# A Unified Scaling Law for Time Series Foundation Models

arXiv: https://arxiv.org/abs/2610.05269 · HTML: https://mlsanity.com/paper/2610.05269/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 55 of its 67 references that are in the map.
- First posted on arXiv 4 Oct 2026.

## Builds on (most cited first, 30 of 55)

- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [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)
- [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)
- [Chronos: Learning the Language of Time Series](https://mlsanity.com/paper/2403.07815/) (2024)
- [A decoder-only foundation model for time-series forecasting](https://mlsanity.com/paper/2310.10688/) (2023)
- [Unified Training of Universal Time Series Forecasting Transformers](https://mlsanity.com/paper/2402.02592/) (2024)
- [Towards Best Practices of Activation Patching in Language Models: Metrics and Methods](https://mlsanity.com/paper/2309.16042/) (2023)
- [Time-MoE: Billion-Scale Time Series Foundation Models with Mixture of Experts](https://mlsanity.com/paper/2409.16040/) (2024)
- [GIFT-Eval: A Benchmark For General Time Series Forecasting Model Evaluation](https://mlsanity.com/paper/2410.10393/) (2024)
- [FITS: Modeling Time Series with 10k Parameters](https://mlsanity.com/paper/2307.03756/) (2023)
- [Chronos-2: From Univariate to Universal Forecasting](https://mlsanity.com/paper/2510.15821/) (2025)
- [Sundial: A Family of Highly Capable Time Series Foundation Models](https://mlsanity.com/paper/2502.00816/) (2025)
- [TiRex: Zero-Shot Forecasting Across Long and Short Horizons with Enhanced In-Context Learning](https://mlsanity.com/paper/2505.23719/) (2025)
- [Moirai-MoE: Empowering Time Series Foundation Models with Sparse Mixture of Experts](https://mlsanity.com/paper/2410.10469/) (2024)
- [Timer-XL: Long-Context Transformers for Unified Time Series Forecasting](https://mlsanity.com/paper/2410.04803/) (2024)
- [Moirai 2.0: When Less Is More for Time Series Forecasting](https://mlsanity.com/paper/2511.11698/) (2025)
- [This Time is Different: An Observability Perspective on Time Series Foundation Models](https://mlsanity.com/paper/2505.14766/) (2025)
- [fev-bench: A Realistic Benchmark for Time Series Forecasting](https://mlsanity.com/paper/2509.26468/) (2025)
- [From Tables to Time: Extending TabPFN-v2 to Time Series Forecasting](https://mlsanity.com/paper/2501.02945/) (2025)
- [In-Context Fine-Tuning for Time-Series Foundation Models](https://mlsanity.com/paper/2410.24087/) (2024)
- [Scaling Law for Time Series Forecasting](https://mlsanity.com/paper/2405.15124/) (2024)
- [Towards Neural Scaling Laws for Time Series Foundation Models](https://mlsanity.com/paper/2410.12360/) (2024)
- [Toto 2.0: Time Series Forecasting Enters the Scaling Era](https://mlsanity.com/paper/2605.20119/) (2026)
- [Zero-shot forecasting of chaotic systems](https://mlsanity.com/paper/2409.15771/) (2024)
- [Timer-S1: A Billion-Scale Time Series Foundation Model with Serial Scaling](https://mlsanity.com/paper/2603.04791/) (2026)
- [Specialized Foundation Models Struggle to Beat Supervised Baselines](https://mlsanity.com/paper/2411.02796/) (2024)
- [Output Scaling: YingLong-Delayed Chain of Thought in a Large Pretrained Time Series Forecasting Model](https://mlsanity.com/paper/2506.11029/) (2025)
- [Foundation Models for Time Series: A Survey](https://mlsanity.com/paper/2504.04011/) (2025)

## Abstract (arXiv)

We develop a Unified Scaling Law and a Unified Theory of Time Series Learning to understand how model capacity and historical information support forecasting. Across different lookback lengths and forecast horizons, we analyze 18,768 experimental cells from 21 checkpoints on 23 dataset-frequency tasks spanning six domains. Our empirical methodology integrates local resource relations into a parsimonious, fitted five-parameter law: capacity gains increase with history, context gains diminish toward saturation, and horizon effects enter as a common shift. Fitted without Toto 2.0, the law predicts its horizon-averaged capacity-scaling curves with mean absolute percentage errors of 1.09% and 1.50% at input lengths 2048 and 4096. To understand how history supports prediction, our learning theory uses Gaussian regression to analyze rule identification and predictive capability. We hypothesize that full-shot models learn by accumulating information in weights, while frozen time series foundation models (TSFMs) use history by extracting information through activations. Matched-history comparisons establish the predictive value of additional history. Controlled parameter exchanges and activation interventions provide evidence that history-derived rule information can be retained, reused across queries, and used to recover a contribution to long-context prediction. Together, these findings inform capacity scaling, context allocation, and the development of models that retain and apply historical rules. Code and main results are available at https://github.com/Fifthky/UniScale.