Reservoir characterization and monitoring represent some of the most ambitious goals for geophysical methods. Several challenges are involved, including sensitivity to the parameter changes and resolution of the obtained results. Electromagnetic (EM) methods are attractive for reservoir applications due to the high sensitivity of the resistivity parameter to oil/water saturations. Crosswell EM and surface-to-borehole EM provide opportunities for reservoir monitoring. The EM inverse problem, however, is highly nonconvex and ill-posed so as to necessitate significant preconditioning in the form of a priori information and regularization that impact resolution. We explore the use of machine-learning (ML) techniques in the form of deep-learning neural networks for implementing EM-based reservoir monitoring coupled with a dynamic fluid flow simulator. A crosswell acquisition setup is modeled in the framework of a realistic water-alternating-gas reservoir simulation scenario for enhanced oil recovery. Several reservoir saturation instances are generated and converted into resistivity, and corresponding crosswell EM data are generated using an electric source and a multicomponent (electric-magnetic) receiver assemblage. The U-Net deep-learning network is modified for the purpose of training and validation in which saturation models and the corresponding EM data are used. We also test the sensitivity of the deep-learning inversion to multiple EM components, noise in the data, generalization problems, and 3D reconstruction ability in which we use 3D convolutional neural network layers. In all cases, ML inversion proves to be robust with good resilience to increased noise levels. Prediction results indicate excellent reconstruction capabilities with resolution comparable to the reservoir models used by the simulator. Our results suggest that ML inversion through deep learning can become an efficient approach to data-driven and physics-constrained reservoir monitoring in which the sensitivity of EM-based techniques to fluid saturations can be fully exploited without compromising the resolution and accuracy of the results.