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For downlink massive multiple-input multiple-output (MIMO) operating in time-division duplex protocol, users can decode the signals effectively by only utilizing the channel statistics as long as channel hardening holds. However, in a reconfigurable intelligent surface (RIS)-assisted massive MIMO system, the propagation channels may be less hardened due to the extra random fluctuations of the effective channel gains. To address this issue, we propose a learning-based method that trains a neural network to learn a mapping between the received downlink signal and the effective channel gains. The proposed method does not require any downlink pilots and statistical information of interfering users. Numerical results show that, in terms of mean-square error of the channel estimation, our proposed learning-based method outperforms the state-of-the-art methods, especially when the light-of-sight (LoS) paths are dominated by non-LoS paths with a low level of channel hardening, e.g., in the cases of small numbers of RIS elements and/or base station antennas.