Innovative Numerical Methods for High-Dimensional Applications

该项目旨在为科学和工程中的一类重要系统开发有效的数值算法，这些系统是用高维偏微分方程（PDE）建模的，如多体薛定谔方程。这类系统的例子包括多体量子力学、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。高维偏微分方程的数值求解一直是计算科学中最大的挑战之一，即使在今天的计算能力和算法的进步仍然是一个艰巨的任务。高效的数值模拟为科学理解的重大突破提供了机会。该项目将采用现代技术为这些重要系统开发新的有效数值算法。本研究计画结合数学分析与演算法设计，在高阶偏微分方程数值方法上取得进展。该研究将借鉴并进一步发展计算物理，量子化学和机器学习的最新进展的想法和工具。特别是，该项目将使用现代技术进行大规模优化和高维函数的非线性参数化。具体而言，PI将（1）开发用于大规模特征值问题的新型高效坐标算法，（2）开发和分析基于高维PDE解的神经网络参数化的高效方法。该奖项反映了NSF的法定使命，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，被认为值得支持。

This project aims to develop efficient numerical algorithms for a class of important systems in science and engineering that are modeled with high-dimensional partial differential equations (PDE) such as the many-body Schrödinger equation. Examples of such systems include 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 PDE has been one of the greatest challenges in computational science and remains a formidable task even with today's computational power and algorithmic advances. Efficient numerical simulations present opportunities for major breakthroughs in scientific understanding. This project will employ modern techniques to develop novel efficient numerical algorithms for such important systems. Graduate students will be trained through involvement in the research.The research project combines mathematical analysis and algorithmic design to make progress in numerical methods for high-dimensional PDE. The research will draw from and further develop ideas and tools from recent advances in computational physics, quantum chemistry, and machine learning. In particular, the project will use modern techniques for large-scale optimization and nonlinear parameterization of high-dimensional functions. Specifically, the PI will (1) develop novel highly efficient coordinate algorithms for large-scale eigenvalue problems, and (2) develop and analyze efficient methods based on neural-network parameterization of the solutions for high-dimensional PDE.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.