Raman amplifiers are crucial for multi-band systems due to their capability of providing wide gain profiles with low noise, making accurate modeling of them an important task. Leveraging neural networks (NN) is a promising approach. Due to the burden of collecting a huge amount of data for training NNs, one of the key issues is how to build data-driven Raman amplifier models with limited data. Based on the physical mechanism of Raman amplification (RA), we propose a grey-box modeling scheme. First, a base model under a certain launch power profile is established based on NN. Then, it is transferred to arbitrary launch power profile cases by linear regression (LR). Physical interpretations of the proposed scheme are detailed with theoretical analyses. In simulations with 5 Raman pumps over a 10-THz signal spectrum, the proposed scheme can reduce the RA model's root mean square error (RMSE) by at least 0.42 dB. Meanwhile, the proposed scheme has stronger generalizability compared to black-box models. Experiments with 4 Raman pumps over an 8.5-THz signal spectrum are conducted for further demonstration. Experimental results prove that the proposed scheme can lower the mean RMSE by up to 0.79 dB compared with the pure NN-based method, while significantly reducing training data requirements.