召集人：陈荣亮（中国科学院深圳先进技术研究院）、蔡小川（澳门大学）、邹军（香港中文大学）

 时间：2024.05.26—2024.06.01

偏微分方程的数值和机器学习方法研讨会

Workshop on Numerical and Machine Learning

Methods for Partial Differential Equations

       Partial Differential Equations (PDEs) are ubiquitous in the modeling of complex physical, biological, and engineering systems. Designing and understanding accurate and efficient methods for solving these equations are critically important for the development of the entire field of sciences and technologies. As we stand on the precipice of a new era, where computational methods and artificial intelligence converge, we are organizing the International Workshop on Numerical and Learning Methods for PDEs to serve as a gathering point for internationally well-known experts, junior researchers, and enthusiasts from diverse domains who are passionate about advancing the frontiers of PDE solving. Over the course of our event, we will explore the interplay between traditional numerical approaches and the emerging power of machine learning techniques, charting new avenues for solving PDEs efficiently on traditional and newer computer architectures.

 会议日程安排

报告摘要

报告一（5月27日上午09:00-09:30）

报告人：杜强  美国哥伦比亚大学

题目：学习动力系统：从实践到数值分析

摘要：如果把给定动力系统的数值积分视为正问题，那么从观测数据中学习隐藏的动力系统则可以被视为逆问题。后者常常出现在多尺度过程的模型简化以及基于机器学习的数据驱动建模。正问题和逆问题的交互研究形成了信息和智能科学计算的循环。我们将试图讨论与其相关的一些实践与理论问题，包括状态变量的识别，维数的确认和数值方法的选择。

报告二（5月27日上午09:30-10:00）

报告人：黄建国  上海交通大学

题目：Friedrichs Learning: Weak Solutions of Partial Differential Equations via Deep Learning

摘要：In this talk, we intend to propose and analyze Friedrichs learning as a novel deep learning methodology that can learn the weak solutions of PDEs via a minimax formulation. The name “Friedrichs learning” is to highlight the close relation between our learning strategy and Friedrichs theory on symmetric systems of PDEs. The weak solution and the test function in the weak formulation are parameterized as deep neural networks in a mesh-free manner, which are alternately updated to approach the optimal solution networks approximating the weak solution and the optimal test function, respectively. Extensive numerical results indicate that our mesh-free Friedrichs learning method can provide reasonably good solutions for a wide range of PDEs defined on regular and irregular domains, where conventional numerical methods such as finite difference methods and finite element methods may be tedious or difficult to be applied, especially for those with discontinuous solutions in high-dimensional problems. The talk is based on a joint work with Fan Chen (Shanghai Jiao Tong University), Chunmei Wang (University of Florida) and Haizhao Yang (University of Maryland College Park).

报告三（5月27日上午10:30-11:00）

报告人：包维柱  新加坡国立大学

题目：A Structure-Preserving Parametric Finite Element Method for Geometric PDEs and Applications

摘要：In this talk, I begin with a review of different geometric flows (PDEs) including mean curvature (curve shortening) flow, surface diffusion flow, Willmore flow, etc., which arise from materials science, interface dynamics in multi-phase flows, biology membrane, computer graphics, geometry, etc. Different mathematical formulations and numerical methods for mean curvature flow are then discussed. In particular, an energy-stable linearly implicit parametric finite element method (PFEM) is presented in details. Then the PFEM is extended to surface diffusion flow and anisotropic surface diffusion flow, and a structure-preserving implicit PFEM is proposed. Finally, sharp interface models and their PFEM approximations are presented for solid-state dewetting. This talk is based on joint works with Harald Garcke, Wei Jiang, Yifei Li, Robert Nuernberg, Yan Wang and Quan Zhao.

报告四（5月27日上午11:00-11:30）

报告人：徐岩  中国科技大学

题目：Well-Balanced Path-Conservative Discontinuous Galerkin Methods with Equilibrium Preserving Space for Two-Layer Shallow Water Equations

摘要：This paper introduces well-balanced path-conservative discontinuous Galerkin (DG) methods for two-layer shallow water equations, ensuring exactness for both still water and moving water equilibrium steady states. The approach involves approximating the equilibrium variables within the DG piecewise polynomial space, while expressing the DG scheme in the form of path-conservative schemes. To robustly handle the nonconservative products governing momentum exchange between the layers, we incorporate the theory of Dal Maso, LeFloch, and Murat (DLM) within the DG method. Additionally, linear segment paths connecting the equilibrium functions are chosen to guarantee the well-balanced property of the resulting scheme. The simple “lake-at-rest” steady state is naturally satisfied without any modification, while a specialized treatment of the numerical flux is crucial for preserving the moving water steady state. Extensive numerical examples in one and two dimensions validate the exact equilibrium preservation of the steady state solutions and demonstrate its high-order accuracy. The performance of the method and high-resolution results further underscore its potential as a robust approach for nonconservative hyperbolic balance laws.

报告五（5月27日上午11:30-12:00）

报告人：刘宏宇  香港城市大学

题目：On a Geometric Principle in Wave Scattering and Applications

摘要：In this talk, I shall discuss a geometric principle to bridge the micro and macro scales in wave scattering theory as well as its applications in super-resolution imaging and invisibility cloaking.

报告六（5月27日下午14:30-15:00）

报告人：刘杰  国防科技大学

题目：An Intelligent Computing Framework for Solving Partial Differential Equations

摘要：Partial differential equations (PDEs) play a crucial role in investigating new physical phenomena and exploring the principles of fluid mechanics. However, traditional PDE solving systems often face the challenges of long research cycles, high costs, and extensive human-computer interactions due to the growing complexity of computational tasks. To meet the burgeoning requirements of contemporary physical sciences, in recent years, the coupling of traditional scientific computing techniques with promising deep neural networks well-known from computer science have emerged as a new research paradigm. The networks utilize multiple layers of interconnected neurons to automatically learn important features from high-dimensional parameter spaces. By performing an optimization process based on the loss function, the network model brings the promise of a powerful approach to approximate complex and nonlinear systems. This talk represents an intelligent computing framework we proposed for solving partial differential equations. More specifically, we will report our recent progress in intelligent mesh generation, surrogate solving, and visualization, anticipating providing a reference source for future research applications in the field of neural network-based PDE solving.

报告七（5月27日下午15:00-15:30）

报告人：武海军  南京大学

题目：Adaptive Finite Element Method for a Nonlinear Helmholtz Equation with High Wave Number

摘要：A nonlinear Helmholtz (NLH) equation with high frequencies and corner singularities is discretized by the linear finite element method (FEM). After deriving some wave-number-explicit stability estimates and the singularity decomposition for the NLH problem, a priori stability and error estimates are proved for the FEM on shape regular meshes including the case of locally refined meshes. Then a posteriori upper and lower bound using a new residual-type error estimator, which is equivalent to the standard one, are derived for the FE solutions to the NLH problem. These a posteriori estimates have confirmed a significant fact that is also valid for the NLH problem, namely the residual-type estimator seriously underestimates the error of the FE solution in the preasymptotic regime, which was first observed by Babuska et al. [Int J Numer Methods Eng 40 (1997)] for a one-dimensional linear problem. Based on the new posteriori error estimator, both the convergence and the quasi-optimality of the resulting adaptive finite element algorithm are proved the first time for the NLH problem, when the initial mesh size lying in the preasymptotic regime. Finally, numerical examples are presented to validate the theoretical findings and demonstrate that applying the continuous interior penalty (CIP) technique with appropriate penalty parameters can reduce the pollution errors efficiently. In particular, the nonlinear phenomenon of optical bistability with Gaussian incident waves is successfully simulated by the adaptive CIPFEM.

报告八（5月27日下午16:00-16:30）

报告人：林平  英国邓迪大学

题目：A Phase-Field Model and Its Computational Method for Vesicle Motions and Interactions Through a 2D Lennard Jones Type Interacting Potential

摘要：Under a thermodynamically consistent phase-field modeling framework for the binary incompressible (quasi-incompressible) fluid, which allows for the different properties (densities, viscosities, and heat conductivities) of each fluid component, we will first show how to derive such a model for motions and deformations of vesicles (e.g. red blood cells or RBCs) in a blood flow passing through a narrowed blood vessel. We will also propose a 2D Lennard-Jones type of interaction potential for vesicle-vesicle and vesicle-vessel wall interactions. Mass conserving and energy law preserving finite element schemes are designed and showed for these models. Many examples including the benchmark RBC deformation under stretching forces, RBC passing through a narrowed vessel wall, cell-vessel wall attraction, cell-cell interactions, and cell aggregation, and how RBCs divide in the blood flow at a vessel bifurcation are computed and will be presented in the talk.

报告九（5月27日下午16:30-17:00）

报告人：李义宝  西安交通大学

题目：3D打印加工成型的多物理场孪生计算

摘要：本报告将介绍3D打印加工成型过程中的热固、热力、液固耦合的机理，并描述宏观-介观-微观耦合的加工成型的模型和计算方法。此外，本报告将介绍模型参数与初值重构的算法，数据与模型耦合修正方法。系统地分享3D打印加工成型的多物理场孪生的建模与计算成果。

报告十（5月27日下午17:00-17:30）

报告人：李步扬  香港理工大学

题目：Convergence of Arbitrary Lagrangian-Eulerian Interface Tracking Methods for Moving Interface Problems

摘要：We report the recent development of matrix-vector formulation of evolving finite element approximations to solution-driven surface evolution, which was originally developed for analyzing evolving finite element approximations to surface evolution under geometric curvature flows. Then we report some recent applications of this approach to shape optimization, two-phase fluid flows (with an unknown sharp interface), and fluid-structure interaction problems (with an unknown sharp interface).

报告十一（5月27日下午17:30-18:00）

报告人：吴树林  东北师范大学

题目：Non-uniform Time Grids for Parareal Computation

摘要：Parareal uses coarse and fine time grids to realize parallel-in-time computation. Many results about this algorithm using equally spaced time grids are available in literature. In this talk we introduce some of the most recent progress for using non-uniform time grids. Using non-uniform time grids is natural for parareal and we do not need to modify anything in practice, but it indeed has some non-negligible influence on the convergence rate.

报告十二（5月28日上午09:00-09:30）

报告人：王奇  美国南卡罗来纳大学

题目：Thermodynamically Consistent Hybrid Computational Models for Fluid-Particle Interactions

摘要：We introduce a novel computational framework designed to explore the dynamic interactions between fluid and solid particles or structures immersed in a viscous fluid medium adhering to the generalized Onsager principle. This innovative framework harnesses the power of the phase-field-embedding method, in which each solid component, whether rigid or elastic, is characterized by a volume-preserving phase field. The unified velocity within the fluid-solid ensemble governs the movement of both solid particles and the surrounding fluid, specifically for passive particles. Active particles, however, are not only influenced by this unified velocity but are also driven by their self-propelling velocities. To capture exclusive volume interactions among particles and between particles and boundaries, we employ repulsive potential forces at a coarser scale. These forces effectively model repulsion and collision effects. Rigid particles maintain structural integrity by enforcing a zero-velocity gradient tensor within their spatial domains, necessitating the introduction of a constraining stress tensor. In contrast, elastic particles are governed by a quasi-linear constitutive equation describing the elastic stress within their domains, allowing for accurate modeling of their deformations. The motion of solid particles is tracked by monitoring the dynamics of their centers of mass. This approach facilitates the development of a hybrid, thermodynamically consistent hydrodynamic model applicable to both rigid and elastic particles. To numerically solve this thermodynamically consistent model for elastic particles, we present a structure-preserving numerical algorithm. Notably, in the limit of an infinite elastic modulus, this algorithm converges to the one employed for modeling rigid particles. Finally, we substantiate the effectiveness, accuracy, and stability of our proposed scheme through a series of numerical experiments. These experiments not only validate the computational framework but also showcase its capabilities, reinforcing the reliability of our approach.

报告十三（5月28日上午09:30-10:00）

报告人：谢小平  四川大学

题目：A Projection-Based Time-Segmented Reduced Order Model for Fluid-Structure Interactions

摘要：A type of novel projection-based, time-segmented reduced order model (ROM) is proposed for dynamic fluid-structure interaction (FSI) problems based upon the arbitrary Lagrangian–Eulerian (ALE)-finite element method (FEM) in a monolithic frame, where spatially, each variable is separated from others in terms of their attribution (fluid/structure), category (velocity/pressure) and component (horizontal/vertical) while temporally, the proper orthogonal decomposition (POD) bases are constructed in some deliberately partitioned time segments tailored through extensive numerical trials. By the combination of spatial and temporal decompositions, the developed ROM approach enables prolonged simulations under prescribed accuracy thresholds. Numerical experiments are carried out to compare numerical performances of the proposed ROM with corresponding full-order model (FOM) by solving a two-dimensional FSI benchmark problem that involves a vibrating elastic beam in the fluid, where the performance of offline ROM on perturbed physical parameters in the online phase is investigated as well. Extensive numerical results demonstrate that the proposed ROM has a comparable accuracy to while much higher efficiency than the FOM. The developed ROM approach is dimension-independent and can be seamlessly extended to solve high dimensional FSI problems.

报告十四（5月28日上午10:30-11:00）

报告人：蔡智强  美国普渡大学

题目：Neural Networks in Scientific Computing

摘要：Neural networks (NNs) have achieved astonishing performance in computer vision, natural language processing, and many other artificial intelligence (AI) tasks. This success encourages wide applications to other fields, such as scientific computing. In this talk, I will first give a brief introduction of NNs from numerical analysis perspective and use a simple example to show why NNs are superior to piecewise polynomials on fixed meshes when approximating discontinuous functions with unknown interface. I will then describe two NN-based methods for solving nonlinear scalar hyperbolic conservation laws. One is a space-time approach (least-squares neural network (LSNN) method), and the other is an explicit approach (evolving neural network (ENN) method) that emulates the underlying physics. Both methods show a great potential to sharply capture shock without oscillation, overshooting, or smearing. The ENN method in one dimension is super accurate and efficient compared with existing, well-developed mesh-based numerical methods. The exceptional approximation powers of NN come with a price: the procedure for determining the values of the nonlinear parameters of NN entails solving a high-dimensional non-convex optimization problem. If time permits, I will describe our newly developed training algorithm for shallow ReLU NN.

报告十五（5月28日上午11:00-11:30）

报告人：徐立伟  电子科技大学

题目：An Investigation on the Structure-Preserving Deep Learning Methods for Solving the Radiative Transport Equations

摘要：In this talk, we present two schemes coupling the neural network methods and the asymptotic-preserving schemes for the solution of the radiative transport equation. The first scheme is based on a micro-macro decomposition scheme, and the second one is designed through an introduction of macroscopic auxiliary equations. The schemes possess advantages on dealing with problems with high dimensionality and multiscale characteristics. Numerical examples are given to demonstrate the efficiency of numerical methods. This is joint work with Hongyan Li, Song Jiang, Wenjun Sun and Guanyu Zhou.

报告十六（5月28日上午11:30-12:00）

报告人：周知  香港理工大学

题目：Neural Network Approximation for PDEs with Nonsmooth Solutions

摘要：Neural network solvers have been successfully applied to many diverse PDEs. However, when applying the method to problems involving singularity, e.g., singular problem data or geometric singularities, the obtained approximations often have low accuracy, due to limited regularity of the exact solution. In this work, we investigate neural networks for solving elliptic equations with nonsmooth solutions. Our approach utilizes the idea of singularity splitting/enrichment, building on known analytic insights into these problems. This closely aligns with the prominent paradigm of incorporating physics into machine learning techniques. The overall approach is as straightforward to implement as the Physics informed neural networks (PINNs), while significantly improving the accuracy of the approximations. Numerical simulations are presented to illustrate the efficiency of the method, and a comparative study is presented with existing neural network-based approaches.

报告十七（5月29日上午09:00-09:30）

报告人：王筱平  香港中文大学（深圳）

题目：Topology Optimization Using Generative Models

摘要：Topology optimization, which aims to find the optimal physical structure that maximizes mechanical performance, is vital in engineering design applications in aerospace, mechanical, and civil engineering. We introduce a deep generative model, based on diffusion models, to address the structure optimization problem. Combine with the threshold dynamics method, we present a successful frame for the topology optimization.

报告十八（5月29日上午09:30-10:00）

报告人：刘颖智  澳门大学

题目：A Learning-based Nonlinear Preconditioner for Modeling Blood Flows with Stochastic Parameters

摘要：As we know, the simulation of blood flow in patient-specific arteries depends on the common parameters including the inflow rate, heart rate, viscosity, density, systolic pressure, and diastolic pressure. The six parameters usually vary within a reasonable range. For complex arteries such as those with severe stenosis or aneurysms, the inexact Newton method can sometimes converge slowly or even diverge when solving the nonlinear discretized systems of blood flows for any a set of parameters within the range. This numerical phenomenon is related to local and strong nonlinearity. In this talk, we introduce a learning-based nonlinear preconditioner for this problem, with these parameters stochastically varied in a given range.

报告十九（5月29日上午10:30-11:00）

报告人：蒋代军  华中师范大学

题目：Convergence Analysis of LMM and DDMs for a Parabolic Inverse Robin Problem

摘要：We study in this talk some numerical methods for solving the highly nonlinear and ill-posed inverse problem of identifying the Robin coefficients in parabolic systems. We first apply the Levenberg-Marquardt method (LMM) to transform the Tikhonov regularized nonlinear non-convex minimizations into convex minimizations. And the quadratic convergence of the LMM is rigorously established for the nonlinear parabolic inverse problems for the first time, under a simple novel adaptive strategy for selecting regularization parameters during the LM iteration. Then the domain decomposition methods (DDMs) are used to solve the convex minimizations. The methods are completely local, and the local minimizers have explicit expressions within the subdomains. Numerical experiments are presented to show the accuracy and efficiency of the methods, in particular, the convergence seems nearly optimal in the sense that the iteration number of the methods is independent of the mesh size.

报告二十（5月29日上午11:00-11:30）

报告人：蒋维  武汉大学

题目：High Order in Time, BGN-based Parametric Finite Element Methods for Solving Geometric Flows

摘要：Geometric flows have recently attracted lots of attention from scientific computing communities. One of the most popular schemes for solving geometric flows is the so-called BGN scheme, which was proposed by Barrett, Garcke, and Nurnberg (J. Comput. Phys., 222 (2007), pp. 441--467). However, the BGN scheme only can attain first-order accuracy in time, and how to design a temporal high-order numerical scheme is challenging. Recently, based on a novel approach, we have successfully proposed temporal high-order, BGN-based parametric finite element method for solving geometric flows of curves/surfaces. Furthermore, we point out that the shape metrics (i.e., manifold distance), instead of the function norms, should be used to measure numerical errors of the proposed schemes. Finally, ample numerical experiments demonstrate that the proposed BGN-based schemes are high-order in time in terms of the shape metric, and much more efficient than the classical BGN schemes.

报告二十一（5月29日上午11:30-12:00）

报告人：董国志  中南大学

题目：Optimal Control of Neural-Network-Informed PDEs

摘要：Neural-network-informed PDEs are a class of partial differential equations with constituents that are in principle unknown and are approximated by (nonsmooth) neural networks. In this talk, I first outline a general framework of using learning-informed models in inverse problems and optimal control. I will present some theoretical and numerical aspects of such optimal control problems. In the end, we shall focus on the case of ReLU-neural-networks. I will propose a descent algorithm to solve the optimal control problem leveraging the structures of the ReLU function. This is a joint work with Michael Hintermueller (WIAS and Humboldt Uni-Berlin) and Kostas Papafitsoros (Queen Mary Uni-London).

报告二十二（5月29日下午14:30-15:00）

报告人：张林波  中国科学院数学与系统科学研究院

题目：Algorithms and Software for Implementation of High-Order Unfitted Finite Element Methods

摘要：Unfitted finite element methods (UFEM), in which the mesh is not required to fit the geometry of the computational domain, have attractive features in dealing with problems involving complex, evolving/changing or curved geometries. Meanwhile, some major issues in both algorithm design/analysis and implementation, such as bad conditioning of the discrete systems and difficulties in designing robust numerical quadrature procedures on curved domains, are still unresolved, especially with high order UFEM. In this talk I will report on our recent progress in developing a parallel software toolbox for high order UFEM. This toolbox is based on the parallel finite element software package PHG (Parallel Hierarchical Grid, http://lsec.cc.ac.cn/phg/index_en.htm). It provides basic building blocks for the implementation of parallel high order UFEM programs, including a general-purpose library for high order numerical quadrature on cut elements, transparent element merging and basis orthonormalization, and an interface to the open-source CAD library OpenCASCADE.

报告二十三（5月29日下午15:00-15:30）

报告人：张智文  香港大学

题目：An Interacting Particle Method for Computing Large Deviation Rate Functions of Entropy Production for Diffusion Processes in the Vanishing-Noise Limit and High Dimensions

摘要：In this talk, we study an interacting particle method (IPM) for computing the large deviation rate function of entropy production in diffusion processes, focusing on the vanishing-noise limit and high dimensions. We show that the principal eigenvalue of elliptic, non-self-adjoint operators, crucial for deriving the rate function, can be approximated by evaluating the spectral radius of a discretized evolution operator. This operator is obtained via an operator splitting and Euler-Maruyama scheme with a small time step. The spectral radius is accessible through iterations of this discretized semigroup, fitting the IPM well. The IPM suits unbounded domains, scales to high dimensions, and adapts to singular behaviors in the vanishing-noise limit. We present numerical examples up to 16 dimensions, showing that our eigenvalue approximations converge to the analytical vanishing-noise limit with fixed particles and time steps.

报告二十四（5月29日下午16:00-16:30）

报告人：涂学民  美国堪萨斯大学

题目：Robust BDDC Algorithms for the Brinkman Problem with HDG Discretizations

摘要：In this talk, the balancing domain decomposition by constraints methods (BDDC) are applied to solve the saddle point problem arising from a hybridizable discontinuous Galerkin (HDG) discretization for the Brinkman equations. In the BDDC algorithms, the edge/face average constraints are enforced across the subdomain interface for each velocity component to ensure that the BDDC preconditioned conjugate gradient (CG) iterations stay in a special subspace, where the reduced system from the original saddle point problem is positive definite. For the cases where the parameters have jumps only across the subdomain interface, the condition number is proved to be uniformly bounded under both Stokes and Darcy dominant cases with a deluxe scaling. When the parameters are highly discontinuous with large jumps inside subdomains, additional adaptively chosen primal constraints, obtained by solving local generalized eigenvalue problems, are introduced to control the condition numbers. Several numerical experiments will be discussed to confirm the theory.

报告二十五（5月29日下午16:30-17:00）

报告人：阳莺  桂林电子科技大学

题目：The Numerical Method on Polygonal/Polyhedron Meshes and Machine Learning Acceleration for the PNP Equations with Applications to Ion Channel

摘要：The Poisson-Nernst-Planck (PNP) equations are a kind of nonlinear coupled partial differential equations, which are widely applied to describe the transport of the charged particles in ion channels, electrochemical systems, semiconductors, etc. This talk includes two parts. First, we introduce a numerical method called virtual element method (VEM) for PNP equations which can be used on very general polygonal/polyhedron. We present both the theoretical and numerical results of the VEM for PNP equations with application to practical biological ion channel problem. Second, we report some numerical results on machine learning acceleration for a nonlinear solver of PNP equations. The Gummel iteration is a commonly used nonlinear solver for PNP equations, the efficiency of which is largely influenced by the relaxation parameter. The machine learning method is used to predict optimal parameter of Gummel iteration, which improves the efficiency of the iteration. Numerical examples including the practical problem show the efficiency of machine learning acceleration.

报告二十六（5月29日下午17:00-17:30）

报告人：许小静  广西大学

题目：Spatial Second-order Positive and Asymptotic Preserving Filtered PN Schemes for Nonlinear Radiative Transfer Equations

摘要：A spatial second-order scheme for the nonlinear radiative transfer equations is introduced in this work. The scheme is based on the filtered spherical harmonics (FPN) method for the angular variable and the unified gas kinetic scheme (UGKS) framework for the spatial and temporal variables respectively. To keep the scheme positive and second-order accuracy, firstly, we use the implicit Monte Carlo (IMC) linearization method in the construction of the UGKS numerical boundary fluxes. Then, by carefully analyzing the constructed second-order fluxes, we establish the sufficient conditions that guarantee the positivity of the radiative energy density and material temperature. Finally, we employ linear scaling limiters for the angular variable in the ���� reconstruction and for the spatial variable in the piecewise linear slopes reconstruction respectively, which are shown to be realizable and reasonable to enforce the sufficient conditions holding. Thus, the desired scheme is obtained. Furthermore, we can show that in the regime ε≪1, and the regime ε=O(1), the second-order fluxes can be simplified. And, a simplified scheme is thus presented, which possesses all the properties of the non-simplified one. Inheriting the merit of UGKS, the proposed schemes are asymptotic preservation. By employing the FPN method for the angular variable, the proposed schemes are almost free of ray effects. Moreover, the above-mentioned way of imposing the positivity would not destroy both AP and second-order accuracy properties. Various numerical experiments are included to validate the properties of the proposed schemes.

报告二十七（5月29日下午17:30-18:00）

报告人：于毅  广西大学

题目：Nonoverlapping Spectral Additive Schwarz Methods (NOSAS)

摘要：To efficiently solve large sparse linear systems arising from elliptic problems with heterogeneous coefficients, many domain decomposition methods (DDMs) arise using the idea of the generalized eigenvalue problems. For example, Adaptive BDDC, GenEO, and Adaptive GDSW are of those types. Nonoverlapping Spectral Additive Schwarz Methods (NOSAS) are a new type of DDMs which use the idea of generalized eigenvalue problems on each subdomain’s interface to construct the bilinear form of the coarse problem. The coarse problem has global and local interaction components which are respectively associated with low-frequency and high-frequency modes obtained locally from the generalized eigenvalue problems. The main complexity of the coarse problem lies in the global components, while the local components can be solved in parallel. In this talk, we propose several variants of the NOSAS methods and compare them with Adaptive BDDC, GenEO, and Adaptive GDSW. We highlight the main difference of our methods compared to traditional DDMs and demonstrate some advantages of this new method.

报告二十八（5月30日上午09:00-09:30）

报告人：舒适  湘潭大学

题目：Fast Solvers for Several PDE Discrete Systems Based on Deep Learning

摘要：This report begins by briefly introducing a reduced order model based on autoencoder and the Fourier neural solver based on meta learning, which are then applied to solve convection-diffusion equations. Next, it focuses on Wave-ADR neural solver designed for Helmholtz equations with high wavenumber and heterogeneous media. The main idea involves addressing the characteristic and non-characteristic components of the iterative error separately, depending on whether the error frequency is close to the angular frequency of Helmholtz equation. Non-characteristic components are eliminated using a standard multigrid V-cycle, known as the wave cycle, employing carefully selected smoothers at each level. Meanwhile, characteristic components are addressed through the solution of an advection-diffusion-reaction equation using another V-cycle at a coarser scale, termed the ADR cycle. Furthermore, we present an efficient implementation utilizing differentiable programming, enabling Wave-ADR as an end-to-end Helmholtz solver with support for backpropagation, batch processing, and GPU acceleration. Numerical experiments showcase the new solver's enhanced adaptability to high wavenumber and improved computational efficiency compared to existing deep learning-based multigrid preconditioners.

报告二十九（5月30日上午09:30-10:00）

报告人：胡齐芽  中国科学院数学与系统科学研究院

题目：A Hybrid Weighted Schwarz Method for Helmholtz Equations

摘要：In this talk we introduce a two-level hybrid weighted Schwarz method with local impedance conditions for Helmholtz equations in two dimensions, which are discretized by the finite element method with conforming nodal finite elements. We design and analyze a new adaptive coarse space for this kind of Schwarz method. This coarse space is spanned by some eigenvalue functions of local generalized eigenvalue problems, which are defined by weighted positive semi-definite bilinear forms on subspaces consisting of local discrete Helmholtz-harmonic functions from impedance boundary data. The two-level hybrid Schwarz preconditioner with the proposed coarse space possesses uniform convergence independent of the mesh size, the subdomain size, and the wave numbers under suitable assumptions. We also report an economic coarse space to avoid solving generalized eigenvalue problems.

报告三十（5月30日上午10:30-11:00）

报告人：蔡勇勇  北京师范大学

题目：Positivity Preserving and Mass Conservative Projection Method for the Poisson-Nernst-Planck Equation

摘要：We present a novel approach to construct structure preserving approximations for the Poisson-Nernst-Planck equations, focusing on the positivity preserving and mass conservation properties. The strategy consists of a standard time marching step with a projection (or correction) step to satisfy the desired physical constraints (positivity and mass conservation). Based on the L² projection, we construct a second order Crank-Nicolson type finite difference scheme, which is linear (excluding the very efficient L² projection part), positivity preserving and mass conserving. Rigorous error estimates in L² norm are established, which are both second order accurate in space and time. The other choice of projection, e.g. H¹ projection will be discussed.

报告三十一（5月30日上午11:00-11:30）

报告人：廖奇峰  上海科技大学

题目：A High-Dimensional Density Estimation Method and Its Application for Solving PDEs

摘要：Probability density estimation remains an open challenging problem in computational science and engineering. By coupling the Knothe-Rosenblatt (KR) rearrangement and the flow-based generative model, we developed an invertible transport map, called KRnet, for high-dimensional density estimation. In this talk, we give an overview of KRnet and its adaptive version for solving high-dimensional PDEs.

报告三十二（5月30日上午11:30-12:00）

报告人：杨海建  湖南大学

题目：Parallel-in-time (PinT) Algorithms for Large-Scale Flow Problems

摘要：As the number of processors on supercomputers has increased dramatically, there is a growing interest in developing scalable algorithms with a high degree of parallelism for large-scale simulation. However, traditional simulators and algorithms for such nonlinear problems are usually based on the family of time-marching methods, where parallelization is restricted to the spatial dimension only. In this talk, we propose a family of parallel-in-time (PinT) algorithms for solving some large-scale flow problems from computational fluid dynamics or reservoir simulation, to fully exploit the parallelism of supercomputers.

报告三十三（5月31日上午09:00-09:30）

报告人：许传炬  厦门大学

题目：Müntz Legendre Polynomials and Applications

摘要：In this talk we discuss the Müntz polynomials and their approximation. The Müntz Legendre polynomials arise by orthogonalizing the Müntz system with respect to Lebesgue measure on [0, 1], which are extension of the classical polynomials. These polynomials were introduced in 90th and defined by the integral of a complex function in the complex plane. However, approximation by the Müntz-Legendre polynomials faces challenges both theoretically and numerically. We discuss recurrence formulas for moments and new Gauss quadratures and interpolation methods based on the Müntz polynomials. These methods are applicable to approximate both smooth functions and singular functions and to solve typical integro-differential equations whose solutions exhibit singularities.

报告三十四（5月31日上午09:30-10:00）

报告人：黄记祖  中国科学院数学与系统科学研究院

题目：APTT: An Accuracy-Preserved Tensor-Train Method for the Boltzmann-BGK Equation

摘要：In this talk, we propose a novel accuracy-preserved tensor-train (APTT) method to solve the Boltzmann-BGK equation. We use the Crank-Nicolson Leap Frog (CNLF) scheme for temporal discretization, based on which the collision term can be evaluated only once at each time step. A second-order scheme based on the upwind rule is employed for spatial discretization to match the accuracy of the CNLF scheme. At each time step, the linear system is constructed with the tensor-train (TT) format, where the matrix, the right-hand side, and the collision term involved in the linear system are all represented using the low-rank TT format. Based on such a representation, an efficient and effective iterative solver can be implemented for solving the linear system, which can reduce two orders of magnitude in terms of both time and memory costs compared with classical methods. We present a theoretical analysis to show that the proposed APTT method can maintain accuracy and an extensive set of 3D3V Boltzmann-BGK test examples to demonstrate its effectiveness.

报告三十五（5月31日上午10:30-11:00）

报告人：周泽慧  美国新泽西州立罗格斯大学

题目：On the Approximation of Bi-Lipschitz Maps by Invertible Neural Networks

摘要：Invertible neural networks (INNs) represent an important class of deep neural network architectures that have been widely used in applications. The universal approximation properties of INNs have been established recently. However, the approximation rate of INNs is largely missing. In this talk, I will first present an analysis of the capacity of a class of coupling-based INNs to approximate bi-Lipschitz continuous mappings on a compact domain. Then I will introduce an approach for approximating bi-Lipschitz maps on infinite-dimensional spaces that simultaneously approximate the forward and inverse maps, by combining model reduction with principal component analysis and INNs for approximating the reduced map and give an overall approximation error of this approach.

报告三十六（5月31日上午11:00-11:30）

报告人：李世顺  信阳师范大学

题目：The Framework of the Overlapping Domain Decomposition Methods for Some Time-Stepping and Time-Parallel Time Integration Schemes

摘要：In this talk, we first introduce some time-stepping and time-parallel time integration schemes. Then we present a family of block implicit methods (BIM) which are described by a tableau including two matrices and two vectors. It shows that the classical time-stepping schemes such as backward Euler, Crank-Nicolson, BDF and the fully implicit Runge-Kutta (based on Gaussian, Radau and Lobatto quadrature formulae) are special cases of BIM. Further, we propose a type of BIM with positive definite matrices and prove that the classical Schwarz theory for parabolic problems can be also extended for high-order BIM. Numerical experiments carried out on a parallel computer with thousands of processors confirm the optimality and scalability of the method.

报告三十七（5月31日上午11:30-12:00）

报告人：邹军  香港中文大学

题目： 求解一般数学物理反问题的两种较为有效的数值方法

摘要： 直接抽样方法和自适应方法是求解一般数学物理反问题的两种较为有效的数值方法。本次报告将回顾这两类数值方法的发展现状、它们在解决一般反问题方面的必要性和有效性以及它们于一般反问题上的应用。

墙报摘要

墙报展示一

展示人：闫争争  中国科学院深圳先进技术研究院

题目：Simulation of Blood Flows in Patient-specific Cardiac-Cerebral System

摘要：Coronary artery disease (CAD) and ischemic stroke, both resulting from hemodynamic abnormalities, show a significant association in clinical studies. The highly patient-specific cardiac-cerebral arterial network, including the unique anatomy of the aortic arch, Circle of Willis, and cerebral vasculature, along with turbulence, pulsatility, and stringent clinical time requirements, poses computational challenges for cardiac-cerebral coupled hemodynamic simulations. This study applies an implicit scheme and an unstructured grid-based Newton-Krylov-Schwarz algorithm to investigate patient-specific cardiac-cerebral blood flow during coronary stenting and successfully predicts the occurrence of ischemic stroke complications. We also present the parallel performance of the algorithm using a large number of processor cores. 

墙报展示二

展示人：闫争争  中国科学院深圳先进技术研究院

题目：Functional Region-based Approach for Patient-specific Cerebral Blood Flow Simulation

摘要： Accurate outflow boundary conditions remain a significant challenge in 3D computational fluid dynamics simulations of patient-specific cerebral blood flow (CBF). In current studies, medical images like magnetic resonance imaging (MRI) are used solely for the 3D geometric reconstruction of cerebral arteries. However, imaging data contains other rich, individualized information highly relevant to hemodynamics, such as personalized brain tissue/function regions. To address this issue, this study proposes a novel functional region-based approach to enhance the accuracy of CBF simulations. The approach partitions cerebral vessels into functional regions based on patient-specific vascular territories derived from medical images. Within each functional region, parameters of Windkessel models for individual outlets are calculated based on their corresponding diameters/areas, accounting for both functional and geometric characteristics. Numerical results demonstrate that the proposed functional region-based approach better agrees with clinical data measured by Transcranial Doppler ultrasonography compared to the current area-based approach, suggesting improved accuracy in predicting cerebral hemodynamics.

墙报展示三

展示人：覃善林  中国科学院深圳先进技术研究院

题目：Numerical Simulation of Blood Flows in A Full-body Artery

摘要： The blood is distributed to different regions of the human body through the complex visceral arterial network. An accurate simulation of blood flows in the full-body artery is important for the understanding of many vascular diseases, such as the high blood pressure, but the computational cost is very high. We introduce a parallel algorithm for the modeling of pulsatile flows in the full-body artery with the consideration of blood distributions to 14 regions.

墙报展示四

展示人：覃善林  中国科学院深圳先进技术研究院

题目：Computational Hemodynamics for Rupture Risk Assessment of AAA

摘要：This study established a methodology for analyzing abdominal aortic aneurysm (AAA) rupture risk using computational fluid dynamics (CFD). We compared hemodynamic parameters between 10 ruptured and 10 unruptured AAA cases. Statistical analysis revealed a significant correlation (p-value < 0.001) between averaged Oscillatory Shear Index (OSI) and AAA rupture. Our findings suggest that hemodynamic factors play a crucial role in AAA rupture and support the potential of CFD analysis for improved diagnosis and treatment strategies.

墙报展示五

展示人：徐磊  中国科学院深圳先进技术研究院

题目：Towards Megacity-scale Wind Flow Simulations on Tianhe new-generation supercomputer

摘要： Urban wind flow simulation, based on numerical methods, serves as a powerful tool for understanding the intricate interactions between urban structures and atmospheric conditions. The Lattice Boltzmann method (LBM) is commonly used for urban wind flow simulations. However, existing LBM simulation approaches have poor scalability on large-scale parallel computing systems and cannot support high-resolution wind flow simulation in megacities spanning hundreds of square kilometers. We present THLB, an LBM simulator specifically designed to facilitate large-scale urban wind flow simulations.

墙报展示六

展示人：徐磊  中国科学院深圳先进技术研究院

题目：A Parallel Discrete Unified Gas Kinetic Scheme on Unstructured Grid for High-speed Compressible Flow Simulation

摘要： The discrete unified gas kinetic scheme (DUGKS) is a recently devised approach to simulate multiscale flows based on the kinetic models, which also shows distinct features for continuum flows. Most of the existing DUGKS are sequential or based on structured grids, thus limiting their scope of application in engineering. In this work, a parallel DUGKS for inviscid high-speed compressible flows on unstructured grids is proposed.

墙报展示七

展示人：蒋毅  中国科学院深圳先进技术研究院

题目：A Highly Parallel Algorithm for Simulating the Elastodynamics of a Patient-specific Human Heart

摘要：In this work, a highly parallel method is developed for simulating the elastodynamics of a four-chamber human heart with patient-specific geometry. The heterogeneous hyperelastic model is discretized by a finite element method in space and a fully implicit adaptive method in time, and the resulting nonlinear algebraic systems are solved by a scalable domain decomposition algorithm. The deformations of the cardiac muscles are quite complex due to the realistic geometry, the heterogenerous hyperelasticity of the cardiac tissue, and the myocardial fibers with active stresses. Moreover, the deformations in different chambers and at different phases of the cardiac cycle are very different. To simulate all the muscle movements, the temporal-spatial mesh needs to be sufficiently fine, but not too fine so that the overall computing time is manageable, we introduce a baseline mesh in space and a two-level time stepping strategy including a uniform baseline time step size to obtain the desired time accuracy and an adaptive time stepping method within a baseline time step to guarantee the convergence of the nonlinear solver. Through numerical experiments, we investigate the performance of the proposed method with respect to the material coefficients, the fiber orientations, as well as the mesh sizes and the time step sizes. For an unstructured tetrahedral mesh with more than 200 million degrees of freedom, the method scales well for up to 16,384 processor cores for all steps of an entire cardiac cycle. 

墙报展示八

展示人：林增  中国科学院深圳先进技术研究院

题目：A Highly Parallel Numerical Method for the Navier-Stokes/Darcy Equations

摘要： In this poster, we establish the coupled Navier-Stokes/Darcy equations to describe the free flow in the hepatic vessels and the porous media flow in the hepatic tissue simultaneously. We construct a scalable parallel finite element method based on the Newton-Krylov-Schwarz algorithms for solving the coupled Navier-Stokes/Darcy equations. Our accurate algorithm is implemented on the supercomputer and provides technical supports for the clinical diagnosis of liver cirrhosis.

墙报展示九

展示人：程载恒  中国科学院深圳先进技术研究院

题目：Hemodynamic Analysis of Pulmonary Artery Hypertension Patients Based on Large-Scale Parallel CFD Method

摘要：Pulmonary artery hypertension (PAH) is a progressive disease characterized by increased blood pressure in the pulmonary arteries. The “gold standard” for diagnosing PAH is right heart catheterization, an invasive procedure that measures pressure by inserting a catheter into the pulmonary artery. We have developed a highly efficient parallel method for accurate blood flow simulation using patient-specific geometry and parameters, offering a potential non-invasive assessment of PAH.

墙报展示十

展示人：宫玉杰  澳门大学

题目：Inexact Newton with Learning-Based Acceleration for Highly Nonlinear Hyperelasticity Problems on Three-Dimensional Unstructured Meshes

摘要：Inexact Newton-type method is widely used in many scientific and engineering applications, but in many situations it converges slowly or even fails to converge, for reasons that are difficult to quantify. We propose a novel nonlinearly preconditioned inexact Newton algorithm with learning capability to improve the convergence and robustness of the method. The proposed method searches through the nonlinear residual and stagnated solution spaces generated during the Newton iterations and identifies the "bad subspaces'' using an unsupervised learning technique, namely principal component analysis. In the nonlinear preconditioner, a learned small-scale projected problem corresponding to the slow subspace of the nonlinear residuals is constructed and solved to provide a much better initial guess for the global inexact Newton method to converge nearly quadratically. We consider a hyperelasticity problem on the human artery with stenosis described by different material parameters. It is nonlinearly difficult because of the discontinuous of the material parameters between the plaques and the healthy parts of the artery. Numerical experiments show that our proposed method offers a significantly reduced number of nonlinear iterations and compute time for this realistic problem.

墙报展示十一

展示人：马天昊  澳门大学

题目：Simulation of Cardiac Electrophysiology of a Human Heart on a Parallel Computer

摘要：In this work, the authors develop parallel solution algorithms on a parallel computer with more than 16000 cores based on domain decomposition methods to enable cardiac simulations of a human heart using a monodomain model with two different ionic models. Numerical simulation of electrophysiology in a full and healthy heart is carried out using a finite element method on an unstructured mesh. The ionic equations and the monodomain equation are solved in a decoupled way. In addition, we devise a sub-iterative approach that uses the newest gating variables computed by the Rush-Larsen scheme to update the ion concentrations, which improves efficiency while ensuring the desired accuracy. The Conjugate Gradient method solves the linear system with an additive Schwarz preconditioner, offering good scalability. These approaches address the computational challenges associated with these comprehensive cardiac electrophysiology models and parallel computation. Besides, the successful implementation of parallel simulation of cardiac electrophysiology on large-scale parallel computers has shortened the computing time extremely. It has brought the clinical application of the monodomain model one step closer.

墙报展示十二

展示人：祁粉粉  澳门大学

题目：Computing the Kidney Hemodynamics with a Stokes-Darcy Model

摘要：Based on the structural morphology of the renal vasculature, the kidney can be considered as a porous medium. In this poster, we simulate the blood flows in a patient-specific kidney with a 3D coupled unsteady Stokes-Darcy model. We use a stabilized P1-P1 finite element method in space and an implicit backward Euler method in time to discretize the coupled system. The resulting linear system at each time step is then solved by GMRES with a novel two-level additive Schwarz preconditioner. We choose different coarse basis functions in different fields and construct the coarse space in a unified framework. Some numerical experiments with simple and realistic geometries are given to verify the convergence rate and performance of the proposed preconditioner.

墙报展示十三

展示人：单钰鑫  中国科学院深圳先进技术研究院

题目：Fast Dense Animated Streamlines Visualization for Interactive Virtual Reality

摘要：Streamlines are essential tools for visualizing computational fluid dynamics simulations. Animated streamlines, especially in Virtual Reality (VR) environments, play a crucial role in unraveling the complexities of fluid flow physics by providing an immersive and interactive platform for the exploration of expansive flow fields. We present a fast animated streamline visualization method based on Motion Map. By leveraging high-performance GPU computing, our method efficiently computes intricate flow behaviors, enabling real-time visualization of the movement of millions of streamlets in VR, provides a versatile tool for gaining deeper insights into fluid dynamics and exploring complex flow phenomena in an immersive and interactive manner.

墙报展示十四

展示人：王菁原  澳门大学

题目：Space-Time Implicit Runge-Kutta Methods and Applications

摘要：The classical implicit Runge-Kutta (IRK) methods offer high accuracy and stability with fewer stages but they are not commonly used due to the high computational cost and difficulties in preconditioning, especially on 3D unstructured grids. In this work, we introduce time-parallel versions of the classical IRK methods, referred to as space-time IRK methods, which enable increased parallelism on large-scale parallel computers. Additionally, we propose one- and two-level overlapping domain decomposition preconditioners for both the classical and space-time IRK methods. These preconditioners maintain the tensor structure by applying the classical additive Schwarz method to a single component of the IRK matrix. We conduct numerical experiments on parabolic and Stokes equations to validate the effectiveness of the methods and analyze the convergence behavior of various parameters, including the time step size, the number of processors, and the window size, with the right choice of the coarse mesh size and the subdomain solver. We compare the proposed method with the classical method and the parallel performance, including the strong/weak scalability and computational time, indicates that the new method outperforms the classical method when the number of processors is large and the space-only parallelization of the classical method is a limiting factor.

计算虚拟现实展示与体验介绍

展示人：宫玉杰、马天昊、祁粉粉、单钰鑫、王菁原

题目：基于高性能计算的虚拟现实框架及应用

摘要：近年来，虚拟现实技术（VR，MR或统称XR）正在迅速崛起，并逐步走进众多科研及健康领域。这些充满未来感和科技感的技术使得虚拟的数字模型与现实场景在物理维度和信息维度上交融共存。这些数字模型可以是人脑或者心脏，也可以城市，从宏观角度来讲，XR实现了人与元宇宙的交互。与此同时，这些技术的应用数量正在蓬勃发展和壮大。通过创建数字模型，我们可以对物理世界进行仿真、推演及规划。当需要实时计算超大规模的数字模型或复杂应用场景时，高性能计算机能够提供所需的计算速度以和精度。

        澳门大学和中国科学院深圳先进技术研究院共同研发了一套新型的基于高性能计算的虚拟现实技术框架。该框架融合了高性能计算，基于Hololens的全息影像技术和手势识别及视线追踪等技术，借助虚拟现实设备实现对于各种尺度问题的精细可视化展示及交互式的数值模拟。该框架可实现小到人体血管中血液流动模拟和人体各个器官的力学仿真模拟，大到超大城市中的风环境深入剖析。我们在此框架下建立的虚拟手术室，利用核磁共振或者CT 技术融合高性能计算技术将扫描得到的患者器官数字模型载入与真实手术室融合，并通过高性能计算技术模拟手术操作过程，可实现在手术前对相关疾病进行预测、诊断、评估及手术规划。在该框架下建立的数字城市模型中，将真实城市环境的数字模型载入虚拟世界，通过高性能计算，可以智能预警城市环境可能产生的潜在灾害，并提供相应的对策建议。  

        这一框架让体验者能够直观地触及基于现实的各种数字模型，交互式地置身于计算数学的应用场景，深入了解复杂系统的可视化分析，同时也让科研或医学工作者得以更加便捷地开展数值模拟并优化手术方案。此平台的构建证实了高性能数值计算与虚拟现实技术的融合可以为很多实际问题提供一套崭新的工具。

参会人员