There has been significant recent work on solving PDEs using neural networks on infinite dimensional spaces. In this talk we consider two examples. First, we prove that transformers can effectively approximate the mean-field dynamics of interacting particle systems exhibiting collective behavior, which are fundamental in modeling phenomena across physics, biology, and engineering. We provide theoretical bounds on the approximation error and validate the findings through numerical simulations.

Second, we show that finite dimensional neural networks can be used to approximate eigenfunction for the Laplace Beltrami operator on manifolds. We provide quantitative insights into the number of neurons needed to learn spectral information and shed light on the non-convex optimization landscape of training.

A fractal uncertainty principle (FUP) roughly says that a function and its Fourier transform cannot both be concentrated on a fractal set. These were introduced to harmonic analysis in order to prove new results in quantum chaos: if eigenfunctions on hyperbolic manifolds concentrated in unexpected ways, that would contradict the FUP. Bourgain and Dyatlov proved FUP over the real numbers, and in this talk I will discuss an extension to higher dimensions. The bulk of the work is constructing certain plurisubharmonic functions on C^n. 

Please join us for pre-seminar Tea

A relaxed-pace seminar on impromptu subjects related to the interests of the audience. Everyone is welcome. The subjects are geometry, probability, combinatorics, dynamics, and more!