Machine Learning for Partial Differential Equations

Partial differential equations (PDEs) have been a cornerstone of mathematical physics and engineering design for over 250 years, since the introduction of the one-dimensional wave equation by d'Alembert in 1752 [20]. PDEs provide a formal mathematical infrastructure for relating how quantities of interest change in several variables, typically space and time. As such, PDEs provide a foundational description of the governing equations of many canonical spatio-temporal physical systems, including electrodynamics, quantum mechanics, fluid mechanics, heat transfer, etc. Today, nearly every aspect of our engineered world is based in some way on the predictive capability of PDEs, from structural modeling of buildings and bridges, to the design of aircraft and other vehicles, to the thermal and electromagnetic management systems in modern portable electronics.