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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 *** first establish some regularity results for the solution of MNSE,which seem to be not available in the ***,we study a semi-implicit time-discrete scheme for the MNSE and prove L2-H1 error estimates for the time discrete ***,certain regularity results for the time discrete solution are establishes *** on these regularity results,we prove the unconditional L2-H1 error estimates for the finite element solution of ***,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

L vector spaces and L fields

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science China mathematics 2024年第10期67卷 2195-2216页

作者： Yinhe Peng Liuzhen Wu academy of mathematics and systems science Chinese Academy of Sciences

We construct in ZFC(the Zermelo-Fraenkel system with choice) an L topological vector space—a topological vector space that is an L space—and an L field—a topological field that is an L space. This generalizes earli... 详细信息

关键词： minimal walk topological vector space topological field L space L vector space L field

Deciphering and integrating invariants for neural operator learning with various physical mechanisms

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National science Review 2024年第4期11卷 111-122页

作者： Rui Zhang Qi Meng Zhi-Ming Ma academy of mathematics and systems science Chinese Academy of Sciences(CAS) Microsoft Research

Neural operators have been explored as surrogate models for simulating physical systems to overcome the limitations of traditional partial differential equation(PDE) solvers. However, most existing operator learning methods assume that the data originate from a single physical mechanism, limiting their applicability and performance in more realistic scenarios. To this end, we propose the physical invariant attention neural operator(PIANO) to decipher and integrate the physical invariants for operator learning from the PDE series with various physical mechanisms. PIANO employs self-supervised learning to extract physical knowledge and attention mechanisms to integrate them into dynamic convolutional *** to existing techniques, PIANO can reduce the relative error by 13.6%–82.2% on PDE forecasting tasks across varying coefficients, forces or boundary conditions. Additionally, varied downstream tasks reveal that the PI embeddings deciphered by PIANO align well with the underlying invariants in the PDE systems, verifying the physical significance of PIANO.

关键词： neural operator PDE solver contrastive learning physical invariants

Kriging Model Averaging Based on Leave-One-Out Cross-Validation Method

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Journal of systems science & Complexity 2024年第5期37卷 2132-2156页

作者： FENG Ziheng ZONG Xianpeng XIE Tianfa ZHANG Xinyu School of mathematics Statistics and Mechanics Beijing University of Technology academy of mathematics and systems science Chinese Academy of Sciences

In recent years, Kriging model has gained wide popularity in various fields such as space geology, econometrics, and computer experiments. As a result, research on this model has *** this paper, the authors propose a model averaging estimation based on the best linear unbiased prediction of Kriging model and the leave-one-out cross-validation method, with consideration for the model uncertainty. The authors present a weight selection criterion for the model averaging estimation and provide two theoretical justifications for the proposed method. First, the estimated weight based on the proposed criterion is asymptotically optimal in achieving the lowest possible prediction ***, the proposed method asymptotically assigns all weights to the correctly specified models when the candidate model set includes these models. The effectiveness of the proposed method is verified through numerical analyses.

关键词： Asymptotic optimality best linear unbiased prediction cross-validation Kriging model model averaging

Forward to the Special Topic on“Solving Differential Equations with Deep Learning”

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Journal of systems science & Complexity 2024年第2期37卷 389-390页

作者： YAN Zhenya academy of mathematics and systems science Chinese Academy of SciencesBeijing 100190China

The concept of artificial intelligence(AI)was first presented at the Dartmouth SummerResearch Project on AI hosted by McCarthy and Minsky in *** that,the developmentof AI has experienced ups and downs due to some limited conditions including intelligentcomputing techniques,computer and hardware and software^([1,2]).In particular,with the rapidof deep artificial neural networks,machine learning(ML)and deep learning(DL)techniques^([3,4])have been applied in many fields of science and engineering,including machine translation,image recognition,genomics,scientific computing and atmospheric science,etc^([5]).

关键词： has artificial hardware

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Density peak clustering using tensor network

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science China(Information sciences) 2024年第3期67卷 321-322页

作者： Xiao SHI Yun SHANG Institute of mathematics Academy of Mathematics and Systems ScienceChinese Academy of Sciences School of Mathematical sciences University of Chinese Academy of Sciences National Center for mathematics and Interdisciplinary sciences Key Laboratory of ManagementDecision and Information SystemsAcademy of Mathematics and Systems ScienceChinese Academy of Sciences

Tensor networks have been a powerful tool in simulating many-body physics and have recently gained recognition in the machine learning community due to their remarkable representation capabilities. However, using tens... 详细信息

关键词： Tensors

Recursive Projected Filter Algorithm with Binary-Valued Observations

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Journal of systems science & Complexity 2024年第5期37卷 1832-1860页

作者： ZHANG Xiaoyan WANG Ying XUE Wenchao ZHAO Yanlong Key Laboratory of systems and Control Institute of Systems Science Academy of Mathematics and Systems Science Chinese Academy of Sciences School of Mathematical sciences University of Chinese Academy of Sciences

This paper focuses on the state estimate for a class of systems with both process noise and measurement noise under binary-valued observations, in which the Gaussian assumption on the predicted density of the state is not required. A recursive projected filter algorithm with time-varying thresholds is constructed to estimate the state under binary-valued observations. The time-varying thresholds are designed as the prediction value of the measurement, which can provide more information about the system state. The convergence property is established with some suitable stability,boundedness and observability conditions. In particular, the estimation error between state and estimate is proved to be asymptotically bounded in the mean-square sense, whose upper bound is related to the variance of process noise. Finally, the theoretical results are demonstrated via numerical examples of first-order and high-order systems.

关键词： Binary-valued observations convergence recursive filter time-varying thresholds

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 *** 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

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Pseudoeffective thresholds and cohomology of twisted symmetric tensor fields on irreducible Hermitian symmetric spaces

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science China mathematics 2024年第11期67卷 2433-2452页

作者： Feng Shao Institute of mathematics Academy of Mathematics and Systems ScienceChinese Academy of Sciences School of Mathematical sciences University of Chinese Academy of Sciences

Let X be an irreducible Hermitian symmetric space of compact type(IHSS for short).In this paper,we give the irreducible decomposition of *** a by-product,we give a cohomological characterization of the rank of ***,we introduce pseudoeffective thresholds to measure the bigness of tangent bundles of smooth complex projective varieties precisely and calculate them for irreducible Hermitian symmetric spaces of compact type.

关键词： tangent bundle pseudoeffective threshold irreducible Hermitian symmetric space of compact type (IHSS) multiplicity free action

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Threshold Selection and Resource Allocation for Quantized Identification

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Journal of systems science & Complexity 2024年第1期37卷 204-229页

作者： WANG Ying LI Xin ZHAO Yanlong ZHANG Ji-Feng Key Laboratory of systems and Control Institute of Systems ScienceAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical sciences University of Chinese Academy of SciencesBeijing 100049China

This paper is concerned with the optimal threshold selection and resource allocation problems of quantized identification, whose aims are improving identification efficiency under limited resources. Firstly, the first-order asymptotically optimal quantized identification theory is extended to the weak persistent excitation condition. Secondly, the characteristics of time and space complexities are established based on the Cramér-Rao lower bound of quantized systems. On these basis, the optimal selection methods of fixed thresholds and adaptive thresholds are established under aperiodic signals, which answer how to achieve the best efficiency of quantized identification under the same time and space complexity. In addition, based on the principle of maximizing the identification efficiency under a given resource, the optimal resource allocation methods of quantized identification are given for the cases of fixed thresholds and adaptive thresholds, respectively, which show how to balance time and space complexity to realize the best identification efficiency of quantized identification.

关键词： Quantized output resource allocation system identification threshold selection.

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