Fokker-Planck equations of neuron networks: numerical simulation and exploring time-periodic solutions
Speaker：Zhennan Zhou
Date：May 12, 2022
Time：10:00-11:30 AM
Tencent Meeting： 836 6547 4971
Abstract：
In this talk, we are concerned with the Fokker-Planck equations associated with the Nonlinear Noisy Leaky Integrate-and-Fire model for neuron networks. Due to the jump mechanism at the microscopic level, such Fokker-Planck equations are endowed with an unconventional structure: transporting the boundary flux to a specific interior point. In the first part of the talk, we present a conservative and positivity preserving scheme for these Fokker-Planck equations, and we show that in the linear case, the semi-discrete scheme satisfies the discrete relative entropy estimate, which essentially matches the only known long time asymptotic solution property. We also provide extensive numerical tests to verify the scheme properties, and carry out several sets of numerical experiments, including finite-time blowup, convergence to equilibrium and capturing time-period solutions of the variant models. Secondly, we are concerned with a kinetic model for neuron networks. Individual neurons are characterized by their voltage and conductance, the dynamics of the voltage is influenced by the conductance and when the voltage is reaching a threshold, it is immediately reset to a lower value. By exploring a series of toy models, we aim to identify the cause of the emergence of time-periodic solutions in such Fokker-Planck equations.
Speaker Introduction：
周珍楠，北京大学北京国际数学研究中心研究员、博士生导师。2014 年在美国威斯康辛大学麦迪逊分校获得博士学位，2014-2017 年在美国杜克大学担任助理研究教授，2017 年加入北京大学北京国际数学研究中心，任研究员、博士生导师。主要研究领域为微分方程的应用分析，微分方程数值解，应用随机分析，随机模拟等，特别是关注来源于自然科学的应用数学问题。入选中组部第十四批“千人计划”青年人才项目（2018），入选北京市科协（2020-2022 年）青年人才托举工程项目。
Machine Learning and Seismic Tomography
Speaker：Xu Yang
Date：April 28, 2022
Time：10:00-11:30 AM
Tencent Meeting： 836 6547 4971
Abstract：
The stochastic gradient descent (SGD) method and deep neural networks (DNN) are two main workhorses in machine learning. In this talk, we present some preliminary results on connecting SGD and DNN to the applications in seismic tomography. On the one hand, motivated by SGD, we propose to use random batch methods to construct the gradient for iterations in seismic tomography. On the other hand, we use deep neural networks to create a reliable PmP database from massive seismic data and study the case in Southern California. The major difficulty lies in that the identifiable PmP waves are rare, making the problem of identifying the PmP waves from a massive seismic database inherently unbalanced.
Speaker Introduction：
Xu Yang got his Ph.D. at the University of Wisconsin-Madison in 2008, and spent two years at Princeton and two years at Courant Institute of NYU as a postdoc. He joined the University of California, Santa Barbara as an assistant professor in 2012, and became a full professor in 2020. His current research focuses on seismic imaging using realistic earthquake data. He has also been working on the applied analysis and numerical computation of scientific problems, including photonic graphene, ferromagnetic materials, and biological modeling.
Quasiperiodic systems: algorithms, analysis and applications
Speaker：Kai Jiang (Xiangtan University)
Organizer：Yi Zhu
Date：April 21, 2022
Time：10:00-11:30 AM
Tencent Meeting： 836 6547 4971
Abstract：
In this talk, we will introduce recent advances in computing quasiperiodic systems. Quasiperiodic systems, related to irrational numbers, are important and widely existing systems, including quasicrystals, incommensurate systems, and interfacial problems. This talk will be split into three parts. The first part will introduce some basic concepts about quasiperiodic systems. The second part will give two methods to address the quasiperiodic systems, including the periodic approximation method (PAM) and the projection method (PM). We will give the approximation theory for these methods. Results demonstrate that the Diophantine approximation error dominates the computational precision of the PAM, which is independent of numerical computation. In contrast, the PM has spectral accuracy. The third part will present some applications, including the thermodynamic stability of quasicrystals, phase transition, quasiperiodic quantum systems, and quasiperiodic interfaces.
Speaker Introduction：
蒋凯，湘潭大学教授，博士生导师。主要研究方向是准周期系统计算、复杂流体建模与计算等，在准晶方面的研究，是博士后合作导师张平文院士在2018年国际数学家大会的主要内容。获湖南省自然科学基金委“杰出青年科学基金”资助，主持国家自然科学基金委面上基金两项。曾获中国计算数学学会优秀青年论文一等奖。目前担任中国工业与应用数学学会“数学与航天交叉学科”专业委员会（筹）副秘书长，“科学工程计算与数值仿真”湖南省重点实验室副主任。
Data Driven Modeling of Dynamics using a Generalized Onsager Principle and Deep Neural Networks with Rectified Power Units
Speaker：Haijun Yu
Organizer：Yi Zhu
Date：April 14, 2022
Time：10:00-11:30 AM
Tencent Meeting： 836 6547 4971
Abstract：
With recent advancements in machine learning and growing availability of data, there is an increasing focus on developing machine-learning-based methods for building dynamical models from observations of natural processes. However, existing machine learning methods usually lead to black-box models, the learned mathematical models may lack a theoretical guarantee of long time stability. We propose a systematic method (OnsagerNet) for learning stable and interpretable low-dimensional dynamical models based on a generalized Onsager principle. The learned dynamics are autonomous ordinary differential equations parameterized by neural networks that retain clear physical structure information. The deep neural networks with rectified power units(RePU) are used to ensure the smoothness of learned dynamics and good approximation capability. For high dimensional problems with a low dimensional slow manifold, an autoencoder with isometric regularization is proposed to find generalized coordinates on which we learn the generalized Onsager dynamics. The method exhibits clear advantages in several benchmark problems for learning ordinary differential equations. We also applied this method as a model reduction tool to learn Lorenz-like low-dimensional models for the Rayleigh-Benard convection problem and derive moment-closure models for the Fokker-Planck equation of liquid crystal polymer dynamics. In those applications, both qualitative and quantitative properties of the underlying dynamics are captured.
Speaker Introduction：
于海军，中国科学院数学与系统科学研究院 研究员。分别于2002年,2007年获得北京大学学士学位和博士学位。2007-2010年曾先后在美国普林斯顿大学和普渡大学从事博士后研究。主要研究方向为高精度数值方法. 在复杂流体的数学建模和计算, 高维偏微分方程稀疏网格谱方法，非梯度系统的相变路径高精度计算等方面取得多项重要成果。先后获得过中科院陈景润之星人才项目, 基金委重大研究计划和国际合作项目等资助。现任北京市计算数学学会理事, 中国工业与应用数学学会大数据与人工智能专委会委员.
Convergence of the Planewave Approximations for Quantum Incommensurate Systems
Speaker：Huajie Chen
Organizer：Yi Zhu
Date：April 07, 2022
Time：10:00-11:30 AM
Tencent Meeting： 836 6547 4971
Abstract：
We study the numerical approximations of the spectrum distribution of Schrodinger operators for incommensurate systems. We provide rigorous analysis to characterize the quantitative of interests, i.e. the density of states, and develop numerical methods to approximate them based on planewave discretizations. In particular, we (i) justify the thermodynamic limit of the density of states in the real space formulation; (ii) propose a planewave approximation for the problem with some novel energy cutoffs specified for incommensurate systems, and provide a convergence analysis and error estimates with respect to the cutoffs; (iii) design an efficient algorithm to evaluate the density of states by sampling the reciprocal space. We present numerical simulations of some typical incommensurate systems to support the reliability and efficiency of our numerical algorithms.
Speaker Introduction：
陈华杰，教授。2005年7月本科毕业于北京师范大学数学科学学院；2010年7月博士毕业于中国科学院数学与系统科学研究院计算数学与科学工程计算研究所；从2010年11月至2016年7月分别在德国柏林工业大学、德国慕尼黑工业大学、英国华威大学做博士后。一直从事量子化学计算中数学问题的研究，包括数学模型的建立和分析，数值算法的设计、分析和实现。
