  Probabilistic forecasting of complex phenomena is paramount to various
scientific disciplines and applications. Despite the generality and importance
of the problem, general mathematical techniques that allow for stable long-term
forecasts with calibrated uncertainty measures are lacking. For most time
series models, the difficulty of obtaining accurate probabilistic future time
step predictions increases with the prediction horizon. In this paper, we
introduce a surprisingly simple approach that characterizes time-varying
distributions and enables reasonably accurate predictions thousands of
timesteps into the future. This technique, which we call Deep Probabilistic
Koopman (DPK), is based on recent advances in linear Koopman operator theory,
and does not require time stepping for future time predictions. Koopman models
also tend to have a small parameter footprint (often less than 10,000
parameters). We demonstrate the long-term forecasting performance of these
models on a diversity of domains, including electricity demand forecasting,
atmospheric chemistry, and neuroscience. For electricity demand modeling, our
domain-agnostic technique outperforms all of 177 domain-specific competitors in
the most recent Global Energy Forecasting Competition.
