The intermittency of solar energy resources has brought a big challenge for the optimization and planning of a future smart grid. To reduce the intermittency, an accurate prediction of photovoltaic (PV) power generation is very important. Therefore, this paper proposes a new forecasting method based on the recurrent neural network (RNN). At first, the entire solar power time series data is divided into inter-day data and intra-day data. Then, we apply RNN to discover the nonlinear features and invariant structures exhibited in the adjacent days and intra-day data. After that, a new point prediction model is proposed, only by taking the previous PV power data as input without weather information. The forecasting horizons are set from 15 to 90 minutes. The proposed forecasting method is tested by using real solar power in Flanders, Belgium. The classical persistence method (Persistence), back propagation neural network (BPNN), radial basis function (RBF) neural network and support vector machine (SVM), and long short-term memory (LSTM) networks are adopted as benchmarks. Extensive results show that the proposed forecasting method exhibits a good forecasting quality on very short-term forecasting, which demonstrates the feasibility and effectiveness of the proposed forecasting model.