Koopman Generative Operators for Efficient Probabilistic Time-Series Forecasting

Koopman Generative Operators for Efficient Probabilistic Time-Series Forecasting. By Raz Marshanski and four others, from Ben-Gurion University of the Negev and Amazon Science. Published on the third of November.

A probabilistic forecast of a time series has to do three things at once. It has to capture the structured dynamics of how the series moves through time. It has to express uncertainty richly, not just as a single error bar. And it has to be cheap enough to run. The existing families each give up one of these. Latent dynamical models impose structure, but they rely on restrictive mechanisms for generating forecasts. Modern generative methods such as diffusion and flow matching are flexible, but they pay for that flexibility with sampling that is iterative and computationally expensive. You can have structure, or you can have expressiveness, and either way speed is not guaranteed.

The Koopman Generative Operator, or KGO, reframes prediction as the evolution of structured uncertainty. Rather than forecasting a value and attaching noise afterwards, the forecast is treated as an uncertain state that moves forward in a structured way. The name points to where that structure lives: a Koopman latent space, in which the method carries out both its extrapolation and its generation.

Three components make this up. The first is Koopman Patch Embedding, called KoPE, which provides temporally consistent extrapolation of latent trajectories. The second, Koopman Flow Matching or KoFM, is where the speed comes from. Instead of stepping a sampler forward many times, it generates in a single step through a closed-form matrix exponential in the Koopman latent space, so the iterative sampling of diffusion and flow matching is bypassed altogether.

The third component is the Adaptive Uncertainty Gate, or AUG. It produces the probabilistic part of the forecast by adapting uncertainty separately for each variable and for each forecast horizon. The abstract describes this as corresponding to a learned decomposition of aleatoric uncertainty, the irreducible noise in the data, and it is obtained without manual tuning.

The evaluation runs on the ProbTS benchmarks. Measured by CRPS, KGO is state of the art on thirteen of the seventeen. Measured by NMAE, it is state of the art on sixteen of the seventeen. Speed is reported against iterative generative models: because there is no iterative sampling to perform, KGO cuts inference time by at least twenty-five times.

The trade the paper sets out to move is between accuracy and efficiency, and it moves on both axes together: leading accuracy on the benchmarks evaluated, with generation done in one step. The authors present KGO as both practical and principled. The single place where that efficiency is won is the closed form in the latent space, which turns a long chain of sampling steps into one calculation.