An approximate dynamic programming (ADP) based near optimal boundary control of distributed parameter systems (DPS) governed by uncertain two dimensional (2D) Burgers equation under Neumann boundary condition is introduced. First, Hamilton-Jacobi-Bellman (HJB) equation is formulated without any model reduction. Next, optimal boundary control policy is derived in terms of value functional which is obtained as the solution to the HJB equation. Subsequently, a novel identifier is developed to estimate the unknown nonlinearity in the partial differential equation (PDE) dynamics. The suboptimal control policy is obtained by forward-in-time approximation of the value functional using a neural network (NN) based online approximator and the identified dynamics. Adaptive weight tuning laws are proposed for online learning of the value functional and identifier. Local ultimate boundedness (UB) of the closed-loop system is verified by using Lyapunov theory.