# MillGNN: Learning Multi-Scale Lead-Lag Dependencies for Multi-Variate Time Series Forecasting

arXiv: https://arxiv.org/abs/2509.03852 · HTML: https://mlsanity.com/paper/2509.03852/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 42 references that are in the map.
- First posted on arXiv 4 Sep 2025.

## Builds on (most cited first, 21 of 21)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Informer: Beyond Efficient Transformer for Long Sequence Time-Series Forecasting](https://mlsanity.com/paper/2012.07436/) (2020)
- [A Time Series is Worth 64 Words: Long-term Forecasting with Transformers](https://mlsanity.com/paper/2211.14730/) (2022)
- [Autoformer: Decomposition Transformers with Auto-Correlation for Long-Term Series Forecasting](https://mlsanity.com/paper/2106.13008/) (2021)
- [Are Transformers Effective for Time Series Forecasting?](https://mlsanity.com/paper/2205.13504/) (2022)
- [FEDformer: Frequency Enhanced Decomposed Transformer for Long-term Series Forecasting](https://mlsanity.com/paper/2201.12740/) (2022)
- [DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks](https://mlsanity.com/paper/1704.04110/) (2017)
- [Non-stationary Transformers: Exploring the Stationarity in Time Series Forecasting](https://mlsanity.com/paper/2205.14415/) (2022)
- [SCINet: Time Series Modeling and Forecasting with Sample Convolution and Interaction](https://mlsanity.com/paper/2106.09305/) (2021)
- [TSMixer: Lightweight MLP-Mixer Model for Multivariate Time Series Forecasting](https://mlsanity.com/paper/2306.09364/) (2023)
- [PDFormer: Propagation Delay-Aware Dynamic Long-Range Transformer for Traffic Flow Prediction](https://mlsanity.com/paper/2301.07945/) (2023)
- [CycleNet: Enhancing Time Series Forecasting through Modeling Periodic Patterns](https://mlsanity.com/paper/2409.18479/) (2024)
- [MSGNet: Learning Multi-Scale Inter-Series Correlations for Multivariate Time Series Forecasting](https://mlsanity.com/paper/2401.00423/) (2023)
- [Fredformer: Frequency Debiased Transformer for Time Series Forecasting](https://mlsanity.com/paper/2406.09009/) (2024)
- [Group-Aware Graph Neural Network for Nationwide City Air Quality Forecasting](https://mlsanity.com/paper/2108.12238/) (2021)
- [Rethinking Channel Dependence for Multivariate Time Series Forecasting: Learning from Leading Indicators](https://mlsanity.com/paper/2401.17548/) (2024)
- [Ada-MSHyper: Adaptive Multi-Scale Hypergraph Transformer for Time Series Forecasting](https://mlsanity.com/paper/2410.23992/) (2024)
- [FilterTS: Comprehensive Frequency Filtering for Multivariate Time Series Forecasting](https://mlsanity.com/paper/2505.04158/) (2025)
- [WeatherGNN: Exploiting Meteo- and Spatial-Dependencies for Local Numerical Weather Prediction Bias-Correction](https://mlsanity.com/paper/2310.05517/) (2023)
- [Unveiling Delay Effects in Traffic Forecasting: A Perspective from Spatial-Temporal Delay Differential Equations](https://mlsanity.com/paper/2402.01231/) (2024)
- [Gaussian Processes for Time Series with Lead-Lag Effects with applications to biology data](https://mlsanity.com/paper/2401.07400/) (2024)

## Built on by (most cited first, 2 of 2)

- [XMatch: Enhancing Covariate-Aware Time Series Forecasting through Tree-Structured Exogenous Matching](https://mlsanity.com/paper/2609.34939/) (2026)
- [CTIA-ODE: Continuous-Time Inter-Variable Attention for Long-Term Time Series Forecasting](https://mlsanity.com/paper/2610.19831/) (2026)

## Abstract (arXiv)

Multi-variate time series (MTS) forecasting is crucial for various applications. Existing methods have shown promising results owing to their strong ability to capture intra- and inter-variate dependencies. However, these methods often overlook lead-lag dependencies at multiple grouping scales, failing to capture hierarchical lead-lag effects in complex systems. To this end, we propose MillGNN, a novel underlinegraph underlineneural underlinenetwork-based method that learns underlinemultunderlineiple grouping scale underlinelead-underlinelag dependencies for MTS forecasting, which can comprehensively capture lead-lag effects considering variate-wise and group-wise dynamics and decays. Specifically, MillGNN introduces two key innovations: (1) a scale-specific lead-lag graph learning module that integrates cross-correlation coefficients and dynamic decaying features derived from real-time inputs and time lags to learn lead-lag dependencies for each scale, which can model evolving lead-lag dependencies with statistical interpretability and data-driven flexibility; (2) a hierarchical lead-lag message passing module that passes lead-lag messages at multiple grouping scales in a structured way to simultaneously propagate intra- and inter-scale lead-lag effects, which can capture multi-scale lead-lag effects with a balance of comprehensiveness and efficiency. Experimental results on 11 datasets demonstrate the superiority of MillGNN for long-term and short-term MTS forecasting, compared with 16 state-of-the-art methods.