The uncertain nonlinear systems can be modeled with fuzzy equations by incorporating the fuzzy set theory. In this paper, the fuzzy equations are applied as the models for the uncertain nonlinear systems. The nonlinear modeling process is to find the coefficients of the fuzzy equations. We use the neural networks to approximate the coefficients of the fuzzy equations. The approximation theory for crisp models is extended into the fuzzy equation model. The upper bounds of the modeling errors are estimated. Numerical experiments along with comparisons demonstrate the excellent behavior of the proposed method.

Many uncertain nonlinear systems can be modeled by the linear-in-parameter model, and the parameters are uncertain in the sense of fuzzy numbers. Fuzzy equations can be used to model these nonlinear systems. The solutions of the fuzzy equations are t

With the fuzzy set theory, the uncertainty of nonlinear systems can be modeled using fuzzy differential equations. The solutions of these equations are the model output, but they are very difficult to obtain. In this paper, we first transform fuzzy differential equations into four ordinary differential equations. Then, we construct neural models with the structure of these ordinary differential equations. Theory analysis and simulation results show that these new models are effective for modeling uncertain nonlinear systems.

Connected and autonomous vehicles, along with the expanding Internet of Vehicles (IoV), are increasingly exposed to complex and evolving cyberattacks. Consequently, Intrusion Detection Systems (IDS) have become a vital component of modern vehicular cybersecurity. Federated Learning (FL) enables multiple vehicles to collaboratively train detection models while keeping their local data private, providing a decentralized alternative to traditional centralized learning. Despite these advantages, FL-based IDS frameworks remain vulnerable to attacks. To address this vulnerability, we propose an explainable federated intrusion detection framework that enhances both the security and interpretability of IDS in connected vehicles. The framework employs a Deep Neural Network (DNN) within a federated setting and integrates explainability through the Shapley Additive Explanations (SHAP) method. This Explainable Artificial Intelligence (XAI) component identifies the most influential network features contributing to detection decisions and assists in recognizing anomalies arising from malicious or corrupted clients. Experimental validation on the CICEVSE2024 and CICIoV2024 vehicular datasets demonstrates that the proposed system achieves high detection accuracy. Moreover, the XAI module improves transparency and enables analysts to verify and understand the model’s decision-making process. Compared with both centralized IDS models and conventional federated approaches without explainability, the proposed system delivers comparable performance, stronger resilience to attacks, and significantly enhanced interpretability. Overall, this work demonstrates that integrating FL with XAI provides a privacy-preserving and trustworthy approach for intrusion detection in connected vehicular networks.