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

Deep Probabilistic Koopman: Long-term time-series forecasting under periodic uncertainties