Deep neural networks have evolved from heuristic pattern recognition tools into mathematically grounded systems whose theoretical understanding spans approximation theory, geometry, dynamical systems, and optimal control. This article develops a unified theoretical framework that interprets classical and modern neural architectures through the lenses of geometric separability, statistical learning, and dynamical systems theory. Drawing exclusively from foundational and contemporary contributions in machine learning, control theory, and high dimensional geometry, we provide a comprehensive synthesis of the evolution from perceptrons and support vector machines to residual networks and neural ordinary differential equations.

The study begins by examining early geometric formulations of classification, including linear threshold units and shattering properties, and situates these within modern capacity analysis. We then analyze multilayer feedforward networks in terms of universal approximation and storage capacity, addressing both width and depth considerations. Special attention is devoted to the power of depth and residual connections, emphasizing the reinterpretation of deep networks as discretized dynamical systems.

A central contribution of this work is an integrative exploration of neural ordinary differential equations and their mean field optimal control formulations. We explain how continuous depth models unify discrete architectures and reveal new insights into controllability, interpolation, and long time behavior. The mean field perspective connects parameter learning with population level dynamics, clarifying the role of measure theoretic interpolation and turnpike phenomena in training trajectories.

We further investigate geometric and topological perspectives, including manifold learning and invertible architectures, demonstrating how controllability conditions determine expressive power in neural ODE frameworks. Stability considerations and identity preserving structures are analyzed to explain empirical success in deep training regimes. Optimization landscape properties, stochastic gradient methods, and automatic differentiation are contextualized within this broader dynamical view.

Finally, the article synthesizes classical statistical learning theory with modern transformer based dynamics and cluster formation in self attention systems, positioning deep learning as a theory of measure evolution under learned flows. By integrating insights from approximation theory, control, geometry, and statistical learning, this work provides a publication ready theoretical narrative that clarifies both the mathematical foundations and the emerging research directions of deep learning as a dynamical systems discipline.