Latent Dynamics · 2026-10-15
CTIA-ODE: Continuous-Time Inter-Variable Attention for Long-Term Time Series Forecasting
Chengyi Pan, Huangqimei Zheng, Hao Xue
AI-assisted summary based on the paper. Consult the linked source for methods and limitations. Benchmark numbers appear after editor verification.
TL;DR
CTIA-ODE is a continuous-time multivariate forecasting framework that embeds Continuous-Time Inter-Variable Attention (CTIA) inside an ODE solver. The continuously evolving endogenous trajectory dynamically queries a static exogenous context along the integration path, avoiding explicit temporal alignment. The authors add Linear Residual Augmentation (LRA) and an Outlier-Robust Hybrid Loss (ORHL) for stable, robust continuous latent dynamics.
Why it matters
The paper targets how to incorporate exogenous variables without rigid discrete-time temporal alignment. CTIA-ODE uses continuous-time cross-variable interaction and adds stability and robustness mechanisms (LRA and ORHL), and the authors report state-of-the-art forecasting results across eight real-world datasets with emphasis on long-horizon settings.
Method
- Embed CTIA in an ODE solver: endogenous hidden state evolves continuously, generating time-varying queries that continuously attend to a static exogenous field without explicit step-by-step temporal alignment.
- Use a dual-stabilization strategy: Linear Residual Augmentation (LRA) as a macroscopic trend anchor and Outlier-Robust Hybrid Loss (ORHL) to mitigate heavy-tailed anomalies via bounded gradients.
- Add Stable Refinement: a dual-loop stable refinement block applying discrete self-attention and an MLP (with residuals and layer norm) to filter numerical noise after ODE integration.
Limitation
A primary limitation is the assumption of regularly sampled historical observations; the current architecture relies on fixed-step 1D convolutional embeddings and adapting the encoder to irregularly sampled data with missing values remains a challenge.
Abstract (from arXiv)
Although multivariate time series forecasting has broad application prospects, effectively incorporating exogenous variables remains challenging. Prevailing discrete-time architectures often impose rigid temporal alignment across heterogeneous variables, which may limit their ability to capture continuously evolving external effects. To address this limitation, we introduce CTIA-ODE, a continuous-time forecasting framework centered on Continuous-Time Inter-Variable Attention (CTIA). Specifically, CTIA is embedded within an ordinary differential equation (ODE) solver, enabling the continuously evolving endogenous trajectory to dynamically query a static exogenous context along the integration path without requiring explicit temporal alignment. To improve the stability of the continuous latent dynamics, we further introduce a dual-stabilization strategy: architecturally, a Linear Residual Augmentation (LRA) module provides a macroscopic trend anchor, while an Outlier-Robust Hybrid Loss (ORHL) mitigates the influence of heavy-tailed anomalies during optimization. Extensive experiments across eight real-world datasets demonstrate that CTIA-ODE achieves state-of-the-art forecasting performance across diverse datasets and prediction horizons, while maintaining a favorable accuracy-efficiency trade-off with a compact memory footprint and low inference latency, particularly under long-horizon forecasting settings. Anonymous code is available at https://anonymous.4open.science/r/CTIA-ODE-15A6/.
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Publication record
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First public release: 2026-10-14 · Indexed by Pub-AI: 2026-10-15 UTC
CTIA-ODE: Continuous-Time Inter-Variable Attention for Long-Term Time Series Forecasting
Preprint posted: 2026-10-14 · Original record ↗
- arxiv
- 2610.19831
Explicit source versions (1)
- v2 · 2026-10-17
- Explicit source version detected · v2
Source date: 2026-10-17 · First detected: 2026-10-25 UTC · Historical baseline
Source evidence (1)
- Official evidence ↗ · atom:id explicit version (official metadata)
- Preprint available
Source date: 2026-10-14 · First detected: 2026-10-25 UTC · Historical baseline
Source evidence (1)
- Official evidence ↗ · atom:published (official metadata)
Abstract · arxiv
Although multivariate time series forecasting has broad application prospects, effectively incorporating exogenous variables remains challenging. Prevailing discrete-time architectures often impose rigid temporal alignment across heterogeneous variables, which may limit their ability to capture continuously evolving external effects. To address this limitation, we introduce CTIA-ODE, a continuous-time forecasting framework centered on Continuous-Time Inter-Variable Attention (CTIA). Specifically, CTIA is embedded within an ordinary differential equation (ODE) solver, enabling the continuously evolving endogenous trajectory to dynamically query a static exogenous context along the integration path without requiring explicit temporal alignment. To improve the stability of the continuous latent dynamics, we further introduce a dual-stabilization strategy: architecturally, a Linear Residual Augmentation (LRA) module provides a macroscopic trend anchor, while an Outlier-Robust Hybrid Loss (ORHL) mitigates the influence of heavy-tailed anomalies during optimization. Extensive experiments across eight real-world datasets demonstrate that CTIA-ODE achieves state-of-the-art forecasting performance across diverse datasets and prediction horizons, while maintaining a favorable accuracy-efficiency trade-off with a compact memory footprint and low inference latency, particularly under long-horizon forecasting settings. Anonymous code is available at https://anonymous.4open.science/r/CTIA-ODE-15A6/.