Date/Time Date(s) - 16/10/202311:30 am - 12:30 pm
Location: Online Zoom – details to be sent in email announcement
Speaker: Samuel Lanthaler (Caltech)
Title: The curse of dimensionality in operator learning
Abstract: Neural networks have proven to be effective approximators of high dimensional functions in a wide variety of applications. In scientific applications the goal is often to approximate an underlying operator, which defines a mapping between infinite-dimensional spaces of input and output functions. Extensions of neural networks to this infinite-dimensional setting, so-called “neural operators”, have been proposed in recent years. This has given rise to the emerging field of operator learning. Our theoretical understanding of these approaches is still in its infancy. In this talk, I will first summarize the proposed methodology and then present recent results on the approximation theory of neural operators. In particular, I will focus on a recent hardness result (joint with Andrew M. Stuart), which identifies fundamental limitations in what these methods can achieve. The identified limitation is an infinite-dimensional analogue of the well-known curse of dimensionality.