Deep Learning is much publicized and has had great empirical success on challenging problems in learning. Yet there is no quantifiable proof of performance and certified guarantees for these methods. This talk will give an overview of Deep Learning from the viewpoint of mathematics and numerical computation.

Deep generative models (DGM) are neural networks with many hidden layers trained to approximate complicated, high-dimensional probability distributions from a finite number of samples. When trained successfully, we can use the DGMs to estimate the likelihood of each observation and to create new samples from the underlying distribution. Developing DGMs has become one of the most hotly researched fields in artificial intelligence in recent years. The literature on DGMs has become vast and is growing rapidly.

Some advances have even reached the public sphere, for example, the recent successes in generating realistic-looking images, voices, or movies; so-called deep fakes.

Despite these successes, several mathematical and practical issues limit the broader use of DGMs: given a specific dataset, it remains challenging to design and train a DGM and even more challenging to find out why a particular model is or is not effective. To help students contribute to this field, this mini course provides an introduction to DGMs and provides a concise mathematical framework.

The course will consist of three 45 minutes blocks each devoted to one of the three most popular approaches: normalizing flows (NF), variational autoencoders (VAE), and generative adversarial networks (GAN). In addition to a mathematical formulation, each block illustrate the advantages and disadvantages of the approaches using numerical experiments that students can use to deepen their understanding after the course. Our goal is to enable and motivate the reader to contribute to this proliferating research area.

Mean-field games play essential roles in AI inference and optimization problems and controlling natural disasters, such as COVID 19. In this talk, we present several results by our MURI team. We designed fast and reliable numerical algorithms with connections to AI and machine learning, controlling COVID 2019 pandemic spreading and planning optimal vaccine distribution. Several numerical examples and engineering experiments will be presented. Future directions will be discussed. This is based on joint works with many people at UCLA, Wuchen Li in University of South Carolina, University of Houston, and Princeton University.