M$^2$FMoE: Multi-Resolution Multi-View Frequency Mixture-of-Experts for Extreme-Adaptive Time Series Forecasting

AI summary

Overview

Research area: time series forecasting, specifically extreme-adaptive forecasting with frequency-domain representation learning and mixture-of-experts architectures, applied to hydrological (reservoir water level) data.

Technical level: Advanced. The paper builds on discrete Fourier transforms, continuous wavelet transforms, spectral band partitioning, routing networks, and mixture-of-experts gating, and it includes a formal theorem on spectral boundary correspondence.

Scope: This summary covers the M²FMoE model, its three modules, its evaluation against nine baselines on five reservoir datasets at prediction horizons of 8 and 72 hours, and the reported ablations and visualizations.

What This Paper Is About

Forecasting models are generally good at capturing dominant, regular patterns in time series but degrade sharply during extreme events such as flash floods, heavy rainfall, or sudden water level rises, which are rare, abrupt, and high variance. The paper argues that regular and extreme events have distinct spectral signatures, so it proposes a model that learns from both Fourier and Wavelet views across multiple temporal resolutions to handle both regimes in one framework, without needing extreme-event labels.

Key Contributions

1. A multi-view frequency mixture-of-experts (MFMoE) module that assigns specialized spectral experts to distinct frequency bands in both the Fourier and Wavelet domains, using routing networks to weight expert contributions adaptively per input.
2. A cross-view shared band splitter (CSS) that learns shared frequency boundaries and maps them into FFT frequency indices and wavelet scale indices, addressing "cross-view spectral misalignment" caused by differences in basis functions and resolution. The mapping is grounded in Theorem 1, which gives a one-to-one correspondence a = γ/f between frequency and wavelet scale (γ = f₀/f_nyq).
3. A multi-resolution adaptive fusion (MAF) module that hierarchically aggregates frequency features from coarse to fine resolutions, plus a temporal gating integration (TGI) module that uses a learnable sigmoid gate to blend recent short-term predictions with a long-range historical context representation.
4. A training objective combining a forecasting loss (MSE) with an expert diversity loss and an expert consistency loss, weighted by hyperparameters λ and μ, to encourage diverse specialization within branches and consistency of the same expert across views.

Main Findings

• Best average rank across all settings: M²FMoE achieves an average rank of 1.4 across five reservoirs and prediction horizons of 8 and 72 hours, versus 1.7 for MCANN, 3.7 for CATS, 4.4 for KAN, 4.9 for CycleNet, 6.3 for TQNet, 7.0 for iTransformer, 7.3 for FreqMoE, 7.9 for Umixer, and 8.5 for DAN.

• Improvement over label-free baselines: M²FMoE reports an average improvement of 22.30% over the best baseline without extreme labels, with a maximum RMSE improvement of 52.86% on the Coyote dataset at a prediction horizon of 8 hours.

• Competitive against label-using baselines: Compared with baselines that use extreme labels (DAN, MCANN), M²FMoE reports an average RMSE improvement of 9.19% across all settings and a maximum improvement of 43.8% on Coyote at horizon 8 hours, despite not using extreme-event labels.

• Statistical significance: RMSE improvements are reported as statistically significant on all reservoirs according to the Wilcoxon signed-rank test (p < 0.05).

• Expert specialization: In the reported visualization with three experts across two resolutions (k=1 and k=24), Fourier-view experts primarily capture low-frequency trends while Wavelet-view experts supply complementary high-frequency detail, and adaptive weighting adjusts expert contributions based on input characteristics.

• Ablation results at horizon 72: Removing the Wavelet view, the Fourier view, the multi-resolution fusion, the CSS, the alignment mechanism, or the dual views all degrade performance on most datasets. For example, on Lexington at horizon 72, RMSE rises from 772.836 (full model) to 827.392 (w/o-WaveletView), 870.735 (w/o-FourierView), 855.236 (w/o-Multi-Res), 916.925 (w/o-CSS), 872.033 (w/o-Alignment), and 789.044 (w/o-DualView). The diversity and consistency losses show mixed effects: with them removed, Coyote RMSE is 448.150 versus 449.944 for the full model, while Lexington RMSE rises to 826.408.

• Recent segment length matters: Reducing the length of the recent segment appropriately improves prediction accuracy, removing it entirely causes a significant performance drop, and overly long segments introduce noise and weaken focus on extremes.

• Moderate expert count is best: Performance is reported as most accurate with 3 or 4 experts; more experts can add noise and overfitting and cause instability.

• t-SNE structure: On the Almaden dataset with three spectral experts, the Fourier view shows low-frequency features forming a well-separated cluster from mid- and high-frequency features, the Wavelet view shows clearer separation between mid- and high-frequency features, and the cross-resolution view shows consistency while retaining local variations.

Methodology in Plain English

The model splits the input series into two pieces: a recent segment used to capture short-term dynamics, and the full input sequence (360 hours, or 15 days) used as historical context. The recent segment is smoothed at several resolutions, and first-order differences are computed at each resolution so that abrupt local changes stand out.

Those differenced sequences are then analyzed in two frequency views. In the Fourier view, a real FFT is applied, the spectrum is split into non-overlapping bands according to learned boundaries, and each band is assigned to its own expert; a small routing network looks at the averaged magnitude spectrum and decides how much each expert should contribute. In the Wavelet view, a continuous wavelet transform using a complex Gaussian wavelet produces a power spectrogram, and the same learned boundaries are converted into wavelet scale indices so that each expert sees the equivalent spectral content. Each wavelet expert processes its masked portion with a convolutional block, and another routing network weights the experts.

The two views' outputs, along with positional encoding, are concatenated and fused, then combined across resolutions by additive accumulation and resolution-specific linear projections. Because everything is learned on differenced sequences, the last observed input slice is added back to restore the original value space. Finally, a gate computed from the recent-feature prediction and a projected version of the historical input blends the two into the final forecast. Training uses MSE plus a term that pushes different experts apart and a term that pulls the same expert's Fourier and Wavelet outputs together.

Why This Matters

The paper targets the observation that forecasting errors in real-world applications come primarily from extreme events, which most models underrepresent because they optimize for dominant periodic and smooth patterns. Framing the problem as frequency heterogeneity, where different bands contribute unequally to regular versus extreme behavior, gives a concrete mechanism for handling rarity without relying on extreme-event labels.

Real-world applications implied by the paper's framing:

• Hydrological early warning for flash floods, heavy rainfall, and sudden water level rises, where delayed warnings can have severe consequences such as widespread flooding.
• Reservoir and water resource management, given the evaluation on hourly water level records from reservoirs in Santa Clara County, California.
• Energy systems, which the paper lists among domains where time series forecasting supports decision-making.
• Transportation and environmental monitoring, also listed among the decision-making domains that depend on time series forecasting.

Industry relevance: the label-free operation is significant because extreme-event annotations are expensive and rare; a model that matches or beats label-using approaches (DAN, MCANN) while requiring no event labels lowers the data cost of deployment. The code is released at https://github.com/Yaohui-Huang/M2FMoE, and the work was supported by the National Natural Science Foundation of China (Grant 62376289), the Natural Science Foundation of Hunan Province (Grant 2024JJ4069), and the Fundamental Research Funds for the Central Universities of Central South University.

Future Directions

• Whether the frequency-band expert design generalizes beyond hydrological data is untested here: the paper evaluates only five reservoir datasets (Almaden, Coyote, Lexington, Stevens Creek, Vasona) from Santa Clara County spanning 1991 to 2019, so behavior on energy, transportation, or other environmental series remains an open question.
• The ablation shows some variants beating the full model on individual datasets (for example, Vasona RMSE of 19.508 for w/o-Multi-Res and 19.280 for w/o-Alignment versus 19.571 for the full model), which raises the question of when each component actually helps.
• The paper reports that moderate expert counts (3 or 4) work best and that more experts introduce noise and overfitting, leaving open how expert count should be selected as data scale or spectral complexity grows.
• The recent segment length is shown to be critical, with too-short, absent, and overly long segments each harming performance; how to set this ratio automatically is not resolved.

Target Audience

Researchers and practitioners working on time series forecasting, particularly those interested in frequency-domain modeling, mixture-of-experts architectures, non-stationary and extreme-event prediction, and hydrological or environmental forecasting. It is also relevant to engineers building early-warning or risk-management systems where extreme events drive the cost of error. Readers need familiarity with Fourier and wavelet transforms, attention and MLP-based forecasting baselines, and mixture-of-experts routing to follow the methodology and the theorem in the appendix.

Authors’ abstract

Forecasting time series with extreme events is critical yet challenging due to their high variance, irregular dynamics, and sparse but high-impact nature. While existing methods excel in modeling dominant regular patterns, their performance degrades significantly during extreme events, constituting the primary source of forecasting errors in real-world applications. Although some approaches incorporate auxiliary signals to improve performance, they still fail to capture extreme events' complex temporal dynamics. To address these limitations, we propose M$^2$FMoE, an extreme-adaptive forecasting model that learns both regular and extreme patterns through multi-resolution and multi-view frequency modeling. It comprises three modules: (1) a multi-view frequency mixture-of-experts module assigns experts to distinct spectral bands in Fourier and Wavelet domains, with cross-view shared band splitter aligning frequency partitions and enabling inter-expert collaboration to capture both dominant and rare fluctuations; (2) a multi-resolution adaptive fusion module that hierarchically aggregates frequency features from coarse to fine resolutions, enhancing sensitivity to both short-term variations and sudden changes; (3) a temporal gating integration module that dynamically balances long-term trends and short-term frequency-aware features, improving adaptability to both regular and extreme temporal patterns. Experiments on real-world hydrological datasets with extreme patterns demonstrate that M$^2$FMoE outperforms state-of-the-art baselines without requiring extreme-event labels.