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This paper investigates a repeated stochastic security game against a non-stationary attacker. Most of the work to date assumes that the defender has a repeated interaction with a fixed type of attacker. In fact, the defender is more likely to encounter changing attackers in multi-round games. A defender faces an attacker whose identity is unknown. The attacker type changes stochastically over time and the defender cannot detect when these changes occur. We adopt the BPR (Bayesian Policy Reuse) algorithm to detect the switches of the attacker, and the defender could play the accurate policy correspondingly. The experiment results show that BPR algorithm could accurately detect switches and help the defender gain more utilities than the EXP3-S algorithm.

This paper introduces an innovative artificial neural networks-based analytical solver for fractional partial differential equations (fPDEs), combining neural networks (NNs) with symbolic computation. Leveraging the powerful function approximation ability of NNs and the exactness of symbolic methods, our approach achieves notable improvements in both computational speed and solution precision. The efficacy of the proposed method is validated through four numerical examples, with results visualized using three-dimensional surface plots, contour mappings, and density distributions. Numerical experiments demonstrate that the proposed framework successfully derives exact solutions for fPDEs without relying on data samples. This research provides a novel methodological framework for solving fPDEs, with broad applicability across scientific and engineering fields.

This paper studies a class of attack behavior in which adversaries assume the role of initiators, orchestrating and implementing attacks by hiring executors. We examine the dynamics of strategic attacks, modeling the initiator as an attack planner and constructing the interaction with the defender within a defender–attack planner framework. The individuals tasked with executing the attacks are identified as attackers. To ensure the attackers’ adherence to the planner’s directives, we concurrently consider the interests of each attacker by formulating a multi-objective problem. Furthermore, acknowledging the information asymmetry where defenders have incomplete knowledge of the planners’ payments and the attackers’ profiles, and recognizing the planner’s potential to exploit this for strategic deception, we develop a defender–attack planner model with deception based on signaling games. Subsequently, through the analysis of the interaction between the defender and planner, we refine the model into a tri-level programming problem. To address this, we introduce an effective decomposition algorithm leveraging genetic algorithms. Ultimately, our numerical experiments substantiate that the attack planner’s deceptive strategy indeed yield greater benefits.

Based on the objective evaluation of the regional ecological quality (Urban Cluster in Mid-inner Zhejiang) by Remote Sensing based Ecological Index (RSEI), it was proposed to study the spatial-temporal response of the regional ecological quality to the urban built-up areas, impervious surface, land use and “production-living-ecological” space under urban settlement change to quantitatively describe the mechanism of human activities’ influence on the regional ecology. The results showed that: (1) From 1985 to 2020, the RSEI of the Urban Cluster in Mid-inner Zhejiang was above 0.50 as a whole, showing a trend of first decreasing and then increasing with a slight decrease in some parts. (2) From the perspective of the urban built-up areas, it was dominated by ecological level 2–3, gradually changing from ecological level 3 to ecological level 2. (3) From the perspective of the impervious surface, it was dominated by ecological level 2–3, accounting for more than 75.00% of the total study area. (4) From the perspective of land use, the cropland, forestland, water and construction land were mainly ecological level 2–4, 3–4, 3–4 and 2–3. (5) From the perspective of “production-living-ecological” space, production space, ecological space and living space were dominated by ecological level 3, 3–4 and 2–3. The results can deepen the understanding of the impact of urban settlement development on the regional ecological quality, avoid the unbalanced situation of blindly pursuing socio-economic development while ignoring the ecological environment, which scientifically helps the sustainable construction and high-quality development of the Urban Cluster in Mid-inner Zhejiang.

Through symbolic computation with Maple, fifty-seven sets of rational wave solutions to the generalized Calogero-Bogoyavlenskii-Schiff equation are presented by employing the generalized bilinear operator when the parameter p=2. Via the three-dimensional plots and contour plots with the help of Maple, the dynamics of these solutions are described very well. These solutions have greatly enriched the exact solutions of the generalized Calogero-Bogoyavlenskii-Schiff equation on the existing literature. The result will be widely used to describe many nonlinear scientific phenomena.

This article focuses on N -solitons and some novel multi-wave interaction phenomena of a (2+1)-dimensional extended Sawada–Kotera equation. Firstly, the bilinear form is given using a Hirota direct method, and the corresponding bilinear equation is presented from Bell polynomial theory of integrable equations. Secondly, the N -solitons of extended equation is constructed by utilizing Hirota perturbation truncation method, the obtained results are displayed in the form of stripe solitons with elastic collision. Finally, three cases of lump-stripe solitons and two new types of periodic rogue waves are obtained through a class of bilinear neural network method based on multi-layer neural network structures. Particularly, the 3D and 2D graphics are used to describe the dynamics and evolutionary collision phenomena of soliton solutions and mixed type multi-wave solutions.

In this work, three novel neural network models are proposed to solve the breaking soliton (gBS) equation, and three types of new trial functions are skillfully constructed via setting the appropriate activation functions in the proposed novel neural network models. The explicit solutions of gBS equation are obtained by exploiting bilinear neural network method.

The Kadomtsev-Petviashvili-Benjamin-Bona-Mahony equation has significant applications in the accurate simulation of wave behavior in physical systems. In recent years, some machine learning methods have been used to obtain numerical solutions, with fewer studies focusing on novel and significant analytical solutions. This paper proposes the improved bilinear neural network method for the analytical solution of the KP-BBM equation. Inspired by Liu et al.’s work, it uses enhanced multi-parameter activation functions to solve a wider range of potential analytical solutions of the KP-BBM equation. In this method, neural network models are configured in both “3-3-1” single-layer and “3-2-3-1” double-layer setups, incorporating a variety of multi-parameter activation functions. By setting specific parameters and employing Maple for symbolic computation, we obtain abundant solutions such as the M-lump solutions, lump-breather solutions, and N-soliton wave solutions. The features of innovative solutions are clarified by a few three-dimensional figures, density plots, and contour maps of three specially presented solutions. These results provide valuable insights into multidimensional wave phenomena and demonstrate the effectiveness of the proposed method in rapidly solving solutions with diverse forms.

In this study, the bilinear neural network method (BNNM) is employed for seeking analytical solutions for the generalized (3+1)-dimensional KP equation, which are subtly constructed by both single-layer “4-3-1” and double-layer “4-2-2-1” neural network architectures. By constructing different activation functions and using Maple software for calculations, we obtained a large number of precise analytical solutions. After conducting a series of experimental assignments, we chose some appropriate parameters. These were then substituted into analytical solutions to highlight the final results more effectively and ensure they comply with physical laws. Ultimately, we obtained lump solutions, fractal soliton solutions, superposed periodic wave solutions, and bright-dark soliton solutions. The dynamic characteristics of these solutions are visualized using three-dimensional graphics, curve plots, density maps, and contour diagrams. These results offer valuable insights into nonlinear phenomena across diverse fields such as optics, acoustics, heat transfer, fluid dynamics, and classical mechanics. At the same time, we have applied BNNM to the generalized (3+1)-dimensional KP equation for the first time and demonstrated its effectiveness. Compared to traditional methods, BNNM exhibits significant advantages, which suggests it will be increasingly utilized in various types of nonlinear research.

This paper aims to present a trilinear neural network method on the basis of the trilinear form of a (3+1)-dimensional p ¯ -GBS equation. This method can be used to construct new exact traveling wave solutions. We set the hidden neurons in three types of tensor functions to some specific functions, and reach a class of periodic bright–dark soliton solutions, two types of breather-like wave solutions and three types of rogue wave solutions for p ¯ -GBS equation successfully. The 3-D and density graphs of those solutions obtained are given to interpret the dynamic characteristics.