# Beyond Static Positional Encoding: Bayesian Time-aware Attention for Non-Stationary Long-term Time Series Forecasting

arXiv: https://arxiv.org/abs/2610.19603 · HTML: https://mlsanity.com/paper/2610.19603/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in MoE for time series and graphs, one of 12 directions in Mixture-of-experts.
- Builds on 42 of its 52 references that are in the map.
- First posted on arXiv 14 Oct 2026.

## Placed in

- Mixture-of-experts / [MoE for time series and graphs](https://mlsanity.com/moe/moe-for-time-series-and-graphs/)

## Builds on (most cited first, 30 of 42)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [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)
- [Transformer-XL: Attentive Language Models Beyond a Fixed-Length Context](https://mlsanity.com/paper/1901.02860/) (2019)
- [Are Transformers Effective for Time Series Forecasting?](https://mlsanity.com/paper/2205.13504/) (2022)
- [Self-Attention with Relative Position Representations](https://mlsanity.com/paper/1803.02155/) (2018)
- [Train Short, Test Long: Attention with Linear Biases Enables Input Length Extrapolation](https://mlsanity.com/paper/2108.12409/) (2021)
- [TimesNet: Temporal 2D-Variation Modeling for General Time Series Analysis](https://mlsanity.com/paper/2210.02186/) (2022)
- [iTransformer: Inverted Transformers Are Effective for Time Series Forecasting](https://mlsanity.com/paper/2310.06625/) (2023)
- [Transformers in Time Series: A Survey](https://mlsanity.com/paper/2202.07125/) (2022)
- [Non-stationary Transformers: Exploring the Stationarity in Time Series Forecasting](https://mlsanity.com/paper/2205.14415/) (2022)
- [TimeMixer: Decomposable Multiscale Mixing for Time Series Forecasting](https://mlsanity.com/paper/2405.14616/) (2024)
- [A Survey on Mixture of Experts in Large Language Models](https://mlsanity.com/paper/2407.06204/) (2024)
- [Transformer Dissection: A Unified Understanding of Transformer's Attention via the Lens of Kernel](https://mlsanity.com/paper/1908.11775/) (2019)
- [AdaRNN: Adaptive Learning and Forecasting of Time Series](https://mlsanity.com/paper/2108.04443/) (2021)
- [DUET: Dual Clustering Enhanced Multivariate Time Series Forecasting](https://mlsanity.com/paper/2412.10859/) (2024)
- [Dish-TS: A General Paradigm for Alleviating Distribution Shift in Time Series Forecasting](https://mlsanity.com/paper/2302.14829/) (2023)
- [Timer-XL: Long-Context Transformers for Unified Time Series Forecasting](https://mlsanity.com/paper/2410.04803/) (2024)
- [KERPLE: Kernelized Relative Positional Embedding for Length Extrapolation](https://mlsanity.com/paper/2205.09921/) (2022)
- [OneNet: Enhancing Time Series Forecasting Models under Concept Drift by Online Ensembling](https://mlsanity.com/paper/2309.12659/) (2023)
- [Frequency Adaptive Normalization For Non-stationary Time Series Forecasting](https://mlsanity.com/paper/2409.20371/) (2024)
- [Learning Fast and Slow for Online Time Series Forecasting](https://mlsanity.com/paper/2202.11672/) (2022)
- [Improving Position Encoding of Transformers for Multivariate Time Series Classification](https://mlsanity.com/paper/2305.16642/) (2023)
- [TimeBridge: Non-Stationarity Matters for Long-term Time Series Forecasting](https://mlsanity.com/paper/2410.04442/) (2024)
- [Relative Positional Encoding for Transformers with Linear Complexity](https://mlsanity.com/paper/2105.08399/) (2021)
- [Koopman Neural Forecaster for Time Series with Temporal Distribution Shifts](https://mlsanity.com/paper/2210.03675/) (2022)
- [Learning Pattern-Specific Experts for Time Series Forecasting Under Patch-level Distribution Shift](https://mlsanity.com/paper/2410.09836/) (2024)
- [Foundation Models for Time Series: A Survey](https://mlsanity.com/paper/2504.04011/) (2025)

## Abstract (arXiv)

Transformers have demonstrated strong performance in long-term time series forecasting (LTSF), yet modeling non-stationary temporal dependencies remains challenging. In this paper, we interpret self-attention as a Bayesian posterior over attended positions, combining a content-based prior induced by scaled dot-product similarity with a temporal-evidence likelihood induced by positional encoding. This formulation provides a probabilistic interpretation of how positional encoding shapes the temporal dependencies captured by self-attention. Building on this perspective, we derive a population-level evidence lower bound (ELBO) that characterizes the role of positional encoding in forecasting weakly stationary time series. Our analysis further highlights a limitation of static positional encoding: when temporal dependence structures vary across input windows, a fixed temporal bias must accommodate heterogeneous dynamics and can therefore be suboptimal for non-stationary forecasting. Motivated by this analysis, we propose Bayesian Time-aware Attention (BTA), which employs a mixture-of-experts (MoE) router to adaptively combine temporal kernels and generate input-conditioned, head-specific positional biases. Extensive experiments on standard LTSF benchmarks demonstrate that BTA consistently improves forecasting performance across multiple Transformer backbones and achieves competitive overall performance with modest computational overhead and stable training.