Multi-time dynamics in neural network optimization: A unified framework bridging game theory and optimal control

We present a novel mathematical framework for neural network optimization based on multi-time dynamics, unifying game-theoretic and optimal control perspectives. Drawing from Udrişte’s geometric multi-time evolution theory, we model each network component as an autonomous agent evolving along its intrinsic time scale, capturing inter-layer dependencies through cross-temporal derivatives. This formulation naturally accommodates the empirically observed heterogeneity in layer-wise learning dynamics, where early convolutional layers, intermediate feature extractors, and final classification layers converge at fundamentally different rates. We establish the existence of Multi-Time Nash Equilibria via Kakutani’s fixed-point theorem under convexity and strong concavity conditions, and prove exponential convergence with explicit geometric rates that depend on the spectral properties of the inter-layer interaction matrix. The resulting algorithm, Multi-Time Nash Learning (MTNL), incorporates strategic interaction terms derived from mixed Hessians and achieves 40% faster convergence and up to 4.5 percentage point accuracy improvement over standard optimizers on benchmark datasets. Extensive experiments on CIFAR-10, Fashion-MNIST, and ImageNet subsets with ResNet-50 and VGG-16 architectures validate our theoretical predictions. Ablation studies confirm that both the multi-time structure and the strategic interaction terms contribute synergistically to the observed gains. This work provides rigorous mathematical foundations for understanding optimization landscapes in deep learning through the lens of differential geometry and multi-agent systems, opening new directions for adaptive learning rate scheduling, federated learning, and distributed neural network training.