Multi-Time Evolution Models in Deep Learning Dynamics

This paper extends the multi-time optimal control theory to deep learning frameworks, developing a mathematical foundation for understanding neural network training as a process evolving across multiple time scales. We formulate neural network optimization as a multi-time evolution problem where different components of the network evolve at different rates. This perspective yields two novel theorems: the first establishes conditions for path-independent convergence in multi-time gradient descent, while the second provides a framework for analyzing the interplay between feature extraction and classification layers in deep networks. Experimental validation on synthetic data demonstrates the practical implications of our theoretical framework, showing improved convergence properties and enabling adaptive learning rate scheduling based on multi-time principles. Our approach opens new avenues for understanding the dynamics of deep learning systems and suggests practical improvements to optimization algorithms.