This paper explores the utilization of machine learning techniques to develop an approximate input-output linearizable neural network model aimed at improving disturbance attenuation. The incorporation of a learning mechanism allows for the synthesis of control laws that effectively address disturbance attenuation by imposing a specific parameterization and an L_2 gain constraint during the learning process. The constraint is subsequently relaxed into a set of diagonally dominant (DD) matrix constraints. This relaxation leads to a series of linear constraints that can be seamlessly incorporated into the loss criterion. Therefore, a log-sum-exp (LSE) function—a smoothed version of the max function— of these linearized constraints is added to the loss criterion which results in an unconstrained problem amenable to training via back-propagation. The proposed methodology is applied to two variants of nonlinear perturbed pendulum systems. The results emphasize the effectiveness of the model, both on its own and as a framework for developing control laws for disturbance attenuation.