While adaptive filtering has been widely used in predictive lossless compression of hyperspectral images, the prediction performance depends heavily on the filtering weights estimated in a step-by-step manner. Traditional filtering methods do not take into account the longer-term dependencies of the data to be predicted. Motivated by the effectiveness of recurrent neural networks in capturing data memory for time series prediction, we design LSTM (long short-term memory) networks that can learn the data dependencies indirectly from filter weight variations. We then use the trained networks to regulate the weights generated by conventional filtering schemes through a close-loop configuration. We compare the proposed method with two other memory-less algorithms, including the popular Least Mean Square (LMS) filtering method, as well as its variant based on the maximum correntropy criterion (MCC). Simulation results on two publicly available datasets show that the proposed LSTM based filtering method can achieve smaller prediction errors.