In statistical mechanics, an approximation technique for complex, inter- active systems called Mean Field Theory (MFT) is now broadly applied to explain why deep learning works. In particular, MFT highlights the dynamical system similarities between a deep network's parameters and interacting particles.

An optimal control problem is defined by the dynamics function f and a cost functional over the entire trajectory x and u: J(x,u)=∫∞0L(x(t),u(t),t)dt. The term functional indicates that this is a function mapping a function to a real number.

To sum up, both problems optimal control and machine learning state a optimization problem in one hand optimal control's goal is to find an optimal policy to control a given process (if exist) where the model exist or could be find in anyway (perhaps modeling technique of control could be applied) while machine ...

General method The optimal control can be derived using Pontryagin's maximum principle (a necessary condition also known as Pontryagin's minimum principle or simply Pontryagin's principle), or by solving the Hamilton–Jacobi–Bellman equation (a sufficient condition).