Flow Matching with Gaussian Process Priors for Probabilistic Time Series Forecasting

TL;DR

This work addresses the challenge of non-i.i.d. priors in probabilistic time-series forecasting by introducing TSFlow, a conditional flow matching model that embeds Gaussian Process priors into the prior $q_0$ to better match temporal structure with the data distribution $q_1$. It integrates MBOT to form data–prior couplings and enables conditional prior sampling and guidance-based conditioning, bridging unconditional generation with probabilistic forecasting. The main contributions are (i) TSFlow with GP priors using SE, OU, and PE kernels, (ii) conditional prior sampling via Langevin dynamics and ALD-based likelihoods, (iii) guided generation to condition generation on observed histories, and (iv) a GP-regression extension for conditional modeling, all validated on eight real-world datasets where TSFlow often achieves state-of-the-art results while requiring fewer neural function evaluations. These results demonstrate improved generative quality and flexible probabilistic forecasting, offering practical benefits for downstream tasks and broader applicability to time-series modeling.

Abstract

Recent advancements in generative modeling, particularly diffusion models, have opened new directions for time series modeling, achieving state-of-the-art performance in forecasting and synthesis. However, the reliance of diffusion-based models on a simple, fixed prior complicates the generative process since the data and prior distributions differ significantly. We introduce TSFlow, a conditional flow matching (CFM) model for time series combining Gaussian processes, optimal transport paths, and data-dependent prior distributions. By incorporating (conditional) Gaussian processes, TSFlow aligns the prior distribution more closely with the temporal structure of the data, enhancing both unconditional and conditional generation. Furthermore, we propose conditional prior sampling to enable probabilistic forecasting with an unconditionally trained model. In our experimental evaluation on eight real-world datasets, we demonstrate the generative capabilities of TSFlow, producing high-quality unconditional samples. Finally, we show that both conditionally and unconditionally trained models achieve competitive results across multiple forecasting benchmarks.

Figures (7)

• Figure 1: Overview of our proposed model TSFlow. To perform forecasting, we first sample $\mathbf{x}_0$ conditioned on our observation $\mathbf{y}^{p}$ from $q_0$. In an unconditional setting, we do this via Langevin dynamics (see \ref{['sec:sps']}), while in the conditional model, we can do this by directly using a conditional prior, e.g., a Gaussian process regression (see \ref{['sec:cond_model']}). Given $\mathbf{x}_0$, we can now sample from $q_1$ by solving its corresponding ODE (see \ref{['eq:flow-ode']}).
• Figure 2: Example forecasts and ground truth of TSFlow-Cond. (OU) of the first three time series in the test set of Traffic.
• Figure 3: Architecture of TSFlow. The architecture consists of three residual blocks containing an S4 layer operating across the time dimension. The input to the model are noisy time series $\mathbf{x}_t$, the timestep $t$, and in the case of the conditional setup, a condition $\mathbf{c}$, which contains $\mathbf{y}^p$ and a binary observation mask. The prediction is the approximated flow field $u_{\bm{\theta}}(t, \mathbf{x}_t)$.
• Figure 4: Examples of samples from the unconditional Gaussian processes for $L=150$ and $p=30$.
• Figure 5: Example forecasts of TSFlow-Cond. on the test set of the two datasets Solar and Electricity.