Spiral waves of electrical activation in cardiac tissue lead to life-threatening ventricular arrhythmias. The tracking of the tip of a spiral wave is a problem of central importance that can play an important role in the elimination of these arrhythmias. We first obtain pseudo-ECGs from our simulations of spiral waves in the two-dimensional, two-variable Aliev-Panfilov model for cardiac tissue. We then show how to use these pseudo ECGs in conjunction with Long Short-Term Memory (LSTM) networks to track the tip trajectories of spiral waves. We demonstrate that our LSTM-based tip-tracking method compares favorably with the Iyer-Gray method, which requires the full spatiotemporal evolution of spiral waves to obtain tip trajectories. By contrast, our method requires only the time series of pseudo-ECGs from three representative spatial locations. We train our LSTM with 20000 trajectories and test it on an additional 5000 spiral-tip trajectories, which are simulated in a range of parameters; our tip-trajectory data include rigid, meandering, and drifting spiral waves. We use the Iyer-Gray method to get the spiral wave trajectories during training. We quantify the errors in our tip-tracking procedure. We discuss extensions of our method to (a) cases with external noise, (b) ensemble-LSTM-based suppression of outliers, and (c) biophysically realistic cardiac-tissue models (e.g., those due to ten Tussher and Panfilov) which use the LSTM model trained only on the Aliev-Panfilov model.