We present PSF-NN, a new approach for time series forecasting. It combines prediction based on sequence similarity with neural networks. PSF-NN first generates predictions using the PSF algorithm that are then refined by the neural network component, which also utilizes additional features. We evaluate the performance of PSF-NN using a time series of hourly electricity demands for the state of New South Wales in Australia for three years. The task is to predict an interval of future values simultaneously, i.e. the 24 demands for the next day, instead of predicting just a single future demand. The results showed that the combined PSF-NN approach provides accurate predictions, outperforming the original PSF algorithm and a number of baselines.