Berkeley, USA

Generative Adversarial Networks (GANs) have celebrated great empirical success, especially in image generation and processing. Meanwhile, Mean-Field Games (MFGs),  as analytically feasible approximations for N-player games, have experienced rapid growth in theory of controls. In this talk, we will discuss a new theoretical connections between GANs and MFGs. Interpreting MFGs as GANs, on one hand, allows us to devise GANs-based algorithm to solve MFGs. Interpreting GANs as MFGs, on the other hand, provides a new and probabilistic foundation for GANs. Moreover, this interpretation helps establish an analytical connection between GANs and Optimal Transport (OT) problems, the connection previously understood mostly from the geometric perspective. We will illustrate by numerical examples of using GANs to solve high dimensional MFGs, demonstrating its superior performance over existing methodology.

Aggregation refers to the thermodynamically favoured coalescence of individual molecular units (monomers) into dense clusters. The formation of liquid drops in oversaturated vapour, or the precipitation of solids from liquid solutions, are 'everyday' examples. A more exotic example, the crystallization of hydrophobic proteins in lipid bilayers, comes from current biophysics.

This talk begins with the basic physics of the simplest classical model, in which clusters grow by absorbing or expelling monomers, and the free monomers are transported by diffusion. Next, comes the description of three successive 'eras' of the aggregation process: NUCLEATION is the initial creation of clusters whose sizes are sufficiently large that they most likely continue to grow, instead of dissolving back into monomers.

The essential physical idea is growth by unlikely fluctuations past a high free energy barrier. The GROWTH of the clusters after nucleation depletes the initial oversaturation of monomer. The free energy barrier against nucleation increases, effectively shutting off any further nucleation. Finally, the oversaturation is so depleted, that the largest clusters grow only by dissolution of the smallest. This final era is called COARSENING.

The initial rate of nucleation and the evolution of the cluster size distribution during coarsening are the subjects of classical, well known models. The 'new meat' of this talk is a 'global' model of aggregation that quantitates the nucleation era, and provides an effective initial condition for the evolution of the cluster size distribution during growth and coarsening. One by-product is the determination of explicit scales of time and cluster size for all three eras. In particular, if G_* is the initial free energy barrier against nucleation, then the characteristic time of the nucleation era is proportional to exp(2G_*/5k_bT), and the characteristic number of monomers in a cluster during the nucleation era is exp(3G_*/5k_bT). Finally, the 'global' model of aggregation informs the selection of the self similar cluster size distribution that characterizes 'mature' coarsening.