2020.05.28 Andrew Stuart - Supervised Learning between Function Spaces
Channel: One World Probability
Date: 2020-05-29

Consider separable Banach spaces X and Y, and equip X with a
probability measure m. Let F: X \to Y be an unknown operator. Given data pairs
{x_j,F(x_j)}  with {x_j} drawn i.i.d. from m, the goal of supervised learning
is to approximate F. The proposed approach is motivated by the recent
successes of neural networks and deep learning in addressing this problem in
settings 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). Algorithms which address the problem for
infinite dimensional spaces X and Y have the potential to speed-up large-scale
computational tasks arising in science and engineering in which F must be
evaluated many times. The talk introduces an overarching approach to this
problem and describes three distinct methodologies which are built from this
approach. Basic theoretical results are explained and numerical results
presented for solution operators arising from elliptic PDEs and from the
semigroup generated by Burgers equation.