Research map / Time series / Long-term forecasting
Attractor Memory for Long-Term Time Series Forecasting: A Chaos Perspective
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Attraos applies chaos theory to long-term time series forecasting by combining non-parametric phase space reconstruction, a multi-resolution polynomial-projection memory (MDMU), and frequency-domain local evolution, matching or beating SOTA with about one-twelfth of PatchTST's parameters.
Problem
Long-term time series forecasting treats discrete samples as data without modeling the continuous, possibly chaotic dynamical system that generated them; existing models do not exploit attractor structure.
Approach
Treat real-world series as low-dimensional observations of an unknown high-dimensional chaotic system: embed via phase space reconstruction, memorize dynamics with a multi-resolution orthogonal-polynomial (HiPPO/SSM-based) memory, and evolve attractor components in the frequency domain with a shared local operator.
Key results
- Attraos achieves the best average MSE/MAE on ETTh1 (0.423/0.420), ETTh2, ETTm1, ETTm2 and Weather (0.246/0.271) among compared models with input length 96.
- On Lorenz96-3d, Attraos averages MSE 0.847 / MAE 0.692, beating the next best (Koopa, 0.919/0.761 MSE/MAE) by a clear margin.
- Attraos uses about 0.2M parameters, one-twelfth of PatchTST's 2.4M in the reported complexity comparison.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use Attraos instead of the strongest listed baseline | Attraos · ETTh1 | MSE: 0.434 -> 0.423 | Helped |
| use Attraos instead of the strongest listed baseline | Attraos · ETTh2 | MSE: 0.375 -> 0.372 | Helped |
| use Attraos instead of the strongest listed baseline | Attraos · ETTm1 | MSE: 0.391 -> 0.382 | Helped |
| use Attraos instead of the strongest listed baseline | Attraos · ETTm2 | MSE: 0.281 -> 0.280 | Helped |
| use Attraos instead of the strongest listed baseline | Attraos · Exchange | MSE: 0.346 -> 0.349 | Hurt |
| use Attraos instead of the strongest listed baseline | Attraos · Crypto | MSE: 0.190 -> 0.187 | Helped |
| use Attraos instead of the strongest listed baseline | Attraos · Weather | MSE: 0.247 -> 0.246 | Helped |
| use Attraos instead of the strongest listed baseline | Attraos · Lorenz96-3d (AVG) | MSE: 0.903 -> 0.847 | Helped |
| use Attraos instead of the strongest listed baseline | Attraos · Air-convection (AVG) | MSE: 0.459 -> 0.454 | Helped |
| add phase space reconstruction (PSR) (ablation) | Attraos · ETTh2 (AVG) | MSE: 0.387 -> 0.376 | Helped |
| add multi-scale hierarchical projection (ablation) | Attraos · ETTh2 (AVG) | MSE: 0.380 -> 0.376 | Helped |
| add time-varying B and Wout (ablation) | Attraos · ETTh2 (AVG) | MSE: 0.380 -> 0.376 | Helped |
Surprising findings
- Direct linear and CNN evolution in the time domain and Hopfield evolution in the time domain underperform, and adding more Hopfield patterns can hurt.
- Injecting Gaussian noise into training data does not degrade and can even improve Attraos performance, and speeds convergence.
Limitations
- Phase space reconstruction (embedding dimension and delay chosen via CC/FNN methods) is described as unstable and non-trainable.
- Multi-scale projection can overfit at certain prediction lengths.
- Validation limited to chaotic datasets of modest size; authors plan larger-scale chaotic evaluation.
Experimental setup
Datasets: ETTh1 ETTh2 ETTm1 ETTm2 Exchange Weather Crypto Lorenz96-3d Air-convection
Benchmarks: ETTh1 ETTh2 ETTm1 ETTm2 Exchange Crypto Weather Lorenz96-3d Air-convection
Metrics: MSE MAE
Baselines: Mamba4TS S-Mamba RWKV-TS GPT-TS Koopa iTransformer PatchTST DLinear FiLM
Builds on: Mamba HiPPO Takens' theorem Modern Hopfield Network
Closest papers by meaning
- Chaos as an interpretable benchmark for forecasting and data-driven modelling (NeurIPS 2021 · similarity 0.90)
- Zero-shot forecasting of chaotic systems (ICLR 2025 · similarity 0.87)
- Chaos Meets Attention: Transformers for Large-Scale Dynamical Prediction (ICML 2025 · similarity 0.87)
- TimeBase: The Power of Minimalism in Efficient Long-term Time Series Forecasting (ICML 2025 · similarity 0.87)
- Learning Pattern-Specific Experts for Time Series Forecasting Under Patch-level Distribution Shift (NeurIPS 2025 · similarity 0.86)
- TimeEmb: A Lightweight Static-Dynamic Disentanglement Framework for Time Series Forecasting (NeurIPS 2025 · similarity 0.85)
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