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As High-Performance Computing (HPC) systems strive towards the exascale goal, failure rates both at the hardware and software levels will increase significantly. Thus, detecting and classifying faults in HPC systems as they occur and initiating corrective actions before they can transform into failures becomes essential for continued operation. Central to this objective is fault injection, which is the deliberate triggering of faults in a system so as to observe their behavior in a controlled environment. In this paper, we propose a fault classification method for HPC systems based on machine learning. The novelty of our approach rests with the fact that it can be operated on streamed data in an online manner, thus opening the possibility to devise and enact control actions on the target system in real-time. We introduce a high-level, easy-to-use fault injection tool called FINJ, with a focus on the management of complex experiments. In order to train and evaluate our machine learning classifiers, we inject faults to an in-house experimental HPC system using FINJ, and generate a fault dataset which we describe extensively. Both FINJ and the dataset are publicly available to facilitate resiliency research in the HPC systems field. Experimental results demonstrate that our approach allows almost perfect classification accuracy to be reached for different fault types with low computational overhead and minimal delay.