Abstract: We focus on the fundamental mathematical structure of score-based generative models (SGMs). We formulate SGMs in terms of the Wasserstein proximal operator (WPO) and demonstrate that, via mean-field games (MFGs), the WPO formulation reveals mathematical structure that describes the inductive bias of diffusion and score-based models. In particular, MFGs yield optimality conditions in the form of a pair of coupled PDEs: a forward-controlled Fokker-Planck (FP) equation, and a backward Hamilton-Jacobi-Bellman (HJB) equation. Via a Cole-Hopf transformation and taking advantage of the fact that the cross-entropy can be related to a linear functional of the density, we show that the HJB equation is an uncontrolled FP equation. Next, with the mathematical structure at hand, we present an interpretable kernel-based model for the score function which dramatically improves the performance of SGMs in terms of training samples and training time. The WPO-informed kernel model is explicitly constructed to avoid the recently studied memorization effects of score-based generative models. The mathematical form of the new kernel-based models in combination with the use of the terminal condition of the MFG reveals new explanations for the manifold learning and generalization properties of SGMs, and provides a resolution to their memorization effects. Our mathematically informed kernel-based model suggests new scalable bespoke neural network architectures for high-dimensional applications. This is a joint work with Benjamin J. Zhang, Markos A. Katsoulakis, Wuchen Li and Stanley J. Osher.
Abstract: We consider networks of oscillator nodes with time delayed, global circulant coupling. We first study the existence of Hopf bifurcations induced by time delayed coupling, and then apply equivariant Hopf bifurcation theory to determine how these bifurcations lead to different patterns of phase-locked oscillations. We apply the theory to a variety of systems inspired by biological neural networks to show how Hopf bifurcations can determine the synchronization state of the network. Finally, we discuss how interaction between two Hopf bifurcations corresponding to different oscillation patterns can induce complex torus solutions in the network.
Abstract: A theorem of Popa and Schnell shows that if a smooth projective variety X admits a 1-form with no zeros it cannot be of general type. However, one expects far more stringent constraints on the geometry of those X actually admitting nonvanishing 1-forms. If X is not uniruled and assuming the conjectures of MMP, we show that X is birational to an isotrivial fibration over an abelian variety. This partially answers conjectures of Hao--Schreieder, Meng--Popa, and Chen--Church--Hao. The proof involves a decomposition result for families of Calabi-Yau varieties surjecting onto a fixed abelian variety. If X is uniruled, we also give a weak structure theorem that relies on using higher direct image Hodge modules in the method of Popa--Schnell.
Abstract:Â In this talk, I will discuss joint work with Levi Haunschmid-Sibitz where we construct the stochastic six-vertex model speed process. We will first define the stochastic six-vertex model with step initial data and show that a second-class particle started at the origin converges almost surely to an asymptotic speed. We will then generalize this by assigning each particle in the model a different class, allowing us to simultaneously track the speeds of all the particles. The speed process is obtained as the joint limit of these speeds. Along the way, we will develop a stochastic domination result for second- and third-class particles, as well as moderate deviation tail bounds for the stochastic six-vertex model.
Abstract: In this talk we will discuss recent results concerningstochastic (and deter- ministic) moving boundary problems, particularly arising in fluid-structure interaction (FSI), where the motion of the boundaryis not known a priori.Fluid-structure interaction refersto physical systems whose behavior is dictated by the interaction of an elasticbody and a fluidmass and it appears in various applications, ranging from aerodynamics to structural engi- neering. Our work is motivated by FSI modelsarising in biofluidic applications that describe the interactions between a viscous fluid,such as human blood, and an elasticstructure, such as a humanartery. To account for theunavoidable numerical and physical uncertainties in applications we analyzethese PDEs underthe influence of external stochastic (random) forces.We will considernonlinearly coupled fluid-structure interaction (FSI) problemsinvolving a viscous fluid in a 2D/3D domain,where part of the fluid domain boundaryconsists of an elastic deformable structure, and wherethe system is perturbed by stochastic effects.The fluid flow is described by the Navier-Stokes equations while the elastodynamics of the thin structure are modeled by shellequations. The fluid and thestructure are coupled via two sets of coupling conditions imposed at thefluid-structure interface. We willconsider the case where the structure is allowed to have unrestricted deformations and exploredifferent kinematic coupling conditions (no-slip and Navier slip)imposed at the randomly moving fluid-structure interface, the displacement ofwhich is not known a priori. We willpresent our results on the existence of (martingale) weak solutions to the(stochastic) FSI models. This is the first body of work that analyzes solutionsof stochastic PDEs posed on random and time-dependent domainsand a first step in the fieldtoward further researchon control problems, singularperturbation problems etc. We willfurther discuss our findings, which reveal a novel hidden regularity in thestructure’s displacement. This resulthas allowed us to address previously open problems in the 3D (deterministic)case involving large vec- torial deformations of the structure. We will discuss both the cases ofcompressible and incompressible fluid.
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