Andrew Stuart: Learning Solution Operators for PDEs
Channel: Oxford ML and Physics Seminars
Date: 2021-06-17

Neural networks have shown great success at learning function approximators between spaces X and Y, in the setting where X is a finite dimensional Euclidean space and where Y is either a finite dimensional Euclidean space (regression) or a set of finite cardinality (classification); the neural networks learn the approximator from N data pairs {x_n, y_n}. In many problems arising in physics it is desirable to learn maps between spaces of functions X and Y; this may be either for the purposes of scientific discovery, or to provide cheap surrogate models which accelerate computations. New ideas are needed to successfully address this learning problem in an accurate, scalable and efficient manner.



In this talk I will describe recent work aimed at addressing the problem of learning operators which map between spaces of functions. I will describe methodological approaches that are being adopted; highlight basic theoretical results to support the approaches being developed; and describe numerical experiments concerned with learning the solution operators. In particular the numerical experiments will focus on mapping initial condition to solution at a positive time for Burgers equation, the Kuramoto-Sivashinsky equation and the 2D incompressible Navier-Stokes equation (Kolmogorov flows).


Slides are available here: https://bit.ly/3gFLRWa