Entropy

18 pages, 712 KB

Open AccessArticle

Lightweight Quantum Authentication and Key Agreement Scheme in the Smart Grid Environment

by Zehui Jiang and Run-Hua Shi

Entropy 2025, 27(9), 957; https://doi.org/10.3390/e27090957 (registering DOI) - 14 Sep 2025

Smart grids leverage smart terminal devices to collect information from the user side, achieving accurate load forecasting and optimized dispatching of power systems, effectively improving power supply efficiency and reliability while reducing energy consumption. However, the development of quantum technology poses severe challenges [...] Read more.

Smart grids leverage smart terminal devices to collect information from the user side, achieving accurate load forecasting and optimized dispatching of power systems, effectively improving power supply efficiency and reliability while reducing energy consumption. However, the development of quantum technology poses severe challenges to the communication security of smart grids that rely on traditional cryptography. To address this security risk in the quantum era, this paper draws on the core idea of quantum private comparison and proposes a quantum-secure identity authentication and key agreement scheme suitable for smart grids. This scheme uses Bell states as quantum resources, combines hash functions and XOR operations, and can adapt to resource-constrained terminal devices. Through a security proof, it verifies the scheme’s ability to resist various attacks; the experimental results further show that the scheme still has good robustness in different noise environments, providing a feasible technical path for the secure communication of smart grids in the quantum environment and having clear practical engineering value. Full article

(This article belongs to the Special Issue Quantum Information Security)

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21 pages, 6576 KB

Open AccessArticle

Fault Diagnosis of Planetary Gearboxes Based on LSTM Improved via Feature Extraction Using VMD, Fusion Entropy, and Random Forest

by Xin Xia, Haoyu Sun and Aiguo Wang

Entropy 2025, 27(9), 956; https://doi.org/10.3390/e27090956 (registering DOI) - 14 Sep 2025

Abstract

Extracting effective fault features from the complex vibration signals of planetary gearboxes is the key to conducting efficient fault diagnosis, and it involves signal processing, feature extraction, and feature selection. In this paper, a novel feature extraction method is proposed using variational mode [...] Read more.

Extracting effective fault features from the complex vibration signals of planetary gearboxes is the key to conducting efficient fault diagnosis, and it involves signal processing, feature extraction, and feature selection. In this paper, a novel feature extraction method is proposed using variational mode decomposition (VMD), fusion entropy, and random forest (RF). Firstly, VMD is employed to process the nonlinear and non-stationary signals of planetary gearboxes, which can effectively address the issues of signal modulation and mode mixing. Additionally, a fusion entropy that incorporates various refined composite multi-scale entropies is proposed; it fully utilizes the signal characteristics reflected by various entropies as features for fault diagnosis. Then, RF is adopted to calculate the importance of each feature, and appropriate features are selected to form a fault diagnosis vector, aiming to solve the problems of feature redundancy and interference in fusion entropy. Finally, long short-term memory (LSTM) is used for fault classification. The experimental results demonstrate that the proposed fusion entropy achieves higher accuracy compared with a single entropy value. The RF-based feature selection can also reduce interference and improve diagnostic efficiency. The proposed fault diagnosis method exhibits high fault diagnosis accuracy under different rotational speeds and environmental noise conditions. Full article

(This article belongs to the Special Issue Entropy-Based Time Series Analysis: Theory and Applications)

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20 pages, 2136 KB

Open AccessArticle

Entropy and Chaos-Based Modeling of Nonlinear Dependencies in Commodity Markets

by Irina Georgescu and Jani Kinnunen

Entropy 2025, 27(9), 955; https://doi.org/10.3390/e27090955 (registering DOI) - 14 Sep 2025

Abstract

This study explores the nonlinear dynamics and interdependencies among major commodity markets—Gold, Oil, Natural Gas, and Silver—by employing advanced chaos theory and information-theoretic tools. Using daily data from 2020 to 2024, we estimate key complexity measures including Lyapunov exponents, correlation dimension, Shannon and [...] Read more.

This study explores the nonlinear dynamics and interdependencies among major commodity markets—Gold, Oil, Natural Gas, and Silver—by employing advanced chaos theory and information-theoretic tools. Using daily data from 2020 to 2024, we estimate key complexity measures including Lyapunov exponents, correlation dimension, Shannon and Rényi entropy, and mutual information. We also apply the stochastic SO(2) Lie group method to model dynamic correlations, and wavelet coherence analysis to detect time-frequency co-movements. Our findings reveal evidence of low-dimensional deterministic chaos and time-varying nonlinear relationships, especially among pairs like Gold–Silver and Oil–Gas. These results highlight the importance of using nontraditional approaches to uncover hidden structure and co-movement dynamics in commodity markets, providing useful insights for portfolio diversification and systemic risk assessment. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

27 pages, 513 KB

Open AccessFeature PaperArticle

On the Monotonicity of Relative Entropy: A Comparative Study of Petz’s and Uhlmann’s Approaches

by Santiago Matheus, Francesco Bottacin and Edoardo Provenzi

Entropy 2025, 27(9), 954; https://doi.org/10.3390/e27090954 (registering DOI) - 14 Sep 2025

Abstract

We revisit the monotonicity of relative entropy under the action of quantum channels, a foundational result in quantum information theory. Among the several available proofs, we focus on those by Petz and Uhlmann, which we reformulate within a unified, finite-dimensional operator-theoretic framework. In [...] Read more.

We revisit the monotonicity of relative entropy under the action of quantum channels, a foundational result in quantum information theory. Among the several available proofs, we focus on those by Petz and Uhlmann, which we reformulate within a unified, finite-dimensional operator-theoretic framework. In the first part, we examine Petz’s strategy, identify a subtle flaw in his original use of Jensen’s contractive operator inequality, and point out how it was corrected to restore the validity of his line of reasoning. In the second part, we develop Uhlmann’s approach, which is based on interpolations of positive sesquilinear forms and applies automatically to non-invertible density operators. By comparing these two approaches, we highlight their complementary strengths: Petz’s method is more direct and clear; Uhlmann’s method is more abstract and general. Our treatment aims to clarify the mathematical structure underlying the monotonicity of relative entropy and to make these proofs more accessible to a broader audience interested in both the foundations and applications of quantum information theory. Full article

(This article belongs to the Section Quantum Information)

16 pages, 1173 KB

Open AccessArticle

Quantum Computing for Transport Network Optimization

by Jiangwei Ju, Zhihang Liu, Yuelin Bai, Yong Wang, Qi Gao, Yin Ma, Chao Zheng and Kai Wen

Entropy 2025, 27(9), 953; https://doi.org/10.3390/e27090953 (registering DOI) - 13 Sep 2025

Abstract

Public transport systems play a crucial role in the development of large cities. Bus network design to optimize passenger flow coverage in a global metropolis is a challenging task. As an essential part of bus travel planning, considering the bus transfer factor in [...] Read more.

Public transport systems play a crucial role in the development of large cities. Bus network design to optimize passenger flow coverage in a global metropolis is a challenging task. As an essential part of bus travel planning, considering the bus transfer factor in the existing extremely complex and extensive public bus network usually leads to a optimization problem characterized by high-dimensionality and non-linearity. While classical computers struggle to deal with this kind of problems, quantum computers shed new light into this field. The coherent Ising machine (CIM), a specialized optical quantum computer using a photonic dissipative architecture, has shown its remarkable computational power in combinatorial optimization problems. We construct the classical model and the quadratic unconstrained binary optimization (QUBO) model of the bus route optimization problem, and solve it using a classical computer and CIM, respectively. Our experimental results demonstrate the significant acceleration capability of CIM over classical computers in finding the optimal or near-optimal solutions, albeit subject to the hardware limitations of the 100-qubit CIM. Full article

(This article belongs to the Special Issue Quantum Information: Working Towards Applications)

17 pages, 1698 KB

Open AccessArticle

A Quasi-Monte Carlo Method Based on Neural Autoregressive Flow

by Yunfan Wei and Wei Xi

Entropy 2025, 27(9), 952; https://doi.org/10.3390/e27090952 (registering DOI) - 13 Sep 2025

Abstract

This paper proposes a novel transport quasi-Monte Carlo framework that combines randomized quasi-Monte Carlo sampling with a neural autoregressive flow architecture for efficient sampling and integration over complex, high-dimensional distributions. The method constructs a sequence of invertible transport maps to approximate the target [...] Read more.

This paper proposes a novel transport quasi-Monte Carlo framework that combines randomized quasi-Monte Carlo sampling with a neural autoregressive flow architecture for efficient sampling and integration over complex, high-dimensional distributions. The method constructs a sequence of invertible transport maps to approximate the target density by decomposing it into a series of lower-dimensional marginals. Each sub-model leverages normalizing flows parameterized via monotonic beta-averaging transformations and is optimized using forward Kullback–Leibler (KL) divergence. To enhance computational efficiency, a hidden-variable mechanism that transfers optimized parameters between sub-models is adopted. Numerical experiments on a banana-shaped distribution demonstrate that this new approach outperforms standard Monte Carlo-based normalizing flows in both sampling accuracy and integral estimation. Further, the model is applied to A-share stock return data and shows reliable predictive performance in semiannual return forecasts, while accurately capturing covariance structures across assets. The results highlight the potential of transport quasi-Monte Carlo (TQMC) in financial modeling and other high-dimensional inference tasks. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

21 pages, 11242 KB

Open AccessArticle

Fault Diagnosis of Wind Turbine Rotating Bearing Based on Multi-mode Signal Enhancement and Fusion

by Shaohu Ding, Guangsheng Zhou, Xinyu Wang and Weibin Li

Entropy 2025, 27(9), 951; https://doi.org/10.3390/e27090951 (registering DOI) - 13 Sep 2025

Abstract

Wind turbines operate under harsh conditions, heightening the risk of rotating bearing failures. While fault diagnosis using acoustic or vibration signals is feasible, single-modal methods are highly vulnerable to environmental noise and system uncertainty, reducing diagnostic accuracy. Existing multi-modal approaches also struggle with [...] Read more.

Wind turbines operate under harsh conditions, heightening the risk of rotating bearing failures. While fault diagnosis using acoustic or vibration signals is feasible, single-modal methods are highly vulnerable to environmental noise and system uncertainty, reducing diagnostic accuracy. Existing multi-modal approaches also struggle with noise interference and lack causal feature exploration, limiting fusion performance and generalization. To address these issues, this paper proposes CAVF-Net—a novel framework integrating bidirectional cross-attention (BCA) and causal inference (CI). It enhances Mel-Frequency Cepstral Coefficients (MFCCs) of acoustic and short-time Fourier transform (STFT) features of vibration via BCA and employs CI to derive adaptive fusion weights, effectively preserving causal relationships and achieving robust cross-modal integration. The fused features are classified for fault diagnosis under real-world conditions. Experiments show that CAVF-Net attains 99.2% accuracy with few iterations on clean data and maintains 95.42% accuracy in high-entropy multi-noise environments—outperforming single-model acoustic and vibration by 16.32% and 8.86%, respectively, while significantly reducing information uncertainty in downstream classification. Full article

(This article belongs to the Special Issue Failure Diagnosis of Complex Systems)

42 pages, 3543 KB

Open AccessReview

Advances of Quantum Key Distribution and Network Nonlocality

by Minming Geng

Entropy 2025, 27(9), 950; https://doi.org/10.3390/e27090950 (registering DOI) - 13 Sep 2025

Abstract

In recent years, quantum network technology has been rapidly developing, with new theories, solutions, and protocols constantly emerging. The breakthrough experiments and achievements are impressive, such as the construction and operation of ultra-long-distance and multi-user quantum key distribution (QKD) networks, the proposal, verification, [...] Read more.

In recent years, quantum network technology has been rapidly developing, with new theories, solutions, and protocols constantly emerging. The breakthrough experiments and achievements are impressive, such as the construction and operation of ultra-long-distance and multi-user quantum key distribution (QKD) networks, the proposal, verification, and experimental demonstration of new network nonlocality characteristics, etc. The results of recent research on QKD and network nonlocality are summarized and analyzed in this paper, including CV-MDI-QKD (continuous-variable measurement-device-independent QKD), TF-QKD (twin-field QKD), AMDI-QKD (asynchronous MDI-QKD), the generalization, sharing, and certification of network nonlocality, as well as the main achievements and related research tools of full network nonlocality and genuine network nonlocality, aiming to identify the current status and future development paths of the QKD and network nonlocality. Full article

(This article belongs to the Special Issue Nonlocality and Entanglement in Quantum Networks)

14 pages, 2971 KB

Open AccessArticle

The Realization of One-to-Two-Port Beam Division in a Five-Channel Acoustic System

by Rui Wang, Zhicheng Xu, Shuai Tang, Wencong Zhang, Jiabin Hou, Haipeng Cui and Yang Liu

Entropy 2025, 27(9), 949; https://doi.org/10.3390/e27090949 - 12 Sep 2025

Viewed by 30

Abstract

In this work, one-to-two-port beam division is achieved in a five-channel acoustic system. The adjacent composing channels are connected by space-varying air slits, thus realizing quantum-like adiabatic energy transfer. Equal-weight beam splitting with opposite phases from two different output ports is obtained in [...] Read more.

In this work, one-to-two-port beam division is achieved in a five-channel acoustic system. The adjacent composing channels are connected by space-varying air slits, thus realizing quantum-like adiabatic energy transfer. Equal-weight beam splitting with opposite phases from two different output ports is obtained in a broadband signal of 6 kHz-10.5 kHz. In addition, owing to the existence of distinct evolution paths, one-way beam division is exhibited when a certain loss is evenly exerted inside the system. Furthermore, one-to-m-port beam division can also be achieved by extending the composing channels, thus making it possible to construct an asymmetric acoustic beam splitter. The simulated results verify that the incident waves can be split into opposite directions unidirectionally, which may have potential applications in concealed information transmission and eavesdropping. Full article

(This article belongs to the Special Issue Shortcut to Adiabaticity in Classical and Quantum Systems)

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33 pages, 12965 KB

Open AccessArticle

Identifying Network Propagation Sources Using Advanced Centrality Measures

by Damian Frąszczak

Entropy 2025, 27(9), 948; https://doi.org/10.3390/e27090948 - 12 Sep 2025

Viewed by 34

Abstract

We live in a time dominated by interconnected networks surrounding us on all fronts. The emergence of social media platforms has driven the expansion of social networks, facilitating fast communication worldwide. Responses to content shared on these platforms can be seen as a [...] Read more.

We live in a time dominated by interconnected networks surrounding us on all fronts. The emergence of social media platforms has driven the expansion of social networks, facilitating fast communication worldwide. Responses to content shared on these platforms can be seen as a propagation process, where information spreads through social networks. Analyzing propagation graphs presents a significant challenge in identifying sources, which is crucial in various fields. This includes detecting the origins of disinformation, identifying patient zero in an epidemic, and tracing the initial sources of viral trends or malware. Numerous studies have attempted to identify these sources using methods similar to centrality measures which assign a value indicating the likelihood of being a source. While centrality measures are a popular topic, with many new measures introduced each year, only a few have been explored in the context of source identification. This article explores a wide range of centrality measures in the context of source identification. The results help identify the most effective measures and pave the way for the development of more efficient detection techniques. Additionally, an analysis was conducted considering multiple hops in the propagation network, providing deeper insights into the impact of extended neighborhood structures on detection performance. Full article

(This article belongs to the Special Issue Spreading Dynamics in Complex Networks)

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23 pages, 348 KB

Open AccessArticle

Two Types of Geometric Jensen–Shannon Divergences

by Frank Nielsen

Entropy 2025, 27(9), 947; https://doi.org/10.3390/e27090947 - 11 Sep 2025

Viewed by 138

Abstract

The geometric Jensen–Shannon divergence (G-JSD) has gained popularity in machine learning and information sciences thanks to its closed-form expression between Gaussian distributions. In this work, we introduce an alternative definition of the geometric Jensen–Shannon divergence tailored to positive densities which does not normalize [...] Read more.

The geometric Jensen–Shannon divergence (G-JSD) has gained popularity in machine learning and information sciences thanks to its closed-form expression between Gaussian distributions. In this work, we introduce an alternative definition of the geometric Jensen–Shannon divergence tailored to positive densities which does not normalize geometric mixtures. This novel divergence is termed the extended G-JSD, as it applies to the more general case of positive measures. We explicitly report the gap between the extended G-JSD and the G-JSD when considering probability densities, and show how to express the G-JSD and extended G-JSD using the Jeffreys divergence and the Bhattacharyya distance or Bhattacharyya coefficient. The extended G-JSD is proven to be an f-divergence, which is a separable divergence satisfying information monotonicity and invariance in information geometry. We derive a corresponding closed-form formula for the two types of G-JSDs when considering the case of multivariate Gaussian distributions that is often met in applications. We consider Monte Carlo stochastic estimations and approximations of the two types of G-JSD using the projective

γ

-divergences. Although the square root of the JSD yields a metric distance, we show that this is no longer the case for the two types of G-JSD. Finally, we explain how these two types of geometric JSDs can be interpreted as regularizations of the ordinary JSD. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

14 pages, 498 KB

Open AccessArticle

Analytic Solutions and Entropy Production of the Double-Diffusive Equation System

by Imre Ferenc Barna and László Mátyás

Entropy 2025, 27(9), 946; https://doi.org/10.3390/e27090946 - 10 Sep 2025

Viewed by 82

Abstract

We investigate the partial differential equation system which describes the double-diffusion convection phenomena with the reduction formalism. Double-diffusion refers to when two scalar quantities with different diffusivity, such as heat and solute concentration, contribute to density gradients within a fluid under the influence [...] Read more.

We investigate the partial differential equation system which describes the double-diffusion convection phenomena with the reduction formalism. Double-diffusion refers to when two scalar quantities with different diffusivity, such as heat and solute concentration, contribute to density gradients within a fluid under the influence of gravity. The time-dependent self-similar trial function is applied and analytic results are presented for the dynamical variables and analyzed in detail. Additionally, the entropy production was derived as well. In the second part of the study we investigate the role of an additional heat source. Full article

(This article belongs to the Special Issue Dissipative Physical Dynamics)

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28 pages, 2519 KB

Open AccessArticle

On the Entropy-Based Localization of Inequality in Probability Distributions

by Rajeev Rajaram, Nathan Ritchey and Brian Castellani

Entropy 2025, 27(9), 945; https://doi.org/10.3390/e27090945 - 10 Sep 2025

Viewed by 135

Abstract

We present a novel method for localizing inequality within probability distributions by applying a recursive Hahn decomposition to the degree of uniformity—a measure derived from the exponential of Shannon entropy. This approach partitions the probability space into disjoint regions exhibiting progressively sharper deviations [...] Read more.

We present a novel method for localizing inequality within probability distributions by applying a recursive Hahn decomposition to the degree of uniformity—a measure derived from the exponential of Shannon entropy. This approach partitions the probability space into disjoint regions exhibiting progressively sharper deviations from uniformity, enabling structural insights into how and where inequality is concentrated. To demonstrate its broad applicability, we apply the method to both standard and contextualized systems: the discrete binomial and continuous exponential distributions serve as canonical cases, while two hypothetical examples illustrate domain-specific applications. In the first, we analyze localized risk concentrations in disease contraction data, revealing targeted zones of epidemiological disparity. In the second, we uncover stress localization in a non-uniformly loaded beam, demonstrating the method’s relevance to physical systems with spatial heterogeneity. This decomposition reveals aspects of structural disparity that are often obscured by scalar summaries. The resulting recursive tree offers a multi-scale representation of informational non-uniformity, capturing the emergence and localization of inequality across the distribution. The framework may have implications for understanding entropy localization, transitions in informational structure, and the dynamics of heterogeneous systems. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

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11 pages, 1243 KB

Open AccessEditorial

Five Fristonian Formulae

by Thomas Parr, Giovanni Pezzulo, Rosalyn Moran, Maxwell Ramstead, Axel Constant and Anjali Bhat

Entropy 2025, 27(9), 944; https://doi.org/10.3390/e27090944 - 10 Sep 2025

Viewed by 202

Abstract

This paper is the contribution of the editorial team for a special issue designed to celebrate the scientific contributions of Karl Friston on his 65th birthday [...] Full article

(This article belongs to the Special Issue From Functional Imaging to Free Energy—Dedicated to Professor Karl Friston on the Occasion of His 65th Birthday)

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27 pages, 1002 KB

Open AccessArticle

Exergy Efficiency of Closed and Unsteady-Flow Systems

by Yunus A. Çengel and Mehmet Kanoğlu

Entropy 2025, 27(9), 943; https://doi.org/10.3390/e27090943 - 10 Sep 2025

Viewed by 96

Abstract

Exergy efficiency is viewed as the degree of approaching reversible operation, with a value of 100 percent for a reversible process characterized by zero entropy generation or equivalently zero exergy destruction since X_destroyed = T₀S_gen. As such, exergy [...] Read more.

Exergy efficiency is viewed as the degree of approaching reversible operation, with a value of 100 percent for a reversible process characterized by zero entropy generation or equivalently zero exergy destruction since X_destroyed = T₀S_gen. As such, exergy efficiency becomes a measure of thermodynamic perfection. There are different conceptual definitions of exergy efficiency, the most common ones being (1) the ratio of exergy output to exergy input η_ex = X_output/X_input = 1 − (X_destroyed + X_loss)/X_input, (2) the ratio of the product exergy to fuel exergy η_ex = X_product/X_fuel = 1 − (X_destroyed + X_loss)/X_fuel, and (3) the ratio of exergy recovered to exergy expended η_ex = X_recovered/X_expended = 1 − X_destroyed/X_expended. Most exergy efficiency definitions are formulated with steady-flow systems in mind, and they are generally applied to systems in steady operation such as power plants and refrigeration systems whose exergy content remains constant. If these definitions are to be used for closed and unsteady-flow systems, the terms need to be interpreted broadly to account for the exergy change of the systems as exergy input or output, as appropriate. In this paper, general exergy efficiency relations are developed for closed and unsteady-flow systems and their use is demonstrated with applications. Also, the practicality of the use of the term exergy loss X_loss is questioned, and limitations on the definition η_ex = W_act,out/W_rev,out are discussed. Full article

(This article belongs to the Special Issue Thermodynamic Optimization of Energy Systems)

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28 pages, 7844 KB

Open AccessArticle

Three-Dimensional Sound Source Localization with Microphone Array Combining Spatial Entropy Quantification and Machine Learning Correction

by Guangneng Li, Feiyu Zhao, Wei Tian and Tong Yang

Entropy 2025, 27(9), 942; https://doi.org/10.3390/e27090942 - 9 Sep 2025

Viewed by 420

Abstract

In recent years, with the popularization of intelligent scene monitoring, sound source localization (SSL) has become a major means for indoor monitoring and target positioning. However, existing sound source localization solutions are difficult to extend to multi-source and three-dimensional scenarios. To address this, [...] Read more.

In recent years, with the popularization of intelligent scene monitoring, sound source localization (SSL) has become a major means for indoor monitoring and target positioning. However, existing sound source localization solutions are difficult to extend to multi-source and three-dimensional scenarios. To address this, this paper proposes a three-dimensional sound source localization technology based on eight microphones. Specifically, the method employs a rectangular eight-microphone array and captures Direction-of-Arrival (DOA) information via the direct path relative transfer function (DP-RTF). It introduces spatial entropy to quantify the uncertainty caused by the exponentially growing DOA combinations as the number of sound sources increases, while further reducing the spatial entropy of sound source localization through geometric intersection. This solves the problem that traditional sound source localization methods cannot be applied to multi-source and three-dimensional scenarios. On the other hand, machine learning is used to eliminate coordinate deviations caused by DOA estimation errors of the direct path relative transfer function (DP-RTF) and deviations in microphone geometric parameters. Both simulation experiments and real-scene experiments show that the positioning error of the proposed method in three-dimensional scenarios is about 10.0 cm. Full article

(This article belongs to the Special Issue Methods in Artificial Intelligence and Information Processing, Third Edition)

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31 pages, 892 KB

Open AccessArticle

Federated Learning over MU-MIMO Vehicular Networks

by Maria Raftopoulou, José Mairton B. da Silva, Jr., Remco Litjens, H. Vincent Poor and Piet Van Mieghem

Entropy 2025, 27(9), 941; https://doi.org/10.3390/e27090941 - 9 Sep 2025

Viewed by 126

Abstract

Many algorithms related to vehicular applications, such as enhanced perception of the environment, benefit from frequent updates and the use of data from multiple vehicles. Federated learning is a promising method to improve the accuracy of algorithms in the context of vehicular networks. [...] Read more.

Many algorithms related to vehicular applications, such as enhanced perception of the environment, benefit from frequent updates and the use of data from multiple vehicles. Federated learning is a promising method to improve the accuracy of algorithms in the context of vehicular networks. However, limited communication bandwidth, varying wireless channel quality, and potential latency requirements may impact the number of vehicles selected for training per communication round and their assigned radio resources. In this work, we characterize the vehicles participating in federated learning based on their importance to the learning process and their use of wireless resources. We then address the joint vehicle selection and resource allocation problem, considering multi-cell networks with multi-user multiple-input multiple-output (MU-MIMO)-capable base stations and vehicles. We propose a “vehicle-beam-iterative” algorithm to approximate the solution to the resulting optimization problem. We then evaluate its performance through extensive simulations, using realistic road and mobility models, for the task of object classification of European traffic signs. Our results indicate that MU-MIMO improves the convergence time of the global model. Moreover, the application-specific accuracy targets are reached faster in scenarios where the vehicles have the same training data set sizes than in scenarios where the data set sizes differ. Full article

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13 pages, 1976 KB

Open AccessArticle

Determining the Upper-Bound on the Code Distance of Quantum Stabilizer Codes Through the Monte Carlo Method Based on Fully Decoupled Belief Propagation

by Zhipeng Liang, Zicheng Wang, Zhengzhong Yi, Fusheng Yang and Xuan Wang

Entropy 2025, 27(9), 940; https://doi.org/10.3390/e27090940 - 9 Sep 2025

Viewed by 315

Abstract

The code distance is a critical parameter of quantum stabilizer codes (QSCs), and determining it—whether exactly or approximately—is known to be an NP-complete problem. However, its upper bound can be determined efficiently by some methods such as the Monte Carlo method. Leveraging the [...] Read more.

The code distance is a critical parameter of quantum stabilizer codes (QSCs), and determining it—whether exactly or approximately—is known to be an NP-complete problem. However, its upper bound can be determined efficiently by some methods such as the Monte Carlo method. Leveraging the Monte Carlo method, we propose an algorithm to compute the upper bound on the code distance of a given QSC using fully decoupled belief propagation combined with ordered statistics decoding (FDBP-OSD). Our algorithm demonstrates high precision: for various QSCs with known distances, the computed upper bounds match the actual values. Additionally, we explore upper bounds for the minimum weight of logical X operators in the Z-type Tanner-graph-recursive-expansion (Z-TGRE) code and the Chamon code—an XYZ product code constructed from three repetition codes. The results on Z-TGRE codes align with theoretical analysis, while the results on Chamon codes suggest that XYZ product codes may achieve a code distance of

O (N^{2 / 3})

, which supports the conjecture of Leverrier et al. Full article

(This article belongs to the Special Issue Quantum Error Correction and Fault-Tolerance)

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20 pages, 2671 KB

Open AccessArticle

Multivariate Time Series Anomaly Detection Based on Inverted Transformer with Multivariate Memory Gate

by Yuan Ma, Weiwei Liu, Changming Xu, Luyi Bai, Ende Zhang and Junwei Wang

Entropy 2025, 27(9), 939; https://doi.org/10.3390/e27090939 - 8 Sep 2025

Viewed by 311

Abstract

In the industrial IoT, it is vital to detect anomalies in multivariate time series, yet it faces numerous challenges, including highly imbalanced datasets, complex and high-dimensional data, and large disparities across variables. Despite the recent surge in proposals for deep learning-based methods, these [...] Read more.

In the industrial IoT, it is vital to detect anomalies in multivariate time series, yet it faces numerous challenges, including highly imbalanced datasets, complex and high-dimensional data, and large disparities across variables. Despite the recent surge in proposals for deep learning-based methods, these approaches typically treat the multivariate data at each point in time as a unique token, weakening the personalized features and dependency relationships between variables. As a result, their performance tends to degrade under highly imbalanced conditions, and reconstruction-based models are prone to overfitting abnormal patterns, leading to excessive reconstruction of anomalous inputs. In this paper, we propose ITMMG, an inverted Transformer with a multivariate memory gate. ITMMG employs an inverted token embedding strategy and multivariate memory to capture deep dependencies among variables and the normal patterns of individual variables. The experimental results obtained demonstrate that the proposed method exhibits superior performance in terms of detection accuracy and robustness compared with existing baseline methods across a range of standard time series anomaly detection datasets. This significantly reduces the probability of misclassifying anomalous samples during reconstruction. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

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24 pages, 3114 KB

Open AccessArticle

GNSS Interference Identification Driven by Eye Pattern Features: ICOA–CNN–ResNet–BiLSTM Optimized Deep Learning Architecture

by Chuanyu Wu, Yuanfa Ji and Xiyan Sun

Entropy 2025, 27(9), 938; https://doi.org/10.3390/e27090938 - 7 Sep 2025

Viewed by 286

Abstract

In this study, the key challenges faced by global navigation satellite systems (GNSSs) in the field of security are addressed, and an eye diagram-based deep learning framework for intelligent classification of interference types is proposed. GNSS signals are first transformed into two-dimensional eye [...] Read more.

In this study, the key challenges faced by global navigation satellite systems (GNSSs) in the field of security are addressed, and an eye diagram-based deep learning framework for intelligent classification of interference types is proposed. GNSS signals are first transformed into two-dimensional eye diagrams, enabling a novel visual representation wherein interference types are distinguished through entropy-centric feature analysis. Specifically, the quantification of information entropy within these diagrams serves as a theoretical foundation for extracting salient discriminative features, reflecting the structural complexity and uncertainty of the underlying signal distortions. We designed a hybrid architecture that integrates spatial feature extraction, gradient stability enhancement, and time dynamics modeling capabilities and combines the advantages of a convolutional neural network, residual network, and bidirectional long short-term memory network. To further improve model performance, we propose an improved coati optimization algorithm (ICOA), which combines chaotic mapping, an elite perturbation mechanism, and an adaptive weighting strategy for hyperparameter optimization. Compared with mainstream optimization methods, this algorithm improves the convergence accuracy by more than 30%. Experimental results on jamming datasets (continuous wave interference, chirp interference, pulse interference, frequency-modulated interference, amplitude-modulated interference, and spoofing interference) demonstrate that our method achieved performance in terms of accuracy, precision, recall, F1 score, and specificity, with values of 98.02%, 97.09%, 97.24%, 97.14%, and 99.65%, respectively, which represent improvements of 1.98%, 2.80%, 6.10%, 4.59%, and 0.33% over the next-best model. This study provides an efficient, entropy-aware, intelligent, and practically feasible solution for GNSS interference identification. Full article

(This article belongs to the Section Signal and Data Analysis)

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20 pages, 1690 KB

Open AccessArticle

3V-GM: A Tri-Layer “Point–Line–Plane” Critical Node Identification Algorithm for New Power Systems

by Yuzhuo Dai, Min Zhao, Gengchen Zhang and Tianze Zhao

Entropy 2025, 27(9), 937; https://doi.org/10.3390/e27090937 - 7 Sep 2025

Viewed by 365

Abstract

With the increasing penetration of renewable energy, the stochastic and intermittent nature of its generation increases operational uncertainty and vulnerability, posing significant challenges for grid stability. However, traditional algorithms typically identify critical nodes by focusing solely on the network topology or power flow, [...] Read more.

With the increasing penetration of renewable energy, the stochastic and intermittent nature of its generation increases operational uncertainty and vulnerability, posing significant challenges for grid stability. However, traditional algorithms typically identify critical nodes by focusing solely on the network topology or power flow, or by combining the two, which leads to the inaccurate and incomplete identification of essential nodes. To address this, we propose the Three-Dimensional Value-Based Gravity Model (3V-GM), which integrates structural and electrical–physical attributes across three layers. In the plane layer, we combine each node’s global topological position with its real-time supply–demand voltage state. In the line layer, we introduce an electrical coupling distance to quantify the strength of electromagnetic interactions between nodes. In the point layer, we apply eigenvector centrality to detect latent hub nodes whose influence is not immediately apparent. The performance of our proposed method was evaluated by examining the change in the load loss rate as nodes were sequentially removed. To assess the effectiveness of the 3V-GM approach, simulations were conducted on the IEEE 39 system, as well as six other benchmark networks. The simulations were performed using Python scripts, with operational parameters such as bus voltages, active and reactive power flows, and branch impedances obtained from standard test cases provided by MATPOWER v7.1. The results consistently show that removing the same number of nodes identified by 3V-GM leads to a greater load loss compared to the six baseline methods. This demonstrates the superior accuracy and stability of our approach. Additionally, an ablation experiment, which decomposed and recombined the three layers, further highlights the unique contribution of each component to the overall performance. Full article

(This article belongs to the Section Complexity)

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22 pages, 415 KB

Open AccessArticle

Infodemic Source Detection with Information Flow: Foundations and Scalable Computation

by Zimeng Wang, Chao Zhao, Qiaoqiao Zhou, Chee Wei Tan and Chung Chan

Entropy 2025, 27(9), 936; https://doi.org/10.3390/e27090936 - 6 Sep 2025

Viewed by 1293

Abstract

We consider the problem of identifying the source of a rumor in a network, given only a snapshot observation of infected nodes after the rumor has spread. Classical approaches, such as the maximum likelihood (ML) and joint maximum likelihood (JML) estimators based on [...] Read more.

We consider the problem of identifying the source of a rumor in a network, given only a snapshot observation of infected nodes after the rumor has spread. Classical approaches, such as the maximum likelihood (ML) and joint maximum likelihood (JML) estimators based on the conventional Susceptible–Infectious (SI) model, exhibit degeneracy, failing to uniquely identify the source even in simple network structures. To address these limitations, we propose a generalized estimator that incorporates independent random observation times. To capture the structure of information flow beyond graphs, our formulations consider rate constraints on the rumor and the multicast capacities for cyclic polylinking networks. Furthermore, we develop forward elimination and backward search algorithms for rate-constrained source detection and validate their effectiveness and scalability through comprehensive simulations. Our study establishes a rigorous and scalable foundation on the infodemic source detection. Full article

(This article belongs to the Special Issue Applications of Information Theory to Machine Learning)

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9 pages, 1589 KB

Open AccessArticle

Application of the Three-Group Model to the 2024 US Elections

by Miron Kaufman, Sanda Kaufman and Hung T. Diep

Entropy 2025, 27(9), 935; https://doi.org/10.3390/e27090935 - 6 Sep 2025

Viewed by 359

Abstract

Political polarization in Western democracies has accelerated in the last decade, with negative social consequences. Research across disciplines on antecedents, manifestations and societal impacts is hindered by social systems’ complexity: their constant flux impedes tracing causes of observed trends and prediction of consequences, [...] Read more.

Political polarization in Western democracies has accelerated in the last decade, with negative social consequences. Research across disciplines on antecedents, manifestations and societal impacts is hindered by social systems’ complexity: their constant flux impedes tracing causes of observed trends and prediction of consequences, hampering their mitigation. Social physics models exploit a characteristic of complex systems: what seems chaotic at one observation level may exhibit patterns at a higher level. Therefore, dynamic modeling of complex systems allows anticipation of possible events. We use this approach to anticipate 2024 US election results. We consider the highly polarized Democrats and Republicans, and Independents fluctuating between them. We generate average group-stance scenarios in time and explore how polarization and depolarization might have affected 2024 voting outcomes. We find that reducing polarization might advantage the larger voting group. We also explore ways to reduce polarization, and their potential effects on election results. The results inform regarding the perils of polarization trends, and on possibilities of changing course. Full article

(This article belongs to the Special Issue Computational and Statistical Physics Approaches for Complex Systems and Social Phenomena, 3rd Edition)

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23 pages, 5122 KB

Open AccessArticle

Time-Varying Autoregressive Models: A Novel Approach Using Physics-Informed Neural Networks

by Zhixuan Jia and Chengcheng Zhang

Entropy 2025, 27(9), 934; https://doi.org/10.3390/e27090934 - 4 Sep 2025

Viewed by 517

Abstract

Time series models are widely used to examine temporal dynamics and uncover patterns across diverse fields. A commonly employed approach for modeling such data is the (Vector) Autoregressive (AR/VAR) model, in which each variable is represented as a linear combination of its own [...] Read more.

Time series models are widely used to examine temporal dynamics and uncover patterns across diverse fields. A commonly employed approach for modeling such data is the (Vector) Autoregressive (AR/VAR) model, in which each variable is represented as a linear combination of its own and others’ lagged values. However, the traditional (V)AR framework relies on the key assumption of stationarity, that autoregressive coefficients remain constant over time, which is often violated in practice, especially in systems affected by structural breaks, seasonal fluctuations, or evolving causal mechanisms. To overcome this limitation, Time-Varying (Vector) Autoregressive (TV-AR/TV-VAR) models have been developed, enabling model parameters to evolve over time and thus better capturing non-stationary behavior. Conventional approaches to estimating such models, including generalized additive modeling and kernel smoothing techniques, often require strong assumptions about basis functions, which can restrict their flexibility and applicability. To address these challenges, we introduce a novel framework that leverages physics-informed neural networks (PINN) to model TV-AR/TV-VAR processes. The proposed method extends the PINN framework to time series analysis by reducing reliance on explicitly defined physical structures, thereby broadening its applicability. Its effectiveness is validated through simulations on synthetic data and an empirical study of real-world health-related time series. Full article

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24 pages, 4429 KB

Open AccessArticle

Ascertaining Susceptibilities in Smart Contracts: A Quantum Machine Learning Approach

by Amulyashree Sridhar, Kalyan Nagaraj, Shambhavi Bangalore Ravi and Sindhu Kurup

Entropy 2025, 27(9), 933; https://doi.org/10.3390/e27090933 - 4 Sep 2025

Viewed by 576

Abstract

The current research aims to discover applications of QML approaches in realizing liabilities within smart contracts. These contracts are essential commodities of the blockchain interface and are also decisive in developing decentralized products. But liabilities in smart contracts could result in unfamiliar system [...] Read more.

The current research aims to discover applications of QML approaches in realizing liabilities within smart contracts. These contracts are essential commodities of the blockchain interface and are also decisive in developing decentralized products. But liabilities in smart contracts could result in unfamiliar system failures. Presently, static detection tools are utilized to discover accountabilities. However, they could result in instances of false narratives due to their dependency on predefined rules. In addition, these policies can often be superseded, failing to generalize on new contracts. The detection of liabilities with ML approaches, correspondingly, has certain limitations with contract size due to storage and performance issues. Nevertheless, employing QML approaches could be beneficial as they do not necessitate any preconceived rules. They often learn from data attributes during the training process and are employed as alternatives to ML approaches in terms of storage and performance. The present study employs four QML approaches, namely, QNN, QSVM, VQC, and QRF, for discovering susceptibilities. Experimentation revealed that the QNN model surpasses other approaches in detecting liabilities, with a performance accuracy of 82.43%. To further validate its feasibility and performance, the model was assessed on a several-partition test dataset, i.e., SolidiFI data, and the outcomes remained consistent. Additionally, the performance of the model was statistically validated using McNemar’s test. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

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20 pages, 7914 KB

Open AccessArticle

Channel Estimation for Intelligent Reflecting Surface Empowered Coal Mine Wireless Communication Systems

by Yang Liu, Kaikai Guo, Xiaoyue Li, Bin Wang and Yanhong Xu

Entropy 2025, 27(9), 932; https://doi.org/10.3390/e27090932 - 4 Sep 2025

Viewed by 449

Abstract

The confined space of coal mines characterized by curved tunnels with rough surfaces and a variety of deployed production equipment induces severe signal attenuation and interruption, which significantly degrades the accuracy of conventional channel estimation algorithms applied in coal mine wireless communication systems. [...] Read more.

The confined space of coal mines characterized by curved tunnels with rough surfaces and a variety of deployed production equipment induces severe signal attenuation and interruption, which significantly degrades the accuracy of conventional channel estimation algorithms applied in coal mine wireless communication systems. To address these challenges, we propose a modified Bilinear Generalized Approximate Message Passing (mBiGAMP) algorithm enhanced by intelligent reflecting surface (IRS) technology to improve channel estimation accuracy in coal mine scenarios. Due to the presence of abundant coal-carrying belt conveyors, we establish a hybrid channel model integrating both fast-varying and quasi-static components to accurately model the unique propagation environment in coal mines. Specifically, the fast-varying channel captures the varying signal paths affected by moving conveyors, while the quasi-static channel represents stable direct links. Since this hybrid structure necessitates an augmented factor graph, we introduce two additional factor nodes and variable nodes to characterize the distinct message-passing behaviors and then rigorously derive the mBiGAMP algorithm. Simulation results demonstrate that the proposed mBiGAMP algorithm achieves superior channel estimation accuracy in dynamic conveyor-affected coal mine scenarios compared with other state-of-the-art methods, showing significant improvements in both separated and cascaded channel estimation. Specifically, when the NMSE is

10^{- 3}

, the SNR of mBiGAMP is improved by approximately 5 dB, 6 dB, and 14 dB compared with the Dual-Structure Orthogonal Matching Pursuit (DS-OMP), Parallel Factor (PARAFAC), and Least Squares (LS) algorithms, respectively. We also verify the convergence behavior of the proposed mBiGAMP algorithm across the operational signal-to-noise ratios range. Furthermore, we investigate the impact of the number of pilots on the channel estimation performance, which reveals that the proposed mBiGAMP algorithm consumes fewer number of pilots to accurately recover channel state information than other methods while preserving estimation fidelity. Full article

(This article belongs to the Special Issue Wireless Communications: Signal Processing Perspectives, 2nd Edition)

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21 pages, 375 KB

Open AccessReview

Sherlock Holmes Doesn’t Play Dice: The Mathematics of Uncertain Reasoning When Something May Happen, That You Are Not Even Able to Figure Out

by Guido Fioretti

Entropy 2025, 27(9), 931; https://doi.org/10.3390/e27090931 - 4 Sep 2025

Viewed by 388

Abstract

While Evidence Theory (also known as Dempster–Shafer Theory, or Belief Functions Theory) is being increasingly used in data fusion, its potentialities in the Social and Life Sciences are often obscured by lack of awareness of its distinctive features. In particular, with this paper [...] Read more.

While Evidence Theory (also known as Dempster–Shafer Theory, or Belief Functions Theory) is being increasingly used in data fusion, its potentialities in the Social and Life Sciences are often obscured by lack of awareness of its distinctive features. In particular, with this paper I stress that an extended version of Evidence Theory can express the uncertainty deriving from the fear that events may materialize, that one is not even able to figure out. By contrast, Probability Theory must limit itself to the possibilities that a decision-maker is currently envisaging. I compare this extended version of Evidence Theory to cutting-edge extensions of Probability Theory, such as imprecise and sub-additive probabilities, as well as unconventional versions of Information Theory that are employed in data fusion and transmission of cultural information. A possible application to creative usage of Large Language Models is outlined, and further extensions to multi-agent interactions are outlined. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

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22 pages, 1017 KB

Open AccessArticle

Optimized Generalized LDPC Convolutional Codes

by Li Deng, Kai Tao, Zhiping Shi, You Zhang, Yinlong Shi, Jian Wang, Tian Liu and Yongben Wang

Entropy 2025, 27(9), 930; https://doi.org/10.3390/e27090930 - 4 Sep 2025

Viewed by 502

Abstract

In this paper, some optimized encoding and decoding schemes are proposed for the generalized LDPC convolutional codes (GLDPC–CCs). In terms of the encoding scheme, a flexible doping method is proposed, which replaces multiple single parity check (SPC) nodes with one generalized check (GC) [...] Read more.

In this paper, some optimized encoding and decoding schemes are proposed for the generalized LDPC convolutional codes (GLDPC–CCs). In terms of the encoding scheme, a flexible doping method is proposed, which replaces multiple single parity check (SPC) nodes with one generalized check (GC) node. Different types of BCH codes can be selected as the GC node by adjusting the number of SPC nodes to be replaced. Moreover, by fine-tuning the truncated bits and the extended parity check bits, or by reasonably adjusting the GC node distribution, the performance of GLDPC–CCs can be further improved. In terms of the decoding scheme, a hybrid layered normalized min-sum (HLNMS) decoding algorithm is proposed, where the layered normalized min-sum (LNMS) decoding is used for SPC nodes, and the Chase–Pyndiah decoding is adopted for GC nodes. Based on analysis of the decoding convergence of GC node and SPC node, an adaptive weight factor is designed for GC nodes that changes as the decoding iterations, aiming to further improve the decoding performance. In addition, an early stop decoding strategy is also proposed based on the minimum amplitude threshold of mutual information in order to reduce the decoding complexity. The simulation results have verified the superiority of the proposed scheme for GLDPC–CCs over the prior art, which has great application potential in optical communication systems. Full article

(This article belongs to the Special Issue LDPC Codes for Communication Systems)

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24 pages, 2585 KB

Open AccessArticle

Comprehensive Examination of Unrolled Networks for Solving Linear Inverse Problems

by Yuxi Chen, Xi Chen, Arian Maleki and Shirin Jalali

Entropy 2025, 27(9), 929; https://doi.org/10.3390/e27090929 - 3 Sep 2025

Viewed by 514

Abstract

Unrolled networks have become prevalent in various computer vision and imaging tasks. Although they have demonstrated remarkable efficacy in solving specific computer vision and computational imaging tasks, their adaptation to other applications presents considerable challenges. This is primarily due to the multitude of [...] Read more.

Unrolled networks have become prevalent in various computer vision and imaging tasks. Although they have demonstrated remarkable efficacy in solving specific computer vision and computational imaging tasks, their adaptation to other applications presents considerable challenges. This is primarily due to the multitude of design decisions that practitioners working on new applications must navigate, each potentially affecting the network’s overall performance. These decisions include selecting the optimization algorithm, defining the loss function, and determining the deep architecture, among others. Compounding the issue, evaluating each design choice requires time-consuming simulations to train, fine-tune the neural network, and optimize its performance. As a result, the process of exploring multiple options and identifying the optimal configuration becomes time-consuming and computationally demanding. The main objectives of this paper are (1) to unify some ideas and methodologies used in unrolled networks to reduce the number of design choices a user has to make, and (2) to report a comprehensive ablation study to discuss the impact of each of the choices involved in designing unrolled networks and present practical recommendations based on our findings. We anticipate that this study will help scientists and engineers to design unrolled networks for their applications and diagnose problems within their networks efficiently. Full article

(This article belongs to the Special Issue Advances in Information Theory and Machine Learning for Computational Imaging)

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13 pages, 952 KB

Open AccessArticle

Sensor Fusion for Target Detection Using LLM-Based Transfer Learning Approach

by Yuval Ziv, Barouch Matzliach and Irad Ben-Gal

Entropy 2025, 27(9), 928; https://doi.org/10.3390/e27090928 - 3 Sep 2025

Viewed by 580

Abstract

This paper introduces a novel sensor fusion approach for the detection of multiple static and mobile targets by autonomous mobile agents. Unlike previous studies that rely on theoretical sensor models, which are considered as independent, the proposed methodology leverages real-world sensor data, which [...] Read more.

This paper introduces a novel sensor fusion approach for the detection of multiple static and mobile targets by autonomous mobile agents. Unlike previous studies that rely on theoretical sensor models, which are considered as independent, the proposed methodology leverages real-world sensor data, which is transformed into sensor-specific probability maps using object detection estimation for optical data and converting averaged point-cloud intensities for LIDAR based on a dedicated deep learning model before being integrated through a large language model (LLM) framework. We introduce a methodology based on LLM transfer learning (LLM-TLFT) to create a robust global probability map enabling efficient swarm management and target detection in challenging environments. The paper focuses on real data obtained from two types of sensors, light detection and ranging (LIDAR) sensors and optical sensors, and it demonstrates significant improvement in performance compared to existing methods (Independent Opinion Pool, CNN, GPT-2 with deep transfer learning) in terms of precision, recall, and computational efficiency, particularly in scenarios with high noise and sensor imperfections. The significant advantage of the proposed approach is the possibility to interpret a dependency between different sensors. In addition, a model compression using knowledge-based distillation was performed (distilled TLFT), which yielded satisfactory results for the deployment of the proposed approach to edge devices. Full article

(This article belongs to the Special Issue Informational Coordinative and Teleological Control of Distributed and Multi Agent Systems)

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Entropy, Volume 27, Issue 9 (September 2025) – 71 articles

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