Flow Matching with Gaussian Process Priors for Probabilistic Time Series Forecasting
Authors: Marcel Kollovieh
Marten Lienen
David Lüdke
Leo Schwinn
Stephan Günnemann
Venue: ICLR 2025 Poster (2025)
Keywords: flow matching
time series forecasting
generative modeling
deep learning
Primary area: learning on time series and dynamical systems
TL;DR: We propose TSFlow, a conditional flow matching model for time series forecasting that leverage domain-specific priors.

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.

Published as a conference paper at ICLR 2025
FLOW MATCHING WITH GAUSSIAN PROCESS PRIORS FOR PROBABILISTIC TIME SERIES FORECASTING
Marcel Kollovieh 1,2.3
Marten Lienen
David Lüdke
Leo Schwinn
Stephan Günnemann 1,2.3
School of Computation, Information and Technology, Technical University of Munich 2 Munich Data Science Institute
3 Munich Center for Machine Learning
Correspondence to: wwl.ooieeeetuum......m
ABSTRACT
Recent advancements in generative modeling, particularly diffusion models, have opened new directions for time series modeling, achieving state-of-the-ard per- formance in forecasting and synthesis. However, the reliance of diffusion-basee 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 (condi- tional) 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 prob- abilistic forecasting with an unconditionally trained model. In our experimental evaluation on eight real-world datasets, we demonstrate the generative capabilities of FlFlo. producing high-quality unconditional samples. Finally, we show that both conditionally and unconditionally trained models achieve competitive results across multiple forecasting benchmarks.
INTRODUCTION
Diffusion (Sohl-Dickstein et al., 2015; Ho et al., 2020) and score-based generative models (Song et al., 2020) have demonstrated strong performance time series analysis, effectively capturing the distribution of time series data in both conditional (Rasul et al., 2021; Tashiro et al., 2021; Biloš et al., 2023; Alcaraz & Strodthoff, 2022) and unconditional (Kollovieh et al., 2023) settings. However, these models typically transform non-i.i.d. distributions of time series data into simple isotropic Gaussian prior by iteratively adding noise leading to long and complex paths. This can hinder the generative process and potentially limit the models' performance.
Conditional Flow Matching (CFM) (Lipman et al., 2022; Tong et al., 2023) provides an efficient alternative to diffusion models and simplifies trajectories by constructing probability paths based on conditional optimal transport. The model is trained by regressing the flow fields of these paths and can accommodate arbitrary prior distributions. Despite its advantages, the application of CFM to time series forecasting remains unexplored.
In this work, we simplify the trajectories of generative time series modeling through informed priors, specifically using Gaussian Processes (GPs) as prior distributions within the CFM framework. We align the prior distributions with the underlying temporal dynamics of real-world data via optimal transport couplings. This approach not only reduces the complexity of the learned transforma- tions but also enhances the model's performance conditional and unconditional generation tasks. Furthermore, we demonstrate how both conditionally and unconditionally trained models can be leveraged for probabilistic forecasting. We propose conditional prior sampling and demonstrate how to use guidance (Dhariwal & Nichol, 2021) to bridge the gap between conditional and unconditional generation, allowing for flexible application of unconditional models without the need for specialized conditional training procedures.

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Our empirical evaluations show that using conditional GPs priors improves generative modeling and forecasting performance, surpassing various baselines from different frameworks on multiple datasets. This validates the effectiivnness of our approach in simplifying the generative problem and enhancing model versatility.
Our key contributions are summarized as follows:
We introduce TSFlow, novel generative model for time series that incorporates conditional Gaussian Processes as informed prior distributions within the Conditional Flow Matching framework.
We demonstrate how these priors align with the data distribution and improve the uncondi- tional generative performance.
We show how to utilize both conditionally trained and unconditionally trained models for forecasting tasks, enhancing the flexibility and applicability of our approach.
Through extensive empirical evaluation, validate the effectiveness of TSFlow, achiev- ing competitive performance and outperforming existing methods on several benchmark datasets.
BACKGROUND: CONDITIONAL FLOW MATCHING
Conditional Flow Matching (CFM) was recently introduced as framework for generative modeling by learning flow field to transform one distribution into another (Lipman al., 2022). The learned flow field then yields transformation Φi that maps sample Xo go from prior distribution go. e.g., an isotropic Gaussian, to a transformed sample Φ1 (Xo) that follows the data distribution @I+ i.e. Φh (xo) (1- The flow field d; is parametrized through vector field ug. which is learned in simple regression task. In the following, we give a short exposition of flow matching. For thorough introduction to flow matching and diffusion, refer the reader to Luo (2022); Lipman et al. (2022).
2.1 PROBABILITY FLOWS
The flow de (x) describes the path of a sample x following the time-dependent vector field u. uy(x). [0, x _ - Rd and is itself the solution to the ordinary differential equation
(x) (x)) dt,
Φo(x) Xg.
For given density po. de induces time-dependent density called a probability path via the push forward operator (x) = ([ot]=Po)(() = Po( (x)) det[der*/an /ax(x)]. Our goal is to find simple, i.., short and straight, probability path pt whose boundary probabilities align with our prior and data distribution. i.e., Po 4a and p1 R 41-
we now consider two flows di and o/. their induced probability paths pt and p/ will be equal if their train generative model for data distribution Qi by matching a model ugg to a flow field it corresponding to a probability path between a prior distribution go and qı-
flow fields u, and ui are equal. Consequently,
Lipman et al. (2022) propose to model the marginal probability paths as a mixture of conditional paths, i.e.,
(x( - Pc z)q(z)dz.
More specifically, they choose low-variance Gaussian paths centered on each data point x i.e., x1) [x(x
= =N(tx, (1 (1 o2 with z = {xi} } and q 91. which result in an isotropic prior distribution, i.c., qo (0, These conditional probability paths have a closed form for their conditional flow field ux(x x1). Furthermore, they show that the marginal flow field arising from these conditional flow fields generates the approximate data distribution di and derive the Conditional Flow Matching objective:
LCFM(O) =
u((x xx)
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Sampling from
Flow Time
Sampling from
X
(N y
(0) (Xo)
Xo (xo yy)
y ωi (xo) (y lyd)
Unconditional: Langevin Dynamics
Prior Conditioning
Forccasting
Figure I: Overview of our proposed model TSFlow. To perform forecasting, we first sample Xo conditioned on our observation y" from qo- In an unconditional setting, do this via Langevin dynamics (see Sec. 3.1.2), while in the conditional model, we can do this by directly using conditional prior, e.g.. a Gaussian process regression (see Sec. 3.2). Given Xp, we can now sample from Q1 by solving its corresponding ODE (see Eq. (1)).
2.2 COUPLINGS
Tong et al. (2023) generalize Conditional Flow Matching to encapsulate arbitrary source distributions. This is achieved by including not only samples from the target distribution but also from the source into the conditioning of the probability paths, i.e., z = (xo, X1 More specifically, they choose
Xo: x1) = N(txx + (1 x)x oh (x
(4)
satisfying po = go and P1 = ạ for iinn - The respective conditional vector field is simply difference of x and i.e., (x(x Xo. x1) x1 and can be learned as a regression task:
()) = [0. xx)x((X))))))> x~ (x x x,)
|| I (4, x) - 4x(x Xn, xI)1
(5)
While the simple choice is to sample Xy and independently, the framework allows for arbitrary joint distributions q(x,. with marginals go and 91-
2.3 MINI-BATCH OPTIMAL TRANSPORT
natural choice for joint distributions is the optimal transport coupling between source and target distributions. In practice, however, finding the optimal transport map is only feasible for small datasets, Instead, Tong et al. propose a mini-batch variant, of the algorithm. For each batch of data (x Ji=1 seen during training, they sample B points x Qo and then compute between these batches. This makes finding я computationally feasible while retaining the benefits of the optimal transport coupling empirically (Tong et al., 2023). Choosing q as the optimal transport map between the prior and data distribution makes both training and sampling more efficient by lowering the variance of the objective and straightening the probability paths.
3 TSFLOW
In this section, we present our main contributions: (1) TSFlow, novel flow matching model tailored for probabilistic time series forecasting (see Fig. 1). TSFlow supports both unconditional and conditional generation, i.e., generation without or with partially observed time series. (2) We explore non-i.i.d. prior distributions within the CFM framework and demonstrate that incorporating a Gaussian process improves unconditional generation by aligning the prior with the data distribution. Further, (3) we show how an unconditional model can be employed for forecasting by conditioning during inference via Langevin dynamics and guidance techniques. Finally, (4) we describe how to use data-da dependent prior distributions in the form of Gaussian process regression to train TSFlow for conditional forecasting directly.

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Problem Statement. Consider a regularly sampled univariate time series y E RR' of length L drawn from the data distribution şn (y) (corresponding to dimension d, as detailed in Sec. 2). We denote the time series y (y" y where y" represents the observed past and y- the unknown future aim to predict. To avoid ambiguity with the flow-matching time parameter I, we introduce T as the time index for positions within the time series data. Here, the flow-matching time f parameterizes the progression along the transformation path from the prior to the target time series distribution, while orthogonal and refers to specific time points in the observed data.
Our objective is to capture the conditional distribution (y y²) using a generative model pa (y y²). By marginalizing over the noisy time series Xo. we decompose the conditional model into a conditional prior and generative process:
Poly yx" =
- - poly Xu, y²) (x y dX)
(6)
Choosing go isotropic Gaussian (as in existing diffusion models) neglects condition the prior on the observed past. In the following, we describe how TSFlow models both the conditional generative model and the conditional prior.
First, we introduce an unconditional model po(y) to capture the data distribution gi (y). During training, this model unaware of any conditioning information, such as past observations. We then explain how to condition this model and its prior on past observations during inference, resulting in po(y y?) and (x y²), respectively.
Then, we introduce a conditionally trained model that directly incorporates y to model the conditional distribution po (y y²). To simplify notation, we denote x1 as y the following. Furthermore, we denote the noisy time series as Xg, and its (temporal) past and future x and x/j. respectively.
3.1 UNCONDITIONAL MODELING
Algorithm 1: Unconditional Training of TSFlow
1: Input: Prior distribution Zo. data distribution Q1, noise levels network tig
2: for iteration = 1,... do
3:
Xo (xo):
q (y)
>> sample batches from the prior and dataset b compute optimal transport mapping sample random time step t b compute mean pe sample from Pe(- z) b regress conditional vector field gradient step
4:
(Xo:y) f- y)
5:
t~ 10,
6:
Pt + ty + +11 (1-12 t)xo
7:
X, N I;
8:
((A) + I (t. xx) u4 (xe 22)
9:
0+ Update 0, VoL(O))
10: end for
Return: up
We begin by modeling the data distribution @1 (y) using an unconditional CFM model po (y), which optimized using mini-batch optimal transport couplings (see Sec. 2.3 and Alg. 1) and does not incorporate any conditioning information during training.
To better capture the temporal correlations the time series y, we explore Gaussian process priors instead of the default isotropic Gaussian distribution. By introducing temporal structure and aligning the prior distribution go more closely with the data distribution q1 via optimal transport couplings, we enhance the model's ability learn temporal patterns (Sec. 3.1. Then, we describe how to condition the unconditional model on observed data y" during inference by incorporating past observations into the generation process via Langevin dynamics and guidance, effectively transforming it into conditional one (Sec. 3.1.2 and 3.1.3).
3.1.1 INFORMED PRIOR DISTRIBUTIONS
Previous work by Tong et al. (2023) has shown that pairing prior and data samples based on the optimal transport map accelerates training, reduces the number of Neural Function Evaluations (NFFs)
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required during inference, and enhances the model's performance by shortening the conditional paths and straightening the learned velocity field.
We can further enhance this effect by choosing domain-specifii prior distribution go closer to the data distribution qı than the standard isotropic Gaussian, yet similarly easy to specify and sample from. In the context of time series, propose to employ Gaussian process (GP) priors (O, K) with a kernel function (TT, T') (Rasmussen & Williams, 2005). Gaussian processes are well-suited for modeling time series data because they naturally capture temporal correlations. By choosing K, we can incorporate dataset-specific structures and temporal patterns into the prior without requiring an extensive training process.
We explore three kernel functions that reflect different types of data characteristics; squared exponen- (OU), and periodic (PE) kernels, defined respectively as:
tial (SE),
de
=exp Kou(T,T)) = exp (-1m) and KPE(T, 7') = =exp exp(- sin (d))
KSE(T, r') = =exp (-20z)
where d 7' and is non-negative parameter that adjusts the length scale of the kernel.
The squared exponential kernel KSE produces infinitely smooth samples, making suitable for modeling time series with high degree of smoothness. The Ornstein-Uhlenlee closely related to Brownian motion and ideal for modeling data with a rougher structure. The periodic kernel KPE captures repeating patterns in the data by modeling the covariance between points as function of their periodic differences, making it suitable for time series with periodic behavior. We provide samples drawn from these priors in App. A.6.
kernel Kou is
3.1.2 CONDITIONAL PRIOR SAMPLING
To adapt the unconditional model for conditional generation post-training, we propose conditional prior sampling, which conditions the prior distribution as (o y²). Given a sample Xo ~ (x y²), we use it as the initial condition for our vector field generate new samples from the distribution y Xg. It is important to note that this only conditions the prior distribution, not the generation process itself, which we discuss Sec. 3.1.3.
Given an observation y", we aim to find a corresponding sample Xg from the prior distribution that aligns with Formally, this entails sampling from the distribution (o y²). If we have access to its score function V xx log (xo y²), we can sample from this distribution via Langevin dynamics:
(1+1) = (6) Vxp log go (x) (8) x
y²) + 22n5 with & ~ I),
where is fixed step size. With sufficient iterations, Eq. (7) converges to the conditional distribution (see Durmus & Moulines (2017) for convergence analysis).
By applying Bayes' rule, we express the conditional score function as:
Vxo log (xo y²) Vx, log q (y² Xo) + Xx log go (Xo).
(8)
Here, the term Vxp xo log & (y Xo) guides Xo towards evolving into sample (after solving the ODE) aligning with the observation y,, while the term Vx, (Xo) ensures adherence to the prior distribution's 's manifold.
As go(x)) is GP, we can compute its likelihood in closed form. However, the term @i (y xn) is unknown and requires more attention. We follow Kollovich et al. (2023) and model it as asymmetric Laplace distribution centered on the output of the flow de,1 learned by our model:
q1 (y Xx) = ALD(y" @0.1 ms).
leading to the quantile loss after applying the logarithm with quantile k. This aligns better with probabilistic forecasting, which is evaluated using distribution-based metrics. To accelerate the evaluation of the score function, we approximate the integration of the flow field in Φø.1 in Eq. (9) using a small number of Euler steps.
By dynamically choosing the number of iterations in the Langevin dynamics, we can control how strongly the sample should resemble the observation. We provide a pseudocode in Alg. 3.
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3.1.3 GUIDED GENERATION
To perform conditional generation more effeettivly, we can employ guidance techniques (Dhariwal & Nichol, 2021; Kollovieh et al., 2023) to modify the score function of the model, directly conditioning the generation process on the observed data y Unlike the previous method, where we only conditioned the prior distribution, we now adjust the dynamics of the generation process itself.
We start by modeling the conditional score function using Bayes' rule, similar to Eq. (8):
Vx, log (Xt y") Vx, Py("
Xxx) + Vx, log (x(x)).
(10)
Here, the term lo Xz) acts as guidance term that steers the generation process toward producing samples consistent with the observed past yP.
To incorporate this guidance into the generation process, we adjust the vector field uo and subtract the guidance score, scaled by factor s:
(t, = (t, - So, V log (y² xx).
(11)
We provide a detailed derivation of üأ [x, t) in App. C. By integrating this modified vector field üg over time, we generate samples that are conditioned on the observed data yP. The parameter allows us to control the influence of the conditioning, with larger values s resulting in samples that more closely resemble the observed past. To model the guidance score, again follow (Kollovieh et al., 2023) and use an asymmetric Laplace distribution centered around the output of the flow as before:
Pr xx) ALD(y") Φ0.1 (x.). K)),
(12)
where k is parameter controlling the asymmetry of the distribution.
By incorporating the guidance term into the vector field and appropriately modeling the guidance score, effectively condition the generation process to produce forecasts consistent with the observed past, enhancing the model's 's ability to generate accurate and coherent time series predictions.
3.2 CONDITIONAL MODELING
TSFlow imposes additional structure on the prior distribution Yo in the form of a Gaussian process, a non parametric time series model. We build upon this and propose an alternative training and inference approach for conditional generation, i.e., forecasting, with TSFlow. Instead of only conditioning the sampling process on an observed sample yP, we additionally condition the prior distribution on the observed past data by factorizing (x y²) with go(x )²) 90(x; y²).
While Tong et al. (2023) couple xo and y via the optimal transport map π during training (Sec. 2.3), exploit the GP prior and define the joint distribution g(xo,y) 41 (y) 9u(Xo y²). We condition the model using equation Eq. (6) during inference. Consequently, our choice of joint distribution (xx,,)) y) effectively trains the model with conditional prior distribution. This aligns the training process with how the model is employed during inference and improves forecasting performance (see Sec, 4.2).
Gaussian Process Regression. Since go is a GP, we can compute (x yy²) analytically. We begin by factorizing the conditional prior as:
(x(xx I 1 = (x I y²) go(x/ I y)))
which follows by assuming independence of x" and xf given y We choose (x y") = 8(x²- y)") to retain the information from the observation. For the unobserved future part, we employ GPR to model the conditional distribution (Rasmussen & Williams, 2005). Specifically,
gx(x |y?)
-N((x
pfpp E5ph,
(13)
with
Hflp = EAp " and Σgpp = E,, - Enp
pp Eps
(14)
where Σ, = f). - K(f,p). Epg. Epp = K(p.p) correspond to covariance matrices computed using the kernel function K.

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Here, [(f.f) represents the covariance matrix among all future points in the series, indicating how each point in the future relates to others and similarly for (p. among past points. K (f. p) describes the covariance between future and past points, illustrating how future values are expected to relate to the observed past. Conversely, (p, f) is the transpose of K K(fp). reflecting the same relationships but from the perspective of past points relating to future points.
Rather than learning the transformation from an isotropic Gaussian to the data distribution, TSFlow leverages an informed prior simplifying the modeling process by utilizing structured, relevant historical data. Note that this approach introduces only minimal computational overhead, as the covariance matrices are computed just once, and only a single vector-matrix multiplication is required for each new instance. We provide the training algorithm in Alg. 2.
4 EXPERIMENTS
In this section, present empirical results and compare TSFlow against various baselines using real-world datasets. Our primary objectives are threefold: First, to determine whether the generative capabilities competitive to other generative frameworks. Second, to investigate the impact of different prior distributions on the performance. Third, to evaluate how TSFlow compares to other models in probabilistic forecasting. Additionally, we aim explore how conditional prior sampling of the unconditional version of TSFlow fares against its conditionally trained counterpart.
Datasets. We conduct experiments on eight univariate time series datasets from various domains and with different frequencies from GluonTS (Alexandrov et al., 2020). Specifically, datasets Electricity (Dheeru & Taniskidou, 2017), Exchange (Lai et al., 2018), KDDCup (Godahewa al., 2021), M4-Hourly (Makridakis al., 2020), Solar (Lai et al., 2018). Traffic (Dheeru & Taniskidou, 2017), UberTI.C-Hourly (FiteThiityEight, 2016), and Wikipedia (Gasthaus et al., 2019). We provide further details about the datasets App. A.I.
the
Baselines. To benchmark the unconditional performance of TSFlow, experiments include Time (Desai et al., 2021) and TSDiff (Kollovich et al., 2023), representing different types of generative models. To assess the forecasting performance of TSFlow, i.e., conditional generation, compare against various established time series forecasting methods. This includes traditional statisti- cal methods such as Seasonal Naive (SN), AutoARIMA, and AutoETS (Hyndman al., 2008). In the domain of deep learning, we evaluate TSFlow against established models including DLinear (Zeng al., 2023), DeepAR (Salinas al., 2020), TFT (Temporal Fusion Transformers) (Lim et al., 2021), WaveNet (Oord et al., 2016), and PatchTST (Nie et al., 2022). Finally, we extend our comparison to include the four diffusion-basee approaches CSDI (Tashiro et al., 2021), TSDiff (Kollovieh et al., 2023). SSSD (Alcaraz& & Strodthoff, 2022), and (Biloš et al., 2023) which represent generative models in probabilistic time series forecasting. We provide more information in App. A.4.
Evaluation Metrics. To assess the generative capabilities of TSFlow in an unconditional setting, we calculate the 2- W-ssssrsttein distance between the synthetic and real samples. Specifically, we generate 10,000 samples using our defined context length for this evaluation. Additionally, to compare the quality of the synthetic samples in a downstream task, we employ the Linear Predictive Score (L.SS) (Kollovieh et al., 2023), which is defined as the (real) test CRPS of linear regression model trained on synthetic samples (see App. A.5 for more details).
To evaluate the probabilistic forecasts, we use the Continuous Ranked Probability Score (CRPS) (Gneiting & Raftery, 2007), defined as
CRPS(F-1.V) /-' (F- 3) dn,
(w- (p<g)) (y- 9) represents the pinball loss at specific quantile level k and F is the cumulative distribution function of the forecast. CRPS is proper scoring function that reaches its minimum when the forecast distribution F coincides with the target value y. As computing the integral is generally not feasible, we follow previous works (Rasul et al., 2021; Kollovich al., 2023; Rasul et al., 2020) and approximate the CRPS using nine uniformly distributed quantile levels ... 0.9). The randomized methods approximate F using 100 samples.
where (n. ()
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Practical Considerations. TSFlow uses DiffWave (Kong et al., 2020) architecture with S4 layers (Gu et al., 2021) similar to previous works (Kollovieh et al., 2023; Alcaraz & Strodthoff, 2022). We depict the architecture in Fig. 3. The conditional model (see Sec. 3.2) gets additional imput c, which contains yP and a binary observation mask indicating which steps are observed. We train the model using Adam (Kingma & Ba, 2014) with learning rate of 10-3 and gradient clipping set to 0.5 and use a standard Euler ODE solver. Furthermore, we do not fit the Gaussian processes to the corresponding datasets keep the prior distributions simple. Finally, we report the mean and standard deviations of five random seeds for all experiments to ensure reproducibility.
4.1 UNCONDITIONAL GENERATION
We test the unconditional generative capability of TSFlow in three experiments. First, we investigate how well the synthetie samples align with the real data. Then, we train downstream model on synthetic data and compare to the performance of the real data. Finally, we test conditional prior sampling (see Sec. 3.1.2) to condition TSFlow during inference time.
4.1.1 GENERATIVE CAPABILITIES
We follow Tong al. (2023) and compute the 2- Wasserstein distance of 10,000 synthetic and real samples for TSFlow using different priors to investigate whether the optimal transport problem is simplified with non-homoscedastic distributions while keeping confounders constant. We report the results in Tab. 1.
Table 1: 2- Wasserstein distance between real and synthetic samples on eight real-world datasets for different generative models and prior distributions. Best scores in bold, second best underlined.
Mothod
NEE
Electr.
Exchange
KSOCop
(8)
Solar
Traftic
UberTLC
MEK12000
TineVAE
a.201.0..0
0.034aa.cco
29.976
7.071
7.537a0
3.826in
79.36.31..00
218.764.1..00
3,112s0.2
0.067
6.095220
7.228a0
67.
T82 ow
Isotropic
2.929aa.001
0.027
27
675
5.19310
7.515ammm
62.019
211.360hu1.54
60
2 2.000:1..0s
0.020:
29. 03..13
6. 6.509
7.253
60 338 =oUcm
203.570
9B
2 7.70Lanune
0.027
23. 781
6.5 6.583em1
388
7,575
80, 00.4522..20 ses
205, 4.--788 L.814
22
2 2.550...00
0.027 anccoo
27 07.0 200
6.70440
0.89210045
7.450
00.901 60.323
205.856 4.504
Isotropic
2. 2.752
m.025e
26 332
5 5.910summ
65, a0.
204 2.0:2.712
25
200.008-+1
2.625-n.002
25.
4.42910
6.991
64.172:0.90
199.8800 190.860-17.43
2. 2.542:0002
0. 12033
25. 23.472asxn0
5. 5.992su0::
4.362
7,051
64.102
200.620
We observe that the three domain-specific Gaussian process priors outperform the isotropic prior, except on the datasets Exchange and UberTLC. Furthermore, on most datasets, the non-isotropic prior distributions with 4 NFEs are competitive with the isotropic prior with 16 NFEs. All variations of TSFlow consistently outperform Time demonstrating its strong generative capabilities. TSFlow also outperforms TSDiff with 16 NFEs and remains competitive when reducing to 4 NFEs. These results align with our observations Fig. 6 and demonstrate that choosing a suitable prior distribution improves unconditional generation.
4.1.2 TRAINING DOWNSTREAM MODELS
In addition to the 2- Wassersttii distances, we compute the Linear Predictive Score (LPS), which measures the performance of a linear regression model trained on synthetic samples and evaluated on the real test set. We present these results in Tab. 2.
Table 2: LPS for different generative models on eight woal-world datasets. Best scores in bold, second best underlined.
Method
Electr.
Exchange
XDDCup
154 (H)
Solar
Treffic
Uber1LC
Wiki2000
TimavAR TSDiÉE
0,161100m
0.0121.000
5.853
0.053
0.8841o.00
0.504 noees
0.754 _o.02
0.891 10.0rs
0.0
0.010
0.651
0.042:0:044
0.63440.c0
0.2245
0.392
0.369:0.27
8 Listropic
0.08 002
0.010
0 0.091+0.00: +0.005
0.6177 600770000
0.360=n.005
a..49-0.005
0.096440 007
0.011+0.000
0.722=0004
0.032+0.001
0.616
0.247=0.00
0.352=o.c0
0.361
34 SE
0.102 o07
1.011=0.00
0.726
0.032g0:00:
0.621 1:0 001
0.241=0.0m 1
0.35 2:0
0.36520.00 0.0
0.083.
0.0121.000:
0.9001o003
0.029_...0
0.590
0.225
0.320_0.006
0.339.
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As observe, all variations of TSFlow consistently outperform Time and, on four out of eight datasets, also surpass TSDiff. The periodic prior outperforms the other priors, achieving the best scores on six datasets. TSDiff only marginally outperforms TSFlow on the Exchange and KDDCup datasets.
4.2 PROBABILISTIC FORECASTING
Table 3: Forecasting results (CRPS) on eight real-world datasets for various statistical and neural methods. Best scores in bold, second best underlined.
Method
Electr.
Exchange
KDDCuP
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We evaluate two variants of TSFlow: a conditional model that uses domain-specific Gaussian process regression prior (TSFlow-Cond) see Sec. 3.2), and an unconditional model that applies conditional prior sampling and guidance (TSFlow- Uncond., see Sec. 3.1.2 and 3.1.3). Tab. 3 shows that TS- Flow is competitive with statistical and neural baselines and achieves state-of-the-art results on various datasets. On 6/8 datasets, TSFlow attains the best scores with up to 14% im- provement to the second-best method. Furthermore, TSFlow- Cond. outperforms or matches the ceffusiin-bassed baselines CSDI, SSSD. (Biloš al., 2023), and TSDiff on 7/8 datasets while requiring fewer NFEs. Finally, we observe that, while TSFlow-Uncond. competitive with other baselines, slightly inferior to the conditional versions of TSFlow, show- ing that the conditional training is an effective method to specialize TSFlow for forecasting. Furthermore, while all kernels yield similar results, the -hieebbbb prior slightly improves over the others. We show example forecasts of TSFlow-Cond on the Traffic dataset in Fig. 2 and for Elec- tricity and Solar in App. A.9. Furthermore, we provide ablation study of TSFlow's different components in App. B.2 and extend the forecasting setting in Apps. B.6 to B.8.
Ground Truth
Forecast
95% CI
0.02
0.00
0.2
0.1
0.0
Jun 13
Jun 14 Jun 15 Jun 16
Figure 2: Example forecasts and ground truth of TSFlow-Cond. (OU) of the first three time series the test set of Traffic.
RELATED WORK
Diffusson Models and Conditional Flow Matching. Diffusson models have been applied to several fields and achieved state-of-the-a performance (Ho et al., 2020; Hoogeboom et al., 2022; Lienen et al., 2024). Various works have demonstrated the effectiveness of unconditional models in conditional tasks through diffusion guidance (Epstein et al., 2023; Dhariwal & Nichol, 2021; Bansal et al., 2023; Nichol et al., 2021; Avrahami et al., 2022), solving inverse problems (Kawar et al., 2022), or iterative resampling (Lugmayr al., 2022). Conditional Flow Matching (Lipman et al., 2022) is a recent approach proposed as an alternative to and generalization framework has been extended to incorporate couplings between data and prior samples, and trained to solve the dynamic optimal transport problem (Tong et al., 2023). Albergo et al. (2023) demonstrated the applicability of data-dependent couplings for stochastic interpolants to improve in-painting and
diffusion models. This

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super-resolution image generation. Lastly, recent work shows the potential for conditional generation by solving an optimization problem and differentiating through the flow (Ben-Hamu et al., 2024).
Generative Models for Time Series. Various generative models have been adapted time series modeling, including Generative Adversarial Networks (GANs) (Yoon et al., 2019), normalizing flows (Rasul al., 2020; Alaa al.. 2020). and Variational Autoencoders (VAEs) (Desai al., 2021). Recent works have successfully applied diffusion models to time series. The first approach, TimeGrad (Rasul et al., 2021), applies diffusion model on top of an LSTM to perform autoregressive multivariate time series forecasting, which was later extended by Biloš et al. (2023) to settings with continuous time and non-isotropic noise distributions. Note that unlike TSFlow, Biloš et al. (2023) neither use optimal transport paths nor couplings. CSDI (Tashiro al., 2021) and SSSD (Alcaraz & Strodthoff, 2022) perform forecasting and imputation via conditional diffusion models. While CSDI uses transformer layers, SSSD makes use of $4 layers (Gu et al., 2021) to model temporal dynamics. Lastly, Diff (Shen Kwok, 2023) explores conditioning mechanisms, and TSDiff (Kollovieh al., 2023) introduced unconditional diffusion model conditioned through diffusion guidance similar to our unconditional model.
Additionally, Tamir al. (2024) model dynamical systems using conditional flow matching (CFM), where the time series and ODE share the same time dimension. In contrast, TSFlow operates on two orthogonal time dimensions, one for the generative process and another within the time series. Moreover, Kerrigan et al. (2023) extend flow-matching to infinitediimensional spaces using Gaussian processes, but their focus differs from ours. While they construct conditional Gaussian measure paths, TSFlow directly parameterizes the joint data and prior distribution.
6 CONCLUSION
In this work, we introduced TSFlow, a novel conditional flow matching model for probabilistic time series forecasting. TSFlow leverages flexible, data-dependent prior distributions and optimal transport paths to enhance unconditional and conditional generative capabilities. Our experiments on eight real-world datasets demonstrated that TSFlow consistently achieves state-of-the seae---f-thh-art-erieeeee art--eefformance.
We found that non-isotropic Gaussian process priors, particularly the periodic kernel, often led to better performance than isotropic priors, even with fewer neural function evaluations (NFEs). The conditional version of TSFlow with Gaussian process regression priors showed improvements over diffusion based approaches, further emphasizing the effectiveness of our proposed methods. Additionally, we demonstrated that the unconditional model can be effectively used in conditional setting via Langevin dynamics, offering additional flexibility in various forecasting scenarios.
Limitations and Future Work. While TSFlow achieves state-of of-the--rtt performance across multi- ple benchmarks and reduces the computational costs compared to its diffusion- predecessors, it has primarily been tested on univariate time series. Furthermore, different datasets and tasks, e.g.. unconditional and conditional generation, require kernel selection, which should be optimized on the validation set. Future work could extend our approach to multivariate time series by incorporating multivariate Gaussian processes that capture dependencies across dimensions. Additionally, TSFlow allows for arbitrary source distributions, enabling the use of more involved priors, such as those based on data statisties (Park al., 2024) neural forecasting methods with intractable likelihoods. Exploring such distributions could further enhance performance.
REPRODUCIBILITY STATEMENT
To ensure the reproducibility of our results, we provide a detailed description of our experimental setup, including the benchmark datasets and evaluation metrics, in Sec. 4 and App. A. Additionally, we outline the hyperparameters used in App. A.2 and include the pseudocodes for our unconditional and conditional model in App. 1.
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A EXPERIMENTAL SETUP AND HYPERPARAMETERS
A.1 DATASETS
We used eight commonly used datasets and train/test splits from GluonTS (Alexandrov et al., 2020)
encompassing various domains and frequencies. In Tab. 4, we show an overview of the datasets.
Table 4: Overview of the datasets and their statistics used in our experiments.
Dataset
Train Size
Test Size
Domain
Freq.
Median Seq. Length
Prediction Length
Electricity"
370
2590
H
5833
24
Exchange
40
R
D
6071
30
KDDCup
270
270
N
H
10850
48
M4 (H)
414
414
H
960
48
Solar
137
959
R
H
7009
24
Traffic
963
6741
(0,
1)
H
4001
24
UberTLC
262
262
N
H
4320
24
Wikipedia"
2000
10000
D
792
30
A.2 HYPERPARAMETERS AND TRAINING DETAILS
Connai
Connh
(t,)))
Sigmoid
tanh
Element-wise mult.
Elementwwise add.
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 Xt, the timestep f, and in the case of the conditional setup, a condition c, which contains y and binary observation mask. The prediction the approximated flow field uo (t,
We trained both the conditional and unconditional variants of TSFlow using the Adam optimizer with learning rate of 10-3 applying gradient clipping with threshold of 0.5. The conditional model was trained for 400 epochs, while the unconditional model was trained for 1000 epochs. Each epoch consists of 128 batches, with each batch containing 64 sequences. To stabilize the training process, we employed an Exponential Moving Average (EMA) of the model parameters with a decay rate of = 0.9999. To parametrize the Gaussian processes, we choose = /1/2, = 1, and /2, for the SE, OU, and PE kernels, respectively, simplifying these further.
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TSFlow uses context length equal to the prediction length and includes additional lag features in the feature vector c. Since "SFlow-Uncone is an unconditional model and does not utilize feature inputs, increase the context window to 336 for hourly data and 210 for daily data.
The architecture of TSFlow is shown in Fig. 3 and consists of three residual blocks with a hidden dimension of 64, resulting in approximately 176,000 trainable parameters. We encode timesteps using 64-dimensional sinusoidal embeddings (Vaswani et al., 2017).
To solve the ordinary differential equation (ODE), utilized Euler solver with 32 steps. The unconditional version, TSFlow-Une -SFlow-Uncond, uses 16 additional NFEs due to the additional conditional prior sampling. which involves 4 iterations with 4 steps each. The unconditional version in Tab. uses 4 steps. TSFlow- USSFoow-Uncond performs four iterations to sample from the prior distribution with step size of 7 = 0.005 and noise scale of 0.5. The guidance scale parameter is selected between 8 and 32 based on the validation set's performance. Finally, we choose к in the guidance score uniformly between 0.1 and 0.9. overview of the hyperparameters and training details is provided in Table 5.
Table 5: Hyperparameters of TSFlow.
Hyperparameter
Value
Learning rate
10-3
Optimizer
Adam
Batch size
64
Gradient clipping threshold
0.5
Epochs
400 / 1000
EMA momentum
0.9999
Residual blocks
3
Residual channels
64
Time Emb. Dim.
64
ODE Steps
32 16/
ODE Solver
Euler
Guidance strength
/ 24 /32 [4 0.9]
Quantile
min
10
10 + /1
O max
Normalization. account for varying magnitudes across time series within the dataset, we normalize each series by dividing its values by the global mean of the training set:
Dorm
menn Xhist)
where mean denotes the mean aggregation function, and Xhist encompasses the entire (train) time series beyond the context window.
A.3 ASYMMETRIC LAPLACE DISTRIBUTION
In Sec. 3.1.2 and 1.3, we parameterize the guidance score using an Asymmetric Laplace Distribution (ALD). The probability density function (PDF) is defined with location parameter a scale parameter X, and a quantile k:
p(r; m. A, k) & exp (- max(k - (m-m), (n 1) (x (x-m))
(15)
By setting 1, the guidance score from Eq. (12) simplifies the quantile loss:
xx ly" xx)) = max(æ: (y é (x:)), (N - 1) (y Φo (x:)))
(16)
which corresponds to a weighted é loss. In practice, we select x uniformly from the range (0.1, 0.9) ensure coverage across different quantiles.
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A.4 BASELINES
We utilize the official implementations provided by the respective authors for CSDI, TSDiff, and Biloš al. (2023). Although the original CSDI paper focuses on multivariate time series experiments. their codebase supports univariate forecasting, allowing us to use it without any modifications with their recommended hyperparameters and training setup. As SSSD targeted primarily multivariate datasets, we adjusted their model to the univariate case. Specifically, to avoid overfitting, we set the hidden dimension to 64 and the number of residual layers to 3, aligning the network sizes with those of TSDiff and TSFlow. For PatchT'T, ARIMA, and ETS, we use AutoGluon (Shchur et al., 2023). For the remaining baselines, we employ the publicly available implementations in GluonTS (Alexandrov et al., 2020) with their recommended hyperparameters.
A.5 METRICS
Linear Predictive Score (LPS) To assess the performance of the unconditional models in down- stream forecasting, use the Linear Predictive Score (LPS) proposed by Kollovieh et al. (2023). The LPS is computed as the test CRPS of a linear ridge regression model trained on synthetic samples generated by the model. Specifically, the linear model is trained to map the observed past of a time series, y E RL", to its future values yd IRLL, The linear model is regressed against the true future values, using the past y" as input, allowing to measure how well the synthetic samples capture the relationship between past and future. Since the linear ridge regression can be fit closed form, it remains robust to random initialization. Following Kollovieh et al. (2023), we generate 10,000 synthetic samples to fit the linear model.
A.6 UNCONDITIONAL PRIORS
Since the observations occur at discrete time steps, i.e., Ti =; for porr=00.... O,..., L- 1, we normalize them to align with the periodicity of the dataset. More specifically, we adjust the time steps using the formula:
norm
Ti π
(17)
P
where p represents the frequency length of the time series; for instance, 24 for hourly data and 30 for daily data. This normalization scales the time steps so that one full period corresponds to æ, ensuring that the periodic kernel accurately captures the cyclical patterns in the data. Examples of the different processes for various kernels are illustrated in Fig. 4. To avoid numerical issues such as ill-conditioned covariance matrices, we extend each kemne K with white noise kernel K (wh 7') = ó(T -T') to K'(T, r') (TT,-') + Kwhite (T, 7'). Adding this white noise ensures that the covariance matrix positive definite and improves numerical stability during computations.
A.7 CONDITIONAL PRIORS
Before applying Gaussian process regression, we apply frequency- based z-score normalization to the data, which standardizes each segment according to its periodicity. For each data point x; the transformation is defined as:
(18) where Ho is the mean computed over segments of length corresponding the periodicity p of the time series. This normalization ensures that each segment of the data, organized by its periodic frequency. has a mean of zero.
norm
X; - Ppp
A.8 RUNTIME
In App. A.8, we provide the runtime of TSFlow the Solar dataset for both training and evaluation on an Nvidia A100 GPU. The evaluation time includes generating 100 sample forecasts for each time series in the dataset, i.e., generating 95900 samples.
We observed minimal differences in runtime between the isotropic and Gaussian process regression (GPR) priors. We attribute this minor difference the computations in Eq. (14). Apart from single
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Ornstein Uhlenheck
Radial Basis Kernel
Periodic
M
2.0
2.0
2.0
0.0
0.0
1.0
2.0
mnn
MMMMWN
M
Am
1.0
-2.0
2.0
Mm
2.0
1.0
0.0-
2.0
1.5
w
0.0
0.0
1.0
2..0
0.5
WW
0.5
0.0-
0.0
0.0
-2.0
50
100
150
50
100
60
100
150
Figure 4: Examples of samples from the unconditional Gaussian processes for T= 150 and p 30.
Table 6: Comparison of different configurations with their corresponding training and evaluation
times.
Configuration
Train Epoch
Eval.
TSFlow-Cond. Isotropic prior
2.94
24.29
TSFlow Cond. GPR prior
3.17
24.65
TSFlow- TSFlow-Uncond. Random
3.66
ASFlow-Unccome Optimal Transport
3.77
CPS Guidance
81.97
matrix-vector multiplication, all computations only need be performed once, which minimizes the overhead associated with the GPR prior.
In the unconditional model, observed neglectable computational overhead when using optimal transport maps during training. For evaluation, however, it takes approximately three times longer than the standard conditional version. We attribute this increase two factors: the use of longer context window (see App. A.2) and the need to differentiate through the ODE integration process.
A.9 ADDITIONAL FORECASTS
We show more example forecasts of Tonlow-Cond. on Solar and Electricity in Fig.
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Published as a conference paper at ICLR 2025
Ground Truth
Forecast
95% C1
Ground Truth Forecast
95%
200
100
100
100
100
100
0
Oct 17
Oct 18
Oct 19
Oct 20
Oct 21
Aug 29 Aug 30
Aug 31 Sop
Sop 02
(a) Solar
(b) Electricity
Figure 5: Example forecasts of TSFlow-Cond. on the test of the datasets Solar and Electricity.
A.10 ALGORITHMS
A.10.1
TRAINING ALGORITHMS FOR TSFLOW
We provide the training algorithm for the unconditional and conditional versions Alg. and Alg. 2.
Algorithm 2: Conditional Training of TSFlow
1: Input: Conditional prior distribution go. data distribution J1, noise level 0t, network 21
2: for iteration 1,... do
3:
< (y);; Xa ~ (x y²) t~ 10,
t sample batches from the conditional prior and dataset sample random time step
4:
5:
ty + +y+(1- t)=0
>> compute mean of pe sample from Pe(- z) D regress conditional vector field gradient step
6:
X, &~~Ni ~ 021)
7:
[(0) + (t. Xt, y²) (x x)]l
8:
0 + Updatacc Vo((O))
9:
end for
10: Return: ug
A.11 EVALUATION ALGORITTMS FOR TSFLOW
We provide formal description of the conditional prior sampling algorithm in Alg. 3.
Algorithm Conditional Prior Sampling (without generation)
Input: Prior distribution fo network ug. observation y", prior sample x
2: for iteration = 1,
do
3: & ~ N0,0
D sample noise b> update xo
(f+1) =x V xo log (x y /
(6)
4: Xg
5: end for ((+1)
6: Return: X6
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Published as a conference paper at ICLR 2025
B ADDITIONAL RESULTS
B.I EFFECT ON THE OPTIMAL TRANSPORT PROBLEM.
To understand the effect of the prior distribution, we measure the Wasserstein distance between batches (x0" go) drawn from the prior and batches {y(0) 91) drawn from the data distribution
across four common benchmark datasets in time series forecasting. Our results Fig. 6 show
Isotropic
Squared Exponential
Omstein-Uhlenheer
Periodic
Relative W/3 Distance
Solar
Electricity
Traffic
Exchange
1.0
0.8
0.6
4
Length (x Freq)
Length (x Freq) Length (x Freq)
Length (x Freq)
Figure 6: Mini-batch Wasserstein W distances between samples from prior and data common benchmark datasets. x-axis shows data dimension in multiples of each dataset's 's base period, e.g., 24h for Solar. Distances relative the transport distance from random transport map as used in CFM.
that Gaussian process priors reduce the Wasserstein distance between the prior and data distribution if the kernel exploits characteristics of the data. When we parametrize the prior with SE and OU kernels, the Wasserstein distance to the data distribution decreases substantially, although it slowly converges to an isotropic prior as the sequence length increases. Notably, the periodic kernel achieves the largest reduction in distance on these datasets with only marginal growth as the sequence length increases because effectively captures the periodicity - a common feature of real-world time series especially well. These findings support our hypothesis that an informed choice of prior distribution can substantially affect the learning problem.
B.2 ABLATION
In the following, we describe and ablate different components of our conditional model. We use the same experimental setup as described in App. A.2. The base model corresponds to a basic Conditional Flow Matching model with an isotropic prior distribution.
Ornstein Uhlenbeck Prior. The CFM framework allows one to accommodate arbitrary prior distributions. Instead of using an isotropic Gaussian, we use a Gaussian Process regression based on the Omstein Uhlenbeck kernel as explained in Sec. 3.2. The results demonstrate improvement of the CRPS on all datasets, demonstrating the advantage of conditional prior distributions.
Lags. To extend information into the past without increasing the context window, we use lagged values as features. More specifically, we use the default lags deployed by Gluon'TS (Alexandrov et al., 2020), which are also used by various baselines.
Bidirectional S4 layers. To allow information flow in both temporal directions, employ bidirectional S4 layers. This allows the model to generate consistent forecasts and slightly improves the results.
Exponential Moving Average. To stabilize training, we employ exponential moving average (EMA), which keeps a copy 0' of the parameters 0 and is updated via a momentum update after each gradient step:
0' - m0' + 01-m)0
where m is a momentum parameter. To generate forecasts, we use the conditional vector field ugr instead of ug-
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Published as a conference paper at ICLR 2025
Table 7: Ablation forecasting results (CRPS) for TSFlow-Cond. on five real-world datasets. Best scores bold, second best underlined.
Mathod
Electr.
KDDCup
Solar
Troffic
UberTIC
Basemodel
0.058=0.002
0.302
0.339=00.00
0.133#+0.001
0.199::0.004
Qu Frior
0.049=u.0e
0.273++0.007
0.333++.000
0.107 0.107=+0.001
0.161+00.004
Lags
0.046-0.0m 001
0.39240 025
0.356=0.011
0.084++00001
0.154400.00
Bidir.
0.046=0.00m 00
0.285 010
0.341 006
0.083:0.000
0.153:10.008 0.05
EMA
0.045=+0.001
0.273:000004
0.343:00.002
0.083+t0.000
0.1533t0.001
B.3 CONDITIONAL PRIOR SAMPLING
We present the effect of conditional prior sampling (CPS) in Table 8 on the benchmark datasets, comparing it to model that employs only guidance without CPS.
Table Effect of the CPS iterations on the forecasting results (CRPS) for TSFlow-Uncond. on eight real-world datasets. Best scores in bold.
R1c.
Electz.
Exchance
KUDCup
84deasly
Solas
Traffic
M1k12000
0
0.049=0.001
0.012=0.001
0 0.294+000002
O. 0.029=0.00
O 00.0000000
0.096+000000
0.154+00.001
0.290
0.049
0.0111..001
09.0000004
0..29.0001
0.46610.004
0.88...0000
0.153L0..00
0.27910.007
As shown in the table, on out of datasets, incorporating CPS yields equal or better CRPS scores compared to the model that uses only guidance. The exception is the Solar dataset, where CPS performs worse than the guided-only model. This suggests that solely using guidance is insufficient for effectively conditioning TSFlow-Uncome during inference.
B.4 GUIDANCE SCORE FUNCTION
In Eqs. (9) and (12), we utilize an asymmetric Laplace distribution (ALD) to model the guidance score. Alternatively, this score could be modeled using a Gaussian distribution, which simplifies to a mean squared error loss after applying the logarithm. As demonstrated in Tab. 9, the asymmetric
Table 9: Effect of the score function on the forecasting results (CRPS) for TSFlow-Uncond. on eight real-world datasets. Best scores in bold.
Disce. Electr.
Exchange
XDDCup
WcHorrl
Teaftio
UbesTLC
Guassian
0.020 0.04910.001
0.05910.001
0.017
0.0311
0.10010.010
0.101=0.001
0.1694:0.005
24.045:16.
0.01110.001
0.200-0.004
0.02910.001
0.46410.0004
0.08010.000
0.1531000002
0.278.10.00
Laplace distribution yields better results across 6 out of 8 datasets. We attribute this to the fact that different quantile parameters take the whole distribution into account rather than focusing single point estimate.
B.5 NON-GAUSSIAN PRIOR DISTRIBUTIONS
In addition Gaussian process priors, TSFlow support non-Gaussian priors for (x yP). Instead of parameterizing the prior via Gaussian process regression, we define it as
go(x y²) = N (x1 g(y²).T)
where g is a base forecaster. We demonstrate this approach using the Seasonal Naive (SN) forecaster and compare it to the Ceustein-Uhlebbed prior in Tab. 10.
B.6 NEURAL FUNCTION EVALUATIONS
In Tab. 1, we present the Wasserstein-2 distances for and 16 Neural Function Evaluations (NFEs). To further illustrate the performance, show Fig. 7 the Wasserstein-2 distances across varying NFEs for selected datasets. As demonstrated in the figure, the non-isotropic priors achieve better performance than the isotropic prior.
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Published as a conference paper at ICLR 2025
Table 10: Forecasting performance (CRPS) eight datasets for the seasonal naive (SN) prior compared to the Omstein-UUlenbecck prior. Best scores are in bold.
Pxios
Ceaffic
UberTSc
Wike
0.044 1.0. 001
0.008 1.0,000
0.321
10.001
0.348 1:0 806
0.082 1000000
0.104 1.0. 000
23333 1.0. 0012 1.0000
0.045
10.001
0.273 10.004
0.025 10.001
0 00.002
0.080 00:000
1.0000
Isotropic
Squared Exponential
-- Constein-Ubleabeee
Periodic
Relative W? Distance
Solar
Electricity
Traffic
Exchange
1.0
0.5
0.0
4
16
16
16
Steps
Steps
Steps
Steps
Figure 7: Wasserstein W3 distances between samples from unconditional generative model and data on common benchmark datasets. x-axis shows NFEs using an Euler solver.
B.7 FORECASTING WITH MISSING VALUES
In our experiments, we focus on forecasting due to the availability standardized benchmarks and to ensure fair comparisons with autoregressive models, which do not benefit from bidirectional information passing. However, TSFlow is not restricted to forecasting setting and can be extended arbitrary conditional generation tasks by adjusting the observation mask.
To demonstrate this, we evaluate forecasting scenarios with missing values in the context window. Following the setup by Kollovieh al. (2023), test three missing values scenarios with 50%
missing rate: (I) random missing (RM), where observed values are randomly masked; (2) blackout missing beginning (BO-B), where values are missing the start of the context window; and (3) blackout missing end (BO-E), where values are missing at the end of the context window just before the forecast. For these experiments, we use the conditional version of TSFlow with the OU kernel and compare it against TSDiff, reporting the best result (conditional or unconditional) for TSDiff in each case. The results for the three setups are presented in Tabs. 11 to 13, respectively.
Table 11: Forecasting results (CRPS) with missing values in the context window (RM). Best scores in bold.
Method Electricity
Exchange
KDDCuP
Solar
Traffic
UberTiC
7SP10w
0.048:10.001
0.009-10.00 001
0.432-10.1
0.362 0.357 023
0.000
0.157#0.000 002
CEDiEE 0.052=0.001
0.012:0.000 001
0. 0..122
0.097 003
0.180=0.015
Overall, TSFlow outperforms both versions of TSDiff in 12 out of 18 cases, demonstrating its robustness across these scenarios.
B.8 MULTIVARIATE FORECASTING
We extend TSFlow to the multivariate setting by incorporating transformer across the feature dimension, following the approach of CSDI (Tashiro et al., 2021). l). To evaluate its effectiveness, we compare this multivariate extension (MV) to chamnel-independent variant (CIDP) in Table 14. For reference, we also report the forecasting performance (CRPS-Sune Sum) of CSDI (Tashiro et al., 2021) and the method proposed by Biloš al. (2023). The results show that yields the best results 3 out of 4 datasets.
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Published as a conference paper at ICLR 2025
Table 12: Forecasting results (CRPS) with missing values in the context window (BO-B). Best scores in bold.
Xethod
Electricity
Exchange
Solar
Traffic
VEerTLC
SSPlow
0.047:00000 0.049
0.008:10.00
0. 305 00.013
0.396:20.028
0.08100000
0.158
7SD1 ff
0.009:10.000
0.441 0.441:10.09
0.377:10.017
0.094.10.005
0.181 1.0009
Table 13: Forecasting results (CRPS) with missing values in the context window (BO-E). Best scores in bold.
Method Electricity
Exchange
KDDCup
Solar
Trafric
UberTIC
CSPlow
0.061 00.011
0.056:00. 004
0.384:0..00
0.418=0.011
0.106#0.002 0.123:40.00
0.165##0.00
DEDiSE
0.063=0.003
0.020
0. 3.44000012
0.37640 036
0.179=0.011
Table 14: Forecasting performance (CRPS-Sum) on four datasets. Best scores are in bold.
Model
Slect
Exchange
Solar
TraCTic
Bilok ct (2023)
0.0260 +0.0011
0.0071 +:0 0009
0.3701 =0.0179
1.2679 +1.0624
CSDI
0.0184 +0.0008
0.0076 0022
0.3179 =0.0197
0.0221
5000012
TSPlow (CTD2)
0.0196 #0.0006
0.0060 40.0004
0.3780
=0.0074
0.0240 #0.0005
rSPlow (VV)
0.0176 +0 0012
0.0073 40.0000
0.2792 =0.0151
0.0217 20 20.0013
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Published as a conference paper at ICLR 2025
C GUIDED GENERATION
To enable guided generation, we use slightly modified conditional probability paths defined as: pr (xt x) - N (x:5 s4.; oi
(19)
where μ4 = - t)xo + tx1, J2 = (1 max + to and max and min are constant standard deviations with Fmxx Ouin- Note that when max minu recover the same probability path as in Sec. 2.2. However, we require max > Jmin to obtain a score function in our vector field.
Using Theorem 3 from Lipman et al. (2022), construct the corresponding vector field:
mmin
maxx
ue(x; z)) = X1 - X0 +
(x: - pe).
(20)
(1 f) + la
We observe that the vector field can be expressed using the score function the conditional probability paths in Eq. (19):
U (x
2) = X1 X
(amin Omax) o Vx, log Pt (xt z)
(21)
Here, we used the fact that the score function of the Gaussian Eq. (19) is:
(1 t)xu tx1
Xx, lx
1)
(22)
((1 t) to mii))
Marginal Vector Field. By integrating over we can rewrite the marginal vector field:
Ue (xe
2x)x((xe 22)(2)
2y (xx)
dz
(23)
(x()
(x1 Xo (Omin Fmax oo V log (x (x) 22)(2)) dz
(24)
Pt (x:)
(x) xo) 2z)(2) dz
(25)
(x(e
Vx, log
Pe log (xe 2 2))e(x: 22)((2) dz
-00min Tmax)ot
(26)
(x.)
(X,) - (amin mmax Je Vx. log Pt (x{);
(27)
where vx(x;) represents the expected drift term Ez x; [x1 - Xo).
Conditional Score Function. with its conditional counterpart:
To incorporate the observed past yP, substitute the score function
Vx, (x(: y²) Vx, (x;;)())) + Xx, (y xx),
(28)
leading to the following modified vector field:
(x,)) vx (x:) (amin 0max Xx, lx,,)(xx) (a(mmin Tmax Jo, Xx, log logpe xx), (29)
(x))
where is scaling parameter introduced to control the strength of the conditioning. We can summarize this as:
ü (x,) (x,) sa, x, logp:(y" xx),
(30)
where we have observed amin Fmax into Thus, to sample from the conditional generative model Pe(;; y"), we use the modified vector field (x,) from Eq. (30) with our trained vector field u (t (t, In our unconditional experiments amin 10* and Tmax 1.
23


Reviews:
[{"review_id": "tBf42IhNBQ", "reviewer": "Reviewer_qDmP", "summary": "The existing diffusion models have problems in the time series generation since the data and prior distributions differ. The authors handle this problem by utilizing conditional flow matching framework. They propose TSFlow, which sets the prior distribution as Gaussian process to make the prior distribution close to the data distribution. Also, they propose conditional prior sampling which makes an unconditionally trained model possible for probabilistic forecasting.", "questions": "•\tThe problem only considers the univariate case. Can the model extend to the multivariate time series problem?\n\n•\tI want some more explanation about the effectiveness of informed prior distributions. Why does closedness of the prior and data distribution imply easy learning. Do you have any experiments about train efficiency or path efficiency? \n\n•\tCan the given prior (Gaussian process) extend to the arbitrary prior ? for example, refer to [1].\n\n•\tDo you have any theoretical evidence about how the selection of kernel functions effect to the model performance? (ex. OU kernel is better when the data follows OU process)\n\n[1] Leveraging Priors via Diffusion Bridge for Time Series Generation, Arxiv24", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "• By utilizing Gaussian process to the conditional flow matching, the model reflects the temporal dependencies of the given time series data better.\n\n• The model enables both unconditional and conditional generations.\n\n• By conditional prior sampling, the unconditionally trained model could follow the given guidance.", "weaknesses": "Please refer to the Questions section.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "r8kTEkO0yD", "reviewer": "Reviewer_jGDf", "summary": "This paper studies the use of flow matching for time series forecasting. The authors first propose the use of flow matching techniques for unconditional generation using Gaussian process priors (section 3.1.1), followed by a technique for conditioning an unconditional model by sampling a relevant prior $x_0$ (section 3.1.2) or guidance (section 3.1.3), and finally the authors discuss a technique which uses a data-dependent Gaussian process prior for conditional sampling (section 3.2). The proposed methodology is empirically validated on several univariate time series datasets.", "questions": "- The exact problem statement was a bit unclear to me. In lines 154-156 the authors describe a time series as a vector in $\\mathbb{R}^L$. Does this mean that the authors only work on time series having a fixed length $L$, or does the method allow for variable-length time series? Similarly, are the authors assuming that the time series all share a fixed discretization (i.e., there are some fixed times $t_1, \\dots, t_L$ corresponding to $y_1, \\dots, y_L$), or can the discretization vary across time series? Is this discretization assumed to be uniform, or can it be irregular as well?\n- It seems to me that the setup is not limited just to forecasting, but could be applied to general conditional generation tasks, e.g., imputation. Have the authors tried anything beyond forecasting?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 2, "strengths": "- The investigation of techniques for conditional sampling is pretty thorough, and I think the proposed methods are quite interesting\n- The proposed method obtains fairly strong empirical results, and the empirical evaluation is convincing \n- Throughout the paper is very clear and well-written", "weaknesses": "- There is some highly relevant related work that the authors do not discuss. [Functional Flow Matching, AISTATS 2024](https://arxiv.org/abs/2305.17209) proposes the use of GP priors in conjunction with flow matching and studies techniques for forecasting with these models.  Similarly, [Conditional Flow Matching for Time Series Modeling, SPIGM@ICML 2024](https://openreview.net/forum?id=Hqn4Aj7xrQ) uses GPs with flow matching for time series. The authors should cite these works and discuss the differences with their proposed method.\n- There are some (relatively minor) clarity issues throughout\n     - The use of equation 9 was a bit unclear to me. Why is this specific form of $q_1$ chosen? Some justification for this modeling choice would be good.\n     - In Section 3.1.2, I am guessing that once we sample $x_0 \\sim q_0(x_0 \\mid y^p)$, then we use $x_0$ as an initial condition for the flow model to generate new samples $y \\mid x_0$. Is this the case? If so, it would help to state this explicitly somewhere in the paper.\n     - In Line 303, the authors write \"approximating q_0(x_0 \\mid y^p)$ with $q_0(x_0 \\mid y^p)$. I think I understand what is meant here, but this seems to be a typo.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Ht8UdSWjui", "reviewer": "Reviewer_dfQz", "summary": "This paper introduces TSFlow, a model for probabilistic time series forecasting that enhances generative modeling by incorporating Gaussian Process (GP) priors within the Conditional Flow Matching (CFM) framework. The use of more informative GP priors helps align the prior distribution with the temporal structure of the data, potentially improving both performance and leading to runtime efficiency by simplifying the probability paths. TSFlow is flexible and can be used for both, conditional and unconditional, time-series generation.\n\nThe model demonstrates strong performance across several benchmark datasets, but there are aspects of the paper that could benefit from further clarification and improvement. Depending on the author's response, I am willing to increase my score from \"weak reject\" to \"weak accept\".", "questions": "See above.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "**Incorporating Gaussian Process Priors:** The main contribution—replacing the typical isotropic Gaussian prior $q(x_0)$ with a data-dependent conditional prior $q(x_0∣y^p)$ is well-motivated. GP priors are naturally suited for time series due to their ability to model temporal dependencies, and this idea is a clear innovation over existing flow matching methods.\n\n**Empirical Performance:** The empirical results show that TSFlow performs well compared to state-of-the-art models across various benchmark tasks.\n\n**Flexibility in Conditional and Unconditional Modeling:** The approach supports both unconditional and conditional generation. While unconditional generation has fewer use cases compared to conditional generation, it is a feature often overlooked in time-series analysis. The ability to use the same model for both tasks—by applying conditioning only during inference—adds versatility to the paper.", "weaknesses": "**Majors**\n\n**Difficulty in Parsing for Non-Experts:** The paper assumes substantial familiarity with flow matching and related generative methods, making it challenging for readers without a deep background in these specific techniques. It took for me considerable time to fully grasp the concepts, which suggests that the paper might also be difficult to read for a broader audience.\n\n**Lack of Runtime Analysis:**  The authors propose replacing the isotropic Gaussian prior with a more complex GP prior. Howeer, they do not address the increased computational cost that comes with using GP priors. Although the GP prior is more suited to time-series tasks, this advantage must be weighed against the computational overhead. The paper should clearly state the theoretical runtime complexity of using GP priors and provide empirical runtime comparisons in the experiments. The reduction in NFEs is a positive, but its trade-off with the computational cost of the GP prior must be considered.\n\n**Baseline Comparison:** A simple baseline for the forecasting tasks could involve using  Eq. 6 but with an isotropic Gaussian prior. Please include this method as a comparison partner in Table 3. If it is not a valid approach, it should be explained why such a baseline is excluded.\n\n**Inconsistent Findings on Kernel Choice:** The periodic kernel minimizes the Wasserstein distance in Figure 2, suggesting it aligns well with the data distribution. However, in Table 1, the periodic kernel does not significantly outperform other kernels, and it is not even considered in Table 4.2. This inconsistency is counterintuitive and warrants further discussion. Moreover, the necessity for different hyperparameter choices across tasks (e.g., generative modeling vs. forecasting) weakens the \"one model for all\" argument, suggesting more task-specific tuning may be required.\n\n**Minors:**\n\n**Missing Experiments on Guided Generation (3.1.3):** I could not find experiments on Guided Generation (3.1.3) in the paper. Where can I find them?\n\n**LPS Score Clarification:** The Linear Predictive Score (LPS) is not sufficiently explained. Please provide more details on how the LPS is calculated, specifically on what the linear model is regressed against and what the output represents.\n\n**GP Hyperparameter Optimization:** You state in your paper that you did not fit the GP hyperparameters. This is somewhat unexpected, I would have had a strong guess that learning the hyperparameters of the GP brings a large benefit to the model. Can you perform a small experiment to evaluate the difference in performance with/without GP hyperparameter optimization?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "oywKq5KOZr", "reviewer": "Reviewer_k8UA", "summary": "This paper proposes a model that uses flow-matching algorithms to generate and forecast univariate time-series. When trained unconditionally, it replaces the isotropic Gaussian prior distribution with an informed one and uses conditional sampling and inferencing. The model itself can also be made conditional. Experiments show that the model surpasses existing ones on benchmark tasks.", "questions": "1. If my understanding is correct, the overall idea of the model is to push an initial GP to the target time-series. Therefore, the sequence length $L$ in the problem formulation section is equivalent to the vector field dimension $d$ in the background section. Can the author(s) kindly confirm if this is true? It would be nice to clarify the setting a little bit. I think the main source of confusion is that there are two time indices in this paper: the time in flow-matching and the time in the time-series, and these are orthogonal to each other. I think this should be made clear somewhere in the paper.\n\n2. Following the first question, it would be nice to indicate in Figure 1 the time-series index and the flow-matching index. Moreover, different notations should be used. (For instance, on line 202-203, $t$ should not be used for the kernel because it has already been used in Eq. (1).)\n\n3. How would the model respond to a growing $L$? That is, there are two subquestions in this query:\n\ta. What is the time complexity of the model and how does it compare to other models that you benchmarked against?\n\tb. When $L$ is large, Eq. (6) seems to suffer from the curse of dimensionality and the integral would be impossible to discretize. How does the model see this issue, or is it not relevant?\n\n4. The section \"Effect on the Optimal Transport Problem\" only considers the distance between sequences but not anything about the training. Can you show some experiments where models that use the periodic kernel are indeed easier to train on tasks that involve periodicity?\n\n5. On line 302-303, you wrote \"we additionally condition the prior distribution on the observed past data by approximating $q_0(\\mathbf{x}_0 | \\mathbf{y}^p)$ with $q_0(\\mathbf{x}_0 | \\mathbf{y}^p)$.\" I assume there is a typo. What would be the intended sentence?", "limitations": "", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The model is flexible with many possible options; the informed prior sampling is intuitive.\n\n2. The model shows promising results on univariate benchmark tasks.", "weaknesses": "1. The model right now is only discussed and validated on univariate time-series.\n\n2. While there are many proposed options in the model, there lacks an ablation or a comprehensive comparison of different choices. Neither is there any theoretical insights in the proposed model.\n\n3. The presentation of the model needs a bit more clarification. The problem formulation can potentially be expanded a bit more (see questions below). The discussion of different training/inference choices is a bit dense. Maybe some concise pseudocode or flowchart in the main text could be helpful.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)