Towards Federated Robust Approximation of Nonlinear Systems with Differential Privacy Guarantee

Nonlinear systems, characterized by their complex and often unpredictable dynamics, are essential in various scientific and engineering applications. However, accurately modeling these systems remains challenging due to their nonlinearity, high-dimensional interactions, and the privacy concerns inherent in data-sensitive domains. Existing federated learning approaches struggle to model such complex behaviors, particularly due to their inability to capture high-dimensional interactions and their failure to maintain privacy while ensuring robust model performance. This paper presents a novel federated learning framework for the robust approximation of nonlinear systems, addressing these challenges by integrating differential privacy to protect sensitive data without compromising model utility. The proposed framework enables decentralized training across multiple clients, ensuring privacy through differential privacy mechanisms that mitigate risks of information leakage via gradient updates. Advanced neural network architectures are employed to effectively approximate nonlinear dynamics, with stability and scalability ensured by rigorous theoretical analysis. We compare our approach with both centralized and decentralized federated models, highlighting the advantages of our framework, particularly in terms of privacy preservation. Comprehensive experiments on benchmark datasets, such as the Lorenz system and real-world climate data, demonstrate that our federated model achieves comparable accuracy to centralized approaches while offering strong privacy guarantees. The system efficiently handles data heterogeneity and dynamic nonlinear behavior, scaling well with both the number of clients and model complexity. These findings demonstrate a pathway for the secure and scalable deployment of machine learning models in nonlinear system modeling, effectively balancing accuracy, privacy, and computational performance.