A nonlinear chaos control method is presented which is based on a feedforward neural network (NN). It represents a nonlinear extension of the linear map based version of the control method by Ott, Grebogi, and Yorke (OGY). We compare the results of the nonlinear NN method with the linear map based version of the OGY method and another linear control method by Pyragas, which uses a continuous time delayed feedback algorithm. All three methods are shown by cross correlation to stabilize the same UFO in the same chaotic attractor. The nonlinear NN method turns out to be the fastest method to control chaos followed by the linear map based method and the linear Pyragas algorithm. Using the NN control, we also demonstrate the suppression of interactive Gaussian noise in a periodic attractor. All numerical simulations are performed with the three- and four-variable model (Montanator) of Gyorgyi and Field.