Modern geosteering is heavily dependent on real-time interpretation of deep electromagnetic (EM) measurements. We present a methodology to construct a deep neural network (DNN) model trained to reproduce a full set of extra-deep EM logs consisting of 22 measurements per logging position. The model is trained in a 1D layered environment consisting of up to seven layers with different resistivity values. A commercial simulator provided by a tool vendor is used to generate a training dataset. The dataset size is limited because the simulator provided by the vendor is optimized for sequential execution. Therefore, we design a training dataset that embraces the geological rules and geosteering specifics supported by the forward model. We use this dataset to produce an EM simulator based on a DNN without access to the proprietary information about the EM tool configuration or the original simulator source code. Despite employing a relatively small training set size, the resulting DNN forward model is quite accurate for the considered examples: a multi-layer synthetic case and a section of a published historical operation from the Goliat Field. The observed average evaluation time of 0.15 ms per logging position makes it also suitable for future use as part of evaluation-hungry statistical and/or Monte-Carlo inversion algorithms within geosteering workflows.