# Quadratic Direct Forecast for Training Multi-Step Time-Series Forecast Models

arXiv: https://arxiv.org/abs/2511.00053 · HTML: https://mlsanity.com/paper/2511.00053/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 56 references that are in the map.
- First posted on arXiv 28 Oct 2025.

## Builds on (most cited first, 30 of 38)

- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (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)
- [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)
- [TimesNet: Temporal 2D-Variation Modeling for General Time Series Analysis](https://mlsanity.com/paper/2210.02186/) (2022)
- [On First-Order Meta-Learning Algorithms](https://mlsanity.com/paper/1803.02999/) (2018)
- [iTransformer: Inverted Transformers Are Effective for Time Series Forecasting](https://mlsanity.com/paper/2310.06625/) (2023)
- [Meta-Learning with Implicit Gradients](https://mlsanity.com/paper/1909.04630/) (2019)
- [TimeMixer: Decomposable Multiscale Mixing for Time Series Forecasting](https://mlsanity.com/paper/2405.14616/) (2024)
- [SCINet: Time Series Modeling and Forecasting with Sample Convolution and Interaction](https://mlsanity.com/paper/2106.09305/) (2021)
- [How to train your MAML](https://mlsanity.com/paper/1810.09502/) (2018)
- [Spectral Temporal Graph Neural Network for Multivariate Time-series Forecasting](https://mlsanity.com/paper/2103.07719/) (2021)
- [Soft-DTW: a Differentiable Loss Function for Time-Series](https://mlsanity.com/paper/1703.01541/) (2017)
- [TFB: Towards Comprehensive and Fair Benchmarking of Time Series Forecasting Methods](https://mlsanity.com/paper/2403.20150/) (2024)
- [Frequency-domain MLPs are More Effective Learners in Time Series Forecasting](https://mlsanity.com/paper/2311.06184/) (2023)
- [Connecting the Dots: Identifying Network Structure via Graph Signal Processing](https://mlsanity.com/paper/1810.13066/) (2018)
- [CycleNet: Enhancing Time Series Forecasting through Modeling Periodic Patterns](https://mlsanity.com/paper/2409.18479/) (2024)
- [FreDF: Learning to Forecast in the Frequency Domain](https://mlsanity.com/paper/2402.02399/) (2024)
- [Shape and Time Distortion Loss for Training Deep Time Series Forecasting Models](https://mlsanity.com/paper/1909.09020/) (2019)
- [Meta-learning framework with applications to zero-shot time-series forecasting](https://mlsanity.com/paper/2002.02887/) (2020)
- [Fredformer: Frequency Debiased Transformer for Time Series Forecasting](https://mlsanity.com/paper/2406.09009/) (2024)
- [DBLoss: Decomposition-based Loss Function for Time Series Forecasting](https://mlsanity.com/paper/2510.23672/) (2025)
- [Enhancing Time Series Forecasting through Selective Representation Spaces: A Patch Perspective](https://mlsanity.com/paper/2510.14510/) (2025)
- [Patch-wise Structural Loss for Time Series Forecasting](https://mlsanity.com/paper/2503.00877/) (2025)
- [Optimal Transport for Treatment Effect Estimation](https://mlsanity.com/paper/2310.18286/) (2023)
- [Temporal Query Network for Efficient Multivariate Time Series Forecasting](https://mlsanity.com/paper/2505.12917/) (2025)
- [MarS: a Financial Market Simulation Engine Powered by Generative Foundation Model](https://mlsanity.com/paper/2409.07486/) (2024)
- [OLinear: A Linear Model for Time Series Forecasting in Orthogonally Transformed Domain](https://mlsanity.com/paper/2505.08550/) (2025)

## Built on by (most cited first, 10 of 10)

- [GCGNet: Graph-Consistent Generative Network for Time Series Forecasting with Exogenous Variables](https://mlsanity.com/paper/2603.08032/) (2026)
- [From Observations to States: Latent Time Series Forecasting](https://mlsanity.com/paper/2602.00297/) (2026)
- [TimeRFT: Stimulating Generalizable Time Series Forecasting for TSFMs via Reinforcement Finetuning](https://mlsanity.com/paper/2605.00015/) (2026)
- [Deep Time-series Forecasting Needs Kernelized Moment Balancing](https://mlsanity.com/paper/2602.00717/) (2026)
- [ST-EVO: Towards Generative Spatio-Temporal Evolution of Multi-Agent Communication Topologies](https://mlsanity.com/paper/2602.14681/) (2026)
- [Multivariate Time Series Forecasting needs Cross Variable Loss](https://mlsanity.com/paper/2608.05742/) (2026)
- [Rethinking the Flow-Based Gradual Domain Adaptation: A Semi-Dual Optimal Transport Perspective](https://mlsanity.com/paper/2602.01179/) (2026)
- [Deep Time-Series Forecasting in 10 Years: A Survey](https://mlsanity.com/paper/2603.19899/) (2026)
- [CoRe: Coherence and Relational Alignment for Multivariate Time Series Forecasting](https://mlsanity.com/paper/2609.19670/) (2026)
- [Rethinking Cross-Channel Importance in Time-Series Forecasting](https://mlsanity.com/paper/2609.32187/) (2026)

## Abstract (arXiv)

The design of learning objectives is central to training time-series forecasting models. Existing learning objectives such as mean squared error mostly treat each future step as an independent, equally weighted task, which leads to the following two challenges: (1) they overlook the label autocorrelation effect among future steps, leading to biased learning objectives; (2) they fail to set heterogeneous task weights for different forecasting tasks corresponding to varying future steps, limiting the forecasting performance. To fill this gap, we propose a novel quadratic-form weighted learning objective, addressing both issues simultaneously. Specifically, the off-diagonal elements of the weighting matrix account for the label autocorrelation effect, whereas the non-uniform diagonals are expected to match the preferred weights of the forecasting tasks with varying future steps. On this basis, we propose a Quadratic Direct Forecast (QDF) learning algorithm, which trains the forecast model using the adaptively updated quadratic-form weighting matrix. Experiments show that our QDF effectively improves the performance of various forecast models, achieving state-of-the-art results. Code is available at https://github.com/Master-PLC/QDF.