The solvability of the nonlinear output regulation problem relies on the existence of a feedforward function defined by a set of mixed nonlinear partial and algebraic equations called regulator equations. Previous approaches to solving the output regulation problem call for the solution of the regulator equations. However, solving the regulator equations is difficult due to the nonlinearity and complexity. This paper proposes a novel approximation approach to solving the output regulation problem by directly approximating the feedforward function using a class of artificial neural networks. Further, a control configuration is developed that allows the reduction of the tracking error by the on-line adjustment of the parameters of the neural networks. The major advantages of our proposed approach over the previous approaches include (1) the precise knowledge of the plant is not needed, and (2) computation complexity is significantly reduced.