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Predicting the Unpredictable: LLM-powered Long-term Chaotic Time Series Forecasting under Short-term Observations
Yuhang Yao, Bohan Jiang
TL;DR
Chaotic forecasting must address long-term unpredictability with potentially short observation histories, while general LLM time-series representations lack dedicated mechanisms for chaotic dynamics. PAC-LLM combines LLM temporal modeling with learned phase-space and coupling features; experiments report improvements over nine baselines across short- and long-term forecasting.
Problem
Chaotic systems are long-term unpredictable, practical settings may lack extended histories, and general time-series methods lack dedicated mechanisms for chaotic dynamics.
Source: §1 · p. 1, §2.2 · p. 2Method
PAC-LLM combines learnable phase-space representations, LLM temporal-pattern modeling, auxiliary coupling information, and gated adaptive fusion.
Source: §1 · p. 1Results
Takeaways & Limitations
Long-term forecasting from limited recent observations addresses settings where extended histories are unavailable and can guide engineering scheduling and reliable operation.
Source: §1 · p. 1For synthetic systems, the input context is set to approximately one Lyapunov time, while the maximum horizon covers multiple future Lyapunov times.
Source: §3 · p. 2
Abstract
from arXiv · showhide
Chaotic time series forecasting is a challenging task due to its sensitivity to initial conditions and long-term unpredictability. Traditional methods typically rely on sufficient temporal trajectories to learn long-term dynamics, which limits their applicability when only short-term observations are available. While recent Large Language Models (LLMs) have shown great potential for time series forecasting, their temporal representations are not explicitly tailored to the phase-space structure and nonlinear evolution of chaotic systems. To address these issues, we propose PAC-LLM, a phase-space-aware adaptive fusion framework for long-term chaotic time series forecasting powered by LLMs. PAC-LLM leverages learned phase-space features and textual information to fully enable LLM's time series forecasting capacity. In particular, we design an auxiliary feature module and a gated weighting mechanism for multivariate coupling information fusion and selection. Extensive experiments on representative chaotic systems demonstrate that our method outperforms existing fine-tuned and zero-shot baselines in both short-term and long-term predictions. Our ablation study further confirms the effectiveness of each key component in PAC-LLM.
1 Introduction
Long-term chaotic forecasting is difficult when only short observations are available, and prior LLM forecasting methods are not specifically tailored to chaotic dynamics. PAC-LLM combines LLM temporal modeling with learned phase-space and coupling features, and outperforms nine fine-tuned and zero-shot baselines on short- and long-term forecasts.
1 Introduction
- Limited sensing duration, newly deployed systems, and online forecasting can leave practitioners without long historical records for long-term chaotic prediction.Source: §1 · p. 1
- Prior approaches use reservoirs, sequence architectures, and coupling models, while LLM effectiveness for chaotic forecasting remains underexplored.Source: §1 · p. 1
- PAC-LLM combines learnable delay-coordinate embeddings, LLM temporal-pattern extraction, auxiliary coupling signals, and gated weighting that limits noise from redundant coupling information.Source: §1 · p. 1
- PAC-LLM outperforms nine state-of-the-art fine-tuned and zero-shot baselines on short- and long-term forecasting across synthetic and real-world chaotic systems.The evaluation also covers accuracy, valid prediction time, and preservation of long-term dynamical structure.Source: §1 · pp. 1–2
2 Related Work
Prior chaotic forecasting work developed reservoir, delayed-feature, deep sequence, and coupling-aware models, while recent LLM forecasting methods largely target general time-series tasks. PAC-LLM is presented as a way to retain LLM contextual modeling while better matching chaotic dynamics.
2.1 Chaotic Time Series Forecasting
- Chaotic forecasting has progressed from reservoir computing and delayed nonlinear features to deep sequence models and CNN/GNN approaches for inter-variable coupling.Source: §2.1 · p. 2
- Existing LLM-based chaotic studies rely on sufficient contexts and use differing training, testing, and evaluation settings, motivating a unified short-observation setting.Source: §2.1 · p. 2
2.2 Large Language Models for Time Series Forecasting
- General-purpose LLM forecasters use prompts, representation mapping, cross-modal alignment, or reasoning, but lack dedicated mechanisms for chaotic sensitivity, coupling, and rolling-error accumulation.Source: §2.2 · p. 2
- PAC-LLM aims to preserve LLM long-range contextual modeling while tailoring representation learning to chaotic dynamics for more accurate and stable long-term prediction with limited histories.Source: §2.2 · p. 2
3 METHODOLOGY
PAC-LLM forecasts chaotic-system trajectories from limited observations by combining learned phase-space representations with LLM temporal modeling and an auxiliary channel-interaction branch. Its weighted fusion regulates how auxiliary features contribute to the forecast.
3 METHODOLOGY
- The task is to forecast multivariate system states over a future horizon from a limited observation window, set here to approximately one Lyapunov time.The maximum horizon covers multiple future Lyapunov times.Source: §3 · p. 2
3.1 Overview of the Proposed Framework
- PAC-LLM combines phase-space-aware representation learning, LLM temporal modeling, auxiliary channel-interaction modeling, and weighted fusion forecasting.The fused features produce the final prediction.Source: §3.1 · p. 2
3.2 Phase-Space-Aware Representation Learning
- Learnable delay embedding reconstructs each variable’s latent phase-space structure from short observations, using continuous delays and differentiable interpolation rather than fixed offline delay choices.The reconstructed representations of all variables are stacked for downstream forecasting.Source: §3.2 · p. 3
- Chaos-aware system statistics, including complexity, kurtosis, and spectral entropy, generate FiLM scale and shift parameters that calibrate the phase-space representation before patch tokenization.The calibrated sequence is divided into overlapping temporal patches, which are projected into latent tokens.Source: §3.2 · pp. 3–4
3.3 Dual-Branch Dynamics Modeling
- The temporal branch aligns patch tokens with the LLM space through reprogramming and combines them with dynamics-aware prompt embeddings, while an auxiliary branch models cross-variable dependencies.The auxiliary interaction branch provides features for subsequent weighted fusion.Source: §3.3 · p. 4
3.4 Weighted Fusion Forecasting with Joint Loss Function
- A similarity-aware gate modulates auxiliary features using their alignment with the main temporal features, and a reliability-aware gate controls how much refined auxiliary information enters the forecast.The stated purpose is to suppress unreliable corrections while preserving beneficial interaction-aware information.Source: §3.4 · pp. 4–5
- Training combines the fused-forecast loss and the auxiliary-forecast loss with a balancing coefficient.The fused forecast is supervised by the ground-truth future trajectory.Source: §3.4 · p. 5
4 Experiment
Across four synthetic chaotic systems, PAC-LLM ranks first in 23 comparisons and second in one, with advantages spanning forecasting accuracy and long-term attractor preservation. Experiments also examine ablations, LLM backbones, noisy real-world data, efficiency, and observation-context sensitivity.
4.1 Experimental Setup
- Experiments use Lorenz63, Rossler, Chua, and 8-dimensional Lorenz96 synthetic data, plus 21 experimental double-pendulum trajectories, with autoregressive evaluation through 300 steps.Synthetic data use one long training trajectory and test trajectories from unseen initial conditions; the double-pendulum trajectories are split 15/3/3 for training, validation, and testing.Source: §4.1 · pp. 5–6
4.2 Main Results
- PAC-LLM ranks first in 23 comparisons and second in one across four systems and metrics spanning short- and long-horizon accuracy and attractor preservation.The reported long-horizon gains become more pronounced at H=120 and H=300; phase-space visualizations also show better attractor reconstruction and closer Lorenz96 trajectories to ground truth.Source: §4.2 · pp. 6–7, Table 1 · p. 6, Figure 2 · p. 7
4.3 Ablation Study
- Removing the auxiliary branch increases average MSE and MAE by 57.9% and 62.2%, while removing the temporal feature module increases them by 44.6% and 49.1% on Lorenz63.Removing weighted fusion raises average MSE and MAE by 17.8% and 20.1%; Takens embedding, chaos-aware modulation, auxiliary supervision, and prompt information also contribute, with prompt gains not primary.Source: §4.3 · pp. 7–8, Table 2 · p. 7
4.4 Analysis of the LLM Component
- PAC-LLM generally outperforms non-LLM variants, while increasing backbone scale does not consistently improve forecasting and fine-tuned GPT-2 achieves lower long-term errors.GPT-2, Qwen2.5-3B, and Llama-3.2-3B can be integrated without changing the forecasting architecture.Source: §4.4 · p. 8, Table 3 · p. 8, Table 4 · p. 8
4.5 Robustness and Real-World Evaluation
- On the experimental double pendulum, PAC-LLM achieves the lowest short-horizon MSE and longest VPT, and it remains relatively stable under noise up to σ=10^-2.Larger observation-noise levels degrade forecasting accuracy and attractor reconstruction; the experimental system includes friction and gradual energy dissipation, so VPT is reported in seconds.Source: §4.5 · p. 8, Figure 4 · p. 8
4.6 Extended Analysis
- PAC-LLM performs best with short observations, whereas Panda benefits substantially from longer context; PAC-LLM also retains lower MSE and MAE across horizons when methods use individually selected contexts.The selected input lengths are 40 for PAC-LLM, 60 for TimesNet, and 20 for Crossformer; PAC-LLM also has an overall memory-usage and inference-speed advantage over Time-LLM and LLM-Mixer.Source: §4.6 · p. 9, Figure 5 · p. 9, Figure 6 · p. 9
5 Conclusion and Future Work
The paper defines a unified setting for long-term chaotic time-series forecasting and proposes PAC-LLM, which combines phase-space representations with gated fusion of LLM and coupling branches. Experiments show consistent improvements across forecasting and attractor-related metrics, while future work will explore physics-informed constraints.
5 Conclusion and Future Work
- PAC-LLM achieves consistent improvements across point-wise forecasting and attractor-related dynamical metrics on multiple chaotic systems.The conclusion presents these results as evidence of effectiveness for long-term chaotic forecasting.Source: §5 · p. 9
- PAC-LLM combines learnable delay embedding and chaos-statistics extraction with an LLM main branch and an auxiliary coupling branch using gated adaptive fusion.The framework also uses auxiliary supervision.Source: §5 · p. 9
- Ablations confirm contributions from phase-space-aware representation learning, auxiliary coupling modeling, adaptive fusion, and auxiliary supervision.Source: §5 · p. 9
- Future work will explore physics-informed constraints to enhance generalization for forecasting more real-world chaotic systems.Source: §5 · p. 9