Probabilistic forecasting skips iterative sampling: KGO's matrix-exponential transport leads CRPS on 13 of 17 ProbTS settings

Probabilistic time-series forecasting gains a generation route that avoids iterative sampling: KGO compresses the generation of future trajectories into a single linear transport in latent space while keeping uncertainty that varies by variable and forecast step. Earlier diffusion and flow-matching methods obtain flexible multimodal predictive distributions through multi-step iterative sampling, with inference cost growing with the number of steps; latent-variable methods instead tie uncertainty to an initial state and induce variability through its propagation. Across 17 ProbTS evaluation settings, KGO attains the lowest CRPS on 13 and the lowest NMAE on 16; in long-term forecasting it ranks first on 7/9 datasets in CRPS and 9/9 in NMAE, and in short-term forecasting on 6/8 and 7/8 respectively.

The model first encodes the historical series into a latent state, then extrapolates that state patch by patch across the forecast horizon with a learned linear operator, yielding a structured latent trajectory. KoPE (Koopman Patch Embedding) repeatedly applies a learned linear operator to the final latent state, normalizing the operator by its spectral norm whenever that norm exceeds 1 so the rollout cannot expand. Across 36 dataset-horizon pairs, KoPE reaches an average CRPS of 0.141, below the 0.165 of both the Last Patch and Context Pooling alternatives; replacing the sLSTM encoder with a Transformer or a Linear RNN changes average CRPS to 0.148 and 0.145.

Generation no longer integrates a time-dependent vector field; a matrix exponential performs one affine transport in latent space, so inference reduces to a matrix multiplication plus a bias. KoFM (Koopman Flow Matching) replaces the time-dependent velocity field of standard flow matching with a time-invariant linear dynamical system whose solution has a closed-form matrix exponential, removing the numerical ODE solve. Against the pretrained flow-matching foundation model Sundial-base-128m at the 50-step Euler budget used in its original paper, KGO averages about 13% lower CRPS and about 7% lower NMAE over 28 dataset-horizon pairs, at 7.8 ms/window versus 387 ms/window.

Evaluation is confined to 17 data configurations under the ProbTS protocol, covering ETT, Electricity, Exchange, Traffic, Weather, Solar and ILI, with separate context lengths for the short-term and long-term regimes; efficiency figures are measured on a single RTX 6000 Ada in FP32 with batch size 1 and 100 trajectories per window.

The matrix exponential exactly captures the homogeneous linear evolution, while the affine term keeps only the first-order term of the corresponding ODE offset integral; AUG's gate weights describe the relative contribution of the two branches and are not a direct estimate of aleatoric uncertainty; the comparison with Sundial differs in architecture and training procedure as well as sampling mechanism.