Series-to-Series Diffusion Bridge Model

## Abstract

Abstract Diffusion models have risen to prominence in time series forecasting, showcasing their robust capability to model complex data distributions. However, their effectiveness in deterministic predictions is often constrained by instability arising from their inherent stochasticity. In this paper, we revisit time series diffusion models and present a comprehensive framework that encompasses most existing diffusion-based methods. Building on this theoretical foundation, we propose a novel diffusion-based time series forecasting model, the Series-to-Series Diffusion Bridge Model ( S 2 ⁢ DBM superscript S 2 DBM \mathrm{S^{2}DBM} roman_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_DBM ), which leverages the Brownian Bridge process to reduce randomness in reverse estimations and improves accuracy by incorporating informative priors and conditions derived from historical time series data. Experimental results demonstrate that S 2 ⁢ DBM superscript S 2 DBM \mathrm{S^{2}DBM} roman_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_DBM delivers superior performance in point-to-point forecasting and competes effectively with other diffusion-based models in probabilistic forecasting.

## 1 Introduction

Diffusion models (Ho et al., ; Song et al., ) have emerged as powerful tools for time series forecasting, offering the capability to model complex data distributions. Building on their success in other domains, such as computer vision (Saharia et al., ; Rombach et al., ) and natural language processing (Reid et al., ; Ye et al., ), researchers have increasingly applied diffusion models to time series prediction. This approach has shown promise in capturing the intricate temporal dependencies and uncertainty in time series data, leading to significant advancements in forecasting accuracy and reliability (Rasul et al., ; Tashiro et al., ; Alcaraz & Strodthoff, ; Li et al., ).

However, the inherent stochasticity of diffusion models makes multivariate time series forecasting challenging. Specifically, most of these methods employ a standard forward diffusion process that gradually corrupts future time series data until it converges to a standard normal distribution. Consequently, their predictions originate from pure noise, lacking temporal structure, with historical time series data merely conditioning the reverse diffusion process and offering limited improvement. This approach often results in forecasting instability and the generation of low-fidelity samples (as shown in Figure 1). While diffusion-based methods perform adequately in probabilistic forecasting, their point-to-point prediction accuracy lags behind that of deterministic models, e.g., Autoformer (Wu et al., ), PatchTST (Nie et al., ), and DLinear (Zeng et al., ).

Figure: (a)
Refer to caption: x1.png

To improve the deterministic estimation performance of diffusion models on time series, we first revisit and consolidate existing non-autoregressive diffusion-based time series forecasting models under a unified framework, demonstrating that these models are fundamentally equivalent, differing primarily in their choice of parameters and network architecture.
Based on this framework, we propose a novel diffusion-based time series forecasting model, Series-to-Series Diffusion Bridge Model ($\mathrm{S^{2}DBM}$).
$\mathrm{S^{2}DBM}$ employs the diffusion bridge as its foundational architecture, which proves effective for multivariate time series forecasting. Specifically, $\mathrm{S^{2}DBM}$ uses the Brownian Bridge to pin down the diffusion process at both ends, reducing the instability caused by noisy input and enabling the accurate generation of future time step features from historical time series. By adjusting the posterior variance, $\mathrm{S^{2}DBM}$ behaves as a deterministic generative model without any Gaussian noise, thereby ensuring stability and precise point-to-point forecasting results.

In our experiments, we employ seven real-world datasets as benchmarks, including Weather, Influenza-like Illness (ILI), Exchange Rate (Lai et al., ), and Electricity Transformer Temperature datasets (ETTh1, ETTh2, ETTm1, ETTm2) (Zhou et al., ). We conduct experiments across various time series forecasting scenarios, covering both point-to-point and probabilistic forecasting. Through extensive testing across these scenarios, our proposed method, $\mathrm{S^{2}DBM}$, demonstrates superior performance over both standard conditional diffusion-based models and a wide range of advanced time series prediction models.

Our main contributions are summarized as follows:

- •
In this paper, we propose a comprehensive framework for non-autoregressive time series diffusion models, into which most existing diffusion-based methods can be integrated. This framework clarifies the interrelationships between these methods and highlights practical implications for diffusion models aimed at point-to-point time series forecasting.
- •
Based on this framework, we introduce the Series-to-Series Diffusion Bridge Model ($\mathrm{S^{2}DBM}$), which utilizes the Brownian Bridge diffusion process to reduce the randomness in reverse process of diffusion estimations. The proposed model uses linear approaches to create informative priors and conditions, thereby improving forecast accuracy by effectively using historical information for multivariate time series.
- •
Extensive experimental results validate the effectiveness of $\mathrm{S^{2}DBM}$, which outperforms state-of-the-art time series diffusion models in point-to-point forecasting tasks. Moreover, $\mathrm{S^{2}DBM}$ achieves forecasting performance on par with probabilistic models.

## 2 Related Works

##### Diffusion-based Time Series Forecasting.

Recently, a range of diffusion-based methods are proposed for time series forecasting. These methods generally adhere to the framework of the standard diffusion model, with their primary distinctions stemming from variations in the denoising network and conditional mechanisms.

TimeGrad (Rasul et al., ) is the pioneer of these diffusion-based methods, integrating diffusion models with an RNN-based encoder to handle historical time series. However, its reliance on autoregressive decoding can lead to error accumulation and slow inference times.
To tackle this problem, CSDI (Tashiro et al., ) employs an entire time series as the target for diffusion and combines it with a binary mask (which denotes missing values) as conditional inputs into two transformers. This masking-based conditional mechanism enables CSDI to generate future time series data in a non-autoregressive fashion.
SSSD (Alcaraz & Strodthoff, ) uses the same conditional mechanism as CSDI, but replaces the transformers in CSDI with a Structured State Space Model (S4) to reduce the computational complexity and is more suited to handling long-term dependencies.
TMDM (Li et al., ) integrates transformers with a conditional diffusion process to improve probabilistic multivariate time series forecasting by effectively capturing covariate dependencies in both the forward and Reverse diffusion processes.
TimeDiff (Shen & Kwok, ) introduces two innovative conditioning mechanisms specifically designed for time series analysis: future mixup and autoregressive initialization, which construct effective conditional embeddings.
To reduce the predictive instability arising from the stochastic nature of the diffusion models, MG-TSD (Fan et al., ) leverages the inherent granularity levels within the data as given targets at intermediate diffusion steps to guide the learning process of diffusion models.
Most of the above diffusion-based methods emphasize their probabilistic forecasting ability; however, their performance in point-to-point forecasting is suboptimal.

##### Diffusion Bridge.

Diffusion bridges (Liu et al., ; Zhou et al., ; Li et al., ) represent a specific class of diffusion models designed to simulate the trajectory of a stochastic process between predetermined initial and final states. They are regarded as conditioned diffusion models subject to particular boundary constraints. These models, stemming from classical stochastic processes like Brownian motion or Ornstein-Uhlenbeck process, have a predetermined terminal value rather than being free.

DDBMs (Zhou et al., ) introduce diffusion bridges, stochastically interpolating between paired distributions to provide smoother transitions and more flexible input handling compared to traditional noise-based diffusion models. Liu et al. () propose I^2SB, which constructs nonlinear diffusion bridges between two domains, making it suitable for tasks like image restoration. BBDM (Li et al., ) models image-to-image translation as a bidirectional diffusion process using a Brownian bridge, directly learning domain translation and achieving competitive benchmark results. GOUB (Yue et al., ) combines the generalized OU process with Doob’s h-transform to create precise diffusion mappings that transform low-quality images into high-quality ones. These diffusion bridge models excel in image restoration by using degraded images as informative priors to facilitate clean image reconstruction.
Bridge-TTS (Chen et al., ) successfully incorporates Schrödinger Bridge diffusion models into text-to-speech (TTS) synthesis task. It leverages the latent representation obtained from text input as a prior and builds a fully tractable Schrödinger bridge between it and the ground-truth mel-spectrogram.
For time series data, Park et al. () introduces TimeBridge, a framework that utilizes diffusion bridges to model transitions between selected prior and data distributions. This framework supports both data- and time-dependent priors, achieving state-of-the-art performance in unconditional and conditional time series generation tasks. However, the TimeBridge uses linear spline interpolation (De Boor, ) to generate priors for imputation tasks, which is unsuitable for time series forecasting.

## 3 Methodology

### 3.1 Preliminaries

Most diffusion-based methods for time series forecasting are designed around conditional Denoising Diffusion Probabilistic Models (DDPMs). The forward process, defined by a fixed Markov chain, progressively transforms the future time series ${\bm{y}}\in\mathbb{R}^{L\times d}$ into a Gaussian noise vector ${\bm{y}}_{T}$ according to a predetermined variance schedule $\left\{\beta_{t}\right\}_{t=1}^{T}$:

$$ $q\left({\bm{y}}_{t}\mid{\bm{y}}_{t-1}\right)=\mathcal{N}\left({\bm{y}}_{t}; \sqrt{1-\beta_{t}}{\bm{y}}_{t-1},\beta_{t}\bm{I}\right),$ $$

where $L$ denotes the length of the forecast window, and $d$ represents the number of distinct features.

With the notation $\alpha_{s}=1-\beta_{s}$ and $\bar{\alpha}_{t}:=\prod_{s=1}^{t}\alpha_{s}$, the forward process can be rewritten as:

$$ ${\bm{y}}_{t}=\sqrt{\bar{\alpha}_{t}}{\bm{y}}_{0}+\sqrt{1-\bar{\alpha}_{t}}\bm{ \epsilon},\bm{\epsilon}\sim\mathcal{N}\left(\mathbf{0},\bm{I}\right).$ $$

During inference, the model reverses the forward process by considering the following distribution:

$$ $p_{\theta}\left({\bm{y}}_{0:T}\mid{\bm{x}}\right)=p_{\theta}\left({\bm{y}}_{T} \right)\prod_{t=1}^{T}p_{\theta}\left({\bm{y}}_{t-1}\mid{\bm{y}}_{t},{\bm{x}} \right),$ $$

where ${\bm{y}}_{T}$ is initially sampled from a standard normal distribution $\mathcal{N}(\mathbf{0},{\bm{I}})$, the subscripts from 0 0 to $T$ denote the diffusion steps. ${\bm{x}}\in\mathbb{R}^{H\times d}$ is the historical data, $H$ represents the length of the lookback window.

Correspondingly, the conditional reverse process at step $t$ is described by:

$$ $p_{\theta}\left({\bm{y}}_{t-1}\mid{\bm{y}}_{t},{\bm{x}}\right):=\mathcal{N} \left(\bm{\mu}_{\theta}\left({\bm{y}}_{t},{\bm{x}},t\right),\bm{\Sigma}_{ \theta}\left({\bm{y}}_{t},t\right)\right),\,\bm{\Sigma}_{\theta}\left({\bm{y}} _{t},t\right)=\tilde{\beta}_{t}=\frac{1-\bar{\alpha}_{t-1}}{1-\bar{\alpha}_{t} }\beta_{t}.$ $$

Following the formulation proposed by  Saharia et al. (), we can parameterize $\bm{\mu}_{\theta}\left({\bm{y}}_{t},{\bm{x}},t\right)$ for either noise or data prediction as follows:

$$ $\bm{\mu}_{\theta}({\bm{y}}_{t},{\bm{x}},t):=\frac{1}{\sqrt{\alpha_{t}}}\left({ \bm{y}}_{t}-\frac{1-\alpha_{t}}{\sqrt{1-\bar{\alpha}_{t}}}\bm{\epsilon}_{ \theta}({\bm{y}}_{t},\mathbf{c},t)\right),$ $$

$$ $\bm{\mu}_{\theta}({\bm{y}}_{t},{\bm{x}},t):=\frac{\sqrt{\alpha_{t}}\left(1- \bar{\alpha}_{t-1}\right)}{1-\bar{\alpha}_{t}}{\bm{y}}_{t}+\frac{\sqrt{\bar{ \alpha}_{t-1}}\beta_{t}}{1-\bar{\alpha}_{t}}{\bm{y}}_{\theta}({\bm{y}}_{t}, \mathbf{c},t),$ $$

where $\mathbf{c}=E\left({\bm{x}}\right)$ represents the condition derived from the historical data ${\bm{x}}$, and $E(\cdot)$ is a conditioning module.

### 3.2 Revisiting Generalized Diffusion Model for Time Series

Most existing diffusion-based time series forecasting methods emphasize their probabilistic forecasting capabilities; however, their performance in point-to-point forecasting remains suboptimal. To develop a specialized diffusion-based model tailored for point-to-point time series forecasting, a deeper understanding of existing approaches is crucial. Therefore, we revisit and consolidate current non-autoregressive diffusion-based time series forecasting models into a unified framework, demonstrating their fundamental equivalence. The primary differences among these models lie in their choice of diffusion-related coefficients and the design of network architectures.

Recognizing components in existing models, diffusion processes can be viewed in a flexible and adaptable manner. As shown in Eq. (1), the diffusion process incorporates historical data and endows the designed models with distinct properties by adjusting the coefficients $\hat{\alpha}_{t}$, $\hat{\beta}_{t}$, $\hat{\gamma}_{t}$, and $\hat{\sigma}_{t}^{2}$.

**Table 1: Comparison between different instances of generalized conditional diffusion framework.**
| Model | $\hat{\alpha}_{t}$ | $\hat{\beta}_{t}$ | $\hat{\gamma}_{t}$ | $\hat{\sigma}_{t}^{2}$ | Estimated Target | ${\bm{E}}(\cdot)$ |
| --- | --- | --- | --- | --- | --- | --- |
| CSDI (Tashiro et al., ) | $\sqrt{\bar{\alpha}_{t}}$ | $\sqrt{1-\bar{\alpha}_{t}}$ | 0 0 | $\frac{1-\bar{\alpha}_{t-1}}{1-\bar{\alpha}_{t}}\beta_{t}$ | $\bm{\epsilon}$ | Transfomer in $\mu_{\theta}$ |
| SSSD (Alcaraz & Strodthoff, ) | $\sqrt{\bar{\alpha}_{t}}$ | $\sqrt{1-\bar{\alpha}_{t}}$ | 0 0 | $\frac{1-\bar{\alpha}_{t-1}}{1-\bar{\alpha}_{t}}\beta_{t}$ | $\bm{\epsilon}$ | S4 in $\mu_{\theta}$ |
| TimeDiff (Shen & Kwok, ) | $\sqrt{\bar{\alpha}_{t}}$ | $\sqrt{1-\bar{\alpha}_{t}}$ | 0 0 | $\frac{1-\bar{\alpha}_{t-1}}{1-\bar{\alpha}_{t}}\beta_{t}$ | ${\bm{y}}_{0}$ | Future mixup + AR model |
| TMDM (Li et al., ) | $\sqrt{\bar{\alpha}_{t}}$ | $\sqrt{1-\bar{\alpha}_{t}}$ | $1-\sqrt{\bar{\alpha}_{t}}$ | $\frac{1-\bar{\alpha}_{t-1}}{1-\bar{\alpha}_{t}}\beta_{t}$ | $\bm{\epsilon}$ | Transformer |
| Ours | $\frac{T-t}{T}$ | $\sqrt{\frac{2t(T-t)}{T^{2}}}$ | $\frac{t}{T}$ | $\frac{2(t-1)}{Tt}$ or 0 | ${\bm{y}}^{*}_{0}$ | Liner Model + Transfomer in $\mu_{\theta}$ |

###### Theorem 1 .

The non-autoregressive diffusion processes in time series can be formalized as follows:

$$ ${\bm{y}}_{t}=\hat{\alpha}_{t}{\bm{y}}_{0}+\hat{\beta}_{t}\bm{\epsilon}+\hat{ \gamma}_{t}{\bm{h}},\quad\bm{\epsilon}\sim\mathcal{N}(\mathbf{0},\bm{I}).$ (1) $$

The reverse diffusion process corresponding to $\hat{\beta}_{t}\neq 0$ can be formulated as:

$$ $\displaystyle p_{\theta}({\bm{y}}_{0:T}\mid{\bm{x}})$ $\displaystyle:=p_{\theta}({\bm{y}}_{T}){\textstyle\prod_{t=1}^{T}}p_{\theta}({ \bm{y}}_{t-1}\mid{\bm{y}}_{t},{\bm{x}}),$ (2) $\displaystyle p_{\theta}({\bm{y}}_{t-1}\mid{\bm{y}}_{t},{\bm{x}})$ $\displaystyle:=\mathcal{N}({\bm{y}}_{t-1};\mu_{\theta}\left({\bm{y}}_{t},{\bm{ h}},\mathbf{c},t\right),\hat{\sigma}_{t}^{2}\bm{I}),$ (3) $$

where $\hat{\alpha}_{t}$, $\hat{\beta}_{t}$, and $\hat{\gamma}t$ are time-dependent scaling factors. The vector ${\bm{h}}=F({\bm{x}})$ acts as the conditional representation incorporating prior knowledge, with $F(\cdot)$ serving as the prior predictor that maps historical time series into a latent space. The initial distribution is given by $p_{\theta}({\bm{y}}_{T})=\mathcal{N}(\hat{\gamma}_{T}{\bm{h}},\hat{\beta}_{T}^
{2}\bm{I})$. The conditioning variable $\mathbf{c}=E({\bm{x}})$ guides the reverse process, where $E(\cdot)$ denotes the conditioning module. The function $\mu_{\theta}$ predicts the mean of ${\bm{y}}_{t-1}$ given inputs ${\bm{y}}_{t}$, ${\bm{h}}$, and $\mathbf{c}$, while $\hat{\sigma}_{t}^{2}$ represents the reverse variance schedule.

Most existing diffusion-based time series forecasting models, including CSDI (Tashiro et al., ), SSSD (Alcaraz & Strodthoff, ), TimeDiff (Shen & Kwok, ), and TMDM (Li et al., ), can be interpreted within our proposed framework, as summarized in Table 1. The key differences lie in the choice of forward variance schedule $\hat{\gamma}_{t}$, the learning objectives of their denoising networks, the architectures of their conditional networks $E(\cdot)$, and their respective conditioning mechanisms.
Specifically, CSDI, SSSD, and TimeDiff utilize identical diffusion coefficients with $\gamma_{t}=0$, aligning with the standard diffusion process. In contrast, TMDM sets $\gamma_{t}=1-\sqrt{\bar{\alpha}_{t}}$, introducing a distinct variance schedule. Regarding the estimation targets, CSDI, SSSD, and TMDM focus on predicting the noise component $\bm{\epsilon}$, whereas TimeDiff directly estimates the data ${\bm{y}}$. The conditioning strategies also differ notably: CSDI and SSSD employ masking with zero-padding to directly condition the denoising network, implemented via Transformer and S4 blocks, respectively. TimeDiff leverages future mixup techniques and incorporates autoregressive models, while TMDM integrates a well-designed Transformer to enhance its conditioning mechanism.

###### Theorem 1 .

### 3.3 Series-to-Series Diffusion Bridge Model

As shown in Table 1, existing diffusion-based time series forecasting methods have been extensively studied using various diffusion paradigms and conditional approaches in the formulation of Theorem 1 and achieve promising predictive ability. However, most of these methods focus on the uncertainty estimation ability and typically rely on a data-to-noise diffusion process due to current conditioning mechanisms. As a result, they are often constrained by the intrinsic stochastic nature and are limited in capturing the inherent complexity and dynamic nature of real-world time series data, leading to suboptimal performance in point-to-point forecasting. To address this gap, we propose the Series-to-Series Diffusion Bridge Model ($\mathrm{S^{2}DBM}$), which uses the Brownian Bridge to pin down the diffusion process at both ends, reducing the instability caused by noisy input and enabling the accurate generation of future time step features from historical time series. By adjusting the posterior variance in Theorem 1, S2DBM behaves as a deterministic generative model without any Gaussian noise, thereby ensuring stability and precise point-to-point forecasting results.

Figure: Figure 2: An illustration of the proposed $\mathrm{S^{2}DBM}$
Refer to caption: x4.png

As shown in Figure 2, $\mathrm{S^{2}DBM}$ employs the diffusion bridge as the foundational architecture by adjusting the coefficient schedules. The diffusion bridge pins down the diffusion process at both ends, enabling the accurate generation of future time step features from historical time series data through a data-to-data process.

###### Corollary 1 (Brownian Bridge between Historical and Predicted Time Series) .

Let the coefficient $\hat{\alpha}_{t}$, constrained to be non-negative and decrease monotonically over time $t$, satisfy the boundary conditions $\hat{\alpha}_{0}=0$ and $\hat{\alpha}_{T}=1$. Additionally, define $\hat{\gamma}_{t}=1-\hat{\alpha}_{t}$ and $\hat{\beta}_{t}=\sqrt{2\hat{\alpha}_{t}(1-\hat{\alpha}_{t})}$ The forward process defined in Eq. (1) can be rewritten in closed form:

$$ $q({\bm{y}}_{t}\mid{\bm{y}}_{0},{\bm{h}})=\mathcal{N}({\bm{y}}_{t};\hat{\alpha} _{t}{\bm{y}}_{0}+(1-\hat{\alpha}_{t}){\bm{h}},2\hat{\alpha}_{t}(1-\hat{\alpha} _{t})\bm{I}).$ (4) $$

Then, the reverse process transition defined in  Eq. (3) turns into:

$$ $p_{\theta}({\bm{y}}_{t-1}\mid{\bm{y}}_{t},{\bm{x}})=\mathcal{N}({\bm{y}}_{t}; \kappa_{t}{\bm{y}}_{t}+\lambda_{t}{\bm{y}}_{\theta}({\bm{y}}_{t},{\bm{h}}, \mathbf{c},t)+\zeta_{t}{\bm{h}},\hat{\sigma}_{t}^{2}\bm{I}),$ (5) $$

where ${\bm{y}}_{\theta}$ is a data prediction model used to predict the ground truth ${\bm{y}}_{0}$. Here, $\kappa_{t}$, $\lambda_{t}$, and $\zeta_{t}$ are scaling factors defined as

$$ $\kappa_{t}=\sqrt{\frac{2\hat{\alpha}_{t-1}(1-\hat{\alpha}_{t-1})-\hat{\sigma}_ {t}^{2}}{2\hat{\alpha}_{t}(1-\hat{\alpha}_{t})}},\quad\lambda_{t}=\hat{\alpha} _{t-1}-\hat{\alpha}_{t}\kappa_{t},\quad\zeta_{t}=1-\hat{\alpha}_{t-1}-\kappa_{ t}(1-\hat{\alpha}_{t}).$ (6) $$

Based on the  Equation 6, $\mathrm{S^{2}DBM}$ constructs a Brownian bridge between the initial state ${\bm{y}}$ and the destination state ${\bm{h}}$, eliminating the need to sample from a noisy Gaussian prior during the sampling process, allowing for the direct assignment of ${\bm{y}}_{T}={\bm{h}}$. This approach captures more structural information about the target time series.

In the reverse process of $\mathrm{S^{2}DBM}$, the diffusion process starts directly from ${\bm{y}}_{T}={\bm{h}}$. According to  Eq. (5), the mean of the reverse transition is determined by both the posterior variance $\hat{\sigma}_{t}^{2}$ and the coefficient $\hat{\alpha}_{t}$. Given $\hat{\alpha}_{t}$, the coefficients $\kappa_{t}$, $\lambda_{t}$, and $\zeta_{t}$ for the reverse process are analytically derived as functions of $\hat{\sigma}_{t}^{2}$.
To control the contributions of $\hat{{\bm{y}}}$, ${\bm{y}}_{t}$, and ${\bm{h}}$ to the predicted mean of $p_{\theta}$, following BBDM (Li et al., ) and I^3SB Wang et al. (), we parameterize $\hat{\sigma}_{t}^{2}$ as follows:

$$ $\hat{\sigma}_{t}^{2}=s\cdot\frac{(1-\hat{\alpha}_{t-1})(\hat{\alpha}_{t-1}- \hat{\alpha}_{t})}{1-\hat{\alpha}_{t}},$ $$

where $s$ is a hyperparameter that scales the variance, and the selection of its numerical value is discussed in the following remark.

###### Remark 1 (The reverse process of $\mathrm{S^{2}DBM}$ ) .

As a consequence of Remark 1, we discuss two instances of the reverse process in $\mathrm{S^{2}DBM}$, both of which employ the same training procedure but are specifically applied to probabilistic and point-to-point forecasting, respectively.

###### Example 1 (Point-to-point forecasting) .

When we set $\hat{\alpha}_{t}=1-\frac{t}{T}$ and $s=0$, the posterior variance $\hat{\sigma}_{t}^{2}$ becomes 0 0, making the sampling process deterministic, akin to the DDIM approach. The reverse process of $\mathrm{S^{2}DBM}$ can be rewritten as:

$$ $\displaystyle{\bm{y}}_{t-1}=$ $\displaystyle\sqrt{\frac{(T-t+1)(t-1)}{(T-t)t}}{\bm{y}}_{t}+\left(\frac{T-t+1} {T}-\sqrt{\frac{(T-t)(T-t+1)(t-1)}{T^{2}t}}\right)\hat{{\bm{y}}}$ $\displaystyle+\left(\frac{t-1}{T}-\sqrt{\frac{t(T-t+1)(t-1)}{T^{2}(T-t)}} \right){\bm{h}}.$ $$

###### Example 2 (Probabilistic forecasting) .

When we set $\hat{\alpha}_{t}=1-\frac{t}{T}$ and $s=1$, the posterior variance $\hat{\sigma}_{t}^{2}$ is defined as $\frac{2(t-1)}{Tt}$. Consequently, the reverse process of  $\mathrm{S^{2}DBM}$ is formulated as:

$$ ${\bm{y}}_{t-1}=\left(1-\frac{1}{t}\right){\bm{y}}_{t}+\frac{1}{t}\hat{{\bm{y}} }+\sqrt{\frac{2(t-1)}{Tt}}\bm{z},\quad\bm{z}\sim\mathcal{N}(\mathbf{0},\bm{I}).$ $$

###### Corollary 1 (Brownian Bridge between Historical and Predicted Time Series) .

###### Remark 1 (The reverse process of S 2 ⁢ DBM superscript S 2 DBM \mathrm{S^{2}DBM} roman_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_DBM ) .

###### Example 1 (Point-to-point forecasting) .

###### Example 2 (Probabilistic forecasting) .

##### Linear Model based Conditioning Method.

The condition $\mathbf{c}$ defined in Eq. (3) represents the useful information extracted from historical data ${\bm{x}}$, guiding the reverse process toward ${\bm{y}}_{0}$. Since the design of the conditioning module $E(\cdot)$ significantly impacts the predictive quality of the denoising network, it is a crucial aspect of time series diffusion models.
In our $\mathrm{S^{2}DBM}$ model, we treat $E(\cdot)$ as independent of the denoising network, allowing $E({\bm{x}})$ to preprocess historical data to provide an initial estimate of the future time series. This estimate is then used as the conditional input for the denoising network $\mu_{\theta}$, thereby simplifying the forecasting task.

The $\mathrm{S^{2}DBM}$ model captures conditional information from historical data not only through the conditioning module $E(\cdot)$, but also via the prior predictor $F(\cdot)$. In time series forecasting, the lookback and forecast windows often differ, and historical sequences cannot directly provide structurally informative priors for prediction targets as damaged images do in image restoration. Therefore, we cannot directly construct a diffusion bridge between historical time series ${\bm{x}}$ and future time series ${\bm{y}}$.
Instead, we use the prior predictor $F(\cdot)$ to transform historical time series into a deterministic conditional representation ${\bm{h}}$, which serves as the endpoint of the diffusion process and provides guidance at the beginning of the reverse process.
Both the conditional encoder network $E$ and the prior predictor $F(\cdot)$ in $\mathrm{S^{2}DBM}$ employ a simple one-layer linear model, chosen for its simplicity, explainability, and efficiency (Toner & Darlow, ).

##### Label-Guided Data Estimation.

The learnable transfer probability $p_{\theta}({\bm{y}}_{t-1}\mid{\bm{y}}_{t},{\bm{x}})$ is an approximation of the posterior distribution $q({\bm{y}}_{t-1}\mid{\bm{y}}_{t},{\bm{y}}_{0},{\bm{x}}):=\mathcal{N}({\bm{y}}_
{t-1};\mu({\bm{y}}_{t},{\bm{y}}_{0},{\bm{x}}),\hat{\sigma}_{t}^{2}\bm{I})$.
In our $\mathrm{S^{2}DBM}$, the denoising network $\mu_{\theta}$ is designed to estimate the data rather than the noise, as we found that estimating the noise introduces more oscillations in the prediction results. Thus, $\mu_{\theta}$ can be expressed as:

$$ $\mu_{\theta}({\bm{y}}_{t},{\bm{h}},\mathbf{c},t)=\kappa_{t}{\bm{y}}_{t}+ \lambda_{t}{\bm{y}}_{\theta}({\bm{y}}_{t},{\bm{h}},\mathbf{c},t)+\zeta_{t}{\bm {h}}.$ (7) $$

In practice, we do not directly estimate the future time series ${\bm{y}}$. Instead, we utilize the labeling strategy employed in some transformer-based time series forecasting models, such as the Informer (Zhou et al., ). Specifically, we treat the terminal portion of the historical data, ${\bm{x}}$, as the label and integrate it with the future time series ${\bm{y}}$ along the time dimension, denoted as ${\bm{y}}^{*}$. Consequently, the denoising network $\mu_{\theta}$ is tasked not only with predicting future time steps but also with reconstructing the known sequence within the label length. This methodology enables the model to more effectively capture underlying patterns in the data. The training loss for $\mathrm{S^{2}DBM}$ is defined as follows:

$$ $\mathcal{L}=\sum_{t=1}^{T}\underset{q\left({\bm{y}}_{t}^{*}\mid{\bm{y}}_{0}^{* },{\bm{h}}\right)}{\mathbb{E}}\left\|{\bm{y}}_{0}^{*}-{\bm{y}}_{\theta}({\bm{y }}_{t}^{*},{\bm{h}},\mathbf{c},t)\right\|^{2}.$ $$

The denoising network of $\mathrm{S^{2}DBM}$ adopts the same architecture as CSDI but removes modules related to its original conditioning mechanism.
The training and sampling procedures of $\mathrm{S^{2}DBM}$ are detailed in Algorithm 1 and Algorithm 2, respectively.

Figure: Algorithm 1 Training of $\mathrm{S^{2}DBM}$

## 4 Experiments

### 4.1 Experimental Settings

##### Datasets.

In this experiment, the time series forecasting benchmark datasets employed encompass several real-world datasets: Weather, Influenza-like Illness (ILI), Exchange-Rate (Lai et al., ), and four Electricity Transformer Temperature datasets (Zhou et al., ) (ETTh1, ETTh2, ETTm1, ETTm2). These datasets are extensively utilized for testing multivariate time-series forecasting models due to their diverse and representative nature, offering insights into the model’s performance across different domains and conditions. Each dataset is normalized using the mean and standard deviation of the training part.

##### Baselines.

We compared our method with several state-of-the-art and representative baseline models. These include Transformer-based methods: Autoformer (Wu et al., ), Informer (Zhou et al., ), and iTransformer (Liu et al., ); linear models: DLinear, NLinear (Zeng et al., ), and RLinear (Li et al., ); as well as diffusion-based time series prediction methods: CSDI (Tashiro et al., ), TMDM (Li et al., ), and TimeDiff (Shen & Kwok, ).

##### Evaluation metrics.

To assess point-to-point forecasting performance, we employ mean squared error (MSE) and mean absolute error (MAE) as primary metrics to quantify discrepancies between forecasted and actual time series values. For evaluating the quality of probabilistic forecasts, we use the continuous ranked probability score (CRPS) (Matheson & Winkler, ) across individual time series dimensions and $\mathrm{CRPS}_{\mathrm{sum}}$ for the aggregate of all dimensions.

##### Implementation details.

We trained our model using the ADAM optimizer, setting the initial learning rate at 0.0001 and parameters $\beta_{1}=0.9$ and $\beta_{2}=0.999$. We configured the number of time steps for the $\mathrm{S^{2}DBM}$ to be T=50 during the training and inference stages. The computational environment comprised a server with an NVIDIA GeForce RTX 3090 24GB GPU.

### 4.2 Main Results

##### Point-to-point forecasting.

**Table 2: Multivariate time series forecasting results in terms of MSE and MAE, lower values mean better performance. The $1^{\text{st}}$ count indicates the numbers of best results.**
|  | Diffusion-based Methods | Transformer-based Methods | Linear Model |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Methods | Ours | CSDI | TMDM | Autoformer | Informer | iTransformer | NLinear | DLinear | RLinear |  |  |  |  |  |  |  |  |  |  |
| Metric | MSE | MAE | MSE | MAE | MSE | MAE | MSE | MAE | MSE | MAE | MSE | MAE | MSE | MAE | MSE | MAE | MSE | MAE |  |
| ETTh1 | 96 | 0.366 | 0.383 | 0.744 | 0.623 | 0.711 | 0.605 | 0.429 | 0.444 | 0.925 | 0.761 | 0.387 | 0.405 | 0.374 | 0.394 | 0.384 | 0.405 | 0.366 | 0.391 |
|  | 192 | 0.405 | 0.407 | 0.952 | 0.715 | 0.922 | 0.720 | 0.440 | 0.451 | 0.995 | 0.778 | 0.441 | 0.436 | 0.408 | 0.415 | 0.443 | 0.450 | 0.403 | 0.412 |
|  | 336 | 0.442 | 0.430 | 1.192 | 0.837 | 0.990 | 0.737 | 0.511 | 0.488 | 1.036 | 0.782 | 0.491 | 0.462 | 0.429 | 0.428 | 0.447 | 0.448 | 0.420 | 0.423 |
|  | 720 | 0.469 | 0.478 | 1.822 | 1.005 | 1.152 | 0.836 | 0.499 | 0.501 | 1.175 | 0.858 | 0.509 | 0.494 | 0.441 | 0.454 | 0.504 | 0.515 | 0.442 | 0.456 |
| ETTh2 | 96 | 0.274 | 0.331 | 1.017 | 0.729 | 0.496 | 0.510 | 0.418 | 0.445 | 3.017 | 1.369 | 0.301 | 0.350 | 0.283 | 0.343 | 0.290 | 0.353 | 0.262 | 0.331 |
|  | 192 | 0.354 | 0.388 | 3.417 | 1.356 | 0.578 | 0.535 | 0.435 | 0.439 | 6.348 | 2.105 | 0.380 | 0.399 | 0.356 | 0.385 | 0.388 | 0.422 | 0.320 | 0.374 |
|  | 336 | 0.433 | 0.454 | 2.642 | 1.216 | 0.715 | 0.598 | 0.480 | 0.481 | 5.628 | 1.998 | 0.424 | 0.432 | 0.362 | 0.403 | 0.463 | 0.473 | 0.326 | 0.388 |
|  | 720 | 0.592 | 0.568 | 3.396 | 1.431 | 0.758 | 0.658 | 0.478 | 0.487 | 4.110 | 1.692 | 0.430 | 0.447 | 0.398 | 0.437 | 0.733 | 0.606 | 0.425 | 0.449 |
| ETTm1 | 96 | 0.293 | 0.333 | 0.556 | 0.509 | 0.547 | 0.512 | 0.471 | 0.463 | 0.621 | 0.557 | 0.342 | 0.377 | 0.306 | 0.348 | 0.301 | 0.345 | 0.301 | 0.343 |
|  | 192 | 0.333 | 0.355 | 0.608 | 0.532 | 0.689 | 0.592 | 0.592 | 0.521 | 0.723 | 0.618 | 0.383 | 0.396 | 0.349 | 0.375 | 0.336 | 0.366 | 0.341 | 0.367 |
|  | 336 | 0.367 | 0.377 | 0.764 | 0.622 | 0.722 | 0.602 | 0.503 | 0.486 | 1.001 | 0.746 | 0.418 | 0.418 | 0.375 | 0.388 | 0.372 | 0.389 | 0.374 | 0.386 |
|  | 720 | 0.442 | 0.422 | 1.071 | 0.792 | 1.072 | 0.785 | 0.751 | 0.582 | 0.980 | 0.747 | 0.487 | 0.457 | 0.433 | 0.422 | 0.427 | 0.423 | 0.430 | 0.418 |
| ETTm2 | 96 | 0.164 | 0.249 | 0.859 | 0.587 | 0.328 | 0.400 | 0.233 | 0.313 | 0.407 | 0.482 | 0.186 | 0.272 | 0.167 | 0.255 | 0.172 | 0.267 | 0.164 | 0.253 |
|  | 192 | 0.219 | 0.292 | 0.907 | 0.614 | 0.415 | 0.423 | 0.278 | 0.336 | 0.807 | 0.706 | 0.254 | 0.314 | 0.221 | 0.293 | 0.237 | 0.314 | 0.219 | 0.290 |
|  | 336 | 0.274 | 0.328 | 1.584 | 0.862 | 0.871 | 0.611 | 0.379 | 0.394 | 1.453 | 0.926 | 0.316 | 0.351 | 0.274 | 0.327 | 0.295 | 0.359 | 0.273 | 0.326 |
|  | 720 | 0.361 | 0.389 | 2.692 | 1.202 | 1.101 | 0.739 | 0.584 | 0.473 | 3.930 | 1.469 | 0.414 | 0.407 | 0.369 | 0.385 | 0.427 | 0.439 | 0.366 | 0.385 |
| ILI | 24 | 2.241 | 0.983 | 3.942 | 1.293 | 4.005 | 1.183 | 3.405 | 1.290 | 5.104 | 1.544 | 2.405 | 0.987 | 2.022 | 0.925 | 2.280 | 1.061 | 2.036 | 0.969 |
|  | 36 | 2.811 | 1.060 | 4.982 | 1.497 | 3.456 | 1.300 | 3.522 | 1.291 | 5.158 | 1.571 | 2.328 | 0.984 | 1.974 | 0.932 | 2.235 | 1.059 | 1.928 | 0.940 |
|  | 48 | 3.024 | 1.084 | 4.164 | 1.331 | 3.059 | 1.124 | 3.478 | 1.294 | 5.101 | 1.565 | 2.330 | 0.990 | 1.979 | 0.955 | 2.298 | 1.079 | 1.880 | 0.931 |
|  | 60 | 3.758 | 1.229 | 5.725 | 1.651 | 2.771 | 1.163 | 2.880 | 1.154 | 5.319 | 1.596 | 2.413 | 1.015 | 1.954 | 0.949 | 2.573 | 1.157 | 2.016 | 0.976 |
| Weather | 96 | 0.172 | 0.210 | 0.251 | 0.235 | 1.048 | 0.300 | 0.269 | 0.339 | 0.335 | 0.406 | 0.176 | 0.216 | 0.181 | 0.232 | 0.174 | 0.233 | 0.175 | 0.225 |
|  | 192 | 0.213 | 0.249 | 0.330 | 0.294 | 2.246 | 0.372 | 0.338 | 0.395 | 0.693 | 0.599 | 0.225 | 0.257 | 0.225 | 0.268 | 0.218 | 0.278 | 0.217 | 0.259 |
|  | 336 | 0.257 | 0.287 | 0.420 | 0.357 | 3.636 | 0.470 | 0.339 | 0.381 | 0.564 | 0.527 | 0.281 | 0.299 | 0.271 | 0.301 | 0.263 | 0.314 | 0.265 | 0.294 |
|  | 720 | 0.343 | 0.353 | 0.538 | 0.423 | 0.795 | 0.541 | 0.429 | 0.433 | 1.105 | 0.771 | 0.358 | 0.350 | 0.339 | 0.349 | 0.332 | 0.374 | 0.329 | 0.339 |
| Exchange | 96 | 0.096 | 0.229 | 0.902 | 0.647 | 0.202 | 0.334 | 0.143 | 0.274 | 0.943 | 0.772 | 0.086 | 0.206 | 0.089 | 0.208 | 0.085 | 0.209 | 0.089 | 0.209 |
|  | 192 | 0.196 | 0.334 | 1.084 | 0.744 | 0.371 | 0.466 | 0.266 | 0.377 | 1.244 | 0.882 | 0.181 | 0.304 | 0.181 | 0.300 | 0.162 | 0.296 | 0.191 | 0.309 |
|  | 336 | 0.886 | 0.733 | 0.775 | 0.678 | 1.122 | 0.852 | 0.465 | 0.509 | 1.790 | 1.070 | 0.338 | 0.422 | 0.330 | 0.415 | 0.333 | 0.441 | 0.363 | 0.434 |
|  | 720 | 2.479 | 1.179 | 1.306 | 0.879 | 1.206 | 0.792 | 1.088 | 0.812 | 2.905 | 1.406 | 0.853 | 0.696 | 0.925 | 0.722 | 0.898 | 0.725 | 0.963 | 0.731 |
| $1^{\text{st}}$ Count | 10 | 11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 2 | 4 | 6 | 3 | 1 | 12 | 9 |  |

Table 2 provides a detailed summary of the point-to-point time series forecasting results for Example 1 of our $\mathrm{S^{2}DBM}$ model, compared to other models. For diffusion-based methods, we evaluate results obtained from one-shot prediction.
The first and second best results are in bold and underlined, respectively. The smaller the value of MSE and MAE, the more accurate the prediction result is. The performance of our $\mathrm{S^{2}DBM}$ surpasses that of other diffusion-based methods in most cases. Compared with the Transformer-based and Linear model-based SOTA methods, our $\mathrm{S^{2}DBM}$ achieves the best performance on most seetings, with the 21 first and 6 second places out of 56 benchmarks in total.

**Table 3: Comparison of multivariate prediction MSE between TimeDiff and $\mathrm{S^{2}DBM}$.**
|  | ETTh1 | ETTm1 | Exchange |
| --- | --- | --- | --- |
| TimeDiff | 0.407 | 0.336 | 0.018 |
| Ours | 0.397 | 0.333 | 0.018 |

Table  3 presents the Mean Squared Error (MSE) results for the diffusion-based method TimeDiff, which employed unique settings for prediction length that differ from other methods. In response, we retrain our model according to these settings and conduct the following comparisons. Experimental results indicate that our method outperforms TimeDiff in terms of MSE.
To complement the quantitative results of diffusion-based methods, Figure 3 provides visualizations of the predictions obtained by CSDI, TMDM, and the proposed $\mathrm{S^{2}DBM}$ on a randomly selected test example from the ETTh1 dataset.
As illustrated, while CSDI delivers accurate short-term predictions (from steps 96-110), its long-term forecasts deviate significantly from the ground truth.
TMDM captures the overall trend of the future time series, but its point-wise prediction accuracy shows significant oscillations, likely influenced by the noise inherent in the diffusion process, leading to fluctuating results.
In contrast, $\mathrm{S^{2}DBM}$ effectively captures the trend and seasonality of time series.

Figure: (a) CSDI.
Refer to caption: x5.png

##### Probabilistic forecasting.

**Table 4: Probabilistic forecasting performance comparisons on ETTh1 and ETTm1 datasets in terms of CRPS and $\mathrm{CRPS}_{\mathrm{sum}}$. The best results are boldfaced. The prediction horizon set to 96.**
| Dataset | ETTh1 | ETTh2 | ETTm1 | ETTm2 | Weather |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Metric | CRPS | $\mathrm{CRPS}_{\mathrm{sum}}$ | CRPS | $\mathrm{CRPS}_{\mathrm{sum}}$ | CRPS | $\mathrm{CRPS}_{\mathrm{sum}}$ | CRPS | $\mathrm{CRPS}_{\mathrm{sum}}$ | CRPS | $\mathrm{CRPS}_{\mathrm{sum}}$ |
| CSDI | 0.512$\pm$0.107 | 2.077$\pm$0.003 | 0.579$\pm$0.096 | 2,985$\pm$0.004 | 0.428$\pm$0.106 | 2.093$\pm$0.002 | 0.490$\pm$0.104 | 2.972$\pm$0.002 | 0.190$\pm$0.026 | 1.747$\pm$0.002 |
| TMDM | 0.385$\pm$0.098 | 1.672$\pm$0.003 | 0.333$\pm$0.094 | 1.546$\pm$0.003 | 0.338$\pm$0.087 | 1.674$\pm$0.002 | 0.241$\pm$0.070 | 1.213$\pm$0.001 | 0.203$\pm$0.027 | 1.623$\pm$0.002 |
| Ours | 0.382$\pm$0.093 | 1.782$\pm$0.003 | 0.328$\pm 0.092$ | 1.554$\pm 0.003$ | 0.333$\pm$0.087 | 1.553$\pm$0.001 | 0.247$\pm$0.069 | 1.219$\pm$0.001 | 0.209$\pm$0.028 | 1.845$\pm$0.002 |

Table 4 summarizes the probabilistic forecasting results for Example 2 of our $\mathrm{S^{2}DBM}$ model, compared with other diffusion-based models. We utilized 100 samples to approximate the probability distribution. The results show that our $\mathrm{S^{2}DBM}$ performs competitively against CSDI and TMDM in terms of CRPS and $\mathrm{CRPS}_{\mathrm{sum}}$, illustrating the capabilities of our $\mathrm{S^{2}DBM}$ in probabilistic forecasting.

### 4.3 Ablation Studies

To validate each component of our proposed $\mathrm{S^{2}DBM}$ model, we performed a comparative analysis of prediction results using five different models on the ETTh1 and ETTm1 datasets. The results are presented in Table 5.
The notation cDDPM indicates that it employs the standard diffusion process instead of the Brownian bridge process used in $\mathrm{S^{2}DBM}$.
The notation w/ CSDI $E$ refers to an operation that utilizes the conditioning mechanism of CSDI. Similarly, w/ CSDI $\mu_{\theta}$ indicates the adoption of the denoising network architecture from CSDI.

**Table 5: Model ablation.We present the MSE and MAE of different variants of the $\mathrm{S^{2}DBM}$ model, with the prediction horizon set to 96.**
| Dataset | ETTh1 | ETTm1 |  |  |
| --- | --- | --- | --- | --- |
| Metric | MSE | MAE | MSE | MAE |
| cDDPM | 0.379 | 0.392 | 0.304 | 0.345 |
| w/ CSDI $E$ | 0.755 | 0.545 | 0.416 | 0.406 |
| w/ CSDI $\mu_{\theta}$ | 0.578 | 0.520 | 0.489 | 0.457 |
| label_len=0 | 0.450 | 0.461 | 0.378 | 0.396 |
| Ours | 0.366 | 0.383 | 0.293 | 0.333 |

Additionally, the notation $\text{label_len}=0$ signifies that $\mathrm{S^{2}DBM}$ no longer reconstructs known data, focusing solely on predicting the future time series.
When comparing our proposed model $\mathrm{S^{2}DBM}$ with cDDPM, we observe notable improvements in both MSE and MAE. Figure 4 visualizes the predictions obtained from both cDDPM and the proposed $\mathrm{S^{2}DBM}$ for a randomly selected test example from the ETTh1 dataset. As illustrated, $\mathrm{S^{2}DBM}$ significantly reduces oscillations in the predictions.
Additionally, comparing w/ CSDI $E$ and w/ CSDI $\mu_{\theta}$ with $\mathrm{S^{2}DBM}$ demonstrates the advantages of the linear model-based conditioning method and the network architecture of $\mathrm{S^{2}DBM}$.
Finally, comparing $\mathrm{S^{2}DBM}$ with $\text{label_len}=0$,
we reveal an average reduction of 21% in MSE and 16% in MAE, indicating the contribution of the labeling strategy.

Figure: (a) Conditional DDPM.
Refer to caption: x8.png

## 5 Conclusion

In this paper, we revisit non-autoregressive time series diffusion models and present a comprehensive framework that integrates most existing diffusion-based methods. Building on this theoretical framework, we propose the Series-to-Series Diffusion Bridge Model ($\mathrm{S^{2}DBM}$). Our $\mathrm{S^{2}DBM}$ utilizes the Brownian Bridge diffusion process to reduce randomness in diffusion estimations, improving forecast accuracy by effectively leveraging historical information through informative priors and conditions. Extensive experimental results demonstrate that $\mathrm{S^{2}DBM}$ achieves superior performance in point-to-point forecasting and performs competitively against other diffusion-based models in probabilistic forecasting.

#### Acknowledgments

Z. Ling is supported by the National Natural Science Foundation of China (via NSFC-62406119), the Natural Science Foundation of Hubei Province (2024AFB074), and the Guangdong Provincial Key Laboratory of Mathematical Foundations for Artificial Intelligence (2023B1212010001).
R. C. Qiu would like to acknowledge the National Natural Science Foundation of China (via NSFC-12141107), the Interdisciplinary Research Program of HUST (2023JCYJ012), the Key Research and Development Program of Guangxi (GuiKe-AB21196034). F. Zhou was supported by the National Natural Science Foundation of China Project (via NSFC-62106121) and the MOE Project of Key Research Institute of Humanities and Social Sciences (22JJD110001).

## Appendix A Appendix

### A.1 Proofs of Theorem 1

The non-autoregressive diffusion processes in time series can be formalized as follows:

$$ ${\bm{y}}_{t}=\hat{\alpha}_{t}{\bm{y}}_{0}+\hat{\beta}_{t}\bm{\epsilon}_{t}+ \hat{\gamma}_{t}{\bm{h}},\quad\bm{\epsilon}_{t}\sim\mathcal{N}(\mathbf{0},\bm{ I}).$ (8) $$

Here, $\hat{\alpha}_{t}$, $\hat{\beta}_{t}$, and $\hat{\gamma}_{t}$ are time-dependent scaling factors, and ${\bm{h}}=F({\bm{x}})$ serves as the conditional representation acting as prior knowledge.

Similarly, the previous state ${\bm{y}}_{t-1}$ can be expressed as:

$$ ${\bm{y}}_{t-1}=\hat{\alpha}_{t-1}{\bm{y}}_{0}+\hat{\beta}_{t-1}\bm{\epsilon}_{ t-1}+\hat{\gamma}_{t-1}{\bm{h}},\quad\bm{\epsilon}_{t-1}\sim\mathcal{N}( \mathbf{0},\bm{I}).$ (9) $$

We are interested in the posterior distribution $q\left({\bm{y}}_{t-1}\mid{\bm{y}}_{t},{\bm{y}}_{0},{\bm{h}}\right)$. According to the properties of Gaussian distributions, this posterior is also Gaussian and can be written as:

$$ $q\left({\bm{y}}_{t-1}\mid{\bm{y}}_{t},{\bm{y}}_{0},{\bm{h}}\right)=\mathcal{N} \left({\bm{y}}_{t-1};\kappa_{t}{\bm{y}}_{t}+\lambda_{t}{\bm{y}}_{0}+\zeta_{t}{ \bm{h}},\hat{\sigma}_{t}^{2}{\bm{I}}\right),$ (10) $$

where $\kappa_{t}$, $\lambda_{t}$, and $\zeta_{t}$ are coefficients to be determined, and $\hat{\sigma}_{t}^{2}$ is the variance.

By substituting Eq. (8) into the expression for ${\bm{y}}_{t-1}$, we obtain:

$$ $\displaystyle{\bm{y}}_{t-1}$ $\displaystyle=\kappa_{t}{\bm{y}}_{t}+\lambda_{t}{\bm{y}}_{0}+\zeta_{t}{\bm{h}} +\hat{\sigma}_{t}\bm{\epsilon}^{\prime}\$ (11) $\displaystyle=\kappa_{t}(\hat{\alpha}_{t}{\bm{y}}_{0}+\hat{\beta}_{t}\bm{ \epsilon}_{t}+\hat{\gamma}_{t}{\bm{h}})+\lambda_{t}{\bm{y}}_{0}+\zeta_{t}{\bm{ h}}+\hat{\sigma}_{t}\bm{\epsilon}^{\prime}\$ $\displaystyle=(\kappa_{t}\hat{\alpha}_{t}+\lambda_{t}){\bm{y}}_{0}+(\kappa_{t} \hat{\gamma}_{t}+\zeta_{t}){\bm{h}}+(\kappa_{t}\hat{\beta}_{t}\bm{\epsilon}_{t }+\hat{\sigma}_{t}\bm{\epsilon}^{\prime}),$ $$

where $\bm{\epsilon}^{\prime}\sim\mathcal{N}(\mathbf{0},\bm{I})$ is independent of $\bm{\epsilon}_{t}$.

Since the sum of two independent Gaussian noises is another Gaussian noise, we have:

$$ $\kappa_{t}\hat{\beta}_{t}\bm{\epsilon}_{t}+\hat{\sigma}_{t}\bm{\epsilon}^{ \prime}=\sqrt{\kappa_{t}^{2}\hat{\beta}_{t}^{2}+\hat{\sigma}_{t}^{2}},\bm{ \epsilon}_{t-1},$ (12) $$

where $\bm{\epsilon}_{t-1}\sim\mathcal{N}(\mathbf{0},\bm{I})$.

Comparing this with Eq. (9), we can equate the coefficients:

$$ $\hat{\alpha}_{t-1}=\kappa_{t}\hat{\alpha}_{t}+\lambda_{t},\quad\hat{\gamma}_{t -1}=\kappa_{t}\hat{\gamma}_{t}+\zeta_{t},\quad\hat{\beta}_{t-1}=\sqrt{\kappa_{ t}^{2}\hat{\beta}_{t}^{2}+\hat{\sigma}_{t}^{2}}.$ (13) $$

Solving for $\kappa_{t}$, $\lambda_{t}$, and $\zeta_{t}$, we get:

$$ $\displaystyle\kappa_{t}=\frac{\sqrt{\hat{\beta}_{t-1}^{2}-\hat{\sigma}_{t}^{2} }}{\hat{\beta}_{t}}$ (14) $\displaystyle\lambda_{t}=\hat{\alpha}_{t-1}-\frac{\hat{\alpha}_{t}\sqrt{\hat{ \beta}_{t-1}^{2}-\hat{\sigma}_{t}^{2}}}{\hat{\beta}_{t}}=\hat{\alpha}_{t-1}- \hat{\alpha}_{t}\kappa_{t}$ $\displaystyle\hat{\zeta}_{t}=\hat{\gamma}_{t-1}-\frac{\hat{\gamma}_{t}\sqrt{ \hat{\beta}_{t-1}^{2}-\hat{\sigma}_{t}^{2}}}{\hat{\beta}_{t}}=\hat{\gamma}_{t- 1}-\hat{\gamma}_{t}\kappa_{t}$ $$

Since ${\bm{h}}$ is completely determined by ${\bm{x}}$, the posterior distribution becomes:

$$ $q\left({\bm{y}}_{t-1}\mid{\bm{y}}_{t},{\bm{y}}_{0},{\bm{x}}\right)=\mathcal{N} \left({\bm{y}}_{t-1};\kappa_{t}{\bm{y}}_{t}+\lambda_{t}{\bm{y}}_{0}+\zeta_{t}{ \bm{h}},\hat{\sigma}_{t}^{2}{\bm{I}}\right).$ (15) $$

However, this posterior depends on the unknown data distribution $q({\bm{y}}_{0})$, making it impractical for direct use. Therefore, we introduce a learnable transition probability $p_{\theta}({\bm{y}}_{t-1}\mid{\bm{y}}_{t},{\bm{x}})$ to approximate $q\left({\bm{y}}_{t-1}\mid{\bm{y}}_{t},{\bm{y}}_{0},{\bm{x}}\right)$ for all $t$. The reverse process is defined as:

$$ $\displaystyle p_{\theta}({\bm{y}}_{0:T}\mid{\bm{x}})$ $\displaystyle:=p_{\theta}({\bm{y}}_{T}){\textstyle\prod_{t=1}^{T}}p_{\theta}({ \bm{y}}_{t-1}\mid{\bm{y}}_{t},{\bm{x}}),$ (16) $\displaystyle p_{\theta}({\bm{y}}_{t-1}\mid{\bm{y}}_{t},{\bm{x}})$ $\displaystyle:=\mathcal{N}({\bm{y}}_{t-1};\mu_{\theta}\left({\bm{y}}_{t},{\bm{ h}},\mathbf{c},t\right),\hat{\sigma}_{t}^{2}\bm{I})$ (17) $$

Here, $\mathbf{c}={\bm{E}}({\bm{x}})$ represents the condition guiding the reverse process, where ${\bm{E}}(\cdot)$ is a conditioning network taking historical data ${\bm{x}}$ as input, and $\theta$ includes all trainable parameters of the model. The mean $\mu_{\theta}$ is trained to predict ${\bm{y}}_{t-1}$ given ${\bm{y}}_{t}$, ${\bm{h}}$, and $\mathbf{c}$, with the reverse variance schedule $\hat{\sigma}_{t}^{2}$ fixed.

When we use ${\bm{y}}_{\theta}$ as the data prediction model to estimate the ground truth ${\bm{y}}_{0}$, the mean $\mu_{\theta}$ can be expressed as:

$$ $\mu_{\theta}({\bm{y}}_{t},{\bm{h}},\mathbf{c},t)=\kappa_{t}{\bm{y}}_{t}+ \lambda_{t}{\bm{y}}_{\theta}({\bm{y}}_{t},{\bm{h}},\mathbf{c},t)+\zeta_{t}{\bm {h}}.$ (18) $$

In this formulation, ${\bm{y}}_{\theta}({\bm{y}}_{t},{\bm{h}},\mathbf{c},t)$ is a neural network that predicts ${\bm{y}}_{0}$ from ${\bm{y}}_{t}$, conditioned on ${\bm{h}}$, $\mathbf{c}$, and time $t$.

### A.2 More forecasting results visualization

To enhance the comprehensive understanding of our forecasting methods, we present additional visualizations of our predictive results in the following sections. These supplemental images delve deeper into the performance variations of our models under different conditions. By exploring these extra results, readers can obtain a more detailed appreciation of the effectiveness and applicability of our forecasting approaches.  Figures 5 and 6 and  Figure 7respectively display partial predictive results of our $\mathrm{S^{2}DBM}$ model on the ETTh1, ETTm1, and Weather datasets.

Figure: Figure 5: The predicted samples by our $\mathrm{S^{2}DBM}$ model for different forecast window lengths on the ETTh1 dataset.
Refer to caption: x10.png

Figure: Figure 6: The predicted samples by our $\mathrm{S^{2}DBM}$ model for different forecast window lengths on the ETTm1 dataset.
Refer to caption: x11.png

Figure: Figure 7: The predicted samples by our $\mathrm{S^{2}DBM}$ model for different forecast window lengths on the Weather dataset.
Refer to caption: x12.png

### A.3 Experimental Details

#### A.3.1 dataset information

We adopt seven real-world benchmarks in the experiments to evaluate the accuracy of multivariate time series forecasting, Table 6 summarizes the statistics of these datasets.

**Table 6: Brief statistics of the datasets.**
| Datasets | Channels | Granularity | Timesteps |
| --- | --- | --- | --- |
| Weather | 21 | 10 min | 59696 |
| ILI | 7 | 1 week | 966 |
| Exchange | 8 | 1 day | 7588 |
| ETTh1&ETTh2 | 7 | 1 hour | 17420 |
| ETTm1&ETTm2 | 7 | 5 min | 69680 |

We adopted the experimental settings from recent studies (Liu et al., ; Zeng et al., ; Li et al., ). Specifically, following the recommendations of Dlinear (Zeng et al., ), we set the input length $H=336$. We assessed the prediction accuracy for lengths $L=\{96,192,336,720\}$ across the Weather, Exchange, ETTh1, ETTh2, ETTm1, and ETTm2 datasets, and $L=\{24,36,48,60\}$ for the ILI dataset.

#### A.3.2 implementation details

As mentioned in Section 3.3, the denoising network of $\mathrm{S^{2}DBM}$ adopts the same architecture as CSDI Tashiro et al. () but removes modules related to its original conditioning mechanism.Both the conditional encoder network $E$ and the prior predictor $F(\cdot)$ in $\mathrm{S^{2}DBM}$ employ a simple one-layer linear model (Zeng et al., ). Table 7 contains the hyperparameters that for $\mathrm{S^{2}DBM}$ training and architecture.

**Table 7: $\mathrm{S^{2}DBM}$ hyperparameters.**
| Hyperparameter | Value |
| --- | --- |
| Residual layers | 4 |
| Residual channels | 8 |
| Diffusion embedding dim | 8 |
| Schedule | Linear |
| Diffusion steps $T$ | 50 |
| Self-attention layers time dim | 1 |
| Self-attention heads time dim | 8 |
| Self-attention layers feature dim | 1 |
| Self-attention layers time dim | 8 |
| EMA decay | 0.995 |
| EMA update interval | 8 |
| Optimizer | Adam |
| Loss function | MAE |
| Max learning rate | $1\times 10^{-4}$ |
| Min learning rate | $5\times 10^{-7}$ |
| Individual channels | False |

## Peer Reviews

confidence: 4
contribution: 2
initial_rating: 3
presentation: 3
questions: NA
soundness: 2
strengths: This paper utilizes the Diffusion Bridge Model to help the reverse process start from a more deterministic state, reducing the instability caused by noise and thereby facilitating better predictions.
summary: This paper presents a comprehensive framework that encompasses most existing diffusion-based methods. Building on this foundation, the authors introduce the Series-to-Series Diffusion Bridge Model (S2DBM). Experimental results demonstrate that S2DBM delivers superior performance.
weaknesses: weakness:
1. There is a notation issue with \(\hat{\gamma}_t\) in line 203; the writing needs to be standardized. Additionally, it needs to be clarified whether the values of \(\hat{\alpha}_t\), \(\hat{\beta}_t\), and \(\gamma_t\) should have a specific relationship to conform to the diffusion model.
2. In line 411, it is mentioned that a comparison with timediff was made, but Table 2 does not include TimeDiff data while other baselines are present.
3. The content of the paper appears to primarily build on existing work by combining the model guidance from TMDM and the non-autoregressive approach of timediff with the existing Brownian Bridge process. This integration seems to lack novelty.

---

confidence: 2
contribution: 2
initial_rating: 6
presentation: 2
questions: See weakness plz.
soundness: 2
strengths: The authors provide a comprehensive summary of existing models, highlighting that their primary differences lie in the formulation of $\hat{\gamma_t}$. By introducing the Brownian bridge process into diffusion-based time series forecasting models, they establish a range of relevant properties.
summary: This paper revisits the application of diffusion models for time series forecasting, presenting a unified framework that consolidates these methods. Building on this framework, the authors incorporate the Brownian bridge process to enhance prediction accuracy. The results demonstrate that the proposed approach outperforms existing diffusion-based forecasting models.
weaknesses: 1. In Line 259, the title of Proposition 1 is "Brownian Bridge between Historical and Predicted Time Series." However, the Brownian bridge’s endpoint is set to $h$, the Prior Predictor’s forecasted value of $x$. The paper provides no explanation as to why $h$ is chosen as the endpoint for the Brownian bridge. 
2. It would be beneficial for the authors to clarify, from a theoretical perspective, why the Brownian bridge is integrated into the diffusion model for time series forecasting. Specifically, how does its ability to "pin down" the diffusion process at both ends help reduce instability from noisy inputs and enable the accurate generation of future features based on historical time series?
3. In line 195, Theorem 1 concerns non-autoregressive diffusion processes. However, the summarized models, CSDI and SSSD, do not appear to be non-autoregressive models. Could there be an issue with the theorem here?

---

confidence: 3
contribution: 2
initial_rating: 5
presentation: 3
questions: Q1.
Can you elaborate on the computational efficiency of $S^2DBM$, particularly in relation to larger datasets or real-time forecasting applications?

Q2.
What specific advantages does the Brownian Bridge process offer over other stochastic processes for time series forecasting in the context of your model?
soundness: 2
strengths: S1.
$S^2DBM$ integrates the Brownian Bridge process into time series forecasting using diffusion models.
By redefining the diffusion framework, the authors introduce a new model and consolidate various non-autoregressive diffusion techniques into a comprehensive framework, elucidating their interrelationships and underlying principles. 

S2.
The authors conduct thorough theoretical groundwork. 
The empirical evaluations are robust, utilizing diverse real-world datasets to benchmark the model's performance against SOTA methods.

S3.
The draft is well-structured and organized. 
The introduction succinctly outlines the problem context and motivates the need for the proposed model.
summary: This paper presents the Series-to-Series Diffusion Bridge Model ($S^2DBM$), a promising approach in time series forecasting using diffusion models. 
Traditional diffusion models often struggle with deterministic point-to-point predictions. 
$S^2DBM$ addresses this by leveraging the Brownian Bridge process to reduce noise and improve accuracy in reverse estimations, effectively capturing temporal dependencies in time series data. 
The model incorporates historical data as informative priors, stabilizing diffusion and enhancing point-to-point prediction capabilities. 
Experimental results on various datasets show that $S^2DBM$ outperforms existing diffusion-based and other state-of-the-art time series models, demonstrating superior accuracy in both deterministic and probabilistic forecasting tasks.
weaknesses: W1. 
Some assumptions and derivations in this work could be better justified. 
For example, the choice of using a Brownian Bridge process warrants a more detailed discussion on why it is preferred over other stochastic processes. 
Providing empirical evidence or theoretical reasoning for the choice of this process could strengthen the argument for its effectiveness in reducing randomness in forecasting.

W2.
While the paper presents strong performance metrics, it lacks robustness testing under adverse conditions, such as noisy inputs or missing data. 
Future experiments should include scenarios with varying levels of data quality to demonstrate how $S^2DBM$ handles real-world challenges.

W3.
Currently, the experiments focus primarily on a single configuration. 
Investigating how different settings of hyperparameters, such as the choice of the prior predictor or conditioning modules, impact performance could provide valuable insights. 
For instance, evaluating the effects of varying the number of diffusion steps or using alternative conditioning mechanisms could highlight the robustness of $S^2DBM$ across diverse scenarios.

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confidence: 3
contribution: 3
initial_rating: 8
presentation: 3
questions: 1. In Table 2, your models do not achieve the best performance across all datasets. Could you explain why linear models perform best in several cases, while your models do not?
2. Since $c = E(x)$ and $h = F(x)$ are two important components of your algorithms, I am curious about the role of F in enhancing performance. What impact does it have? An ablation study of F may help.
soundness: 3
strengths: 1. The presentation of this paper is clear and easy to follow, with algorithms well articulated.
2. The paper provides a unified framework for several existing diffusion-based time series models, which is well-supported by theoretical foundations. The insights offered are impressive.
3. Both point-to-point and probabilistic forecasting are addressed in the reverse process, broadening the algorithm's application scope.
summary: This paper addresses point-to-point time-series forecasting through the use of diffusion models. The authors propose a unified non-autoregressive framework that encompasses most existing diffusion-based time-series models. Building on this framework, they introduce S2DBM, which incorporates the Brownian Bridge process. By integrating historical information via informative priors and conditions, S2DBM effectively reduces randomness and enhances forecasting accuracy.
weaknesses: 1. Some expressions reference prior work without sufficient explanations, which can hinder comprehension. For instance, in the last equation of Section 3.1, the introduction of \(y_\theta\) relies solely on a citation, making it time-consuming for readers to fully grasp its meaning.
2. Table 4 presents results only for a horizon of 96, omitting long horizon settings. This limits the analysis and may leave important insights unaddressed.

Meta-review: The paper introduces a unified framework for diffusion models for time series forecasting and, building on this framework, it introduces the Brownian bridge process to enhance prediction accuracy. In terms of overall ratings, one reviewer is positive, two reviewers are mildly positive and one reviewer is negative. However, the two mildly positive reviews do appear lukewarm -- one reviewer increased their score during the rebuttal period as an "encouragement" -- and they do share some of the same concerns raised by the reviewer with an overall negative rating. The main issues comprise the significance of the contribution, the justification/motivation to use the Brownian bridge, and the experimental evaluation. As a consequence, I am unable to recommend acceptance.

Decision: Reject