ABSTRACT

In this talk, I plan to cover two different topics. In the first part, I will discuss the convergence rate of second order mean-field games to first order ones, motivated by numerical challenges in first order mean field PDEs arising from transportation and crypto modeling, and the weak noise theory in KPZ universality. When the Hamiltonian and the coupling function have a certain growth, the rate is independent of the dimension; on the other hand, the rate decays in dimension (curse of dimensionality) when the Hamiltonian and the coupling function have small growth. The is based on joint work with Yuming Paul Zhang.

The past decade has witnessed the success of generative modeling in creating high quality samples in a wide variety of data modalities. The second part of this talk is concerned with the recently developed diffusion models, the key idea of which is to reverse a certain stochastic dynamics. I will first take a continuous-time perspective, and examine the performance of different SDE schemes including VE and VP. The discretization is more subtle, and our idea is to "contract" the reversed dynamics leading to possible new diffusion model designs. I will also highlight the difference between the ideal continuous time framework, and more realistic discrete modeling. This is based on joint work with Hanyang Zhao.