The boundary control problem is a non-convex optimization and control problem in many scientific domains, including fluid mechanics, structural engineering, and heat transfer optimization. The aim is to find the optimal values for the domain boundaries such that the enclosed domain adhering to the governing equations attains the desired state values. Traditionally, non-linear optimization methods, such as the Interior-Point method (IPM), are used to solve such problems. This project explores the possibilities of using deep learning and reinforcement learning to solve boundary control problems. We adhere to the framework of iterative optimization strategies, employing a spatial neural network to construct well-informed initial guesses, and a spatio-temporal neural network learns the iterative optimization algorithm using policy gradients. Synthetic data, generated from the problems formulated in the literature, is used for training, testing and validation. The numerical experiments indicate that the proposed method can rival the speed and accuracy of existing solvers. In our preliminary results, the network attains costs lower than IPOPT, a state-of-the-art non-linear IPM, in 51\% cases. The overall number of floating point operations in the proposed method is similar to that of IPOPT. Additionally, the informed initial guess method and the learned momentum-like behaviour in the optimizer method are incorporated to avoid convergence to local minima.