Fall 2022

Abstract: I will discuss two recent applications of reversed diffusions to sampling: the proximal sampler and score-based generative models (SGMs). First, the proximal sampler, which is the sampling analogue of the proximal point method from optimization, is an attractive alternative to the well-studied Langevin algorithm. Second, SGMs such as DALL-E 2 have led to spectacular empirical successes for audio and image generation, and they can provably sample from arbitrarily non-log-concave data distributions given an accurate approximation of the score function. This is based on joint work with Sitan Chen, Yongxin Chen, Jerry Li, Yuanzhi Li, Adil Salim, Andre Wibisono, and Anru Zhang.

Host person: Katy

Abstract: We consider the incompressible 2D Navier-Stokes equations on the torus driven by a deterministic time quasi-periodic force and a noise that is white in time and extremely degenerate in Fourier space. We show that the asymptotic statistical behavior is characterized by a uniquely ergodic and exponentially mixing quasi-periodic invariant measure. The result is true for any value of the viscosity and does not depend on the strength of the external forces.

By utilizing this quasi-periodic invariant measure, we are able show the strong law of large numbers and central limit theorem for the continuous time inhomogeneous solution processes. Estimates of the corresponding rate of convergence are also obtained, which is the same as in the time homogeneous case for the strong law of large numbers, while the convergence rate in the central limit theorem depends on the Diophantine approximation property on the quasi-periodic frequency and the mixing rate of the quasi-periodic invariant measure. We also prove the existence of a stable quasi-periodic solution in the laminar case (when the viscosity is large).

Host person: Quyuan

Abstract: Symmetry is ubiquitous in machine learning and scientific computing. Robust incorporation of symmetry prior into the learning process has shown to achieve significant model improvement for various learning tasks, especially in the small data regime. In the first part of the talk, I will explain a principled framework of deformation-robust symmetry-preserving machine learning. The key idea is the spectral regularization of the (group) convolutional filters, which ensures that symmetry is robustly preserved in the model even if the symmetry transformation is “contaminated” by nuisance data deformation. In the second part of the talk, I will demonstrate how to incorporate additional structural information (such as group symmetry) into generative adversarial networks (GANs) for data-efficient distribution learning. This is accomplished by developing new variational representations for divergences between probability measures with embedded structures. We study, both theoretically and empirically, the effect of structural priors in the two GAN players. The resulting structure-preserving GAN is able to achieve significantly improved sample fidelity and diversity—almost an order of magnitude measured in Fréchet Inception Distance—especially in the limited data regime.

Host person: Sui

Abstract: We provide new proofs based on the Myers--Steenrod theorem to confirm that travel time data, travel time difference data and the broken scattering relations determine a simple Riemannian metric on a disc up to the natural gauge of a boundary fixing diffeomorphism. Our method of the proof leads to a Lipschitz-type stability estimate for the first two data sets in the class of simple metrics.

Host person: Hanming

Abstract: I will talk about a deep unfitted Nitsche method for computing elliptic interface problems with high contrasts in high dimensions. To capture discontinuities of the solution caused by interfaces, we reformulate the problem as an energy minimization involving two weakly coupled components. This enables us to train two deep neural networks to represent two components of the solution in high-dimensional. The curse of dimensionality is alleviated by using the Monte-Carlo method to discretize the unfitted Nitsche energy function. We present several numerical examples to show the performance of the proposed method. 

Host person: Xu

Abstract: The perspectives and opinions of people change and spread through social interactions on a daily basis. In the study of opinion dynamics on networks, one often models entities as nodes and their social relationships as edges, and examines how opinions evolve as dynamical processes on networks, including graphs, hypergraphs, multi-layer networks, etc. In this talk, I will introduce a model of opinion dynamics and derive its mean-field limit, where the opinion density satisfies a kinetic equation of Kac type. We prove properties of the solution of this equation, including nonnegativity, conservativity, and steady-state convergence. 

The parameters of such opinion models play a nontrivial role in shaping the dynamics. However, in reality, these parameters often can't be measured directly. In the second part of the talk, I will approach the problem from an `inverse' perspective and present how to infer the interaction kernel from limited partial observations. I will provide sufficient conditions of measurement for two scenarios, such that one is able to reconstruct the kernel uniquely. I will also provide a numerical algorithm of the inference when the data set only has a limited number of data points.

Host person: Carlos

Abstract: Mean-field game (MFG) is a framework for modeling large systems of interacting agents that play non-cooperative differential games. In a PDE form, an MFG system comprises a Hamilton-Jacobi PDE coupled with a Kolmogorov-Fokker-Planck or continuity equation. Such systems are challenging to analyze and solve due to their non-linear structure and strong coupling. Especially challenging are the first-order systems which lack a priori regularization mechanisms such as viscosity. This talk will discuss variational techniques for analyzing and solving MFG systems. The workhorse of such methods is the extension of the celebrated Benamou-Brenier formulation of the optimal transportation problem to a class of MFG systems called potential; that is, systems representing first-order optimality conditions for a suitable energy functional. Extensions of the Benamou-Brenier technique beyond these systems were unknown. I'll show that, surprisingly, under appropriate monotonicity conditions, non-potential MFG systems also admit an analog of the Benamou-Brenier formulation.

Host person: Davit

Abstract: This talk is devoted to the Relativistic Vlasov-Maxwell (RVM) system in space dimension three. After reviewing the main known results about local and global well-posedness of weak and strong solutions, I will present a new representation formula of the momentum increment. As a consequence, I will show how the domain of influence in momentum is controlled by mild information on the initial data. In particular, this allows us to construct smooth solutions to the RVM system in the regime of dense, hot and strongly magnetized plasmas, like in fusion reactors.

This is a joint work with C. Cheverry.

Host person: Quyuan

Abstract: TBD

Host person: Carlos