目涉及数学分析和数值算法开发的交叉，以解决在解决这些非线性，高维方程的挑战。这些研究成果将促进我们对量子多体问题和其他高维偏微分方程组问题的数学理解和数值算法的改进。该项目涉及新课程开发和应用数学和计算科学研究生的培训。该研究项目旨在通过借鉴和进一步开发计算物理、量子化学和机器学习的最新进展的思想和工具，来创新高维PDE的数值策略。特别是，高维函数的非线性参数化和高维分布的抽样的现代技术。具体地说，研究人员将(1)设计和分析具有对称约束的高维函数的神经网络参数化，以及(2)开发和分析用于训练高维偏微分方程组的神经网络解的有效的自适应采样策略。这项研究结合了数学分析和算法设计，在高维偏微分方程的数值方法方面取得了进展。这一奖项反映了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。

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.