We present the Inverse Drum Machine, a novel approach to Drum Source Separation that leverages an analysis-by-synthesis framework combined with deep learning. Unlike recent supervised methods that require isolated stem recordings for training, our approach is trained on drum mixtures with only transcription annotations. IDM integrates Automatic Drum Transcription and One-shot Drum Sample Synthesis, jointly optimizing these tasks in an end-to-end manner. By convolving synthesized one-shot samples with estimated onsets, akin to a drum machine, we reconstruct the individual drum stems and train a Deep Neural Network on the reconstruction of the mixture. Experiments on the StemGMD dataset demonstrate that IDM achieves separation quality comparable to state-of-the-art supervised methods that require isolated stems data.