Variational Streaming Flow: Probabilistic Forecasting in Physical Time

Hans Hao-Hsun Hsu; Minseon Gwak; Soon Hoe Lim; Pan Li; N. Benjamin Erichson

cs.LGOct 1, 2026

Variational Streaming Flow: Probabilistic Forecasting in Physical Time

Authors: Hans Hao-Hsun Hsu, Minseon Gwak, Soon Hoe Lim, Pan Li, N. Benjamin Erichson

Organizations: Georgia Institute of Technology · Lawrence Berkeley National Lab · International Computer Science Institute · KTH Royal Institute of Technology · Nordita

Abstract

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.

Figures & tables

Figure 1: Our variational streaming flow approach obtains lower forecasting error over long rollouts than standard streaming flow. Left: continuous ranked probability score (CRPS) on the deterministic double pendulum. Right: energy distance (ED) on the partially observed Rössler system.

Figure 2: VSF can serve as a predictor in a JEPA-based world model. We replace the deterministic autoregressive predictor with an action-conditioned VSF predictor while keeping the JEPA encoder and CEM planner fixed. The encoder maps the current and goal observations into latent space. For each action sequence, VSF samples multiple latents and rolls out future trajectories. CEM compares the predicted terminal states with the goal representation and uses these costs to select a plan.

Figure 3: Deterministic and stochastic systems pose different forecasting problems. (A) When deterministic dynamics are fully observed, the future is unique. With partial observations and stochastic dynamics, the same observed history can lead to different future trajectories. (B) SF produces a single forecast from the observed state, while VSF can generate multiple plausible futures.

Deterministic dynamics
	Double pendulum			Rössler			Lorenz
Model	CRPS $\downarrow$	RMSE $\downarrow$	MAE $\downarrow$	CRPS $\downarrow$	RMSE $\downarrow$	MAE $\downarrow$	CRPS $\downarrow$	RMSE $\downarrow$	MAE $\downarrow$	RT
Block diffusion	$0.46\mathbin{\pm}0.07$	$0.91\mathbin{\pm}0.14$	$0.47\mathbin{\pm}0.07$	$0.29\mathbin{\pm}0.13$	$0.54\mathbin{\pm}0.26$	$0.29\mathbin{\pm}0.13$	$0.31\mathbin{\pm}0.12$	$0.42\mathbin{\pm}0.18$	$0.33\mathbin{\pm}0.12$	30.2
AR diffusion	$0.43\mathbin{\pm}0.00$	$0.96\mathbin{\pm}0.01$	$0.59\mathbin{\pm}0.01$	$0.35\mathbin{\pm}0.06$	$1.00\mathbin{\pm}0.13$	$0.53\mathbin{\pm}0.08$	$0.86\mathbin{\pm}0.04$	$1.96\mathbin{\pm}0.06$	$1.50\mathbin{\pm}0.06$	78.2
Rectified flow	$0.53\mathbin{\pm}0.07$	$1.08\mathbin{\pm}0.11$	$0.67\mathbin{\pm}0.07$	$0.40\mathbin{\pm}0.07$	$0.85\mathbin{\pm}0.20$	$0.48\mathbin{\pm}0.08$	$1.46\mathbin{\pm}0.89$	$2.69\mathbin{\pm}1.46$	$1.94\mathbin{\pm}1.14$	41.1
DeepAR	$0.58\mathbin{\pm}0.00$	$1.07\mathbin{\pm}0.00$	$0.68\mathbin{\pm}0.00$	$0.17\mathbin{\pm}0.04$	$0.54\mathbin{\pm}0.05$	$0.21\mathbin{\pm}0.05$	$10.30\mathbin{\pm}0.22$	$12.42\mathbin{\pm}0.13$	$10.88\mathbin{\pm}0.13$	3.9

Table 1: VSF achieves the lowest forecasting error across all deterministic and stochastic systems over a 1,000-step rollout. We report CRPS, RMSE, and MAE for deterministic systems and CME, CSE,

W_{1}

, and ED for stochastic systems. Runtime (RT), in seconds, is averaged across tasks.

Figure 4: VSF preserves the Rössler geometry while capturing variability across plausible futures. We show

10

trajectories from each method. Block diffusion produces less diverse forecasts, while AR diffusion is more prone to rollout drift.

Figure 5: VSF samples remain consistent with the stochastic Lorenz dynamics, while AR diffusion drifts from the attractor. Two futures are shown.

Stochastic dynamics
	Lorenz
Model	CME $\downarrow$	CSE $\downarrow$	$W_{1}$ $\downarrow$	ED $\downarrow$
Block diffusion	$0.79\mathbin{\pm}0.09$	$0.66\mathbin{\pm}0.11$	$\bm{76.48\mathbin{\pm}3.20}$	$34.79\mathbin{\pm}11.90$
AR diffusion	$0.62\mathbin{\pm}0.03$	$0.53\mathbin{\pm}0.03$	$96.60\mathbin{\pm}2.48$	$12.43\mathbin{\pm}0.52$
Rectified flow	$0.77\mathbin{\pm}0.16$	$0.59\mathbin{\pm}0.21$	$90.41\mathbin{\pm}1.75$	$39.60\mathbin{\pm}16.90$
SF (perturb.)	$1.13\mathbin{\pm}0.33$	$0.70\mathbin{\pm}0.21$	$111.95\mathbin{\pm}22.66$	$39.81\mathbin{\pm}14.86$

Table 2: VSF achieves the lowest CME, CSE, and ED on stochastic Lorenz over a 2,500-step rollout. AR diffusion produces diverse samples but can drift from the system dynamics, as shown in Fig. 5 .

	PushT		Cube		TwoRoom		Reacher
Method	Success	RMSE $\downarrow$	Success	RMSE $\downarrow$	Success	RMSE $\downarrow$	Success	RMSE $\downarrow$
AR predictor	$84.7\mathbin{\pm}4.1$	$0.28\mathbin{\pm}0.01$	$69.3\mathbin{\pm}2.5$	$0.80\mathbin{\pm}0.05$	$78.7\mathbin{\pm}9.0$	$0.67\mathbin{\pm}0.03$	$61.3\mathbin{\pm}8.2$	$1.06\mathbin{\pm}0.01$
SF predictor	$88.0\mathbin{\pm}2.8$	$0.30\mathbin{\pm}0.04$	$76.7\mathbin{\pm}0.9$	$\bm{0.64\mathbin{\pm}0.02}$	$76.7\mathbin{\pm}8.2$	$0.59\mathbin{\pm}0.05$	$49.3\mathbin{\pm}5.0$	$1.13\mathbin{\pm}0.03$
VSF predictor	$\bm{92.7\mathbin{\pm}2.5}$	$\bm{0.27\mathbin{\pm}0.03}$	$\bm{80.0\mathbin{\pm}1.6}$	$0.65\mathbin{\pm}0.08$	$\bm{82.7\mathbin{\pm}5.0}$	$\bm{0.58\mathbin{\pm}0.01}$	$\bm{68.0\mathbin{\pm}0.0}$	$\bm{1.03\mathbin{\pm}0.02}$

Table 3: As a world-model predictor, VSF achieves the highest planning success on all four tasks and the lowest predictor RMSE on three of four tasks. RMSE is measured between predicted and ground-truth latent trajectories along the CEM-selected action sequence.

Panel of Figure 6 — Figure 6: VSF serves as an effective world-model predictor on Cube. The top row shows ground-truth rollout frames, and the bottom row compares the VSF latent trajectory (red) with the ground truth (dashed). The prediction closely follows the ground truth and leads to a successful plan.

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Physical time	History-cond.	Latent	Variational	Future-informed latent
FM/Rectified Flow	$\times$	$\times$	$\times$	$\times$	$\times$
Variational Rectified FM	$\times$	$\times$	$\checkmark$	$\checkmark$	$\checkmark$
SFP	$\checkmark$	$\checkmark$	$\times$	$\times$	$\times$
SFP + auxiliary latent	$\checkmark$	$\checkmark$	$\checkmark$	$\times$	$\times$
VSF (ours)	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$

Appendix

Table 4: Positioning of VSF relative to closely related flow-based formulations. “Physical time” means that flow progression is identified with the temporal evolution of the modeled sequence, rather than an artificial generative-time coordinate. “History-conditioned” denotes explicit conditioning on an observed history. “Variational” denotes the use of a training-time recognition model for latent inference. “Future-informed latent” indicates that the latent representation is inferred during training from the realized target dynamics, rather than introduced solely as an auxiliary stochastic coordinate.

Figure 7: (a) Two futures share the same physical state up to the bifurcation time

\tau

. (b) In non-latent SF, branch identity must be encoded by small differences in the physical state

x

, making the branch signal increasingly sharp as the Gaussian tube narrows. (c) An ideal latent realization instead carries branch identity in

Z

, allowing the same physical state

X_{\tau}=0

to generate either future. (d) Accordingly, the standard-SF physical-state sensitivity grows as

J_{\tau,t}^{\mathrm{SF}}\sim\exp(C/\varepsilon_{\tau}^{2})

as the fidelity error

\varepsilon_{\tau}\to 0

, while the ideal latent realization retains

J_{\tau,t}^{\mathrm{ideal}}=e^{-k(t-\tau)}

Dataset	Regime	State dimension	Initial conditions	Trajectories per initial condition	Time steps	$\Delta t$	History length
Double pendulum	Deterministic	8	200	1	1,200	0.01	4
Rössler	Deterministic	3	200	1	1,500	0.02	10
Lorenz	Deterministic	3	200	1	3,000	0.01	5
Rössler	Parameter mixture	3	40	100	1,500	0.02	10
Lorenz	Parameter mixture	3	40	100	3,000	0.01	5

Appendix

Table 5: Forecasting dataset statistics. For deterministic datasets, each sampled initial condition produces one trajectory. For parameter-mixture datasets, each initial condition is paired with 100 independently sampled dynamical parameters, producing 100 trajectories from that same initial state. We verify that these trajectories remain closely aligned for at least the full conditioning history: across all within-group trajectory pairs and conditioning time points, the mean pointwise RMSE is

0.008

for Rössler and

0.047

for Lorenz. Their subsequent futures are allowed to diverge. The integration intervals are

[0,12]

for double pendulum and

[0,30]

for Rössler and Lorenz.

Figure 8: Energy-score exponent.

\alpha

ablation on (a)–(b) deterministic double pendulum and (c)–(e) stochastic Rössler. Dashed lines show the best overall baseline for each system; dotted lines show MSE-trained VSF. Three-seed means.

Figure 9: KL regularization.

\beta_{\mathrm{KL}}

ablation on (a)–(b) deterministic double pendulum and (c)–(e) stochastic Rössler. Dashed lines show the best overall baseline for each system. Three-seed means.

Figure 10: Latent contribution. Bottleneck latent-weight ablation on (a)–(b) deterministic double pendulum and (c)–(e) stochastic Rössler. Dashed lines show the best overall baseline for each system. Three-seed means.

Figure 11: Latent resampling. Fixed rollout latent versus per-step resampling on (a)–(b) deterministic double pendulum and (c)–(e) stochastic Rössler. Three-seed means.

Variant	Training latent	Posterior encoder	$\beta_{\mathrm{KL}}$	Forecast latent
SF	None	No	–	None
Prior-latent SF	${\bm{z}}^{(1:M)}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}p({\bm{z}})$	No	–	${\bm{z}}\sim p({\bm{z}})$ , fixed
Posterior VSF (no KL)	${\bm{z}}^{(1:M)}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}q_{\bm{\phi}}$	Yes	$0$	${\bm{z}}\sim p({\bm{z}})$ , fixed
Full VSF	${\bm{z}}^{(1:M)}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}q_{\bm{\phi}}$	Yes	$2\times 10^{-3}$	${\bm{z}}\sim p({\bm{z}})$ , fixed

Appendix

Table 6: Definition of the latent-source and prior-alignment variants.

Figure 12: Latent source and prior alignment. SF, Prior-latent SF, Posterior no-KL, and Full VSF on (a)–(b) deterministic double pendulum and (c)–(e) stochastic Rössler. Three-seed means; breaks mark labeled out-of-range values.

Figure 13: Per-timestep error accumulation for 1,000-step deterministic forecasts. Columns show double pendulum, Rössler, and Lorenz; rows show CRPS, RMSE, and MAE. Curves and shaded bands report the mean and one standard deviation.

Figure 14: Per-timestep distributional error for stochastic Rössler over 1,000 steps and stochastic Lorenz over 2,500 steps. Columns show the two systems; rows show CME, CSE, marginal $W_{1}$ , and marginal energy distance. Curves and shaded bands report the mean and one standard deviation. These per-timestep scores compare marginal standardized state distributions. They provide a different view from the joint whole-trajectory metrics in Table 1 and 2 .

Figure 15: Qualitative forecasts on the deterministic systems. Columns show ground truth (GT), VSF samples, the VSF ensemble predictive mean, SF, and block diffusion. Sample panels contain

N=10

trajectories; the VSF ensemble is the mean of the samples. The last three rows magnify highlighted regions of the chaotic double-pendulum, Rössler, and Lorenz forecasts.

Figure 16: Visualization samples from the ground truth and generated trajectories for the partially observed stochastic Lorenz.

Panel of Figure 17 — Figure 17: The top row shows the ground-truth observations during rollout. The bottom row compares the predicted and ground-truth latent trajectories after joint-PCA projections. (a) The AR Predictor trajectory diverges and the rollout fails. (b) The VSF Predictor trajectory follows the ground-truth trajectory and the rollout succeeds.

Figure 18: Qualitative PushT result. VSF action-conditioned predicted latent trajectory follows the ground-truth trajectory in a successful rollout.

Figure 19: Qualitative TwoRoom result. VSF action-conditioned predicted latent trajectory follows the ground-truth trajectory in a successful rollout.

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ProtoFlow: Prototype-Guided Flow Matching for Multivariate Time Series Forecasting

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Nanyang Technological University · Tsinghua University · Shanghai Jiao Tong University

Multivariate Time Series Forecasting Vector-Quantized Variational Autoencoder Multivariate Time-Series Autoregressive Model Time Series

Open paper AskPDF arXiv

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.

Abstract

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