We apply an artificial neural network (ANN) of 20 hidden layers and backpropagation regression to the forecast of experimental time series from a Kerr lens mode locking (KLM) Ti:sapphire laser and a Nd:vanadate with modulation losses. In both cases the neural network is able to predict up to 10 steps ahead. In the Ti:sapphire laser the prediction in pulse amplitude is accurate even when the pulse is an extreme event. In the Nd:vanadate laser we forecast both pulse amplitude and pulse-to-pulse time separation. In both cases the prediction goes beyond the Lyapunov prediction horizon.