一周活动预告（11.13-11.20）
请查收。
目录：
Subdivision modeling method and its application （邓重阳）
An Integrated Framework of Isogeometric Modeling and Simulation Optimization Based on Volume Subdivision （徐岗）
Deterministic-statistical approach for inverse problems with partial data （孙继广）
Numerical methods using and for interacting particle systems（李磊）
Physically consistent numerical methods for flows in porous media（陈黄鑫 ）
Energy stability and error analysis of a maximum bound principle preserving scheme for the dynamical Ginzburg-Landau equations of Superconductivity (乔中华)
Deep image prior for inverse problems: acceleration and probabilistic treatment （Bangti Jin）
Conforming Finite Element Gradgrad and Divdiv Complexes（胡俊）
Mathematical Theory of Structured Deep Neural Networks（Ding-Xuan Zhou）
Efficient and Trustworthy Al and Its Applications to 5G Networks (Prof. Zhiquan Luo)
A weakly nonlinear, energy stable scheme for the strongly anisotropic Cahn-Hilliard system (Prof. Cheng Wang)
A Stochastic Neural Network for uncertainty quantification of deep neural networks (Professor Yanzhao Cao)
Reinforced Inverse Scattering（Haizhao Yang）
Weak intermittency for the stochastic heat equation and its discretizations（陈楚楚）
Recent advances in dynamical low-rank approximation for kinetic equations (Lukas EINKEMMER)
Finite Element Approximation of a Membrane Model for Liquid Crystal Polymeric Networks ( Lucas Bouck)
The mathematical foundations of deep learning: from rating impossibility to practical existence theorems (Simone Brugiapaglia)
Learning Nonlocal Constitutive Models with Neural Operators（Jiequn Han）
Feature Screening with Latent Responses（Cui Hengjian）
Structure-preserving integrators for dissipative systems based on reversible–irreversible splitting（商晓成）
Iterative regularization for nonsmooth inverse problems (Prof. Christian CLASON)
Overdamped generalized Langevin equations with fractional noise: Euler-type methods（代新杰）
Symmetry-preserving machine learning for computer vision, scientific computing, and distribution learning (Wei Zhu)
Relaxation oscillations in slow-fast Rosenzweig-MacArthur predator-prey model with additional food（王成）
1. Subdivision modeling method and its application
报告人: 邓重阳（杭州电子科技大学）
报告时间: 2022-11-13 9:00-10:30
报告地点: 腾讯会议ID: 579 202 373
信息来源:
http://tianyuan.xmu.edu.cn/cn/letures/960.html
报告摘要:
The subdivision modeling method has been widely used in computer graphics and related fields. With the rise of isogeometric analysis and other technologies, the subdivision modeling method has attracted extensive attention due to its excellent properties. This report first briefly introduces the basic knowledge of subdivision modeling and its latest progress, and then looks forward to its application prospects in the fields of isogeometric analysis and volume modeling.
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2. An integrated framework of isogeometric modeling and simulation optimization based on volume subdivision
报告人: 徐岗（杭州电子科技大学）
报告时间: 2022-11-13 9:45-11:15
报告地点: 腾讯会议ID: 579 202 373
信息来源:
http://tianyuan.xmu.edu.cn/cn/letures/959.html
报告摘要:
The research focus of isogeometric analysis has gradually focused on 3D CAD models with complex shapes to meet the needs of practical applications. This report will report some explorations of the research group in the theory of the limit point of volume subdivision, and based on the theory of volume subdivision, in complex curved body modeling suitable for IGA, high-precision IGA simulation, complex model multi-resolution shape and topology optimization. Research work, so as to truly realize the seamless integration of geometric modeling, physical simulation and structural optimization.
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3. Deterministic-statistical approach for inverse problems with partial data
报告人: 孙继广（Michigan technological university）
报告时间: 2022-11-14 9:00-10:00
报告地点: 腾讯会议ID: 418 975 887
信息来源:
http://math.xjtu.edu.cn/info/1089/11700.htm
报告摘要:
We propose a deterministic-statistical approach for inverse problems with partial data. Certain deterministic method is first used to obtain useful (qualitative) information for the unknowns. Then the inverse problem is recast as a statistical inference problem and the Bayesian inversion is employed to obtain more (quantitative) information of the unknowns. Several examples are presented for demonstration. Furthermore, we introduce new statistical estimators to characterize the non-unique solutions of several inverse problems.
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4. Numerical methods using and for interacting particle systems
报告人: 李磊（上海交通大学）
报告时间: 2022-11-14 10:00-12:40
报告地点:腾讯会议ID: 960 293 673
信息来源:
https://math.csu.edu.cn/info/1736/11523.htm
报告摘要:
I will give a brief introduction to our recent works regarding the applications of interacting particle systems in scientific computing. In the first part, I will talk about several algorithms we proposed for sampling and solving PDEs using interacting particle systems, which can be efficiently implemented with the random batch approximation. In particular, the random batch Monte Carlo method, the random batch Ewald method and a particle method for PNP equation will be talked about. The second part goes to the fluctuation suppression and enhancement phenomena in interacting particle systems, which may give some evidence why sampling based on particle systems may be preferred sometimes.
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5. Physically consistent numerical methods for flows in porous media
报告人: 陈黄鑫 教授（厦门大学数学科学学院）
报告时间: 2022-11-14 15:30-17:30
报告地点: 腾讯会议ID: 184631176 密码: 832586
信息来源: https://www.math.sjtu.edu.cn/Default/seminarshow/tags/MDAwMDAwMDAwMLGelJiG2qF2
报告摘要:
Simulation of multi-phase flow in porous media has wide applications such as petroleum fields and water resources. In this talk we will introduce physically consistent numerical methods for the simulation of incompressible and immiscible two-phase flow in heterogeneous porous media with capillary pressure. The new algorithm is unbiased and locally mass conservative for both of phases. We will also introduce an efficient energy stable numerical method for the Maxwell-Stefan-Darcy two-phase flow model in porous media, which can preserve multiple important physical properties of the model. Moreover, a fully discrete scheme for the stochastic Stokes-Darcy equations with multiplicative noise and its numerical analysis will also be discussed.
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6. Energy stability and error analysis of a maximum bound principle preserving scheme for the dynamical Ginzburg-Landau equations of Superconductivity
报告人: 乔中华（香港理工大学）
报告时间: 2022-11-15 9:30-10:30
报告地点: 腾讯会议ID: 336 891 606
信息来源:
http://tianyuan.scu.edu.cn/portal/article/index/id/763/cid/3/p/7.html
报告摘要:
We focus on numerical study of the dynamical Ginzburg-Landau equations under the temporal gauge, and propose a decoupled numerical scheme based on the finite element method. For the variable A, the second type Nedelec element is employed for the space discretization and the backward Euler is applied for the time discretization where the nonlinear term is treated explicitly. For the order parameter, the first order exponential time differencing method is employed with the linear operator generated by the linear element method with lumping. The proposed numerical scheme is proved to preserve the discrete maximum bound principle for the order parameter and admit an unconditional energy decay property. An optimal error estimate is also given for the scheme which is verified by the numerical examples.
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7. Deep image prior for inverse problems: acceleration and probabilistic treatment
报告人: Bangti Jin（The Chinese university of Hong Kong）
报告时间: 2022-11-15 10:00-11:00
报告地点: 腾讯会议ID: 427 400 798
信息来源:
http://www.amss.cas.cn/mzxsbg/202211/t20221108_6545930.html
报告摘要:
Since its first proposal in 2018, deep image prior has emerged as a very powerful unsupervised deep learning technique for solving inverse problems. The approach has demonstrated very encouraging empirical success in image denoising, deblurring, super-resolution etc. However, there are also several known drawbacks of the approach, notably high computational expense. In this talk, we describe some of our efforts: we propose to accelerate the training process by pretraining on synthetic dataset and further we propose a novel probabilistic treatment of deep image prior to facilitate uncertainty quantification.
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8. Conforming Finite Element Gradgrad and Divdiv Complexes
报告人: 胡俊（北京大学）
报告时间: 2022-11-15 14:00-15:00
报告地点: 腾讯会议ID: 918-7000-7748
信息来源:
http://math.ustc.edu.cn/2022/1107/c18822a579741/page.htm
报告摘要:
We focus on two families of Hilbert complexes which are related to the stable mixed finite element method for the linearized Einstein-Bianchi system: the gradgrad complex and the divdiv complex. We first construct the exact discrete gradgrad complex. There are three main results of this part: (i) an intrinsic structure of the symmetric H(curl) elements; (ii) an intrinsic structure of the traceless H(div) elements; (iii) explicit expressions of the H(curl) and H(div) bubble spaces. We also present the first family of conforming finite element divdiv complexes. In these complexes, the symmetric H(divdiv) elements are from existing literature, while the finite elements of both H(sym curl) and H1 are newly constructed here. It is proved that these finite element complexes are exact. As a results, these spaces can be used in the mixed form of the linearized Einstein-Bianchi system.
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9. Mathematical Theory of Structured Deep Neural Networks
Speaker: Ding-Xuan Zhou（University of Sydney）
Time: 2022-11-15 14:00-15:00
Venue:
Zoom ID: 865 5557 4541 Password: PSJAS1115
Tencent Meeting ID: 281-254-841
Info Source:
https://ins.sjtu.edu.cn/seminars/2184
Abstract:
Deep learning has been widely applied and brought breakthroughs in speech recognition, computer vision, natural language processing, and many other domains. The involved deep neural network architectures and computational issues have been well studied in machine learning. But there lacks a theoretical foundation for understanding the modelling, approximation or generalization ability of deep learning models with network architectures. One family of structured neural networks is deep convolutional neural networks (CNNs) with convolutional structures. The convolutional architecture gives essential differences between deep CNNs and fully-connected neural networks, and the classical approximation theory for fully-connected networks developed around 30 years ago does not apply. We describe a mathematical theory for deep CNNs associated with the rectified linear unit activation. In particular, we discuss approximation and learning abilities of deep CNNs dealing with functions of many variables and nonlinear functionals on infinite dimensional spaces.
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10. Efficient and Trustworthy Al and Its Applications to 5G Networks
Info Source:
https://www.polyu.edu.hk/ama/news-and-events/events/2022/11/20221115-ama-50th-anniversary-efficient-and-trustworthy-ai-and-its-applications-to-5g-networks/
11. A weakly nonlinear, energy stable scheme for the strongly anisotropic Cahn-Hilliard system
Venue:
https://polyu.zoom.us/j/95722862096?pwd=QWMxVHdlQllSQklwTXlmeldoNTA1UT09
Info Source:
https://www.polyu.edu.hk/ama/news-and-events/events/2022/11/20221116-a-weakly-nonlinear-energy-stable-scheme-for-the-strongly-anisotropic-cahn-hilliard-system/
12. A Stochastic Neural Network for uncertainty quantification of deep neural networks
Info Source:
https://hkumath.hku.hk/web/event/event-seminar.php
13. Reinforced Inverse Scattering
报告人: Prof. Haizhao Yang（Department of Mathematics , University of Maryland College Park）
报告时间: 2022-11-16 10:00-11:30
报告地点: 腾讯会议ID: 429-575-915 密码: 516815
信息来源: https://www.math.sjtu.edu.cn/Default/seminarshow/tags/MDAwMDAwMDAwMLGelJiFtKF2
报告摘要:
Artificial intelligence (AI) has been a powerful tool in science and engineering. Developing AI-aided tools in computational science is a recently popularized research topic. Along this direction, this talk discusses an example of designing intelligent machine that can learn a strategy from data to better solve inverse problems than human-designed algorithms. Especially, we apply reinforcement learning to achieve this goal for solving inverse scattering problems.
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14. Weak intermittency for the stochastic heat equation and its discretizations
Speaker: 陈楚楚（中国科学院数学与系统科学研究院）
Time: 2022-11-16 14:00-15:00
Venue: Tencent Meeting ID: 793-257-207 Meeting Password: 221116
Info Source:
https://ins.sjtu.edu.cn/seminars/2226
Abstract:
Intermittency is a universal phenomenon in random fields of multiplicative type. Does a discretization actually reflect the dynamical behavior in particular the weak intermittency of the original equation? To investigate this problem, this talk focuses on the stochastic heat equation with multiplicative noise under discretizations. We show that a class of discretizations of stochastic heat equation can preserve the weak intermittency, and even the index of Lyapunov exponents of the original equation.
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15. Recent advances in dynamical low-rank approximation for kinetic equations
Venue:
https://polyu.zoom.us/j/95287900375?pwd=Z3RuTWtNZjM2QmZnRElnTXZMNGVyZz09(密码1116)
Info Source:
https://www.polyu.edu.hk/ama/news-and-events/events/2022/11/20221116-recent-advances-in-dynamical-low-rank-approximation-for-kinetic-equations/
16. Finite Element Approximation of a Membrane Model for Liquid Crystal Polymeric Networks
Speaker: Lucas Bouck, University of Maryland
Time: 2022-11-16, 4:10PM-5PM Berkeley time
Registration Link and Info Source:
Venue:
https://berkeley.zoom.us/j/186935273
Info Source:
https://berkeleyams.lbl.gov/fall22/bouck.html
Abstract:
Liquid crystal polymeric networks are materials where a nematic liquid crystal is coupled with a rubbery material. When actuated with heat or light, the interaction of the liquid crystal with the rubber creates complex shapes. Starting from the classical 3D trace formula energy of Bladon, Warner and Terentjev (1994), we derive a 2D membrane energy as the formal asymptotic limit of the 3D energy. The derivation is similar to derivations in Ozenda, Sonnet, and Virga (2020) and Cirak et. al. (2014). We characterize the zero energy deformations and prove that the energy lacks certain convexity properties. We propose a finite element method to discretize the problem. To address the lack of convexity of the membrane energy, we regularize with a term that mimics a higher order bending energy. We prove that minimizers of the discrete energy converge to minimizers of the continuous energy. For minimizing the discrete problem, we employ a nonlinear gradient flow scheme, which is energy stable. Additionally, we present computations showing the geometric effects that arise from liquid crystal defects. Computations of configurations from nonisometric origami are also presented.
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17. The mathematical foundations of deep learning: from rating impossibility to practical existence theorems
Speaker: Simone Brugiapaglia
Time: 2022-11-16, 12:00 noon ET
Registration Link and Info Source:
https://www.oneworldml.org
Abstract:
Deep learning is having a profound impact on scientific research. Yet, while deep neural networks continue show impressive performance in a wide variety of fields, their mathematical foundations are far from being well established. In this talk, we will present recent developments in this area by illustrating two case studies.
First, motivated by applications in cognitive science, we will present “rating impossibility" theorems. These theorems identify frameworks where neural networks are provably unable to generalize outside the training set while performing the seemingly simple task of learning identity effects, i.e. classifying pairs of objects as identical or not.
Second, motivated by applications in scientific computing, we will illustrate “practical existence" theorems. These theorems combine universal approximation results for deep neural networks with compressed sensing and high-dimensional polynomial approximation theory. As a result, they yield sufficient conditions on the network architecture, the training strategy, and the number of samples able to guarantee accurate approximation of smooth functions of many variables.
Time permitting, we will also discuss work in progress and open questions in this research area.
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18. Learning Nonlocal Constitutive Models with Neural Operators
Speaker: Jiequn Han（Flatiron Institute）
Time: 2022-11-17 10:00-11:00
Venue: Tencent Meeting ID: 943861934 Password: 407546
Info Source:
https://ins.sjtu.edu.cn/seminars/2224
Abstract:
Constitutive models are widely used for modeling complex systems in science and engineering, when first-principle-based, well-resolved simulations are prohibitively expensive. For example, in fluid dynamics, constitutive models are required to describe nonlocal, unresolved physics such as turbulence and laminar-turbulent transition. However, traditional constitutive models based on PDEs often lack robustness and are too rigid to accommodate diverse calibration datasets. We propose a frame-independent, nonlocal constitutive model based on a vector-cloud neural network that represents the physics of PDEs and meanwhile can be learned with data. The proposed model can be interpreted as a neural operator, which by design, respects all the invariance properties desired by constitutive models, faithfully reflects the region of influence in physics, and is applicable to different spatial resolutions. We demonstrate its performance on modeling the Reynolds stress for Reynolds-averaged Navier--Stokes (RANS) equations in two situations: (1) emulating the Reynolds stress transport model through synthetic data and (2) calibrating the Reynolds stress through data from direct numerical simulations. Our results show that the proposed neural operator is a promising alternative to traditional nonlocal constitutive models and paves the way for developing robust and nonlocal, non-equilibrium closure models for the RANS equations.
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19. Feature Screening with Latent Responses
报告人: Cui Hengjian（Capital Normal University）
报告时间: 2022-11-17 10:00- 11:00
报告地点: Tencent Meeting ID: 475-277-586
信息来源: https://www.math.pku.edu.cn/kxyj/xsbg/tlb/probabilityandstatistics/145031.htm
报告摘要:
A novel feature screening method is proposed to examine the correlation between latent responses and potential predictors in ultrahigh dimensional data analysis. First, a confirmatory factor analysis (CFA) model is used to characterize latent responses through multiple observed variables. The expectation-maximization algorithm is employed to estimate the parameters in the CFA model. Second, R-Vector (RV) correlation is used to measure the dependence between the multivariate latent responses and covariates of interest. Third, a feature screening procedure is proposed on the basis of an unbiased estimator of the RV coefficient. The sure screening property of the proposed screening procedure is established under certain mild conditions. Monte Carlo simulations are conducted to assess the finite sample performance of the feature screening procedure. The proposed method is applied to an investigation of the relationship between psychological well-being and the human genome.
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20. Structure-preserving integrators for dissipative systems based on reversible–irreversible splitting
报告人: 商晓成（英国伯明翰大学）
报告时间: 2022-11-17 14:00-16:30
报告地点: 腾讯会议 ID: 774375728 密码: 771376
信息来源: https://www.math.sjtu.edu.cn/Default/seminarshow/tags/MDAwMDAwMDAwMLGelJiFpKF2
报告摘要:
We study the optimal design of numerical integrators for dissipative systems, for which there exists an underlying thermodynamic structure known as GENERIC (general equation for the nonequilibrium reversible–irreversible coupling). We present a framework to construct structure-preserving integrators by splitting the system into reversible and irreversible dynamics. The reversible part, which is often degenerate and reduces to a Hamiltonian form on its symplectic leaves, is solved by using a symplectic method (e.g. Verlet) with degenerate variables being left unchanged, for which an associated modified Hamiltonian (and subsequently a modified energy) in the form of a series expansion can be obtained by using backward error analysis. The modified energy is then used to construct a modified friction matrix associated with the irreversible part in such a way that a modified degeneracy condition is satisfied. The modified irreversible dynamics can be further solved by an explicit midpoint method if not exactly solvable. Our findings are verified by various numerical experiments, demonstrating the superiority of structure-preserving integrators over alternative schemes in terms of not only the accuracy control of both energy conservation and entropy production but also the preservation of the conformal symplectic structure in the case of linearly damped systems.
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21. Iterative regularization for nonsmooth inverse problems
Info Source:
https://www.math.cuhk.edu.hk/seminars/iterative-regularization-nonsmooth-inverse-problems
22. Overdamped generalized Langevin equations with fractional noise: Euler-type methods
Speaker: 代新杰（中国科学院数学与系统科学研究院）
Time: 2022-11-17 20:00-21:00
Venue: Tencent Meeting ID: 108-542-336 Password: 221117
Info Source:
https://ins.sjtu.edu.cn/seminars/2225
Abstract:
This talk considers the strong error analysis of the Euler and fast Euler methods for nonlinear overdamped generalized Langevin equations driven by the fractional noise. The main difficulty lies in handling the interaction between the fractional Brownian motion and the singular kernel, which is overcome by means of the Malliavin calculus and fine estimates of several multiple singular integrals. Consequently, these two methods are proved to be strongly convergent with order nearly min{2(H+α-1),α}, where H ∈ (1/2,1) and α ∈ (1-H,1) respectively characterize the singularity levels of fractional noises and singular kernels in the underlying equation. As an application of the theoretical findings, we further investigate the complexity of the multilevel Monte Carlo simulation based on the fast Euler method.
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23. Symmetry-preserving machine learning for computer vision, scientific computing, and distribution learning
Speaker: Wei Zhu (University of Massachusetts Amherst)
Time: 2022-11-17, 2:30PM ET
Registration Link and Info Source:
https://sites.google.com/view/minds-seminar/home
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.
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24. Relaxation oscillations in slow-fast Rosenzweig-MacArthur predator-prey model with additional food
报告人: 王成（南京财经大学）
报告时间: 2022-11-18 15:30-16:15
报告地点: 腾讯会议ID: 370-495-153 会议密码: 712450
信息来源: https://www.math.sjtu.edu.cn/Default/seminarshow/tags/MDAwMDAwMDAwMLGelJiGyqF2
报告摘要:
Motivated by real-world considerations, we propose and study a slow-fast predator-prey model in which the predator functional response is Holling type II and the predators are provided with additional food. Using geometric singular perturbation theory, we prove the existence and uniqueness of relaxation oscillations for the model. We corroborate our theoretical analysis by numerical simulations. Biological interpretation of relaxation oscillations is also presented.
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