Learning solution operators of PDEs from input-output data is an important task in scientific machine learning. While current approaches can be quite successful for diffusion-dominated equations, advection-dominated equations prove much more difficult. I'll show how statistical learning theory allows one to derive sample complexity lower bounds for learning solution operators of certain classes of advective equations from data, using the transport equation as a toy model. At the end, I'll give some results showing that learning the solution operator of some advective equations can be impossible, even with added diffusion, if the input data are too smooth.
About the Speaker: 
Chris is a PhD candidate at Cornell University being advised by Alex Townsend. Hisprimary research interests are in randomized numerical linear algebra and in PDEoperator learning.
