TimeBridge: Non-Stationarity in Forecasting

• TimeBridge is a multivariate time series forecasting framework that separates short-term de-trending from long-term cointegration to exploit non-stationary properties.
• It employs Integrated Attention to remove spurious short-term fluctuations and Cointegrated Attention to preserve long-term dependencies across variables.
• Empirical evaluations on benchmark datasets show state-of-the-art performance with over 10% relative gains on metrics compared to competitive models.

TimeBridge most commonly denotes a framework for long-term multivariate time series forecasting that treats non-stationarity as having different roles in short-term and long-term dependency modeling. In that formulation, short-term non-stationarity is removed to avoid spurious regressions and stabilize intra-variate modeling, whereas long-term non-stationarity is preserved to expose inter-variate cointegration. The same name is also used in other research areas for bridge-based constructions over diffusion priors, timing-distribution stacks, and temporally evolving networks, so the term is best understood as a family of domain-specific concepts rather than a single uniform system (Liu et al., 2024).

1. Scope and principal usages

The term is used in several distinct technical senses.

Context	Meaning	arXiv id
Long-term forecasting	Non-stationarity-aware multivariate forecasting framework	(Liu et al., 2024)
Time series generation	Diffusion-bridge framework with task-specific priors	(Park et al., 2024)
ALICE Run 3 timing	CRU-based TTC-PON to GBT communication bridge	(Mitra et al., 2018)
Temporal-network analysis	Temporal bridge role exposed by temporal betweenness	(Rad et al., 2015)

In forecasting, TimeBridge is the title of “TimeBridge: Non-Stationarity Matters for Long-term Time Series Forecasting,” which defines the best-known usage in the supplied literature. In generation, the same name is assigned to a diffusion-bridge framework that replaces the fixed standard-Gaussian prior with data-dependent, time-dependent, or scale-preserving priors. In high-energy physics, “TimeBridge” refers to a firmware-defined bridge inside the Common Readout Unit that forwards timing, trigger, and control information from TTC-PON to GBT links. In temporal-network analysis, the term is not a product name but a useful descriptor for nodes whose importance emerges only when time ordering is respected (Park et al., 2024, Mitra et al., 2018, Rad et al., 2015).

A related but explicitly unnamed case appears in distributed quantum networking: a modular Time Tagging agent bridges local detector time tags to a White Rabbit absolute timeline and onward to centralized coincidence analysis, but the paper states that it does not name a “TimeBridge” system (Amlou et al., 17 May 2025).

2. Forecasting formulation and theoretical motivation

In the forecasting literature, TimeBridge addresses multivariate time series forecasting with historical observations

$X = [x_1,\cdots,x_I] \in \mathbb{R}^{C\times I},$

and future targets

$$Y = [x_{I+1},\cdots,x_{I+O}] \in \mathbb{R}^{C\times O},$$

where $C$ is the number of variables, $I$ the input length, and $O$ the forecast horizon. Its central claim is that non-stationarity should neither be uniformly eliminated nor uniformly retained. Eliminating non-stationarity is described as essential for avoiding spurious regressions and capturing local dependencies in short-term modeling, while preserving it is described as crucial for revealing long-term cointegration across variates (Liu et al., 2024).

The theoretical vocabulary is drawn from integrated processes and cointegration. The appendix defines integration with the Augmented Dickey-Fuller test:

$$\Delta X_t = \alpha + \beta t + \gamma X_{t-1} + \sum_{i=1}^{p} \delta_i \Delta X_{t-i} + \epsilon_t,$$

where failure to reject the unit-root null indicates non-stationarity. Cointegration is introduced through a stationary linear combination of non-stationary series, for example

$Z_t = X_t - \beta Y_t,$

and tested with the Engle-Granger procedure:

$$X_t = \alpha + \beta Y_t + \epsilon_t,\qquad \Delta \epsilon_t = \gamma \epsilon_{t-1} + \sum_{i=1}^{p} \delta_i \Delta \epsilon_{t-i} + \nu_t.$$

If the residuals are stationary, the variables are cointegrated. TimeBridge operationalizes this distinction architecturally: local intra-variate modeling is made more stationary, whereas long-range inter-variate modeling preserves the shared non-stationary structure from which cointegration can be inferred.

This yields a specific correction to a common misconception in forecasting research: non-stationarity is not treated as a uniform nuisance. The paper argues that the same statistical property can be harmful for one dependency class and informative for another.

3. Architecture: patching, Integrated Attention, and Cointegrated Attention

The model begins by splitting each variate into non-overlapping patches of length $S$:

$$\{p_1,\cdots,p_N\} = \text{Patching}(X), \qquad P = \text{Embedding}(p_1,\cdots,p_N),$$

with $$Y = [x_{I+1},\cdots,x_{I+O}] \in \mathbb{R}^{C\times O},$$0. The patch embedding maps each patch to a $$Y = [x_{I+1},\cdots,x_{I+O}] \in \mathbb{R}^{C\times O},$$1-dimensional token, yielding $$Y = [x_{I+1},\cdots,x_{I+O}] \in \mathbb{R}^{C\times O},$$2. This patchwise representation is the structural basis for treating short-term and long-term effects differently (Liu et al., 2024).

The first major block is Integrated Attention, designed for short-term dependencies within each variate after removing short-term non-stationarity. Each patch is de-trended by a moving average:

$$Y = [x_{I+1},\cdots,x_{I+O}] \in \mathbb{R}^{C\times O},$$3

Attention is then computed with detrended patches as Query and Key, while the original patch tokens remain as Value:

$$Y = [x_{I+1},\cdots,x_{I+O}] \in \mathbb{R}^{C\times O},$$4

$$Y = [x_{I+1},\cdots,x_{I+O}] \in \mathbb{R}^{C\times O},$$5

The design choice is explicit: the attention map is made stationary by using $$Y = [x_{I+1},\cdots,x_{I+O}] \in \mathbb{R}^{C\times O},$$6 for Query and Key, so short-term fluctuations do not dominate local dependency modeling.

The second major block is Cointegrated Attention, designed for long-term dependencies across variates. Before it is applied, TimeBridge performs patch downsampling:

$$Y = [x_{I+1},\cdots,x_{I+O}] \in \mathbb{R}^{C\times O},$$7

Here $$Y = [x_{I+1},\cdots,x_{I+O}] \in \mathbb{R}^{C\times O},$$8 reduces the patch count from $$Y = [x_{I+1},\cdots,x_{I+O}] \in \mathbb{R}^{C\times O},$$9 to $C$0 with $C$1. After downsampling, attention is applied over the channel dimension at fixed time index $C$2:

$C$3

$C$4

This stage preserves non-stationarity, because the long-term equilibrium structure across variables is often expressed through shared non-stationary trends. The paper explicitly interprets the resulting attention weights as reflecting cointegration strength.

The ordering of the blocks is part of the method rather than an incidental implementation detail. The reported ablations show that Integrated Attention followed by Cointegrated Attention works best, while the reverse order works worst. The paper also states that for the ETT datasets, where long-term cointegration is weaker, the model focuses on short-term intra-variate variation only.

4. Experimental evaluation and ablation evidence

TimeBridge is evaluated in three settings: long-term forecasting, short-term forecasting, and financial forecasting. For long-term forecasting it uses ETTm1, ETTm2, ETTh1, ETTh2, Weather, Electricity, Traffic, and Solar, with prediction lengths $C$5. For short-term forecasting it uses PeMS03, PeMS04, PeMS07, and PeMS08, with input length 96 and forecast length 12. For financial forecasting it uses CSI 500 and S&P 500, predicting next-day returns and evaluating portfolio performance with ARR, AVol, MDD, ASR, CR, and IR. The implementation uses PyTorch, Adam, two RTX 3090 GPUs, batch size 16 for Electricity and Traffic and 32 otherwise, 10 training epochs, and $C$6 patches; dataset-specific downsampling values $C$7 are tuned, with Weather and Solar using $C$8 and Electricity and Traffic using $C$9 (Liu et al., 2024).

On long-term forecasting, the paper reports state-of-the-art or near-state-of-the-art results across all eight datasets. In the “original papers” comparison table, TimeBridge has the best average performance on every dataset except that it is very close on some small cases, and the paper summarizes its improvement as averaging over 10% relative gains versus baselines. Against strong competitors, it reduces MSE/MAE by 3.10% / 1.64% versus PDF, 3.55% / 0.81% versus ModernTCN, and 6.92% / 4.54% versus TimeMixer. In the hyperparameter-search setting, representative averages are reported as ETTm1: $I$0, ETTm2: $I$1, ETTh1: $I$2, ETTh2: $I$3, Weather: $I$4, Electricity: $I$5, Traffic: $I$6, and Solar: $I$7. The appendix reports 99% confidence intervals.

On short-term forecasting over PeMS traffic networks, TimeBridge is reported as best across all four datasets, with PeMS03: MAE 14.52, MAPE 14.21, RMSE 23.10, PeMS04: MAE 19.24, MAPE 12.42, RMSE 31.12, PeMS07: MAE 20.43, MAPE 8.42, RMSE 33.44, and PeMS08: MAE 14.98, MAPE 9.56, RMSE 23.77. The paper attributes this to the de-noising effect of Integrated Attention together with the patchwise handling of strong spatiotemporal dependence.

The financial results are also reported as best in the table. On CSI 500, TimeBridge achieves

$I$8

On S&P 500, it achieves

$I$9

The appendix reports 95% confidence intervals. The stated interpretation is that stock markets are highly non-stationary and cross-sector cointegrated, which matches the model’s design assumptions.

The ablations reinforce the methodological thesis. Performance is best when non-stationarity is removed in Integrated Attention and preserved in Cointegrated Attention; using either block alone is worse than the full model; and channel-independent modeling is preferable for short-term intra-variate dynamics, while channel-dependent modeling is beneficial for long-term cointegrated relationships, especially when the number of channels is large.

5. Position in subsequent forecasting literature

Subsequent work treats TimeBridge as a strong modern baseline. In “Beyond MSE: Ordinal Cross-Entropy for Probabilistic Time Series Forecasting,” it appears alongside Autoformer, DLinear, iTransformer, and TimeXer as one of five baseline forecasting models. The paper identifies it as a 2024 long-term time series forecasting model, trains the baselines with MSE loss, and evaluates on ETTh1, ETTh2, ETTm1, ETTm2, Exchange, ILI, and Weather using MSE and MAE. The setup uses batch size 32, 15 epochs, Adam, learning rate 0.005, early stopping patience 5, a lookback window usually 336 and 104 for ILI, with ETT series normalized to [0, 1] and other datasets normalized to [-1, 1] (Wang et al., 13 Nov 2025).

The reported comparison is not uniformly favorable to either method. TimeBridge is better than OCE-TS on ETTh1 at horizons 192, 336, and 720 for both MSE and MAE, and on ETTm1 at 336 and 720 for both metrics. OCE-TS is substantially better on ETTh2, better on every ETTm2 horizon, strongly better on Exchange, dramatically better on ILI, and consistently better on Weather. The paper’s own interpretation is that TimeBridge is a strong baseline on some smoother ETT-style series, especially ETTh1 and ETTm1 at longer horizons, whereas OCE-TS is more robust overall on noisier or more difficult datasets. TimeBridge’s runtime is reported as 21 ms/iter, faster than iTransformer and much faster than Autoformer, though somewhat faster than OCE-TS. The same paper explicitly states that there are no probabilistic metrics, no calibration tables, and no TimeBridge-specific ablations.

A later line of work studies TimeBridge as a backbone to be corrected rather than replaced. “Reviving Error Correction in Modern Deep Time-Series Forecasting” proposes UEC-STD, an architecture-agnostic post-hoc corrector that can be integrated with existing forecasters without retraining. The paper evaluates four major backbones—TimeMixer, TimesNet, TimeXer, and TimeBridge—and reports average improvements of 2.32% in MSE and 0.94% in MAE across backbones on seven common benchmarks, with larger gains on the Monash datasets. It also states that UEC-STD can improve even strong backbones such as TimeBridge, and in a dedicated TimeBridge setting with tuned lookback windows it matches or outperforms the backbone on most datasets while still yielding meaningful improvements on already saturated benchmarks like ETTm1 and ETTh2 (Nguyen et al., 20 May 2026).

These later studies place TimeBridge in a specific methodological position: it is treated neither as an obsolete baseline nor as a probabilistic forecaster, but as a competitive deterministic architecture whose strengths are clear enough that later work either benchmarks against it or wraps it with corrective modules.

6. Other TimeBridge concepts beyond long-term forecasting

A separate machine-learning usage appears in “TimeBridge: Better Diffusion Prior Design with Bridge Models for Time Series Generation.” There, TimeBridge is a framework for time series generation that replaces the fixed terminal prior $O$0 with priors tailored to the task. For unconditional generation it introduces data-dependent priors

$O$1

and time-dependent Gaussian-process priors

$O$2

with RBF kernel

$O$3

For conditional generation it uses scale-preserving priors such as $O$4 for trend-conditioned synthesis and interpolated priors for imputation. The framework employs a denoiser based on Diffusion-TS, a VP bridge by default, direct $O$5-matching, Fourier loss, and a second-order Heun sampler with 40 steps and about 119 NFEs; the paper reports that VP is better than VE, and that TimeBridge-GP often ranks best on unconditional generation metrics such as Context-FID, correlational score, discriminative score, and predictive score (Park et al., 2024).

In high-energy physics, “TimeBridge” denotes the CRU-based communication bridge between the TTC-PON timing network and the GBT links in the ALICE Run 3 upgrade. Its purpose is to forward timing, trigger, and control information with deterministic, low-jitter, constant-latency behavior in a trigger-less readout architecture. The paper evaluates four metrics: firmware latency, PLL jitter-cleaning behavior, BER, and optical transceiver parameter effects. Reported values include 151 ns TTC-PON latency between transmit and receive flags; GBT round-trip latency values from 150 ns to 550 ns depending on loopback and optimization mode; and RMS jitter values such as 8.786 ps for the VLDB clock out with the TTC-PON + GBT bridge at 40 MHz, 200 Hz. The paper concludes that the 200 Hz loop bandwidth gives the best jitter cleaning performance and that the bridge meets ALICE timing requirements with stable latency and no points of uncertainty over repeated PFR rounds (Mitra et al., 2018).

In temporal-network analysis, the relevant notion is not a named system but a temporal bridge role. “Tempus Fugit: The Impact of Time in Knowledge Mobilization Networks” argues that static betweenness can miss nodes that are crucial for early transmission and introduces a temporal betweenness measure over foremost increasing journeys:

$O$6

This identifies nodes with low static centrality but high temporal importance—called “invisible rapids”—as distinct from nodes with high static but low temporal importance, called “invisible brooks.” In that literature, a “time bridge” is a node that accelerates knowledge flow because it connects earlier and later activity in the right temporal order (Rad et al., 2015).

A related systems interpretation appears in distributed quantum networking. “Scalable Time-Tagged Data Acquisition for Entanglement Distribution in Quantum Networks” does not name a TimeBridge system, but it describes a bridge-like service in which a modular Time Tagging agent uses a White Rabbit 1 PPS reference to align one-second data blocks across remote labs, applies real-time calibration and overflow mitigation, compresses the data with Blosc, and forwards the results for live coincidence analysis. The reported storage reduction is from 14.32 bytes per tag to 3.80 bytes per tag, a 73.5% reduction, and the two-lab entanglement experiment achieves synchronized coincidence detection at approximately 25,000 counts per second with a 10 ns coincidence window (Amlou et al., 17 May 2025).

Taken together, these usages show that “TimeBridge” functions less as a single canonically fixed architecture than as a recurring bridge metaphor in technical systems: bridging short-term and long-term statistical structure, chosen priors and data distributions, upstream and downstream timing networks, or static topology and temporal causality.