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

Yaohui Huang, Runmin Zou, Yun Wang, Laeeq Aslam, Ruipeng Dong

TL;DR

This work introduces M$^2$FMoE, a multi-resolution, multi-view frequency mixture-of-experts model for extreme-adaptive time series forecasting. It combines a dual-view (Fourier and Wavelet) expert design with a cross-view alignment mechanism, a hierarchical multi-resolution fusion, and a temporal gating strategy to capture both regular patterns and rare, high-impact events. Evaluated on five reservoir datasets, the approach outperforms state-of-the-art baselines without requiring extreme-event labels, demonstrating strong practical value for hydrological forecasting and other domains with extreme dynamics. The combination of frequency-aware modeling, multi-resolution integration, and adaptive gating offers a robust framework for accurate forecasting in the presence of irregular and extreme temporal behavior.

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.

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

TL;DR

This work introduces MFMoE, a multi-resolution, multi-view frequency mixture-of-experts model for extreme-adaptive time series forecasting. It combines a dual-view (Fourier and Wavelet) expert design with a cross-view alignment mechanism, a hierarchical multi-resolution fusion, and a temporal gating strategy to capture both regular patterns and rare, high-impact events. Evaluated on five reservoir datasets, the approach outperforms state-of-the-art baselines without requiring extreme-event labels, demonstrating strong practical value for hydrological forecasting and other domains with extreme dynamics. The combination of frequency-aware modeling, multi-resolution integration, and adaptive gating offers a robust framework for accurate forecasting in the presence of irregular and extreme temporal behavior.

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 MFMoE, 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 MFMoE outperforms state-of-the-art baselines without requiring extreme-event labels.
Paper Structure (44 sections, 1 theorem, 30 equations, 11 figures, 8 tables)

This paper contains 44 sections, 1 theorem, 30 equations, 11 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Problem Statement
  5. Discrete Fourier Transform (DFT)
  6. Continuous Wavelet Transform (CWT)
  7. Methodology
  8. Hierarchical Temporal Segmentation
  9. Multi-Resolution Sequence Generation
  10. Temporal Embedding
  11. Multi-View Frequency Mixture-of-Experts Module
  12. Cross-View Shared Band Splitter
  13. Fourier-View Expert Branch
  14. Wavelet-View Expert Branch
  15. Multi-Resolution Adaptive Fusion Module
  16. ...and 29 more sections

Key Result

Theorem 1

Let $f$ denote the normalized frequency, $a$ is the scale in the CWT, and $\gamma = f_0 / f_{\mathrm{nyq}}$ is a wavelet-dependent constant. The mapping $a = \gamma / f$ establishes a one-to-one correspondence between frequency and scale boundaries mallat2002theory, such that $f_{\max} \mapsto a_{\m

Figures (11)

  • Figure 1: Comparison of frequency spectra between regular and extreme events.
  • Figure 2: The proposed M$^2$FMoE with three experts per branch for capturing high-, mid-, and low-frequency patterns.
  • Figure 3: Prediction results and expert weights of M$^2$FMoE.
  • Figure 4: Impact of the length of recent segment $T_r/T_{in}$.
  • Figure 5: Impact of the number of experts $E$.
  • ...and 6 more figures

Theorems & Definitions (1)

  • Theorem 1: Spectral Boundary Correspondence