Spin-torque transfer magnetic random access memory (STT-MRAM) is a promising emerging non-volatile memory (NVM) technology with wide applications. However, the data recovery of STT-MRAM is affected by the diversity of channel raw bit error rate (BER) across different dies caused by process variations, as well as the unknown resistance offset due to temperature change. Therefore, it is critical to develop effective decoding algorithms of error correction codes (ECCs) for STT-MRAM. In this article, we first propose a neural bit-flipping (BF) decoding algorithm, which can share the same trellis representation as the state-of-the-art neural decoding algorithms, such as the neural belief propagation (NBP) and neural offset min-sum (NOMS) algorithm. Hence, a neural network (NN) decoder with a uniform architecture but different NN parameters can realize all these neural decoding algorithms. Based on such a unified NN decoder architecture, we further propose a novel deep-learning (DL)-based adaptive decoding algorithm whose decoding complexity can be adjusted according to the change of the channel conditions of STT-MRAM. Extensive experimental evaluation results demonstrate that the proposed neural decoders can greatly improve the performance over the standard decoders, with similar decoding latency and energy consumption. Moreover, the DL-based adaptive decoder can work well over different channel conditions of STT-MRAM irrespective of the unknown resistance offset, with a 50% reduction of the decoding latency and energy consumption compared to the fixed decoder.