PACM Colloquium, PACM Graduate Students, Princeton University

^{Fizay Noah Lee, Year: G5}

^{On Nonequilibrium Electroconvection}

^{Abstract: We consider the Nernst-Planck-Stokes (NPS) system on a three dimensional bounded domain, which models electroconvection of ions in a fluid in the presence of boundaries. We consider nonequilibrium boundary conditions, which are empirically associated with the onset of electrokinetic instability, whereby vortical and/or chaotic flow patterns are observed in a boundary layer. In this talk, we give a brief overview of recent mathematical results concerning the long time behavior of solutions of NPS in the context of nonequilibrium boundary conditions. Topics include 1) stability of weak currents 2) existence of finite dimensional global attractor and 3) space-time averaged electroneutrality in the singular limit of Debye length going to zero. This talk includes joint work with Peter Constantin and Mihaela Ignatova.}

^{Bio: Noah Lee is a 5th year PhD student in the Program in Applied & Computational Mathematics at Princeton University, where he is advised by Professor Peter Constantin. Noah’s research focuses on the analysis of PDE models that arise in fluid dynamics and electrochemistry.}

^{Hongkang Yang, Year: G3}

^{Generalization ability of distribution learning models}

^{Abstract: The modeling of probability distributions, specifically generative modeling and density estimation, has become a popular subject in recent years by virtue of its outstanding performance on sophisticated data such as images and texts. Nevertheless, a theoretical understanding of its success is still incomplete. One mystery is the paradox between memorization and generalization: In theory, the model is trained to be exactly the same as the empirical distribution of the finite samples, whereas in practice, the trained model can generate new samples or estimate the likelihood of unseen samples. In this talk, we discuss our results that resolve this paradox. Specifically, it is established that distribution learning models enjoy implicit regularization during training, so that the generalization error at early-stopping escapes from the curse of dimensionality. The three models we analyze are the bias-potential model, normalizing flow with stochastic interpolants, and a simplified version of the generative adversarial network, which cover the main approaches to distribution representation.
Bio: Hongkang Yang received his B.A. and M.S. degrees in Mathematics from New York University in 2019 and 2020. He is currently a graduate student at Princeton PACM advised by Professor Weinan E. His research focuses on the theoretical analysis of machine learning models and the representation of unstructured data.}

^{Scander Mustapha, Year: G4}

^{Well-posedness of the supercooled Stefan problem with oscillatory initial condition}

^{Abstract: We consider the one-dimensional supercooled Stefan problem and prove uniqueness of solutions for oscillatory initial conditions.
The proof of the main result is based on a new iteration argument which allows extending [Delarue, Nadtochiy, Shkolnikov, 2022, Global solutions to the supercooled Stefan problem with blow-ups: regularity and uniqueness] by weakening their monotonicity condition in an averaged version. We prove that this weaker condition is satisfied by fairly general oscillating densities, like those obtained from sample trajectories of random processes.
In particular, we prove uniqueness for an initial data given near the origin by $f(x) = (1 + W_x - \sqrt{2x|\log{|\log{x}|}|})_{+}\wedge 1$, where $W$ is a standard Brownian motion. We also consider deterministic densities of the form $f(x) = \frac{1}{2}\left(1 + \sin{1/x}\right)$. Finally, we highlight examples of densities where it is possible to go beyond our main assumption and to establish uniqueness via further complementary arguments.}

^{Bio: Scander Mustapha is a 4th year PhD student in the Program in Applied & Computational Mathematics at Princeton University, where he is advised by Professor Mykhaylo Shkolnikov. His interests are the analysis of probabilistic systems of particles.}