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Fourier-KAN-Mamba: A Novel State-Space Equation Approach for Time-Series Anomaly Detection

Xiancheng Wang, Lin Wang, Rui Wang, Zhibo Zhang, Minghang Zhao

TL;DR

This paper tackles time-series anomaly detection by leveraging a novel state-space backbone (Mamba) augmented with a learnable Fourier-KAN front-end and a temperature-controlled gating mechanism to model periodic and nonlinear dynamics. Introduces an energy-based LEH anomaly score and two unsupervised differentiable losses to maximize separation between normal and anomalous patterns, enabling end-to-end training without abundant labeled anomalies. Evaluated on five public TSAD datasets, the method achieves current $F_1$-score state-of-the-art on SMD, MSL, SMAP, SWaT, and PSM and demonstrates cross-domain robustness and early fault sensing in industrial deployment. The results, together with ablations and interpretability analyses, indicate practical potential for scalable, reliable industrial anomaly detection with interpretable LEH scoring.

Abstract

Time-series anomaly detection plays a critical role in numerous real-world applications, including industrial monitoring and fault diagnosis. Recently, Mamba-based state-space models have shown remarkable efficiency in long-sequence modeling. However, directly applying Mamba to anomaly detection tasks still faces challenges in capturing complex temporal patterns and nonlinear dynamics. In this paper, we propose Fourier-KAN-Mamba, a novel hybrid architecture that integrates Fourier layer, Kolmogorov-Arnold Networks (KAN), and Mamba selective state-space model. The Fourier layer extracts multi-scale frequency features, KAN enhances nonlinear representation capability, and a temporal gating control mechanism further improves the model's ability to distinguish normal and anomalous patterns. Extensive experiments on MSL, SMAP, and SWaT datasets demonstrate that our method significantly outperforms existing state-of-the-art approaches. Keywords: time-series anomaly detection, state-space model, Mamba, Fourier transform, Kolmogorov-Arnold Network

Fourier-KAN-Mamba: A Novel State-Space Equation Approach for Time-Series Anomaly Detection

TL;DR

This paper tackles time-series anomaly detection by leveraging a novel state-space backbone (Mamba) augmented with a learnable Fourier-KAN front-end and a temperature-controlled gating mechanism to model periodic and nonlinear dynamics. Introduces an energy-based LEH anomaly score and two unsupervised differentiable losses to maximize separation between normal and anomalous patterns, enabling end-to-end training without abundant labeled anomalies. Evaluated on five public TSAD datasets, the method achieves current -score state-of-the-art on SMD, MSL, SMAP, SWaT, and PSM and demonstrates cross-domain robustness and early fault sensing in industrial deployment. The results, together with ablations and interpretability analyses, indicate practical potential for scalable, reliable industrial anomaly detection with interpretable LEH scoring.

Abstract

Time-series anomaly detection plays a critical role in numerous real-world applications, including industrial monitoring and fault diagnosis. Recently, Mamba-based state-space models have shown remarkable efficiency in long-sequence modeling. However, directly applying Mamba to anomaly detection tasks still faces challenges in capturing complex temporal patterns and nonlinear dynamics. In this paper, we propose Fourier-KAN-Mamba, a novel hybrid architecture that integrates Fourier layer, Kolmogorov-Arnold Networks (KAN), and Mamba selective state-space model. The Fourier layer extracts multi-scale frequency features, KAN enhances nonlinear representation capability, and a temporal gating control mechanism further improves the model's ability to distinguish normal and anomalous patterns. Extensive experiments on MSL, SMAP, and SWaT datasets demonstrate that our method significantly outperforms existing state-of-the-art approaches. Keywords: time-series anomaly detection, state-space model, Mamba, Fourier transform, Kolmogorov-Arnold Network

Paper Structure

This paper contains 21 sections, 48 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Technical Description
  4. Overview of FKM-AD
  5. Hybrid Linear Fourier Series KAN
  6. Temperature-Controlled Sharpening Mechanism
  7. Selective Enhancement of the Main Branch Input $X$
  8. Anomaly Score
  9. Locality
  10. Energy
  11. High-Frequency Ratio (HFR)
  12. Complete Anomaly Score
  13. Interpretability of Anomaly Scores
  14. Theoretical Explanation of Energy, Frequency Domain, and Locality
  15. Two Complementary Unsupervised, End-to-End Differentiable Training Losses
  16. ...and 6 more sections

Figures (2)

  • Figure 1: KMA-AD Architecture: (a) Description of the main structure of the KMA-AD model; (b) Sources of the SSM parameters, the loss functions, and the error suppression module.
  • Figure 2: Model Hyperparameter Sensitivity Experiments: (a) Sensitivity Experiment of Batch Size Hyperparameter; (b) Sensitivity Experiment of Number of Frequencies Hyperparameter; (c) Sensitivity Experiment of Lambda Pass Hyperparameter; (d) Sensitivity Experiment of Lambda Margin Hyperparameter; (e) Sensitivity Experiment of D State Hyperparameter; (f) Sensitivity Experiment of Window Size Hyperparameter.