QFLOW: Deterministic Quantile Transport for Probabilistic Time Series Forecasting

Recent diffusion forecasters have shown that long-horizon prediction can be reduced to a linear-time recursion over physical time, with noise and scale schedules estimated directly from data. However, these deterministic forecasts do not by themselves provide a principled way to turn the state evolution into a predictive distribution, while resorting to Monte Carlo sampling would reintroduce much of the cost that efficient diffusion forecasting seeks to avoid. We present QFLOW, which extends such forecasters with deterministic quantile transport for probabilistic time series forecasting. We use the Monge–Ampère equation to relate probability density to the local stretching and compression of a vector quantile map. A diagonal approximation of this relation then yields a one-step recursion that transports an entire quantile curve under the same probability-flow field as the underlying state update. At inference, QFLOW advances a dense transport grid, centers the resulting curve on a separately estimated conditional mean, and computes probabilistic forecast metrics directly from its quantiles, without sample ensembles. Across nine long-horizon forecasting datasets, QFLOW ranks among the top three recent diffusion forecasters in 33 of 36 dataset–metric comparisons. It achieves this performance using only a temporal linear layer for the conditional-mean head and either a temporal linear layer or a single cross-attention block for the denoiser.