We consider near maximum-likelihood (ML) decoding of short linear block codes based on neural belief propagation (BP) decoding recently introduced by Nachmani et al.. While this method significantly outperforms conventional BP decoding, the underlying parity-check matrix may still limit the overall performance. In this paper, we introduce a method to tailor an overcomplete parity-check matrix to (neural) BP decoding using machine learning. We consider the weights in the Tanner graph as an indication of the importance of the connected check nodes (CNs) to decoding and use them to prune unimportant CNs. As the pruning is not tied over iterations, the final decoder uses a different parity-check matrix in each iteration. For ReedMuller and short low-density parity-check codes, we achieve performance within 0.27dB and 1.5dB of the ML performance while reducing the complexity of the decoder.