Fractal-domain Transformer Based on Learnable Multifractal Spectrum for Chaotic Systems Classification

Abstract

Conventional deep learning in the spatiotemporal-frequency domain frequently encounter challenges in terms of slow convergence rates and limited generalization, particularly for classification of chaotic systems. To address these limitations, this paper introduces a novel fractal-inspired deep network model, specifically, the Multifractal Spectrum Transformer (MFS-Transformer), grounded in learnable multifractal analysis. Initially, we put forward the conceptual framework of fractal learning, compared with traditional fractal signal processing methodologies and spatial-temporal domain learning paradigms. Subsequently, a Learnable Multifractal Spectrum (LMFS) derived from 3D spatial gridding, coupled with fractal-domain filtering, is proposed to construct the iterative learning process within the fractal domain. Further, we formulate the MFS-Transformer, an innovative architecture that integrates multi-channel embedding, LMFS, fractal-domain filtering, residual fusion mechanisms, a mixer module, and a classifier, tailored for chaotic system classification. Ultimately, we evaluate the efficacy of our model in classifying 3D chaotic systems under stringent conditions of short-term sequences and low Signal-to-Noise Ratio (SNR). Experimental outcomes underscore the substantial performance gains achieved by the MFS-Transformer, with classification accuracy enhancements of 13.34% and 5.00% over existing Convolutional Neural Networks (CNNs) and Recurrent Neural Networks (RNNs), respectively, under SNR=0dB and 32-sample sequences. These findings validate the superiority of the MFS-Transformer in addressing the complexities of chaotic system classification under complex scenarios. This research not only advances the frontier of fractal deep learning but also presents a novel perspective and methodology for tackling intricate spatiotemporal classification problems.