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Unbroken- and broken-spiral waves, in partial-differential-equation (PDE) models for cardiac tissue, are the mathematical analogs of life-threatening cardiac arrhythmias, namely, ventricular tachycardia and ventricular-fibrillation. We develop (a) a deep-learning method for the detection of unbroken- and broken-spiral waves and (b) the elimination of such waves, e.g., by the application of low-amplitude control currents in the cardiac-tissue context. Our method is based on a convolutional neural network (CNN) that we train to distinguish between patterns with spiral-waves S and without spiral-waves NS. We obtain these patterns by carrying out extensive direct numerical simulations of PDE models for cardiac tissue in which the transmembrane potential V, when portrayed via pseudocolor plots, displays patterns of electrical activation of types S and NS. We then utilize our trained CNN to obtain, for a given pseudocolor image of V, a heatmap that has high intensity in the regions where this image shows the cores of spiral waves and the associated wavefronts. Given this heatmap, we show how to apply low-amplitude currents of a two-dimensional Gaussian profile to eliminate spiral-waves efficiently. Our in silico results are of direct relevance to the detection and elimination of these arrhythmias because our elimination of unbroken or broken-spiral waves is the mathematical analog of low-amplitude defibrillation.