Editorial for the Special Issue “Modeling and Optimization of Complex Engineering Systems Under Uncertainties”

1. Introduction to This Special Issue

The contemporary engineering frontier is defined by a paradigm shift toward hyper-integrated systems operating within increasingly volatile environments, ranging from precision micro-electronics to critical large-scale infrastructure and sustainable energy ecosystems [1,2]. A pivotal challenge in this domain is the ubiquity of uncertainty—spanning inherent physical stochasticity to epistemic modeling deficits—which poses severe risks to structural integrity and operational continuity [3]. As traditional deterministic approaches struggle to cope with these complexities, often leading to over-conservative designs or catastrophic vulnerabilities, the development of robust modeling and optimization strategies has become paramount [4]. This Special Issue underscores the critical need to transition from static safety margins to dynamic uncertainty management, thereby guaranteeing the long-term resilience, cost-effectiveness, and sustainability of complex engineering systems facing real-world unpredictability.

This Special Issue, titled “Modeling and Optimization of Complex Engineering Systems under Uncertainties”, was curated to address this critical gap by assembling a collection of high-quality research that bridges advanced mathematical theories with practical engineering applications. The primary objective of this volume is to explore novel methodologies that can quantify, manage, and optimize systems despite the presence of stochastic variables, material degradation, and environmental fluctuations. The contributions selected for this issue demonstrate a convergence of disciplines, merging reliability-based design optimization, scientific machine learning, swarm intelligence, and rigorous mechanical analysis to solve problems that were previously intractable. By integrating probabilistic methods with deep learning and heuristic algorithms, the researchers featured herein provide a roadmap for the next generation of resilient engineering systems. This editorial synthesizes the significant findings of the eight published papers, categorizing them into design strategies, control methodologies, and practical applications, thereby highlighting the multidisciplinary nature of the solutions required to navigate the complexities of modern engineering under uncertainty.

2. Designs Proposed in This Special Issue

The establishment of robust design frameworks is the foundational step in mitigating the risks associated with engineering uncertainties [5,6,7,8]. Two significant contributions to this Special Issue address the fundamental challenge of design optimization and threshold determination under conditions where parameters are not fixed but fluctuate within probabilistic or interval bounds.

The work by Yang et al. (Contribution 1) introduces a sophisticated Reliability-Based Design Optimization strategy capable of handling hybrid uncertainties. Their research confronts the computational inefficiency often plaguing traditional double-loop reliability methods by constructing a decoupled framework that integrates an improved particle swarm optimization algorithm with a simulated annealing algorithm. This hybrid approach allows for the efficient identification of the most probable point of failure without the prohibitive computational cost usually associated with high-dimensional nonlinear problems. By validating their method on complex mechanical components such as gear reducers and composite cylinders, they demonstrate that decoupling the reliability analysis from the optimization loop can significantly enhance both solution speed and design accuracy. Complementing this algorithmic approach to design, the study by Hao et al. (Contribution 2) focuses on the material-to-mechanical analysis required to establish safety thresholds for critical infrastructure. Their investigation into the GINA gaskets used in immersed tunnels provides a rigorous method for determining warning thresholds for deformation. By analyzing the hyperelastic constitutive models of rubber materials and considering the uneven settlement patterns of tunnel elements, they derive a graded warning mechanism. This design-centric research connects the material properties of the gasket, specifically its hardness and cross-sectional shape, to the macroscopic mechanical behavior of the tunnel joint under shear and bending loads.

These two papers collectively illustrate that effective design under uncertainty requires a dual approach that encompasses both the algorithmic efficiency of optimization solvers and the rigorous mechanical definition of safety limits for physical components.

3. Control Methods Discussed in This Special Issue

Beyond the initial design phase, the operational phase of complex systems requires advanced control methods and algorithmic tools to manage uncertainties that arise during real-time performance. The contributions in this section highlight the growing role of artificial intelligence and deep learning in creating control mechanisms that are robust to noise and data imperfections. Erbet et al. (Contribution 3) propose a comprehensive methodology for uncertainty-aware Scientific Machine Learning. Their work addresses a critical gap in the application of neural networks to physical systems by integrating Bayesian inference via Markov Chain Monte Carlo methods with the training of soft sensors. By rigorously quantifying the epistemic and aleatory uncertainties in the model identification process, they provide a framework that ensures machine learning models remain reliable even when trained on synthetic or limited data, as demonstrated in their polymerization reactor case study. In a similar vein of enhancing algorithmic performance, Yang et al. (Contribution 4) presents a method for the parameter optimization of analog circuits. Their approach couples a bidirectional Long Short-Term Memory network with an enhanced whale optimization algorithm. This combination allows for the precise tuning of circuit parameters by leveraging the sequence processing capabilities of the neural network to model complex dependencies, while the swarm intelligence algorithm efficiently navigates the hyperparameter space. This method proves particularly effective in handling the nonlinearities inherent in analog circuit design. Furthermore, the work by Ander et al. (Contribution 5) addresses the specific control challenge of simultaneous object localization and classification in robotic systems. They introduce a novel loss function within a custom training loop for Convolutional Neural Networks. By balancing the cross-entropy loss for classification with the mean squared error for regression, their method improves the ability of robotic vision systems to accurately locate objects in unstructured environments. This contribution is vital for the development of collaborative robots that must operate safely alongside humans in manufacturing settings where visual data is often noisy or ambiguous. Together, these papers underscore the necessity of embedding uncertainty quantification and robust optimization directly into the control algorithms that govern modern engineering systems [9,10,11].

4. Applications Discussed in This Special Issue

The theoretical models and control strategies discussed above find their ultimate validation when applied to specific, high-stakes engineering problems ranging from renewable energy to civil infrastructure and anomaly detection. The Special Issue features several papers that demonstrate the practical utility of these advanced methods in diagnosing faults and ensuring operational continuity. Olivier and Jung (Contribution 6) tackle the challenge of structural health monitoring in laminated composite plates, materials that are critical to the aerospace and automotive industries. They propose an unsupervised anomaly detection framework that combines a self-attention autoencoder with a Gaussian mixture model. This architecture is designed to learn the latent representations of healthy Lamb wave signals and identify damages as statistical outliers. By utilizing a generative adversarial network to augment their training data, they successfully address the common issue of data scarcity in damage scenarios, proving that deep learning can effectively detect delamination and cracks without extensive labeled datasets. In the renewable energy sector, Gao et al. (Contribution 7) investigate the reliability of wind turbine gears under multistage random loadings. Their research develops a residual strength degradation model that accounts for the interaction between loads and the sequence of loading events. By integrating the rain flow counting method with the Goodman correction, they map the seasonal variations in wind load to the fatigue life of the gear, providing a dynamic reliability assessment that is crucial for the preventive maintenance of wind farms. Finally, Eric et al. (Contribution 8) applies swarm intelligence to the construction industry, a sector rife with uncertainties related to human logistics and safety. Their study utilizes an Artificial Bee Colony algorithm to optimize the movement paths of workers on construction sites to maintain safe physical distancing. This application demonstrates that mathematical optimization tools can be effectively deployed to solve logistical challenges in real time, ensuring worker safety during complex operations such as those necessitated by the COVID-19 pandemic. These diverse applications highlight the versatility of the proposed mathematical frameworks, showing that whether the system is a composite plate, a wind turbine gearbox, or a construction crew, the core principles of modeling uncertainty remain applicable and effective.

5. Conclusions

The collection of articles presented in this Special Issue “Modeling and Optimization of Complex Engineering Systems under Uncertainties” offers a comprehensive overview of the current state of the art in reliability engineering and applied mathematics. The research highlights a fundamental transition in engineering design and control, moving away from static safety factors toward dynamic, probabilistic, and data-driven frameworks. We have observed that hybrid optimization algorithms, such as the coupled Simulated Annealing and Particle Swarm Optimization, offer superior efficiency in navigating high-dimensional design spaces. We have seen that Scientific Machine Learning can be made robust to uncertainty through Bayesian inference, ensuring that soft sensors and control models remain trustworthy in critical applications. Furthermore, the successful application of these methods to diverse fields, including tunnel engineering, wind energy, structural health monitoring, and construction safety, confirms the universal relevance of uncertainty quantification. However, challenges remain, particularly in the computational cost of uncertainty propagation for large-scale systems and the interpretability of deep learning models in safety-critical contexts. Future research must continue to refine these methodologies, perhaps by exploring physics-informed neural networks that enforce physical laws to reduce the data requirements for uncertainty quantification. We hope that the methodologies and findings presented in this volume will serve as a foundation for future inquiries and inspire researchers to further explore the intricate dynamics of complex engineering systems. We express our sincere gratitude to all the authors for their valuable contributions and to the reviewers for their rigorous assessment, which ensured the high scientific quality of this Special Issue.