This paper proposes a transfer learning neural network (TLNN) approach for digital pre-distortion (DPD) of mm-Wave active phased arrays (APA) operated under variable signal bandwidth regimes. Compared with the conventional artificial neural network (ANN) method, the proposed approach can achieve similar linearization performance with much lower computational complexity by transferring part of a trained model from one bandwidth to another bandwidth. In the recently introduced 5G, the increased signal bandwidth triggers considerable memory effects in the APA. Moreover, dealing with different signal bandwidths typically requires a time-consuming recalculation of the predistorter parameters. In this paper, the authors propose to have those challenges solved by using a DPD model based on the transfer learning method. The proposed approach was validated with over-the-air (OTA) measurements on an APA excited with signals of varying bandwidth, namely from 20 MHz to 100 MHz. Experimental results show a significant reduction in the training time while ensuring good linearization performance. With the applied TLNN DPD, an 8.5 dB improvement of adjacent channel leakage ratio (ACLR) and 8.6% points improvement of error vector magnitude (EVM) is achieved. Under the variable bandwidth regime, the complexity of the DPD model in terms of the number of multiplications is reduced from 199168 to 160. The proposed TLNN DPD proved to be robust concerning variation in the bandwidth of the APA excitation signal.