How do I measure the accuracy of a LSTM regression model for a timeseries? Given the stochastic nature of the model, it will produce different results for RMSE, MAPE etc each time you run it. Do I just build the model n times and I get the average of the metrics?

Since your LSTM is stochastic, there is no "the" accuracy. One possibility is to train it multiple times and average the resulting accuracies (which estimated the expected accuracy). An alternative, as Dave comments, is to set an RNG seed, so your LSTM is not stochastic any more, and then you indeed can report "the" accuracy.
