Date/Time : 28 Feb 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Jiequn Han, Flatiron Institute, USA

Partial differential equations (PDEs) are among the most ubiquitous tools in modeling natural phenomena, and various numerical methods have been developed for decades to solve PDE problems. While deep learning has introduced new techniques to the field, the limited accuracy of deep neural networks hinders their application to scientific problems that require high precision.

This talk will present a warm-start approach that combines the strengths of deep neural networks and classical numerical solvers. The approach uses neural networks to provide an initial guess, enabling classical numerical solvers to achieve a good solution more efficiently. We will demonstrate the advantages of the proposed method through two examples. In the first example, we will demonstrate how the initial guess provided by neural networks helps identify the basin of attraction of the true solution in the inverse scattering problem. In the second example, we will show how the initial guess provided by neural networks leads to faster convergence in solving the Reynolds-averaged Navier-Stokes equations. In both examples, the combination of the classical PDE solver and the neural network outperforms either approach alone. The potential of this approach will be discussed, along with the new challenges that must be tackled to solve challenging scientific problems using this new paradigm

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Date/Time : 28 Feb 2024 16:30
Venue : S17 #05-12 (SR4)
Speaker : Guo Ruchi, The Chinese University of Hong Kong

In physics and mathematics, a large class of PDE systems can be formulated as energy minimization problems subject to specific constraints. Lagrange multipliers are widely used for solving such problems, which however leads to min-max optimization problems known as saddle point systems. The development of fast solvers for saddle point systems, especially those nonlinear ones, presents considerable challenges in the sense that

(i) one has to consider the preconditioning in the primal and dual directions collectively;

(ii) the preconditioners have to evolve in iterations to accommodate the nonlinearity;

(iii) all the subsystems and subproblems cannot be solved exactly due to the high computational cost involved.

In this work, we develop an efficient transformed primal-dual (TPD) algorithm, and this is done by modifying the original gradient flow from the perspective of matrix factorization. We prove the global optimal linear convergence in terms of the condition numbers measured by variable metrics. To demonstrate the effectiveness of our approach, we apply the algorithm to a Darcy-Forchheimer model and a nonlinear ferromagnetic model, showing it is much more efficient than some traditional methods.

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Date/Time : 27 Feb 2024 14:00
Venue : S17 #05-12 (SR4)
Speaker : Shilin Yu, Xiamen University

Unipotent representations are believed to be the building blocks of unitary dual of semisimple/reductive Lie groups. In their study of special unipotent representations for complex semisimple groups, Barbasch and Vogan constructed a duality map between the nilpotent orbits of $G$ and that of its Langlands dual group $G^\vee$ (also discovered by Lusztig and Spaltenstein), which allows them to describe the unipotent ideals and representations of $G$ in terms of $G^\vee$. Later Sommers and Achar extended this duality map by considering pairs $(\mathbb{O}, C)$ consisting of a nilpotent orbit $\mathbb{O}$ and a conjugacy class $C$ in its Lusztig canonical quotient group.

In the joint work with Lucas Mason-Brown and Dmytro Matvieievskyi (arXiv:2309.14853), we have extended the Sommers’ duality map in another direction by considering covers of nilpotent orbits, which turns out to be equivalent to Achar’s duality map (in a nontrivial way). This extended duality map allows us to describe the (generalized) unipotent ideals and bimodules attached to nilpotent covers, proposed previously by Ivan Losev and my other two coauthors, in terms of the Langlands dual group.

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Date/Time : 22 Feb 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Bin Xu, Tsinghua University

The irreducible admissible representations of Arthur class are the local components of automorphic representations. They are conjectured to be parametrized by the Arthur parameters, and the set of irreducible representations associated with a single Arthur parameter is called an Arthur packet. For quasisplit symplectic and orthogonal groups, the Arthur packets have been determined by Arthur, and their structure in the p-adic case can be understood in a very complicated way by the works of Moeglin, Atobe and myself. In this talk, we would like to introduce some conjectures on their structure in the real case, which are motivated by the results in the p-adic case. For simplicity we will only consider the symplectic groups. This is an ongoing project with Taiwang Deng and Chang Huang.

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Date/Time : 22 Feb 2024 14:00
Venue : S17 #05-11 (SR5)
Speaker : Sebastian Pokutta, Zuse Institute Berlin (ZIB) and TU Berlin

Conditional Gradient methods are an important class of methods to minimize (non-)smooth convex functions over (combinatorial) polytopes. Recently these methods received a lot of attention as they allow for structured optimization and hence learning, incorporating the underlying polyhedral structure into solutions. In this talk I will give a broad overview of these methods, their applications, as well as present some recent results both in traditional optimization and learning as well as in deep learning.

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Date/Time : 21 Feb 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Chengyang Shao, University of Chicago

The speaker aims to report some advances in the study of irrotational oscillation of a water droplet under zero gravity. The governing physical laws, resembling the well-studied capillary water waves equation, are converted to a quasilinear dispersive para-differential system defined on the 2-sphere. Regarding this conversion, a coordinate-independent, global para-differential calculus defined on compact Lie groups and homogeneous spaces is developed as a toolbox. After discussing Cauchy theory under this novel formalism, the speaker will propose several problems concerning existence of periodic solutions, normal form reduction and generic lifespan estimates. It is pointed out that these problems are closely related to certain Diophantine equations.

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Date/Time : 21 Feb 2024 15:00
Venue : S17 #06-11 (SR6)
Speaker : Zhaojie Yang, Fudan University

In this talk, I will present a backward uniqueness result of bounded mild solutions of 3D Navier-Stokes Equations in the whole space with non-trivial final data. A direct consequence is that a solution must be axi-symmetric in [0, T] if it is so at time T. The proof is based on a new weighted estimate which enables to treat terms involving Calderon-Zygmund operators. The new weighted estimate is expected to have certain applications in control theory when classical Carleman-type inequality is not applicable. This is a joint work with Prof. Zhen Lei and Dr. Cheng Yuan.

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Date/Time : 21 Feb 2024 17:00
Venue : S17 #04-05 (SR2)
Speaker : Neil Barton, University of Oslo

A popular view in the philosophy of set theory is that of *potentialism*: the position that the set-theoretic universe unfolds as more sets come into existence or become accessible to us. This often gets formalised using *modal logic*, but there is always a question of how to move to *non-modal* theories. In this latter regard, a difficult question for the potentialist is to explain how *intensional entities* (entities individuated by an application condition rather than an extension) behave, and in particular what logic governs them. This talk will discuss some work in progress on this issue. We’ll see how to motivate acceptance of different propositional logics for different flavours of potentialism, and discuss the prospects for proving results about the kinds of first-order theories validated.

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Date/Time : 20 Feb 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Yin Zhang, Chinese University of Hong Kong, Shenzhen

The Transformer architecture, in which the key innovation is a Self-Attention (SA) mechanism, has revolutionized AI fields since its introduction in 2017. However, it is difficult to extend the original architecture to deeper models with 10 or more layers. In this work, we show that this depth problem is caused by a phenomenon called token similarity escalation (TSE) which is driven by spectral properties of attention matrices in the SA mechanism. Based on the gained insights, we propose a simple de-escalation strategy to surgically removes excessive token similarity without discounting SA as a whole. Empirical evidence is presented to demonstrate the effectiveness of our scheme on small-scale models.

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