中国科学院数学与系统科学研究院期刊网

25 April 2024, Volume 37 Issue 2
    

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  • YAN Zhenya
    Journal of Systems Science & Complexity. 2024, 37(2): 389-390. https://doi.org/10.1007/s11424-024-4002-6
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  • GUO Yixiao, MING Pingbing
    Journal of Systems Science & Complexity. 2024, 37(2): 391-412. https://doi.org/10.1007/s11424-024-3250-9
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    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.
  • CHEN Fukai, LIU Ziyang, LIN Guochang, CHEN Junqing, SHI Zuoqiang
    Journal of Systems Science & Complexity. 2024, 37(2): 413-440. https://doi.org/10.1007/s11424-024-3294-x
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    In this paper, the authors propose Neumann series neural operator (NSNO) to learn the solution operator of Helmholtz equation from inhomogeneity coefficients and source terms to solutions. Helmholtz equation is a crucial partial differential equation (PDE) with applications in various scientific and engineering fields. However, efficient solver of Helmholtz equation is still a big challenge especially in the case of high wavenumber. Recently, deep learning has shown great potential in solving PDEs especially in learning solution operators. Inspired by Neumann series in Helmholtz equation, the authors design a novel network architecture in which U-Net is embedded inside to capture the multiscale feature. Extensive experiments show that the proposed NSNO significantly outperforms the state-ofthe-art FNO with at least 60% lower relative L2-error, especially in the large wavenumber case, and has 50% lower computational cost and less data requirement. Moreover, NSNO can be used as the surrogate model in inverse scattering problems. Numerical tests show that NSNO is able to give comparable results with traditional finite difference forward solver while the computational cost is reduced tremendously.
  • XIAO Shanshan, CHEN Mengyi, ZHANG Ruili, TANG Yifa
    Journal of Systems Science & Complexity. 2024, 37(2): 441-462. https://doi.org/10.1007/s11424-024-3252-7
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    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.
  • WANG Zhen, CUI Shikun
    Journal of Systems Science & Complexity. 2024, 37(2): 463-479. https://doi.org/10.1007/s11424-024-3337-3
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    The soliton resolution conjecture proposes that the initial value problem can evolve into a dispersion part and a soliton part. However, the problem of determining the number of solitons that form in a given initial profile remains unsolved, except for a few specific cases. In this paper, the authors use the deep learning method to predict the number of solitons in a given initial value of the Korteweg-de Vries (KdV) equation. By leveraging the analytical relationship between Asech2(x) initial values and the number of solitons, the authors train a Convolutional Neural Network (CNN) that can accurately identify the soliton count from spatio-temporal data. The trained neural network is capable of predicting the number of solitons with other given initial values without any additional assistance. Through extensive calculations, the authors demonstrate the effectiveness and high performance of the proposed method.
  • SUN Jiuyun, DONG Huanhe, FANG Yong
    Journal of Systems Science & Complexity. 2024, 37(2): 480-493. https://doi.org/10.1007/s11424-024-3349-z
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    In this paper, physics-informed liquid networks (PILNs) are proposed based on liquid timeconstant networks (LTC) for solving nonlinear partial differential equations (PDEs). In this approach, the network state is controlled via ordinary differential equations (ODEs). The significant advantage is that neurons controlled by ODEs are more expressive compared to simple activation functions. In addition, the PILNs use difference schemes instead of automatic differentiation to construct the residuals of PDEs, which avoid information loss in the neighborhood of sampling points. As this method draws on both the traveling wave method and physics-informed neural networks (PINNs), it has a better physical interpretation. Finally, the KdV equation and the nonlinear Schrödinger equation are solved to test the generalization ability of the PILNs. To the best of the authors’ knowledge, this is the first deep learning method that uses ODEs to simulate the numerical solutions of PDEs.
  • LIU Haiyi, ZHANG Yabin, WANG Lei
    Journal of Systems Science & Complexity. 2024, 37(2): 494-510. https://doi.org/10.1007/s11424-024-3321-y
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    Recently, the physics-informed neural network shows remarkable ability in the context of solving the low-dimensional nonlinear partial differential equations. However, for some cases of highdimensional systems, such technique may be time-consuming and inaccurate. In this paper, the authors put forward a pre-training physics-informed neural network with mixed sampling (pPINN) to address these issues. Just based on the initial and boundary conditions, the authors design the pre-training stage to filter out the set of the misfitting points, which is regarded as part of the training points in the next stage. The authors further take the parameters of the neural network in Stage 1 as the initialization in Stage 2. The advantage of the proposed approach is that it takes less time to transfer the valuable information from the first stage to the second one to improve the calculation accuracy, especially for the high-dimensional systems. To verify the performance of the pPINN algorithm, the authors first focus on the growing-and-decaying mode of line rogue wave in the Davey-Stewartson I equation. Another case is the accelerated motion of lump in the inhomogeneous Kadomtsev-Petviashvili equation, which admits a more complex evolution than the uniform equation. The exact solution provides a perfect sample for data experiments, and can also be used as a reference frame to identify the performance of the algorithm. The experiments confirm that the pPINN algorithm can improve the prediction accuracy and training efficiency well, and reduce the training time to a large extent for simulating nonlinear waves of high-dimensional equations.
  • ZHOU Huijuan
    Journal of Systems Science & Complexity. 2024, 37(2): 511-544. https://doi.org/10.1007/s11424-024-3467-7
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    This paper mainly introduces the parallel physics-informed neural networks (PPINNs) method with regularization strategies to solve the data-driven forward-inverse problems of the variable coefficient modified Korteweg-de Vries (VC-MKdV) equation. For the forward problem of the VCMKdV equation, the authors use the traditional PINN method to obtain satisfactory data-driven soliton solutions and provide a detailed analysis of the impact of network width and depth on solving accuracy and speed. Furthermore, the author finds that the traditional PINN method outperforms the one with locally adaptive activation functions in solving the data-driven forward problems of the VC-MKdV equation. As for the data-driven inverse problem of the VC-MKdV equation, the author introduces a parallel neural networks to separately train the solution function and coefficient function, successfully addressing the function discovery problem of the VC-MKdV equation. To further enhance the network’s generalization ability and noise robustness, the author incorporates two regularization strategies into the PPINNs. An amount of numerical experimental data in this paper demonstrates that the PPINNs method can effectively address the function discovery problem of the VC-MKdV equation, and the inclusion of appropriate regularization strategies in the PPINNs can improves its performance.
  • SUN Junchao, CHEN Yong, TANG Xiaoyan
    Journal of Systems Science & Complexity. 2024, 37(2): 545-566. https://doi.org/10.1007/s11424-024-3500-x
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    The multiple patterns of internal solitary wave interactions (ISWI) are a complex oceanic phenomenon. Satellite remote sensing techniques indirectly detect these ISWI, but do not provide information on their detailed structure and dynamics. Recently, the authors considered a three-layer fluid with shear flow and developed a (2+1) Kadomtsev-Petviashvili (KP) model that is capable of describing five types of oceanic ISWI, including O-type, P-type, TO-type, TP-type, and Y-shaped. Deep learning models, particularly physics-informed neural networks (PINN), are widely used in the field of fluids and internal solitary waves. However, the authors find that the amplitude of internal solitary waves is much smaller than the wavelength and the ISWI occur at relatively large spatial scales, and these characteristics lead to an imbalance in the loss function of the PINN model. To solve this problem, the authors introduce two weighted loss function methods, the fixed weighing and the adaptive weighting methods, to improve the PINN model. This successfully simulated the detailed structure and dynamics of ISWI, with simulation results corresponding to the satellite images. In particular, the adaptive weighting method can automatically update the weights of different terms in the loss function and outperforms the fixed weighting method in terms of generalization ability.
  • FENG Shuang, SHEN Liyong
    Journal of Systems Science & Complexity. 2024, 37(2): 567-580. https://doi.org/10.1007/s11424-024-2238-9
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    Let $f(t,y,y')=\sum_{i=0}^n a_i(t,y)y'^i=0$ be an irreducible first order ordinary differential equation with polynomial coefficients. Eremenko in 1998 proved that there exists a constant $C$ such that every rational solution of $f(t,y,y')=0$ is of degree not greater than $C$. Examples show that this degree bound $C$ depends not only on the degrees of $f$ in $t,y,y'$ but also on the coefficients of $f$ viewed as the polynomial in $t,y,y'$. In this paper, the authors show that if $f$ satisfies $\deg(f,y)<\deg(f,y')$ or $$ \max_{i=0}^n \{ \deg(a_i,y)-2(n-i)\}>0, $$ then the degree bound $C$ only depends on the degrees of $f$ in $t,y,y'$, and furthermore we present an explicit expression for $C$ in terms of the degrees of $f$ in $t,y,y'$.
  • ZHANG Yijia, FEI Qing, YAO Xiaolan, SUN Jian, ZHANG Yanjun, CHEN Zhen
    Journal of Systems Science & Complexity. 2024, 37(2): 581-608. https://doi.org/10.1007/s11424-024-3021-7
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    This paper presents a new study on modeling and optimization of trajectory and posture for the super-giant (SG) slalom of alpine skiing. It is the first time that a Three-Rigid-Body-Particle model based on rigorous derivations and stability analysis is established to represent skiers trajectory and posture characteristics, as it is more accurate than the single-rigid-body model which is commonly used in existing studies. In addition, the Radau pseudospectral method is applied to solve the trajectory and posture optimization problem in order to obtain better skiing trajectory, skiing posture, and some key kinematic parameters of skiers. Moreover, this paper analyzes the effects of different body types, minimum turning radii, and flexor and extensor strength of knees and hip joint on skiing performance. Finally, based on the findings of the study, some advice about how to improve the performance of the SG slalom in view of science and technology is given to skiers and coaches for reference.
  • ZHU Xinghua, GAN Die, LIU Zhixin
    Journal of Systems Science & Complexity. 2024, 37(2): 609-628. https://doi.org/10.1007/s11424-024-3016-4
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    In this paper, the authors consider the distributed adaptive identification problem over sensor networks using sampled data, where the dynamics of each sensor is described by a stochastic differential equation. By minimizing a local objective function at sampling time instants, the authors propose an online distributed least squares algorithm based on sampled data. A cooperative nonpersistent excitation condition is introduced, under which the convergence results of the proposed algorithm are established by properly choosing the sampling time interval. The upper bound on the accumulative regret of the adaptive predictor can also be provided. Finally, the authors demonstrate the cooperative effect of multiple sensors in the estimation of unknown parameters by computer simulations.
  • DAI Chaoqun, GUO Yuqian, GUI Weihua
    Journal of Systems Science & Complexity. 2024, 37(2): 629-646. https://doi.org/10.1007/s11424-024-2376-0
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    This study investigates the robust feedback set stabilization of switched logic control networks (SLCNs) with state-dependent uncertain switching and control constraints. First, based on the properties of the semi-tensor product of matrices and the vector representation of logic, an SLCN with state-dependent uncertain switching and control constraints is expressed in algebraic form. Second, an input transformation and a switching model are constructed to transfer the original SLCN into one with a free control input and arbitrary switching. The equivalence between the set stabilizability of the original SLCN and that of the resulting SLCN is established. Based on such equivalence, the authors propose a necessary and sufficient condition for robust feedback set stabilizability. Finally, an example is presented to demonstrate the application of the results obtained.
  • ZHAO Ping, LI Xuerong, WANG Shouyang
    Journal of Systems Science & Complexity. 2024, 37(2): 647-667. https://doi.org/10.1007/s11424-024-1450-y
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    The authors aim to interpret human and AI interactions from the decision perspective. The authors decompose the interaction analysis into the following main components in the context of interactions: Individual behavior patterns, interaction relationships, and comprehensive analysis. The authors interpret intertemporal decisions from a physical perspective and employ cross-discipline concepts and methodologies to extract the behavior characteristics of players in the empirical case study. About the individual behavior patterns, the authors find that human players prefer short-term periods to AI in decision-making. The interaction relationship analysis reveals a dynamic relationship between possible short-term co-movement and nearly counter-movement in the long run. The authors apply principal component analysis to descriptive indicators and discover a regular decision hierarchy. The main behavior pattern of players in the game of Go is switching between careful and daring behaviors. The differences in the decision hierarchies imply a discrepancy of patience between humans and AI.
  • LI Jianbin, HANG Zhouxin, CHEN Zhiyuan, XIAO Shan
    Journal of Systems Science & Complexity. 2024, 37(2): 668-691. https://doi.org/10.1007/s11424-024-2164-x
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    E-commerce platform financing is a new service pattern of supply chain finance. However, this pattern may bring some new issues when considering the problem of cash flow shortage and financing difficulties of small and medium-sized enterprises. When enterprises use this service, they worry about the leakage of preferential wholesale price when applying the full loan amount and providing the true transaction information. Based on the model consisting of a supplier, a retailer and a crossborder e-commerce platform, the authors design a price masking strategy to prevent the retailer’s preferential wholesale price information from leakage. The authors analyze the profit of the retailer and the platform before and after adopting the price masking strategy. The authors find that the price masking strategy always benefits the retailer. Besides, the optimal profit of the retailer and the platform are both affected by the loan interest rate. Moreover, there exists a range of loan interest rates that can benefit both the retailer and the platform if the price masking strategy is adopted. The research emphasizes that platform can expand the total business volume by allowing retailers to use price masking strategy. In other words, there will be more and more retailers attracted by the strategy, which benefits the long-term growth of cross-border e-commerce platform financing.
  • MENG Jing, FENG Long, ZOU Changliang, WANG Zhaojun
    Journal of Systems Science & Complexity. 2024, 37(2): 692-728. https://doi.org/10.1007/s11424-023-2342-2
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    In matrix completion, additional covariates often provide valuable information for completing the unobserved entries of a high-dimensional low-rank matrix ${\bm A}$. In this paper, the authors consider the matrix recovery problem when there are multiple structural breaks in the coefficient matrix $\bm \beta$ under the column-space-decomposition model $\bm A=\bm X \bm \beta+\bm B$. A cumulative sum (CUSUM) statistic is constructed based on the penalized estimation of $\bm \beta$. Then the CUSUM is incorporated into the Wild Binary Segmentation (WBS) algorithm to consistently estimate the location of breaks. Consequently, a nearly-optimal recovery of ${\bm A}$ is fulfilled. Theoretical findings are further corroborated via numerical experiments and a real-data application.
  • YUE Dequan, ZHANG Yuying, XU Xiuli, YUE Wuyi
    Journal of Systems Science & Complexity. 2024, 37(2): 729-758. https://doi.org/10.1007/s11424-024-1207-7
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    In this study, the authors consider an M/M/1 queuing system with attached inventory under an (s, S) control policy. The server takes multiple vacations whenever the inventory is depleted. It is assumed that the lead time and the vacation time follow exponential distributions. The authors formulate the model as a quasi-birth-and-dearth (QBD) process and derive the stability condition of the system. Then, the stationary distribution in product form for the joint process of the queue length, the inventory level, and the server’s status is obtained. Furthermore, the conditional distributions of the inventory level when the server is on and operational, and when it is off due to a vacation, are derived. Using the stationary distribution, the authors obtain some performance measures of the system. The authors investigate analytically the effect of the server’s vacation on the performance measures. Finally, several numerical examples are presented to investigate the effects of some parameters on the performance measures, the optimal policy, and the optimal cost.
  • SHEN Tingting, TAO Zhifu, CHEN Huayou
    Journal of Systems Science & Complexity. 2024, 37(2): 759-775. https://doi.org/10.1007/s11424-024-2112-9
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    Long-memory process has been widely studied in classical financial time series analysis, which has merely been reported in the field of interval-valued financial time series. The aim of this paper is to explore long-memory process in the prediction of interval-valued time series (IvTS). To model the long-memory process, two novel interval-valued time series prediction models named as intervalvalued vector autoregressive fractionally integrated moving average (IV-VARFIMA) and ARFIMAXFIGARCH were established. In the developed long-memory pattern, both of the short term and long-term influences contained in IvTS can be included. As an application of the proposed models, interval-valued form of WTI crude oil futures price series is predicted. Compared to current IvTS prediction models, IV-VARFIMA and ARFIMAX-FIGARCH can provide better in-sample and out-ofsample forecasts.
  • GU Enguo
    Journal of Systems Science & Complexity. 2024, 37(2): 776-804. https://doi.org/10.1007/s11424-024-1198-4
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    This paper aims at understanding the price dynamics generated by the interaction of traders relying on heterogeneous expectations in an asset pricing model. In the present work the authors analyze a financial market populated by five types of boundedly rational speculators-two types of fundamentalists, two types of chartists and trend followers which submit buying/selling orders according to different trading rules. The authors formulate a stock market model represented as a 2 dimensional piecewise linear discontinuous map. The proposed contribution to the existing financial literature is two aspects. First, the authors perform study of the model involving a 2 dimensional piecewise linear discontinuous map through a combination of qualitative and quantitative methods. The authors focus on the existence conditions of chaos and the multi-stability regions in parameter plane. Related border collision bifurcation curves and basins of multi-attractors are also given. The authors find that chaos or quasi-period exists only in the case of fixed point being a saddle (regular or flip) and that the coexistence of multiple attractors may exist when the fixed point is an attractor, but it is common for spiral and flip fixed points.
  • LI Shiyong, LIU Huan, LI Wenzhe, SUN Wei
    Journal of Systems Science & Complexity. 2024, 37(2): 805-840. https://doi.org/10.1007/s11424-024-2038-2
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    Fog computing can deliver low delay and advanced IT services to end users with substantially reduced energy consumption. Nevertheless, with soaring demands for resource service and the limited capability of fog nodes, how to allocate and manage fog computing resources properly and stably has become the bottleneck. Therefore, the paper investigates the utility optimization-based resource allocation problem between fog nodes and end users in fog computing. The authors first introduce four types of utility functions due to the diverse tasks executed by end users and build the resource allocation model aiming at utility maximization. Then, for only the elastic tasks, the convex optimization method is applied to obtain the optimal results; for the elastic and inelastic tasks, with the assistance of Jensen’s inequality, the primal non-convex model is approximated to a sequence of equivalent convex optimization problems using successive approximation method. Moreover, a two-layer algorithm is proposed that globally converges to an optimal solution of the original problem. Finally, numerical simulation results demonstrate its superior performance and effectiveness. Comparing with other works, the authors emphasize the analysis for non-convex optimization problems and the diversity of tasks in fog computing resource allocation.
  • CHATTERJEE Kashinath, LIU Min-Qian, QIN Hong, YANG Liuqing
    Journal of Systems Science & Complexity. 2024, 37(2): 841-862. https://doi.org/10.1007/s11424-024-2379-x
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    The theory of uniform design has received increasing interest because of its wide application in the field of computer experiments. The generalized discrete discrepancy is proposed to evaluate the uniformity of the mixed-level factorial design. In this paper, the authors give a lower bound of the generalized discrete discrepancy and provide some construction methods of optimal mixed-level uniform designs which can achieve this lower bound. These methods are all deterministic construction methods which can avoid the complexity of stochastic algorithms. Both saturated mixed-level uniform designs and supersaturated mixed-level uniform designs can be obtained with these methods. Moreover, the resulting designs are also χ2-optimal and minimum moment aberration designs.
  • LI Na, FEI Yu, ZHANG Xinyu
    Journal of Systems Science & Complexity. 2024, 37(2): 863-885. https://doi.org/10.1007/s11424-024-2187-3
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    Prediction plays an important role in data analysis. Model averaging method generally provides better prediction than using any of its components. Even though model averaging has been extensively investigated under independent errors, few authors have considered model averaging for semiparametric models with correlated errors. In this paper, the authors offer an optimal model averaging method to improve the prediction in partially linear model for longitudinal data. The model averaging weights are obtained by minimizing criterion, which is an unbiased estimator of the expected in-sample squared error loss plus a constant. Asymptotic properties, including asymptotic optimality and consistency of averaging weights, are established under two scenarios: (i) All candidate models are misspecified; (ii) Correct models are available in the candidate set. Simulation studies and an empirical example show that the promise of the proposed procedure over other competitive methods.
  • WU Yi, YU Wei, WANG Xuejun, PRAKASA RAO B L S
    Journal of Systems Science & Complexity. 2024, 37(2): 886-906. https://doi.org/10.1007/s11424-024-2054-2
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    In this paper, the authors study a class of weighted version of probability density estimator. It is shown that the weighted estimator contains some existing estimators of probability density, no matter they are recursive or non-recursive. Some statistical results including weak consistency, strong consistency, rate of strong consistency, and asymptotic normality are established under some mild conditions. Moreover, the random weighted estimator is also investigated. Some numerical simulations and a real data analysis are presented to study the numerical performances of the estimators.