Innovation of Numerical Methods for High-Dimensional Partial Differential Equations

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.