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Symmetry

Symmetry is an international, peer-reviewed, open access journal covering research on symmetry/asymmetry phenomena wherever they occur in all aspects of natural sciences. Symmetry is published monthly online by MDPI.

Open Access— free for readers, with article processing charges (APC) paid by authors or their institutions.
High Visibility: indexed within SCIE (Web of Science), Scopus, CAPlus / SciFinder, Inspec, Astrophysics Data System, and other databases.
Journal Rank: JCR - Q2 (Multidisciplinary Sciences) / CiteScore - Q1 (General Mathematics )
Rapid Publication: manuscripts are peer-reviewed and a first decision is provided to authors approximately 17.1 days after submission; acceptance to publication is undertaken in 2.8 days (median values for papers published in this journal in the first half of 2025).
Recognition of Reviewers: reviewers who provide timely, thorough peer-review reports receive vouchers entitling them to a discount on the APC of their next publication in any MDPI journal, in appreciation of the work done.
Testimonials: See what our editors and authors say about Symmetry.

Impact Factor: 2.2 (2024); 5-Year Impact Factor: 2.1 (2024)

Imprint Information Journal Flyer Open Access ISSN: 2073-8994

Latest Articles

20 pages, 49600 KiB

Open AccessArticle

An Improved Symmetric Network with Feature Difference and Receptive Field for Change Detection

by Botao Zhang, Yixuan Wang, Jia Lu and Qin Wang

Symmetry 2025, 17(7), 1095; https://doi.org/10.3390/sym17071095 - 8 Jul 2025

Change detection (CD) is essential for Earth observation tasks, as it identifies alterations in specific geographic areas over time. The advancement of deep learning has significantly improved the accuracy of CD. However, encoder–decoder architectures often struggle to effectively capture temporal differences. Encoders may [...] Read more.

Change detection (CD) is essential for Earth observation tasks, as it identifies alterations in specific geographic areas over time. The advancement of deep learning has significantly improved the accuracy of CD. However, encoder–decoder architectures often struggle to effectively capture temporal differences. Encoders may lose critical spatial details, while decoders can introduce bias due to inconsistent receptive fields across layers. To address these limitations, this paper proposes an enhanced symmetric network, termed FDRF (feature difference and receptive field), which incorporates two novel components: the multibranch feature difference extraction (MFDE) module and the adaptive ensemble decision (AED) module. MFDE independently extracts differential features from bitemporal images at each encoder layer, using multiscale fusion to retain image content and improve the quality of feature difference modeling. AED assigns confidence weights to predictions from different decoder layers based on their receptive field sizes and then combines them adaptively to reduce scale-related bias. To validate the effectiveness and robustness of FDRF, experiments were conducted on five public datasets (SYSU, LEVIR-CD, WHU, NJDS, and CLCD), as well as a UAV-based dataset collected from two national coastal nature reserves in Guangxi Beihai, China. The results demonstrate that FDRF consistently outperforms existing methods in accuracy and robustness across diverse scenarios. Full article

(This article belongs to the Section Computer)

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27 pages, 13752 KiB

Open AccessArticle

Robust Watermarking of Tiny Neural Networks by Fine-Tuning and Post-Training Approaches

by Riccardo Adorante, Alessandro Carra, Marco Lattuada and Danilo Pietro Pau

Symmetry 2025, 17(7), 1094; https://doi.org/10.3390/sym17071094 - 8 Jul 2025

Because neural networks pervade many industrial domains and are increasingly complex and accurate, the trained models themselves have become valuable intellectual properties. Developing highly accurate models demands increasingly higher investments of time, capital, and expertise. Many of these models are commonly deployed in [...] Read more.

Because neural networks pervade many industrial domains and are increasingly complex and accurate, the trained models themselves have become valuable intellectual properties. Developing highly accurate models demands increasingly higher investments of time, capital, and expertise. Many of these models are commonly deployed in cloud services and on resource-constrained edge devices. Consequently, safeguarding them is critically important. Neural network watermarking offers a practical solution to address this need by embedding a unique signature, either as a hidden bit-string or as a distinctive response to specially crafted “trigger” inputs. This allows owners to subsequently prove model ownership even if an adversary attempts to remove the watermark through attacks. In this manuscript, we adapt three state-of-the-art watermarking methods to “tiny” neural networks deployed on edge platforms by exploiting symmetry-related properties that ensure robustness and efficiency. In the context of machine learning, “tiny” is broadly used as a term referring to artificial intelligence techniques deployed in low-energy systems in the mW range and below, e.g., sensors and microcontrollers. We evaluate the robustness of the selected techniques by simulating attacks aimed at erasing the watermark while preserving the model’s original performances. The results before and after attacks demonstrate the effectiveness of these watermarking schemes in protecting neural network intellectual property without degrading the original accuracy. Full article

(This article belongs to the Section Computer)

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25 pages, 906 KiB

Open AccessArticle

Query-Efficient Two-Phase Reinforcement Learning Framework for Black-Box Adversarial Attacks

by Zerou Ma and Tao Feng

Symmetry 2025, 17(7), 1093; https://doi.org/10.3390/sym17071093 - 8 Jul 2025

Generating adversarial examples under black-box settings poses significant challenges due to the inaccessibility of internal model information. This complexity is further exacerbated when attempting to achieve a balance between the attack success rate and perceptual quality. In this paper, we propose QTRL, a [...] Read more.

Generating adversarial examples under black-box settings poses significant challenges due to the inaccessibility of internal model information. This complexity is further exacerbated when attempting to achieve a balance between the attack success rate and perceptual quality. In this paper, we propose QTRL, a query-efficient two-phase reinforcement learning framework for generating high-quality black-box adversarial examples. Unlike existing approaches that treat adversarial generation as a single-step optimization problem, QTRL introduces a progressive two-phase learning strategy. The initial phase focuses on training the agent to develop effective adversarial strategies, while the second phase refines the perturbations to improve visual quality without sacrificing attack performance. To compensate for the unavailability of gradient information inherent in black-box settings, QTRL designs distinct reward functions for the two phases: the first prioritizes attack success, whereas the second incorporates perceptual similarity metrics to guide refinement. Furthermore, a hard sample mining mechanism is introduced to revisit previously failed attacks, significantly enhancing the robustness and generalization capabilities of the learned policy. Experimental results on the MNIST and CIFAR-10 datasets demonstrate that QTRL achieves attack success rates comparable to those of state-of-the-art methods while substantially reducing query overhead, offering a practical and extensible solution for adversarial research in black-box scenarios. Full article

(This article belongs to the Section Computer)

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19 pages, 9926 KiB

Open AccessArticle

Deep Learning-Based Optimal Condition Monitoring System for Plant Growth in an Indoor Smart Hydroponic Greenhouse

by Oybek Eraliev Maripjon Ugli and Chul-Hee Lee

Symmetry 2025, 17(7), 1092; https://doi.org/10.3390/sym17071092 - 8 Jul 2025

This study introduces a deep learning (DL)-based optimal condition monitoring and control system tailored to indoor smart greenhouses, with a novel focus on maintaining symmetry—defined as a dynamic equilibrium among temperature, humidity, and CO₂ levels—critical in plant growth. A hydroponic greenhouse prototype [...] Read more.

This study introduces a deep learning (DL)-based optimal condition monitoring and control system tailored to indoor smart greenhouses, with a novel focus on maintaining symmetry—defined as a dynamic equilibrium among temperature, humidity, and CO₂ levels—critical in plant growth. A hydroponic greenhouse prototype was developed to capture real-time climate data at high temporal resolution. A custom 1D convolutional neural network (1D-CNN) optimized via a genetic algorithm (GA) was employed to predict environmental fluctuations, achieving R² scores up to 0.99 and a standard error of prediction (SEP) as low as 0.35%. The system then actuated climate control mechanisms to restore and maintain symmetry. Experimental validation revealed that plants grown under the symmetry-aware control system exhibited significantly improved growth metrics. The results underscore the potential of integrating symmetry-aware DL strategies into precision agriculture in achieving sustainable and resilient plant production systems. Full article

(This article belongs to the Section Computer)

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12 pages, 1601 KiB

Open AccessArticle

Effects of Residual Stresses on the Structures and Mechanical Behavior of ZrO_xN_y/V₂O₃ Nano-Multilayers

by Wenjie Cheng, Lingran Wang and Zhiming Li

Symmetry 2025, 17(7), 1091; https://doi.org/10.3390/sym17071091 - 8 Jul 2025

Residual stress plays a crucial role in determining the structural reliability and mechanical performance of nano-multilayers. In the present study, nano-multilayers composed of ZrO_xN_y and V₂O₃ were deposited via magnetron sputtering, with the N:Ar flow ratio systematically [...] Read more.

Residual stress plays a crucial role in determining the structural reliability and mechanical performance of nano-multilayers. In the present study, nano-multilayers composed of ZrO_xN_y and V₂O₃ were deposited via magnetron sputtering, with the N:Ar flow ratio systematically varied during the process. Through the precise control of the deposition conditions, the compressive residual stress within the films was effectively reduced to approximately 0 GPa, thereby improving their mechanical robustness. It was observed that the optimization of the stress distribution was strongly influenced by the structural symmetry of the multilayer configuration. This symmetrical design not only mitigated stress accumulation but also ensured uniform mechanical response throughout the multilayer structure. The results from nanoindentation testing revealed a steady hardness value near 10.6 GPa. Furthermore, the maximum H³/E² and H/E ratios recorded were 0.054 GPa and 0.073, respectively, suggesting enhanced resistance to both plastic deformation and cracking. Full article

(This article belongs to the Section Engineering and Materials)

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19 pages, 1256 KiB

Open AccessArticle

More Details on Two Solutions with Ordered Sequences for Binomial Confidence Intervals

by Lorentz Jäntschi

Symmetry 2025, 17(7), 1090; https://doi.org/10.3390/sym17071090 - 8 Jul 2025

While many continuous distributions are known, the list of discrete ones (usually derived from counting) often reported is relatively short. This list becomes even shorter when dealing with dichotomous observables: binomial, hypergeometric, negative binomial, and uniform. Binomial distribution is important for medical studies, [...] Read more.

While many continuous distributions are known, the list of discrete ones (usually derived from counting) often reported is relatively short. This list becomes even shorter when dealing with dichotomous observables: binomial, hypergeometric, negative binomial, and uniform. Binomial distribution is important for medical studies, since a finite sample from a population included in a medical study with yes/no outcome resembles a series of independent Bernoulli trials. The problem of calculating the confidence interval (CI, with conventional risk of 5% or otherwise) is revealed to be a problem of combinatorics. Several algorithms dispute the exact calculation, each according to a formal definition of its exactness. For two algorithms, four previously proposed case studies are provided, for sample sizes of 30, 50, 100, 150, and 300. In these cases, at 1% significance level, ordered sequences defining the confidence bounds were generated for two formal definitions. Images of the error’s alternation are provided and discussed. Both algorithms propose symmetric solutions in terms of both CIs and actual coverage probabilities. The CIs are not symmetric relative to the observed variable, but are mirrored symmetric relative to the middle of the observed variable domain. When comparing the solutions proposed by the algorithms, with the increase in the sample size, the ratio of identical confidence levels is increased and the difference between actual and imposed coverage is shrunk. Full article

(This article belongs to the Section Mathematics)

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26 pages, 1884 KiB

Open AccessArticle

A Symmetry-Based Spherical Fuzzy MCDM Approach for the Strategic Assessment of Alternative Fuels Toward Sustainable Energy Policies

by Adnan Abdulvahitoğlu

Symmetry 2025, 17(7), 1089; https://doi.org/10.3390/sym17071089 - 8 Jul 2025

Alternative fuels obtained from renewable sources, providing low greenhouse gas emissions and high energy efficiency, offer significant advantages in terms of sustainability. In addition, the wide applicability of these fuel types in sectors such as housing, transportation, and industry creates significant opportunities in [...] Read more.

Alternative fuels obtained from renewable sources, providing low greenhouse gas emissions and high energy efficiency, offer significant advantages in terms of sustainability. In addition, the wide applicability of these fuel types in sectors such as housing, transportation, and industry creates significant opportunities in terms of reducing dependence on fossil fuels. Alternative fuels should be evaluated not only according to their environmental contributions but also based on multi-dimensional criteria such as economic cost, technical suitability, sustainability level, fuel properties, infrastructure requirements, and social acceptance. In this context, a comparative analysis of alternative fuel types in terms of various basic parameters is no longer optional, but a necessity. These parameters generally include symmetrical relationships such as balanced trade-offs between economic and environmental dimensions or mutual effects between technical and social criteria. However, they also show variability and uncertainty depending on the fuel type. Therefore, Spherical Fuzzy Multi-Criteria Decision Making (SF-MCDM) methods, which can effectively represent symmetry in membership and hesitation degrees, have been used to achieve more realistic and reliable results in uncertain decision environments. The proposed model provides a systematic and flexible evaluation structure that helps decision makers determine the most appropriate alternative fuel options and contributes to the formation of sustainable energy policies. Full article

(This article belongs to the Section Mathematics)

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23 pages, 3045 KiB

Open AccessArticle

Type-2 Fuzzy-Controlled Air-Cleaning Mobile Robot

by Chian-Song Chiu, Shu-Yen Yao and Carlo Santiago

Symmetry 2025, 17(7), 1088; https://doi.org/10.3390/sym17071088 - 8 Jul 2025

This research presents the development of a type-2 fuzzy-controlled autonomous mobile robot specifically designed for monitoring and actively maintaining indoor air quality. The core of this system is the proposed type-2 fuzzy PID dual-mode controller used for stably patrolling rooms along the walls [...] Read more.

This research presents the development of a type-2 fuzzy-controlled autonomous mobile robot specifically designed for monitoring and actively maintaining indoor air quality. The core of this system is the proposed type-2 fuzzy PID dual-mode controller used for stably patrolling rooms along the walls of the environment. The design method ingeniously merges the fast error correction capability of PID control with the robust adaptability of type-2 fuzzy logic control, which utilizes interval type-2 fuzzy sets. Furthermore, the type-2 fuzzy rule table of the right wall-following controller can be extended from the first designed fuzzy left wall-following controller in a symmetrical design manner. As a result, this study eliminates the drawbacks of excessive oscillations arising from PID control and sluggish response to large initial errors in typical traditional fuzzy control. The following of the stable wall and obstacle is facilitated with ensured accuracy and easy implementation so that effective air quality monitoring and active PM2.5 filtering are achieved in a movable manner. Furthermore, the augmented reality (AR) interface overlays real-time PM2.5 data directly onto a user’s visual field, enhancing situational awareness and enabling an immediate and intuitive assessment of air quality. As this type of control is different from that used in traditional fixed sensor networks, both broader area coverage and efficient air filtering are achieved. Finally, the experimental results demonstrate the controller’s superior performance and its potential to significantly improve indoor air quality. Full article

(This article belongs to the Special Issue Applications Based on Symmetry in Control Systems and Robotics)

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26 pages, 1729 KiB

Open AccessArticle

AnnCoder: A Mti-Agent-Based Code Generation and Optimization Model

by Zhenhua Zhang, Jianfeng Wang, Zhengyang Li, Yunpeng Wang and Jiayun Zheng

Symmetry 2025, 17(7), 1087; https://doi.org/10.3390/sym17071087 - 7 Jul 2025

The rapid progress of Large Language Models (LLMs) has greatly improved natural language tasks like code generation, boosting developer productivity. However, challenges persist. Generated code often appears “pseudo-correct”—passing functional tests but plagued by inefficiency or redundant structures. Many models rely on outdated methods [...] Read more.

The rapid progress of Large Language Models (LLMs) has greatly improved natural language tasks like code generation, boosting developer productivity. However, challenges persist. Generated code often appears “pseudo-correct”—passing functional tests but plagued by inefficiency or redundant structures. Many models rely on outdated methods like greedy selection, which trap them in local optima, limiting their ability to explore better solutions. We propose AnnCoder, a multi-agent framework that mimics the human “try-fix-adapt” cycle through closed-loop optimization. By combining the exploratory power of simulated annealing with the targeted evolution of genetic algorithms, AnnCoder balances wide-ranging searches and local refinements, dramatically increasing the likelihood of finding globally optimal solutions. We speculate that traditional approaches may struggle due to narrow optimization focuses. AnnCoder addresses this by introducing dynamic multi-criteria scoring, weighing functional correctness, efficiency (e.g., runtime/memory), and readability. Its adaptive temperature control dynamically modulates the cooling schedule, slowing cooling when solutions are diverse to encourage exploration, then accelerating convergence as they stabilize. This design elegantly avoids the pitfalls of earlier models by synergistically combining global exploration with local optimization capabilities. After conducting thorough experiments with multiple LLMs analyses across four problem-solving and program synthesis benchmarks—AnnCoder showcased remarkable code generation capabilities—HumanEval 90.85%, MBPP 90.68%, HumanEval-ET 85.37%, and EvalPlus 84.8%. AnnCoder has outstanding advantages in solving general programming problems. Moreover, our method consistently delivers superior performance across various programming languages. Full article

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20 pages, 1261 KiB

Open AccessArticle

Risk Analysis of Five-Axis CNC Water Jet Machining Using Fuzzy Risk Priority Numbers

by Ufuk Cebeci, Ugur Simsir and Onur Dogan

Symmetry 2025, 17(7), 1086; https://doi.org/10.3390/sym17071086 - 7 Jul 2025

The reliability and safety of five-axis CNC abrasive water jet machining are critical for many industries. This study employs Failure Mode and Effects Analysis (FMEA) to identify and mitigate potential failures in this machining system. Traditional FMEA, which relies on crisp numerical values, [...] Read more.

The reliability and safety of five-axis CNC abrasive water jet machining are critical for many industries. This study employs Failure Mode and Effects Analysis (FMEA) to identify and mitigate potential failures in this machining system. Traditional FMEA, which relies on crisp numerical values, often struggles with handling uncertainty in risk assessment. To address this limitation, this paper introduces an Interval-Valued Spherical Fuzzy FMEA (IVSF-FMEA) approach, which enhances risk evaluation by incorporating membership, non-membership, and hesitancy degrees. The IVSF-FMEA method leverages the inherent rotational symmetry of interval-valued spherical fuzzy sets and the permutation symmetry among severity, occurrence, and detectability criteria, resulting in a transformation-invariant and unbiased risk assessment framework. Applying IVSF-FMEA to seven periodic failure (PF) modes in five-axis CNC water jet machining achieves a more precise prioritization of risks, leading to improved decision-making and resource allocation. The findings highlight improper fixturing of the workpiece (PF6) as the most critical failure mode, with the highest RPN value of −0.54, followed by mechanical vibrations (PF2) and tool wear and breakage (PF1). This indicates that ensuring proper fixturing stability is essential for maintaining machining accuracy and preventing defects. Comparative analysis with traditional FMEA demonstrates the superiority of the proposed fuzzy-based approach in handling subjective assessments and reducing ambiguity. The findings highlight improper fixturing, mechanical vibrations, and tool wear as the most critical failure modes, necessitating targeted risk mitigation strategies. This research contributes to advancing risk assessment methodologies in complex manufacturing environments. Full article

(This article belongs to the Special Issue Recent Developments on Fuzzy Sets Extensions)

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21 pages, 5416 KiB

Open AccessArticle

Optimized Multiple Regression Prediction Strategies with Applications

by Yi-Ming Zhao, Shu-Chuan Chu, Ali Riza Yildiz and Jeng-Shyang Pan

Symmetry 2025, 17(7), 1085; https://doi.org/10.3390/sym17071085 - 7 Jul 2025

As a classical statistical method, multiple regression is widely used for forecasting tasks in power, medicine, finance, and other fields. The rise of machine learning has led to the adoption of neural networks, particularly Long Short-Term Memory (LSTM) models, for handling complex forecasting [...] Read more.

As a classical statistical method, multiple regression is widely used for forecasting tasks in power, medicine, finance, and other fields. The rise of machine learning has led to the adoption of neural networks, particularly Long Short-Term Memory (LSTM) models, for handling complex forecasting problems, owing to their strong ability to capture temporal dependencies in sequential data. Nevertheless, the performance of LSTM models is highly sensitive to hyperparameter configuration. Traditional manual tuning methods suffer from inefficiency, excessive reliance on expert experience, and poor generalization. Aiming to address the challenges of complex hyperparameter spaces and the limitations of manual adjustment, an enhanced sparrow search algorithm (ISSA) with adaptive parameter configuration was developed for LSTM-based multivariate regression frameworks, where systematic optimization of hidden layer dimensionality, learning rate scheduling, and iterative training thresholds enhances its model generalization capability. In terms of SSA improvement, first, the population is initialized by the reverse learning strategy to increase the diversity of the population. Second, the mechanism for updating the positions of producer sparrows is improved, and different update formulas are selected based on the sizes of random numbers to avoid convergence to the origin and improve search flexibility. Then, the step factor is dynamically adjusted to improve the accuracy of the solution. To improve the algorithm’s global search capability and escape local optima, the sparrow search algorithm’s position update mechanism integrates Lévy flight for detection and early warning. Experimental evaluations using benchmark functions from the CEC2005 test set demonstrated that the ISSA outperforms PSO, the SSA, and other algorithms in optimization performance. Further validation with power load and real estate datasets revealed that the ISSA-LSTM model achieves superior prediction accuracy compared to existing approaches, achieving an RMSE of 83.102 and an

R^{2}

of 0.550 during electric load forecasting and an RMSE of 18.822 and an

R^{2}

of 0.522 during real estate price prediction. Future research will explore the integration of the ISSA with alternative neural architectures such as GRUs and Transformers to assess its flexibility and effectiveness across different sequence modeling paradigms. Full article

(This article belongs to the Section Computer)

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23 pages, 3268 KiB

Open AccessArticle

Symmetry-Informed Optimization and Verification of Loader Working Device Based on Improved Genetic Algorithm

by Zhikui Dong, Lingchao Meng, Ding Song, Zixian Wang, Peng Gao, Long Ma, Yongkuan Sun, Huibin Liu and Menglong Zhang

Symmetry 2025, 17(7), 1084; https://doi.org/10.3390/sym17071084 - 7 Jul 2025

The translation of motion lift, as an important performance metric of a reversing six-link loader working device, is influenced by multiple factors, such as the mechanical structure, system components, and operational experience. To ensure that the loader’s motion lift performance is optimized, this [...] Read more.

The translation of motion lift, as an important performance metric of a reversing six-link loader working device, is influenced by multiple factors, such as the mechanical structure, system components, and operational experience. To ensure that the loader’s motion lift performance is optimized, this paper takes the fork trajectory and the horizontal angle between the bucket cylinder and the ground as the main optimization objectives. Kinematic modeling and multi-objective optimization are conducted to reduce the influence of external factors on the motion lift process. Firstly, a parametric model of the reversing six-link mechanism is established based on its geometric and symmetric characteristics, and the expressions for the fork’s motion trajectory and the cylinder–ground angle are derived. Then, an optimization model is constructed with the aim of minimizing both the translational error during fork lifting and the horizontal angle of the bucket cylinder. An improved multi-objective genetic algorithm is employed for the global search and optimization. Inspired by the principle of symmetry, the algorithm incorporates a structured search strategy that enhances convergence efficiency and solution balance. A multi-criteria decision function is further applied to identify the optimal solution from the Pareto front. Finally, a real-vehicle experiment validates the optimization results. The findings confirm that the proposed method significantly improves the translational performance of the fork and effectively controls the horizontal angle of the cylinder while also enhancing the driver’s visibility and coordination of the entire system. These results provide a theoretical and engineering basis for the symmetry-informed multi-objective performance optimization of loader working devices. Full article

(This article belongs to the Section Engineering and Materials)

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19 pages, 1289 KiB

Open AccessArticle

Adaptive Control of Nonlinear Non-Minimum Phase Systems Using Actor–Critic Reinforcement Learning

by Monia Charfeddine, Khalil Jouili and Mongi Ben Moussa

Symmetry 2025, 17(7), 1083; https://doi.org/10.3390/sym17071083 - 7 Jul 2025

This study introduces a novel control strategy tailored to nonlinear systems with non-minimum phase (NMP) characteristics. The framework leverages reinforcement learning within a cascade control architecture that integrates an Actor–Critic structure. Controlling NMP systems poses significant challenges due to the inherent instability of [...] Read more.

This study introduces a novel control strategy tailored to nonlinear systems with non-minimum phase (NMP) characteristics. The framework leverages reinforcement learning within a cascade control architecture that integrates an Actor–Critic structure. Controlling NMP systems poses significant challenges due to the inherent instability of their internal dynamics, which hinders effective output tracking. To address this, the system is reformulated using the Byrnes–Isidori normal form, allowing the decoupling of the input–output pathway from the internal system behavior. The proposed control architecture consists of two nested loops: an inner loop that applies input–output feedback linearization to ensure accurate tracking performance, and an outer loop that constructs reference signals to stabilize the internal dynamics. A key innovation in this design lies in the incorporation of symmetry principles observed in both system behavior and control objectives. By identifying and utilizing these symmetrical structures, the learning algorithm can be guided toward more efficient and generalized policy solutions, enhancing robustness. Rather than relying on classical static optimization techniques, the method employs a learning-based strategy inspired by previous gradient-based approaches. In this setup, the Actor—modeled as a multilayer perceptron (MLP)—learns a time-varying control policy for generating intermediate reference signals, while the Critic evaluates the policy’s performance using Temporal Difference (TD) learning. The proposed methodology is validated through simulations on the well-known Inverted Pendulum system. The results demonstrate significant improvements in tracking accuracy, smoother control signals, and enhanced internal stability compared to conventional methods. These findings highlight the potential of Actor–Critic reinforcement learning, especially when symmetry is exploited, to enable intelligent and adaptive control of complex nonlinear systems. Full article

(This article belongs to the Section Engineering and Materials)

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30 pages, 918 KiB

Open AccessSystematic Review

Symmetric Therapeutic Frameworks and Ethical Dimensions in AI-Based Mental Health Chatbots (2020–2025): A Systematic Review of Design Patterns, Cultural Balance, and Structural Symmetry

by Ali Algumaei, Noorayisahbe Mohd Yaacob, Mohamed Doheir, Mohammed Nasser Al-Andoli and Mohammed Algumaie

Symmetry 2025, 17(7), 1082; https://doi.org/10.3390/sym17071082 - 7 Jul 2025

Artificial intelligence (AI)-powered mental health chatbots have evolved quickly as scalable means for psychological support, bringing novel solutions through natural language processing (NLP), mobile accessibility, and generative AI. This systematic literature review (SLR), following PRISMA 2020 guidelines, collates evidence from 25 published, peer-reviewed [...] Read more.

Artificial intelligence (AI)-powered mental health chatbots have evolved quickly as scalable means for psychological support, bringing novel solutions through natural language processing (NLP), mobile accessibility, and generative AI. This systematic literature review (SLR), following PRISMA 2020 guidelines, collates evidence from 25 published, peer-reviewed studies between 2020 and 2025 and reviews therapeutic techniques, cultural adaptation, technical design, system assessment, and ethics. Studies were extracted from seven academic databases, screened against specific inclusion criteria, and thematically analyzed. Cognitive behavioral therapy (CBT) was the most common therapeutic model, featured in 15 systems, frequently being used jointly with journaling, mindfulness, and behavioral activation, followed by emotion-based approaches, which were featured in seven systems. Innovative techniques like GPT-based emotional processing, multimodal interaction (e.g., AR/VR), and LSTM-SVM classification models (greater than 94% accuracy) showed increased conversation flexibility but missed long-term clinical validation. Cultural adaptability was varied, and effective localization was seen in systems like XiaoE, okBot, and Luda Lee, while Western-oriented systems had restricted contextual adaptability. Accessibility and inclusivity are still major challenges, especially within low-resource settings, since digital literacy, support for multiple languages, and infrastructure deficits are still challenges. Ethical aspects—data privacy, explainability, and crisis plans—were under-evidenced for most deployments. This review is different from previous ones since it focuses on cultural adaptability, ethics, and hybrid public health incorporation and proposes a comprehensive approach for deploying AI mental health chatbots safely, effectively, and inclusively. Central to this review, symmetry is emphasized as a fundamental idea incorporated into frameworks for cultural adaptation, decision-making processes, and therapeutic structures. In particular, symmetry ensures equal cultural responsiveness, balanced user–chatbot interactions, and ethically aligned AI systems, all of which enhance the efficacy and dependability of mental health services. Recognizing these benefits, the review further underscores the necessity for more rigorous academic research into the development, deployment, and evaluation of mental health chatbots and apps, particularly to address cultural sensitivity, ethical accountability, and long-term clinical outcomes. Full article

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21 pages, 661 KiB

Open AccessArticle

Semi-Analytical Solutions of the Rayleigh Oscillator Using Laplace–Adomian Decomposition and Homotopy Perturbation Methods: Insights into Symmetric and Asymmetric Dynamics

by Emad K. Jaradat, Omar Alomari, Audai A. Al-Zgool and Omar K. Jaradat

Symmetry 2025, 17(7), 1081; https://doi.org/10.3390/sym17071081 - 7 Jul 2025

This study investigates the solution structure of the nonlinear Rayleigh oscillator equation through two widely used semi-analytical techniques: the Laplace–Adomian Decomposition Method (LADM) and the Homotopy Perturbation Method (HPM). The Rayleigh oscillator exhibits inherent asymmetry in its nonlinear damping term, which disrupts the [...] Read more.

This study investigates the solution structure of the nonlinear Rayleigh oscillator equation through two widely used semi-analytical techniques: the Laplace–Adomian Decomposition Method (LADM) and the Homotopy Perturbation Method (HPM). The Rayleigh oscillator exhibits inherent asymmetry in its nonlinear damping term, which disrupts the time-reversal symmetry present in linear oscillatory systems. Applying the LADM and HPM, we derive approximate solutions for the Rayleigh oscillator. Due to the absence of exact analytical solutions in the literature, these approximations are benchmarked against high-precision numerical results obtained using Mathematica’s NDSolve function. We perform a detailed error analysis across different damping parameter values ε and time intervals. Our results reveal how the asymmetric damping influences the accuracy and convergence behavior of each method. This study highlights the role of nonlinear asymmetry in shaping the solution dynamics and provides insight into the suitability of the LADM and HPM under varying conditions. Full article

(This article belongs to the Section Physics)

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12 pages, 263 KiB

Open AccessArticle

Generalized Statistical Convergence in Sequence Spaces: A Modulus Function Perspective

by Gülcan Atıcı Turan

Symmetry 2025, 17(7), 1080; https://doi.org/10.3390/sym17071080 - 7 Jul 2025

This study introduces a flexible framework for analyzing how sequences of numbers approach a limit, even when traditional convergence criteria fail. By incorporating modulus function mathematical tools that quantify growth rates, this research extends the concept of statistical convergence to handle sequences with [...] Read more.

This study introduces a flexible framework for analyzing how sequences of numbers approach a limit, even when traditional convergence criteria fail. By incorporating modulus function mathematical tools that quantify growth rates, this research extends the concept of statistical convergence to handle sequences with irregular or sparse behavior. Key results establish connections between this generalized convergence theory and related properties, like boundedness, providing a unified approach to understanding sequence dynamics. The findings enhance our ability to model and analyze complex data patterns in mathematics and beyond. Full article

(This article belongs to the Section Mathematics)

16 pages, 256 KiB

Open AccessArticle

On t-Dimensional Gersenne Sequences and Their Symmetry Properties

by Paula M. M. C. Catarino, Douglas C. Santos and Eudes A. Costa

Symmetry 2025, 17(7), 1079; https://doi.org/10.3390/sym17071079 - 7 Jul 2025

In this paper, we introduce a new class of sequences of integers called t-dimensional Gersenne sequences, with

2 \leq t < \infty

, which extends the classical Mersenne-type sequences into higher dimensions by incorporating complex and hyper-complex structures. One of our main [...] Read more.

In this paper, we introduce a new class of sequences of integers called t-dimensional Gersenne sequences, with

2 \leq t < \infty

, which extends the classical Mersenne-type sequences into higher dimensions by incorporating complex and hyper-complex structures. One of our main goals is to establish analogous or symmetrical properties for the t-dimensional Gersenne sequences that are similar to those we know for the classical Gersenne sequence. We investigate some algebraic properties of this new sequences, and the Binet formula is given so as to allow us to study several identities, such as the Tagiuri–Vajda identity, among others. Furthermore, some results concerning the partial sums involving terms of these new sequences are presented. Full article

(This article belongs to the Section Mathematics)

28 pages, 7407 KiB

Open AccessArticle

WaveAtten: A Symmetry-Aware Sparse-Attention Framework for Non-Stationary Vibration Signal Processing

by Xingyu Chen and Monan Wang

Symmetry 2025, 17(7), 1078; https://doi.org/10.3390/sym17071078 - 7 Jul 2025

This study addresses the long-standing difficulty of predicting the remaining useful life (RUL) of rolling bearings from highly non-stationary vibration signals by proposing WaveAtten, a symmetry-aware deep learning framework. First, mirror-symmetric and bi-orthogonal Daubechies wavelet filters are applied to decompose each raw signal [...] Read more.

This study addresses the long-standing difficulty of predicting the remaining useful life (RUL) of rolling bearings from highly non-stationary vibration signals by proposing WaveAtten, a symmetry-aware deep learning framework. First, mirror-symmetric and bi-orthogonal Daubechies wavelet filters are applied to decompose each raw signal into multi-scale approximation/detail pairs, explicitly preserving the left–right symmetry that characterizes periodic mechanical responses while isolating asymmetric transient faults. Next, a bidirectional sparse-attention module reinforces this structural symmetry by selecting query–key pairs in a forward/backward balanced fashion, allowing the network to weight homologous spectral patterns and suppress non-symmetric noise. Finally, the symmetry-enhanced features—augmented with temperature and other auxiliary sensor data—are fed into a long short-term memory (LSTM) network that models the symmetric progression of degradation over time. Experiments on the IEEE PHM2012 bearing dataset showed that WaveAtten achieved superior mean squared error, mean absolute error, and R² scores compared with both classical signal-processing pipelines and state-of-the-art deep models, while ablation revealed a 6–8% performance drop when the symmetry-oriented components were removed. By systematically exploiting the intrinsic symmetry of vibration phenomena, WaveAtten offers a robust and efficient route to RUL prediction, paving the way for intelligent, condition-based maintenance of industrial machinery. Full article

(This article belongs to the Section Computer)

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16 pages, 632 KiB

Open AccessArticle

Structured Stability of Hybrid Stochastic Differential Equations with Superlinear Coefficients and Infinite Memory

by Chunhui Mei and Mingxuan Shen

Symmetry 2025, 17(7), 1077; https://doi.org/10.3390/sym17071077 - 7 Jul 2025

The stability of hybrid stochastic differential equations (SDEs in short) depends on multiple factors, such as the structures and parameters of subsystems, switching rules, delay, etc. Regarding stability analysis for hybrid stochastic systems incorporating subsystems with diverse structures, existing research results require the [...] Read more.

The stability of hybrid stochastic differential equations (SDEs in short) depends on multiple factors, such as the structures and parameters of subsystems, switching rules, delay, etc. Regarding stability analysis for hybrid stochastic systems incorporating subsystems with diverse structures, existing research results require the system to possess either Markovian properties or finite memory characteristics. However, the stability problem remains unresolved for hybrid stochastic differential equations with infinite memory (hybrid IMSDEs in short), as no systematic theoretical framework currently exists for such systems. To bridge this gap, this paper develops a rigorous stability analysis for a class of hybrid IMSDEs by introducing a suitably chosen phase space and leveraging the theory of fading memory spaces. We establish sufficient conditions for exponential stability, extending the existing results to systems with unbounded memory effects. Finally, a numerical example is provided to illustrate the effectiveness of the proposed criteria. Full article

(This article belongs to the Section Mathematics)

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20 pages, 407 KiB

Open AccessArticle

Leveraging Asymmetric Adaptation with Dynamic Sparse LoRA for Enhanced Nuance in LLM-Based Offensive Language Detection

by Yanzhe Wang, Bingquan Chen and Jingchao Sun

Symmetry 2025, 17(7), 1076; https://doi.org/10.3390/sym17071076 - 7 Jul 2025

The challenge of detecting nuanced, context-dependent offensive language highlights the need for Large Language Model (LLM) adaptation strategies that can effectively address inherent data and task asymmetries. Standard Parameter-Efficient Finetuning (PEFT) methods like Low-Rank Adaptation (LoRA), while efficient, often employ a more uniform, [...] Read more.

The challenge of detecting nuanced, context-dependent offensive language highlights the need for Large Language Model (LLM) adaptation strategies that can effectively address inherent data and task asymmetries. Standard Parameter-Efficient Finetuning (PEFT) methods like Low-Rank Adaptation (LoRA), while efficient, often employ a more uniform, or symmetric, update mechanism that can be suboptimal for capturing such linguistic subtleties. In this paper, we propose Dynamic Sparse LoRA (DS-LoRA), a novel technique that leverages asymmetric adaptation to enhance LLM finetuning for nuanced offensive language detection. DS-LoRA achieves this by (1) incorporating input-dependent gating mechanisms, enabling the asymmetric modulation of LoRA module contributions based on instance-specific characteristics, and (2) promoting asymmetric sparsity within LoRA update matrices via L1 regularization. This dual asymmetric strategy empowers the model to selectively engage and refine only the most pertinent parameters for a given input, fostering a more parsimonious and contextually aware adaptation. Extensive experiments on benchmark datasets demonstrate DS-LoRA’s significant overperformance over standard LoRA and other strong baselines, particularly in identifying subtle and contextually ambiguous offensive content, underscoring the benefits of its asymmetric adaptive capabilities. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry Studies in Data Mining & Machine Learning of Large Language Models)

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