# STEPS: A Temporal Smooth Error Propagation Solver on the Manifolds for Test-Time Adaptation in Time Series Forecasting

arXiv: https://arxiv.org/abs/2605.08005 · HTML: https://mlsanity.com/paper/2605.08005/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 43 references that are in the map.
- First posted on arXiv 8 May 2026.

## Builds on (most cited first, 26 of 26)

- [A Time Series is Worth 64 Words: Long-term Forecasting with Transformers](https://mlsanity.com/paper/2211.14730/) (2022)
- [Are Transformers Effective for Time Series Forecasting?](https://mlsanity.com/paper/2205.13504/) (2022)
- [Do We Really Need to Access the Source Data? Source Hypothesis Transfer for Unsupervised Domain Adaptation](https://mlsanity.com/paper/2002.08546/) (2020)
- [Unified Training of Universal Time Series Forecasting Transformers](https://mlsanity.com/paper/2402.02592/) (2024)
- [Connecting the Dots: Multivariate Time Series Forecasting with Graph Neural Networks](https://mlsanity.com/paper/2005.11650/) (2020)
- [Continual Test-Time Domain Adaptation](https://mlsanity.com/paper/2203.13591/) (2022)
- [Efficient Test-Time Model Adaptation without Forgetting](https://mlsanity.com/paper/2204.02610/) (2022)
- [Non-stationary Transformers: Exploring the Stationarity in Time Series Forecasting](https://mlsanity.com/paper/2205.14415/) (2022)
- [Robust Test-Time Adaptation in Dynamic Scenarios](https://mlsanity.com/paper/2303.13899/) (2023)
- [NOTE: Robust Continual Test-time Adaptation Against Temporal Correlation](https://mlsanity.com/paper/2208.05117/) (2022)
- [Efficient Test-Time Adaptation of Vision-Language Models](https://mlsanity.com/paper/2403.18293/) (2024)
- [AdaRNN: Adaptive Learning and Forecasting of Time Series](https://mlsanity.com/paper/2108.04443/) (2021)
- [SoTTA: Robust Test-Time Adaptation on Noisy Data Streams](https://mlsanity.com/paper/2310.10074/) (2023)
- [Poisson Learning: Graph Based Semi-Supervised Learning At Very Low Label Rates](https://mlsanity.com/paper/2006.11184/) (2020)
- [Frequency Adaptive Normalization For Non-stationary Time Series Forecasting](https://mlsanity.com/paper/2409.20371/) (2024)
- [An Analysis of Linear Time Series Forecasting Models](https://mlsanity.com/paper/2403.14587/) (2024)
- [Battling the Non-stationarity in Time Series Forecasting via Test-time Adaptation](https://mlsanity.com/paper/2501.04970/) (2025)
- [Time-Series Forecasting for Out-of-Distribution Generalization Using Invariant Learning](https://mlsanity.com/paper/2406.09130/) (2024)
- [TEA: Test-time Energy Adaptation](https://mlsanity.com/paper/2311.14402/) (2023)
- [Are Self-Attentions Effective for Time Series Forecasting?](https://mlsanity.com/paper/2405.16877/) (2024)
- [Proactive Model Adaptation Against Concept Drift for Online Time Series Forecasting](https://mlsanity.com/paper/2412.08435/) (2024)
- [Graph-based Time Series Clustering for End-to-End Hierarchical Forecasting](https://mlsanity.com/paper/2305.19183/) (2023)
- [Graph-based Forecasting with Missing Data through Spatiotemporal Downsampling](https://mlsanity.com/paper/2402.10634/) (2024)
- [Accurate Parameter-Efficient Test-Time Adaptation for Time Series Forecasting](https://mlsanity.com/paper/2506.23424/) (2025)
- [Evolving Multi-Scale Normalization for Time Series Forecasting under Distribution Shifts](https://mlsanity.com/paper/2409.19718/) (2024)
- [Test-Time Adaptation for Non-stationary Time Series: From Synthetic Regime Shifts to Financial Markets](https://mlsanity.com/paper/2602.00073/) (2026)

## Abstract (arXiv)

Test-Time Adaptation (TTA) aims to improve time series forecasting under distribution shifts by using limited observations revealed during inference. However, forecasting TTA must operate in a source-free online setting, where the adaptation signal is short, temporally correlated, and potentially noisy. Existing methods can therefore suffer from weak identifiability, error accumulation, and unstable long-horizon corrections when the revealed prefix is sparse or contaminated. To address these issues, we propose STEPS, a Smooth Temporal Error Propagation Solver for TTA in time-series forecasting. STEPS reformulates forecasting TTA as a Dirichlet Boundary Value Problem on a temporal manifold, where the revealed prefix error serves as the boundary condition for the unknown future error field. Then, STEPS solves a smooth and bounded correction field in prediction space: a Local Solver propagates prefix errors under temporal smoothness, a Global Solver retrieves stable cross-window error memory and Spatiotemporal Manifold Fusion (SMF) integrates both solutions into the final correction. Across six standard benchmarks and four frozen backbones, STEPS achieves an average relative MSE reduction of 26.82% over the zero-shot backbone, exceeding the strongest compared TTA baseline by 12.77%. Additional sparse prefix and contamination tests confirm the robustness of STEPS under limited and noisy prefixes.