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Structure-Preserving Recurrent Neural Networks for a Class of Birkhoffian Systems

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Journal of Systems Science & Complexity 2024年第2期37卷 441-462页

作者： XIAO Shanshan CHEN Mengyi ZHANG Ruili TANG Yifa lsec ICMSECAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China School of Mathematics and Statistics Beijing Jiaotong UniversityBeijing 100044China

In this paper,the authors propose a neural network architecture designed specifically for a class of Birkhoffian systems—The Newtonian system.The proposed model utilizes recurrent neural networks(RNNs)and is based on a mathematical framework that ensures the preservation of the Birkhoffian structure.The authors demonstrate the effectiveness of the proposed model on a variety of problems for which preserving the Birkhoffian structure is important,including the linear damped oscillator,the Van der Pol equation,and a high-dimensional example.Compared with the unstructured baseline models,the Newtonian neural network(NNN)is more data efficient,and exhibits superior generalization ability.

关键词： Birkhoffian system k(z,t)-symplectic neural networks recurrent neural network

Multiscale analysis-based peridynamic simulation of fracture in porous media

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Frontiers of Structural and Civil Engineering 2024年第1期18卷 1-13页

作者： Zihao YANG Shangkun SHEN Xiaofei GUAN Xindang HE Junzhi CUI School of Mathematics and Statistics Northwestern Polytechnical UniversityXi’an 710072China School of Mathematical Sciences Tongji UniversityShanghai 200092China School of Mechanics Civil Engineering and ArchitectureNorthwestern Polytechnical UniversityXi’an 710072China lsec ICMSECAcademy of Mathematics and Systems ScienceCASBeijing 100190China

The simulation of fracture in large-scale structures made of porous media remains a challenging task.Current techniques either assume a homogeneous model,disregarding the microstructure characteristics,or adopt a micro-mechanical model,which incurs an intractable computational cost due to its complex stochastic geometry and physical properties,as well as its nonlinear and multiscale features.In this study,we propose a multiscale analysis-based dual-variable-horizon peridynamics(PD)model to efficiently simulate macroscopic structural fracture.The influence of microstructures in porous media on macroscopic structural failure is represented by two PD parameters:the equivalent critical stretch and micro-modulus.The equivalent critical stretch is calculated using the microscale PD model,while the equivalent micro-modulus is obtained through the homogenization method and energy density equivalence between classical continuum mechanics and PD models.Numerical examples of porous media with various microstructures demonstrate the validity,accuracy,and efficiency of the proposed method.

关键词： porous media multiscale variable-horizon peridynamic equivalent critical stretch equivalent micro-modulus

SEMI-PROXIMAL POINT METHOD FOR NONSMOOTH CONVEX-CONCAVE MINIMAX OPTIMIZATION

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Journal of Computational Mathematics 2024年第3期42卷 617-637页

作者： Yuhong Dai Jiani Wang Liwei Zhang lsec ICMSECAMSSChinese Academy of SciencesBeijing 100190China School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijing 100049China Academy of Mathematics and Systems Science Chinese Academy of SciencesBeijing 100190China School of Mathematical Sciences Dalian University of TechnologyDalian 116024China

Minimax optimization problems are an important class of optimization problems arising from modern machine learning and traditional research areas.While there have been many numerical algorithms for solving smooth convex-concave minimax problems,numerical algorithms for nonsmooth convex-concave minimax problems are rare.This paper aims to develop an efficient numerical algorithm for a structured nonsmooth convex-concave minimax problem.A semi-proximal point method(SPP)is proposed,in which a quadratic convex-concave function is adopted for approximating the smooth part of the objective function and semi-proximal terms are added in each subproblem.This construction enables the subproblems at each iteration are solvable and even easily solved when the semiproximal terms are cleverly chosen.We prove the global convergence of our algorithm under mild assumptions,without requiring strong convexity-concavity condition.Under the locally metrical subregularity of the solution mapping,we prove that our algorithm has the linear rate of convergence.Preliminary numerical results are reported to verify the efficiency of our algorithm.

关键词： Minimax optimization Convexity-concavity Global convergence Rate of con-vergence Locally metrical subregularity

A Deep Learning Method for Computing Eigenvalues of the Fractional Schrödinger Operator

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Journal of Systems Science & Complexity 2024年第2期37卷 391-412页

作者： GUO Yixiao MING Pingbing lsec Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China

The authors present a novel deep learning method for computing eigenvalues of the fractional Schrödinger operator.The proposed approach combines a newly developed loss function with an innovative neural network architecture that incorporates prior knowledge of the problem.These improvements enable the proposed method to handle both high-dimensional problems and problems posed on irregular bounded domains.The authors successfully compute up to the first 30 eigenvalues for various fractional Schrödinger operators.As an application,the authors share a conjecture to the fractional order isospectral problem that has not yet been studied.

关键词： Eigenvalue problem deep learning fractional Schrödinger operator isospectral problem

UNCONDITIONAL CONVERGENCE AND ERROR ESTIMATES OF A FULLY DISCRETE FINITE ELEMENT METHOD FOR THE MICROPOLAR NAVIER-STOKES EQUATIONS

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Journal of Computational Mathematics 2024年第1期42卷 71-110页

作者： Shipeng Mao Jiaao Sun Wendong Xue NCMIS LSECInstitute of Computational Mathematics and Scientific/Enginnering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China

In this paper,we consider the initial-boundary value problem(IBVP)for the micropolar Naviers-Stokes equations(MNSE)and analyze a first order fully discrete mixed finite element scheme.We first establish some regularity results for the solution of MNSE,which seem to be not available in the literature.Next,we study a semi-implicit time-discrete scheme for the MNSE and prove L2-H1 error estimates for the time discrete solution.Furthermore,certain regularity results for the time discrete solution are establishes rigorously.Based on these regularity results,we prove the unconditional L2-H1 error estimates for the finite element solution of MNSE.Finally,some numerical examples are carried out to demonstrate both accuracy and efficiency of the fully discrete finite element scheme.

关键词： Micropolar fluids Regularity estimates Euler semi-implicit scheme Mixed finite element methods Unconditional convergence

Positive definiteness of real quadratic forms resulting from the variable-step L1-type approximations of convolution operators

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Science China Mathematics 2024年第2期67卷 237-252页

作者： Hong-Lin Liao Tao Tang Tao Zhou School of Mathematics Nanjing University of Aeronautics and AstronauticsNanjing 211106China Division of Science and Technology BNU-HKBU United International CollegeZhuhai 519087China SUSTech International Center for Mathematics Shenzhen 518055China NCMIS&lsec Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China

The positive definiteness of real quadratic forms with convolution structures plays an important rolein stability analysis for time-stepping schemes for nonlocal operators. In this work, we present a novel analysistool to handle discrete convolution kernels resulting from variable-step approximations for convolution operators.More precisely, for a class of discrete convolution kernels relevant to variable-step L1-type time discretizations, weshow that the associated quadratic form is positive definite under some easy-to-check algebraic conditions. Ourproof is based on an elementary constructing strategy by using the properties of discrete orthogonal convolutionkernels and discrete complementary convolution kernels. To our knowledge, this is the first general result onsimple algebraic conditions for the positive definiteness of variable-step discrete convolution kernels. Using theunified theory, we obtain the stability for some simple nonuniform time-stepping schemes straightforwardly.

关键词： discrete convolution kernels positive definiteness variable time-stepping orthogonal convolution kernels complementary convolution kernels

Energy Stable Nodal DG Methods for Maxwell’s Equations of Mixed-Order Form in Nonlinear Optical Media

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Communications on Applied Mathematics and Computation 2024年第1期6卷 30-63页

作者： Maohui Lyu Vrushali A.Bokil Yingda Cheng Fengyan Li State Key Laboratory of Scientific and Engineering Computing(lsec. Academy of Mathematics and System SciencesChinese Academy of SciencesBeijing100190China Department of Mathematics College of ScienceOregon State UniversityCorvallisOR97331USA Department of Mathematics Michigan State UniversityEast LansingMI48824USA Department of Computational Mathematics Science and EngineeringMichigan State UniversityEast LansingMI48824USA Department of Mathematical Sciences Rensselaer Polytechnic InstituteTroyNY12180USA

In this work,we develop energy stable numerical methods to simulate electromagnetic waves propagating in optical media where the media responses include the linear Lorentz dispersion,the instantaneous nonlinear cubic Kerr response,and the nonlinear delayed Raman molecular vibrational response.Unlike the first-order PDE-ODE governing equations considered previously in Bokil et al.(J Comput Phys 350:420–452,2017)and Lyu et al.(J Sci Comput 89:1–42,2021),a model of mixed-order form is adopted here that consists of the first-order PDE part for Maxwell’s equations coupled with the second-order ODE part(i.e.,the auxiliary differential equations)modeling the linear and nonlinear dispersion in the material.The main contribution is a new numerical strategy to treat the Kerr and Raman nonlinearities to achieve provable energy stability property within a second-order temporal discretization.A nodal discontinuous Galerkin(DG)method is further applied in space for efficiently handling nonlinear terms at the algebraic level,while preserving the energy stability and achieving high-order accuracy.Indeed with d_(E)as the number of the components of the electric field,only a d_(E)×d_(E)nonlinear algebraic system needs to be solved at each interpolation node,and more importantly,all these small nonlinear systems are completely decoupled over one time step,rendering very high parallel efficiency.We evaluate the proposed schemes by comparing them with the methods in Bokil et al.(2017)and Lyu et al.(2021)(implemented in nodal form)regarding the accuracy,computational efficiency,and energy stability,by a parallel scalability study,and also through the simulations of the soliton-like wave propagation in one dimension,as well as the spatial-soliton propagation and two-beam interactions modeled by the two-dimensional transverse electric(TE)mode of the equations.

关键词： Maxwell’s equations Kerr and Raman Discontinuous Galerkin method Energy stability

HYBRID REGULARIZED CONE-BEAM RECONSTRUCTION FOR AXIALLY SYMMETRIC OBJECT TOMOGRAPHY

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Acta Mathematica Scientia 2022年第1期42卷 403-419页

作者：李兴娥魏素花许海波陈冲 Institute of Applied Physics and Computational Mathematics Beijing 100094China lsec ICMSECAcademy of Mathematics and Systems ScienceCASBeijing 100190China

In this paper,we consider 3 D tomographic reconstruction for axially symmetric objects from a single radiograph formed by cone-beam X-rays.All contemporary density reconstruction methods in high-energy X-ray radiography are based on the assumption that the cone beam can be treated as fan beams located at parallel planes perpendicular to the symmetric axis,so that the density of the whole object can be recovered layer by layer.Considering the relationship between different layers,we undertake the cone-beam global reconstruction to solve the ambiguity effect at the material interfaces of the reconstruction results.In view of the anisotropy of classical discrete total variations,a new discretization of total variation which yields sharp edges and has better isotropy is introduced in our reconstruction model.Furthermore,considering that the object density consists of continually changing parts and jumps,a high-order regularization term is introduced.The final hybrid regularization model is solved using the alternating proximal gradient method,which was recently applied in image processing.Density reconstruction results are presented for simulated radiographs,which shows that the proposed method has led to an improvement in terms of the preservation of edge location.

关键词： high-energy X-ray radiography cone-beam global reconstruction inverse problem total variation alternating proximal gradient method

INTERPOLATION BY LOOP＇S SUBDIVISION FUNCTIONS

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Journal of Computational Mathematics 2005年第3期23卷 247-260页

作者： Guo-liangXu lsec ICMSECAcademyofMathematicsandSystemSciencesChineseAcademyofSciencesBeijing100080China

For the problem of constructing smooth functions over arbitrary surfaces from discretedata, we propose to use Loop's subdivision functions as the interpolants. Results on theexistence, uniqueness and error bound of the interpolants are established. An efficientprogressive computation algorithm for the interpolants is also presented.

关键词： Interpolation Loop's subdivision function