Interpretable Wavelet Transforms: A Unified Framework for Frequency-Aware Learning and Dynamical System Identification

Abstract

Wavelet transforms provide simultaneous time–frequency localisation through a mathematically rigorous multi-resolution framework, yet their classical formulations fix filter coefficients independently of any learning objective. This paper makes three original contributions. First, we provide a unified mathematical treatment of learnable wavelet decomposition, establishing precise conditions under which trainable filters retain or forfeit perfect reconstruction guarantees. Second, we derive a gradient-based layer importance metric that quantifies which frequency bands drive model decisions, and demonstrate its application to physiological signal classification with reproducible experimental details. Third, we show that the multi-resolution signal decomposition principle underlying wavelets can serve as a structural prior for governing equation discovery in complex network dynamics, creating an explicit bridge between classical wavelet theory and modern neural symbolic regression. Worked examples on the ECGFiveDays benchmark and SIS epidemic dynamics illustrate the unified framework.