Designing feedback controllers for high-dimensional optimal control problems is a central challenge in optimal control, with applications ranging from large-scale engineered systems to networked and distributed dynamics. While optimal feedback laws are characterized by the associated optimal value function, computing this function in high-dimensional state spaces is notoriously difficult due to the curse of dimensionality. This talk shows how an inherent decaying sensitivity property of optimal control problems enables efficient representations of high-dimensional optimal value functions using deep neural networks. We first demonstrate that neural networks can approximate separable functions with a number of neurons growing only polynomially in the state dimension, thereby mitigating the curse of dimensionality for this class of functions. Building on this observation, we explain how decaying sensitivity gives rise to approximately separable structures in optimal value functions, allowing them to be represented as sums of localized contributions that reflect the diminishing influence of distant state variables. We discuss how such sensitivity decay arises under uniform stabilizability and detectability assumptions in linear-quadratic optimal control problems and present initial results toward extending this framework to nonlinear dynamics. The theoretical insights are supported by numerical experiments demonstrating the effectiveness of deep neural networks for approximating high-dimensional optimal value functions.