Some of the key tasks in seismic processing involve suppressing multiples and noise that interfere with primary events. Conventional multiple attenuation on seismic prestack data is time-consuming and subjective. As an alternative, we propose model-driven processing using a convolutional neural network trained on synthetically modeled training data. The crucial part of our approach is to generate appropriate training data. Here, we compute a generic data set with pairs of synthetic gathers with and without multiples. Because we generate the primaries first and then add multiples, we ensure that we have perfect target data without any multiple energy. To compute generic and realistic training data, we include elements of wave propagation physics and implement a randomized flexibility of settings such as the wavelet, frequency content, degree of random noise, and amplitude variation with offset effects with each gather pair. A fully convolutional neural network is trained on the synthetic data in order to learn to suppress the noise and multiples. Evaluations of the approach on benchmark data indicate that our trained network is faster than conventional multiple attenuation because it can be run efficiently on a modern GPU, and it has the potential to better preserve primary amplitudes. Multiple removal with model-driven processing is demonstrated on seismic field data, and the results are compared to conventional multiple attenuation using a commercial Radon algorithm. The model-driven approach performs well when applied to real common-depth point gathers, and it successfully removes multiples, even where the multiples interfere with the primary signals on the near offsets.