Deterministic Prediction as Global Velocity: Flow Matching for Probabilistic Time-Series Forecast

Probabilistic forecasting is important for predicting complex dynamical systems because intrinsic randomness and incomplete observations can cause the same observed state to evolve into multiple plausible futures. While flow matching is a flexible approach for probabilistic forecasting, it is computationally expensive. Streaming flow (SF) reformulates this approach to model temporal evolution efficiently by learning a continuous velocity field directly in physical time. However, SF learns a deterministic velocity field. Thus, it provides only a single future trajectory for a given fixed initial state and observation history. To overcome this limitation, we introduce Variational Streaming Flow (VSF). Our approach learns a latent distribution that is conditioned on the dynamics of interest. In turn, this enables probabilistic forecasting. Importantly, we retain the computational efficiency of SF by generating in physical time. Across deterministic and stochastic dynamical systems, VSF demonstrates superior predictive accuracy and distributional fidelity. We demonstrate the advantage for both long-horizon rollouts exceeding 1,000 steps, and settings with bifurcating dynamics. Moreover, VSF can be integrated into existing Joint-Embedding Predictive Architecture (JEPA)-based world models as a plug-and-play predictor to improve temporal dynamics and goal-directed success rate in navigation, motion planning, and manipulation.

Generative modeling has shown strong promise for multivariate time mseries (MTS) forecasting, especially scale to high-dimensional settings. Diffusion-based methods achieve competitive performance but typically require many sampling steps at inference. VAE-based non-iterative forecasting frameworks have therefore emerged as an efficient alternative. Within this line of work, vector quantization (VQ) enables controllable latent space modeling by mapping multivariate series into compact discrete representations. Existing VQ-based forecasting methods, however, typically rely on autoregressive (AR) token generation, which suffers from exposure bias and training-inference mismatch. Flow matching provides an efficient non-autoregressive alternative for latent forecasting, but existing formulations usually initialize transport from a generic Gaussian prior. We instead observe that the trained VQ codebook already captures representative latent prototypes and can thus serve as a more informative prior for flow matching. Based on this insight, we propose ProtoFlow, a forecasting framework that combines vector-quantized autoencoding with Prototype-prior Flow matching. Our method first maps multivariate sequences into a discrete latent space, then constructs a structured prior from the learned codebook, and finally learns a DiT-based rectified flow to transport samples from this prior to future latent representations conditioned on historical observations. By replacing generic noise initialization with a learned prototype prior, ProtoFlow avoids the rollout mismatch of AR token prediction and promotes faster training convergence. Extensive experiments on benchmark datasets show that it consistently achieves superior forecasting performance with efficient inference.

Real-world time series exhibit temporally structured uncertainty: volatility clusters in turbulent regimes, dissipates in stable periods, and shifts abruptly around structural breaks. Yet many probabilistic forecasting methods estimate predictive uncertainty as an independent per-step quantity, leaving the evolution and persistence of volatility regimes under-modeled. We formalize this missing dimension as temporal uncertainty dynamics and instantiate it in the Volatility Dynamics Variational Autoencoder (VolDy-VAE), a non-autoregressive generative forecaster with a location-scale decoder. VolDy-VAE combines a location path for mean prediction with a recurrent scale path that transfers and evolves a volatility hidden state from the look-back window to the forecasting horizon, enabling temporally coherent predictive variances. This design yields an adaptive attenuation mechanism: high-variance observations receive lower influence on the location estimate while their uncertainty is preserved through explicit scale predictions. We further provide a simplified regime-switching analysis showing that, when variances are known or consistently estimated, the volatility-aware objective reduces to inverse-variance weighting, whereas MSE-based estimators remain unbiased but statistically inefficient. Experiments on nine benchmarks show that VolDy-VAE improves forecasting accuracy and uncertainty calibration over competitive probabilistic and point-forecasting baselines while maintaining low inference latency; plug-in studies further indicate that the VolDy principle can benefit GAN, Koopman VAE, and Transformer backbones. The source code is publicly available at https://github.com/wangyijunlyy/VolDy-VAE.