Seminar: Numerical Methods for Partial Differential Equations

Building 2, 255

Speaker: Chris Rackauckas (JuliaHub)Title: Building a General PDE Solving Framework with Symbolic-Numeric Scientific Machine LearningAbstract:The dream is to be able to write down a symbolic expression of a PDE and just get a solution. Not only get a solution, but get that solution efficiently. The question is, how do we effectively build a software ecosystem to achieve this goal, and what are the remaining mathematical problems that are necessary to solve in order to fill the gaps? In this talk we will discuss the Julia SciML ecosystem's approach to a general PDE solving framework. We will focus on 5 aspects: (1) a symbolic structural hierarchical classification of PDEs for shuttling high level PDE descriptions to appropriate solution approaches, (2) new time-stepping methods for accelerating the solution of semi-discretized equations and generalizing approaches from PDEs to structured ODE forms, (3) symbolic-numeric approaches for automating the transformation of semi-discretizations to simpler and more numerically stable equations before solving, and (4) new improvements to adjoint methods to decrease the memory requirements for differentiation of PDE solutions, and (5) scientific machine learning approaches to generate accelerated approximations (surrogates) to PDE simulators. We will demonstrate these methods with open source software that starts from symbolic expressions and solves industrial-scale PDEs with minimal lines of code.https://mit.zoom.us/j/93945001205