Our understanding and ability to compute the solutions to nonlinear partial differential equations has been strongly curtailed by our inability to effectively parameterize the inertial manifold of their solutions. I will discuss our ongoing efforts for using machine learning to advance the state of the art, both for developing a qualitative understanding of "turbulent" solutions and for efficient computational approaches. We aim to learn parameterizations of the solutions that give more insight into the dynamics and/or increase computational efficiency. I will discuss our recent work using machine learning to develop models of the small scale behavior of spatio-temporal complex solutions, with the goal of maintaining accuracy albeit at a highly reduced computational cost relative to a full simulation.
References: https://www.pnas.org/content/116/31/15344 and https://arxiv.org/pdf/2102.01010.pdf
Michael P. Brenner is the Michael F. Cronin Professor of Applied Mathematics & Applied Physics and a Professor of Physics at Harvard University. Additionally, Brenner is a Research Scientist at Google Research. Brenner earned a Bachelor’s of Science degree at the University of Pennsylvania and obtained a doctorate at the University of Chicago. Brenner’s research focuses on methods and ideas of applied mathematics to address a wide variety of problems in science and engineering. His current research projects range from efforts to understand the design rules for creating synthetic materials with life-like properties, to efforts to use machine learning to accelerate scientific discovery, to specific problems in fluid mechanics, material science and biology. He recently wrote a book on Science and Cooking, published by WH Norton, based on a class he developed at Harvard.