Some Results on Neural Network Stability, Consistency, and Convergence: Insights into Non-IID Data, High-Dimensional Settings, and Physics-Informed Neural Networks

This paper addresses critical challenges in machine learning, particularly the stability, consistency, and convergence of neural networks under non-IID data, distribution shifts, and high-dimensional settings. We provide new theoretical results on uniform stability for neural networks with dynamic learning rates in non-convex settings. Further, we establish consistency bounds for federated learning models in non-Euclidean spaces, accounting for distribution shifts and curvature effects. For Physics-Informed Neural Networks (PINNs), we derive stability, consistency, and convergence guarantees for solving Partial Differential Equations (PDEs) in noisy environments. These results fill significant gaps in understanding model behavior in complex, non-ideal conditions, paving the way for more robust and reliable machine learning applications.