Learn from the Core, Fit to the Data: Deep Koopman Learning of the Atmosphere

Deep Koopman learning typically consists of dictionary learning followed by operator fitting, but we find that this pipeline can severely overfit, especially when the rank of the fitted operator is overspecified, because the same stochastic transitions select the features and fit the operator. We study two ways to mitigate this issue: separating the data used for dictionary learning and operator fitting, and using the forecasts of an approximate prediction model, which we call a core model, to train the dictionary. Our main finding is that core-trained dictionaries can substantially improve the held-out VAMP-E score over data-trained ones. On ERA5 and a controlled stochastic shallow-water benchmark, the advantage becomes more pronounced at long lags, and in the benchmark also under stronger transition noise.