In order to improve prediction and forecast accuracy for the sunspot number, two process neural network(PNN) training algorithms of fixed extreme learning machine PNN(FELM-PNN) and incremental extreme learning machine PNN(IELM-PNN) are proposed. The FELM-PNN has fixed numbers of hidden layer nodes, and uses singular value decomposition(SVD) to compute Moore-Penrose generalized inverse of the hidden layer output matrix. The hidden layer output weights are solved by using the least squares method. For the IELM-PNN, the hidden layer nodes are added to the model one by one. The output weights for the added node are computed by according to the hidden layer output matrix and the network output error. The effectiveness of the two proposed methods is verified by Henon time series prediction. The two proposed methods are applied to the 24th cycle sunspot smoothed monthly mean Mid-and-long forecasting problem. The experimental results show that the prediction accuracy of the two methods increased at certain degree, and the training convergence of the IELM-PNN is better than that of the FELM-PNN.