Approximate solutions to the Witsenhausen counterexample are derived by constraining the unknown control functions to take on fixed structures containing "free" parameters to be optimized. Such structures are given by "nonlinear approximating networks," i.e., linear combinations of parametrized basis functions that benefit by density properties in normed linear spaces. This reduces the original functional problem to a nonlinear programming one which is solved via stochastic approximation. The method yields lower values of the costs than the ones achieved so far in the literature, and, most of all, provides rather a complete overview of the shapes of the optimal control functions when the two parameters that characterize the Witsenhausen counterexample vary. One-hidden-layer neural networks are chosen as approximating networks.