PKUAMLSemniar

PKU Applied Math Lunch Seminar

应用数学青年讨论班（午餐会）日程表

本学期（2023秋）应用数学拔尖计划将开展午餐会，本学期由豆旭桉和谢中林同学负责。

本学期日程如下：

Sep 28

贾富琦（中国科学院大学）

可满足性与深度学习

具体信息

摘要: 本次报告将以可满足性问题为中心，包括命题可满足性问题（SAT）、可满足性模理论（SMT）以及求解算法等，通过一系列自动推理与深度学习交叉的工作，探讨深度学习在自动推理和约束求解领域的定位与意义。作为计算机领域的根本性问题，可满足性问题在模型检测、硬件和软件形式化验证、数学证明和推理等广泛应用中发挥着重要作用。随着深度学习的快速发展，它为可满足性算法中专家策略等的设计提供了重要的参考，为求解复杂问题提供了新的思路和方法。当前，将深度学习与可满足性求解算法相结合的研究也成为自动推理领域的重要前沿方向。

报告人信息: 贾富琦，目前就读于中国科学院大学，2020级博士研究生。其研究方向包括自动推理、约束求解、神经符号方法，具体方向为SAT/SMT求解算法研究、深度学习与自动推理的交叉研究。研究成果发表于NeurIPS, AAAI, ISSTA, ASE等国际会议，并获得了2023年ACM Sigsoft杰出论文奖。目前独立开发的SMT求解器支持非线性整数理论，在SMT-COMP中取得了Single Query和Model Validation Track的第二名的成绩。
Oct 12

张鼎怀（Mila - Quebec AI Institute）

GFlowNets: Exploratory Probabilistic Inference for Scientific Discovery

具体信息

摘要: This talk will introduce and discuss generative flow networks (GFlowNets), a learning framework for amortized sampling, from a sequential decision-making perspective. Different from reinforcement learning, which optimizes trajectory-level statistics, GFlowNet targets sampling proportional to the reward function over the terminal states in a Markov decision process. A family of algorithms could thus be derived (as in RL). Fruitful connection with previous probabilistic methods and control can be drawn. We would also talk about its wide application in science in domains such as causal discovery, and drug discovery, and combinatorial optimization.

报告人信息: Dinghuai Zhang is a PhD candidate at Mila, advised by Prof. Aaron Courville and Prof. Yoshua Bengio. His research focuses on the intersection of probabilistic inference and scientific discovery. From a methodology perspective, he studies how to incorporate structured exploration into inference problems such as sampling, leveraging the power of the generative flow network (GFlowNet) framework which revolves around active learning, Bayesian inference, black box optimization, and reinforcement learning. He develops methods for applications on different sorts of scientific discovery tasks, including sequence design, molecule synthesis, and combinatorial optimization. Dinghuai also has spent time in FAIR lab (Meta AI). Dinghuai obtained a bachelor's degree in math from Peking University.
Oct 18

韦静蓉（UC Irvine）

Transformed Primal-Dual Methods for Nonlinear Partial Differential Equations

具体信息

摘要: Steady-state nonlinear partial differential equations can be understood as finding the minimum of some smooth convex energy with equality constraints. After introducing the Lagrange multiplier, we are seeking the saddle point of a nonlinear system. A transformed primal-dual (TPD) flow is developed for such a nonlinear saddle point system. The flow for the dual variable contains a Schur complement which is strongly convex. Exponential stability of the saddle point is obtained by showing the strong Lyapunov property. A TPD iteration is derived by time discretization of the TPD flow. Under mild assumption, the algorithm is global linearly convergent, and the convergence rate depends on the relative condition number of the objective function and the Schur complement under variant metric as preconditioners. The developed algorithm is then applied to partial differential equations: Darcy–Forchheimer model and a nonlinear electromagnetic model. Numerical results demonstrate the efficiency of the method. This is joint work with Long Chen (UC Irvine) and Ruchi Guo (CUHK).

报告人信息: 韦静蓉是加利福尼亚大学欧文分校数学系博士生，师从陈龙教授。她于2019年获得北京大学数学科学学院学士学位。她的研究兴趣主要集中在科学计算和数值分析领域，包括非线性偏微分方程、鞍点问题和约束优化算法。
Oct 25

任潇（北京国际数学研究中心）

Steady-state Navier-Stokes solutions in 2D exterior domains

具体信息

摘要: Consider the steady-state Navier-Stokes system in planar exterior domains. With no-slip boundary condition and a prescribed constant limit velocity at infinity, this problem describes steady-state flows around cylindrical obstacles. An open problem since Leray (1933) asks whether solutions to this problem exist for arbitrary limit velocities. We recall the classical theory concerning the invading domain method, asymptotic behaviour of D-solutions, and purtubative analysis based on Oseen asymptotics. Then we present some recent progress. Based on joint works with M. Korobkov and J. Guillod.

报告人信息: 任潇是北京国际数学研究中心博士后，2023年博士毕业于复旦大学上海数学中心，师从雷震教授。他博士期间主要研究流体力学方程组，特别是不可压 Navier-Stokes 方程组中的分析问题。
Nov 1

黄政宇（BICMR-PKU）

Efficient Derivative-Free Bayesian Inference for Large-Scale Inverse Problems

具体信息

摘要: We consider Bayesian inference for large-scale inverse problems, where computational challenges arise from the need for repeated evaluations of an expensive forward model, which is often given as a black box or is impractical to differentiate. We propose a framework, which is built on Kalman methodology and Fisher-Rao Gradient flow, to efficiently calibrate and provide uncertainty estimations of such models with noisy observation data. In this talk, I will first explain some basics of variational inference under general metric tensor. In particular, under the Fisher-Rao metric, the gradient flow of the KL divergence has the form of a birth-death process, which has both exponential convergence O(e^-t) and the affine invariant property. The Gaussian approximation of it leads to the natural gradient descent. Next, I will discuss two different derivative-free approximations of the Fisher-Rao gradient flow. The Gaussian approximation leads to unscented/ensemble Kalman Inversion algorithms. They can also be obtained from a Gaussian approximation of the filtering distribution of a novel mean-field dynamical system. Theoretical guarantees for linear inverse problems are provided. The Gaussian mixture approximation leads to an efficient derivative-free Bayesian inference approach capable of capturing multiple modes. Finally, I will demonstrate the effectiveness of these approaches in several numerical experiments: learning permeability parameters in subsurface flow; and learning subgrid-scale parameters in a global climate model.

报告人信息: 黄政宇是北京大学北京国际数学研究中心助理教授，之前在加州理工学院从事博士后研究、在斯坦福大学获得博士学位、在北京大学获得本科学位。
Nov 22

张栋（北京大学数学科学学院）

New trends for p-Laplacian eigenproblems on graphs

具体信息

摘要: The spectrum of the graph p-Laplacian is closely related to many combinatorial properties of the graph itself; and its eigenpairs, reveal important information about the topology and geometry of the graph. In particular, when p is equal to 1 and infinity, the p-Laplacian eigenvalues approximate the multi-way Cheeger constants and the packing radii of the graph, respectively. However, the eigenvalue problem for graph p-Laplacians is a challenging topic both theoretically and computationally. In this talk, I will present some new ingredients for studying the graph p-Laplacian eigenproblem. These include homological eigenvalues, spectral duality, and tools from signed graphs.

报告人信息: 张栋是北京大学数学科学学院助理教授，之前在马普数学科学所从事博士后研究。他的研究兴趣是离散非线性分析，包括(非线性)谱图理论、组合临界点理论和离散-连续扩展方法。
Nov 29

周沛劼（CMLR-PKU）

Discovery and analysis of cell-fate landscape underlying single-cell transcriptome data

具体信息

摘要: Single-cell sequencing technologies provide unprecedented resolution for studying the dynamic process of cell-state transitions during development and complex disease. These transitions can be modeled and understood as a multi-scale, stochastic dynamical system on a cell-fate landscape. This could significantly enhance the interpretability and robustness of data mining and biological discovery. However, analyzing high-dimensional, static single-cell RNA-sequencing (scRNA-seq) data with such models can be challenging due to the curse of dimensionality. In this talk, I will discuss how machine learning has enabled us to overcome the challenge and use dynamical system tools to analyze scRNA-seq data. I will first introduce the MuTrans algorithm, which uses a low-dimensional dynamical manifold to identify attractor basins and transition probabilities in snapshot data. I will also present the spliceJAC algorithms, which use non-equilibrium dynamical theory to analyze attractor stability within the landscape and identify transition-driving genes in gene expression and splicing processes. Finally, I will discuss our efforts to construct a time-varying landscape, which interpolates non-stationary time-series scRNA-seq data using Wasserstein-Fisher-Rao metric, unbalanced optimal transport and its neural network-based partial differential equation implementations. Overall, these approaches bridge the model-based and data-driven methods in the modeling and analysis of single-cell biology.

报告人信息: 周沛劼，北京大学前沿交叉学科研究院国际机器学习研究中心助理教授，2010-2019年就读于北大数学学院计算数学专业，获得学士和博士学位，师从李铁军教授。毕业后在UC Irvine做博士后研究，任访问助理教授，合作导师Qing Nie。他的研究兴趣为数据驱动的计算系统生物学。
Dec 11

黄桢（加州大学伯克利数学系）

On Matsubara Green’s functions in quantum many body physics: fast Feynman diagrams evaluations and robust analytic continuation

具体信息

摘要: SThis talk assumes no prior knowledge in physics or chemistry, but will discuss several algorithmic contributions in such fields. We will first introduce a fast deterministic solver for Anderson impurity problems. The bottleneck of this problem is to accurately evaluate n-th order Feynman diagrams. Mathematically, this corresponds to computing MANY 2n-dimensional integrals, which is computationally non trivial for n>2. Then we will discuss how to solve the analytic continuation problem in condensed matter physics, which could be seen as a very ill-posed inverse problem from an applied mathematics perspective. Finally, we will discuss some recent trends in the algorithmic development of many-body physics, focusing on Green’s function approaches and diagrammatics.

报告人信息: 黄桢, 2021年毕业于北京大学数学科学学院计算数学系，现为加州大学伯克利数学系的博士生。
Dec 13

曹立元（BICMR-PKU）

Model-Based Derivative-Free Optimization

具体信息

摘要: Derivative-free optimization concerns the optimization of black-box functions. It is a diverse field studied under multiple disciplines, which leads to an array of methods with fundamental distinctions. Among them, algorithms based on polynomial interpolation and trust-region methods are capable of finding solutions without the need for a large number of function queries, perfect for handling problems base on expensive computer simulations, while gradient descent methods coupled with finite differences are commonly used for solving machine learning problems such as reinforcement learning and neural adversarial attack. In this talk, Dr. Liyuan Cao presents his theoretical work on these two types of algorithms, including the analysis of approximation methods such as (randomized) finite differences and linear interpolation, and the analysis of adaptive optimization algorithms such as line search and trust-region methods under noise.

报告人信息: Liyuan Cao is a postdoctoral researcher at BICMR, Peking University under the supervision of Prof. Zaiwen Wen. He received his Ph.D. from the Industrial and Systems Engineering Department at Lehigh University in 2021 under Prof. Katya Scheinberg. His research focuses on the analysis and development of algorithms in nonlinear optimization and derivative-free optimization.
Dec 20

邵宇秀（北京师范大学系统科学学院）

Identifying the computational roles of local connectivity structures in Excitatory-Inhibitory networks

具体信息

摘要: In both neuroscience and machine learning, understanding how network connectivity shapes dynamics and computation is crucial. Past studies have approached connectivity analysis either through biological experiments, focusing on local connectivity motifs in small neuron groups, or via artificial neural networks, examining network-wide low-rank connectivity patterns influencing low-dimensional dynamics. While both approaches suggest network connectivity's influence on activity patterns and dynamics, there's a gap in understanding how local connectivity statistics relate to global connectivity and impact network dynamics. To address this, we introduce an analytical method mapping locally-defined biological connectivity statistics to an approximate global low-rank structure (Shao & Ostojic, 2023). Leveraging perturbation theory for random matrices, we approximate the global connectivity matrix using dominant eigenvectors. This method demonstrates that networks with local connectivity statistics, under the central limit theorem, can be approximated by low-rank connectivity with Gaussian-mixture statistics. Applied to excitatory-inhibitory networks, our method accurately predicts low-dimensional dynamics, the balance between excitation and inhibition, and population activity statistics. All in all, our approach allows us to disentangle the effects of mean connectivity and multiple types of second-order motifs on global recurrent feedback and feedforward propagation, providing an intuitive picture of how local connectivity shapes global network dynamics.

报告人信息: 邵宇秀于 2023 年 11 月加入北京师范大学系统科学学院，现任助理研究员。她的研究关注于生物神经元和人工神经网络的交叉方向，重点是揭示生物/人工神经网络系统背后的统一框架。在巴黎高等师范学院做博士后期间，她师从理论神经科学家Srdjan Ostojic，采用数学方法研究网络连接性与动力学之间的复杂关系。在此之前，她在北京大学生命科学学院攻读计算神经科学方向博士学位，师从Louis Tao教授。

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