Wavelet neural networks have been widely applied to dynamical system identification fields. The most difficult issue lies in selecting the optimal control parameters (the wavelet base type and corresponding resolution level) of the network structure. This paper utilizes the advantages of Legendre multiwavelet (LW) bases to construct a Legendre multiwavelet neural network (LWNN), whose simple structure consists of an input layer, hidden layer, and output layer. It is noted that the activation functions in the hidden layer are adopted as LW bases. This selection if based on the its rich properties of LW bases, such as piecewise polynomials, orthogonality, various regularities, and more. These properties contribute to making LWNNs more effective in approximating the complex characteristics exhibited by uncertainties, step, nonlinear, and ramp in the dynamical systems compared to traditional wavelet neural networks. Then, the number of selection LW bases and the corresponding resolution level are effectively optimized by the simple Genetic Algorithm, and the improved gradient descent algorithm is implemented to learn the weight coefficients of LWNN. Finally, four nonlinear dynamical system identification problems are applied to validate the efficiency and feasibility of the proposed LWNN-GA method. The experiment results indicate that the LWNN-GA method achieves better identification accuracy with a simpler network structure than other existing methods.

Strong background noise hinders partial discharge (PD) detection and degrades deep learning models. This study proposes LWDenoiser, an adaptive Legendre multi-wavelet (LW) denoising module that exploits the multi-regularity of LW to match the sharp transient singularities of PD pulses. LWDenoiser comprises LW Threshold Denoising (LWTD) and a Multi-wavelet Frequency Channel Attention Mechanism (MFCAM). LWTD leverages LW multi-wavelet multi-scale representations and applies wavelet specific and band specific thresholds to suppress noise while preserving PD details. MFCAM recalibrates high frequency channels to enhance salient PD features. Experiments show that inserting LWDenoiser into ResNet18 improves the Matthews Correlation Coefficient (MCC) by 25.7%, Recall by 12.8%, and F1 Score by 8.8%, indicating greater noise robustness and higher detection accuracy.

Partial discharge (PD) detection is critical for power system reliability, yet existing deep learning methods often operate as black boxes, failing to leverage the inherent physical properties of PD signals. Specifically, they overlook that a PD pulse manifests as a local singularity—an abrupt, non-smooth transient. This paper introduces a physics-prior-guided detection framework that explicitly embeds this singularity prior into a deep learning model. We propose a novel wavelet Hölder regularity-enhanced convolutional (WHRE-Conv) module, which quantifies local signal regularity via the Hölder exponent and uses this physical prior to guide feature extraction through a channel attention mechanism (CAM). This module is integrated into a new unsupervised convolutional autoencoder (WHRE-CAE), designed to learn deep features that are highly sensitive to PD-related singularities. Furthermore, a genetic algorithm (GA) systematically optimizes the hyperparameters of the WHRE-Conv module, ensuring peak performance. Extensive experiments on a public dataset demonstrate that our method achieves state-of-the-art performance with a Matthews correlation coefficient (MCC) of 0.752, showing that physics-prior-guided deep learning enables more accurate and reliable PD detection.

Partial discharge (PD) detection is essential for evaluating insulation health in overhead lines with covered conductors (CC), particularly within smart grid systems. However, since transient PD signals require high-sampling-rate equipment for capture and analysis, PD detection faces the challenge of efficiently extracting sparse PD features from long signals. To address these challenges, this paper proposes a novel LWPET framework for PD detection. Specifically, we design the Legendre multi-wavelet channel attention (LWCA) module and the Legendre multi-wavelet spatial attention (LWSA) module in the multi-wavelet multi-scale frequency domain to effectively extract PD features. This approach fully leverages the multi-regularity of LW, which closely matches the physical characteristics of PD faults, to enable in-depth PD pattern mining and also enhances the analytical capacity of the feature extraction model through attention mechanisms. Additionally, the Legendre multi-wavelet patch embedding (LWPE) module is introduced to address the difficulty of long-sequence representation learning. This module extracts key patch-level features using LWCA and LWSA, and then feeds the refined features into a Transformer Encoder for further PD feature extraction. Experiments on the VSB dataset demonstrate that LWPET achieves high accuracy (MCC 0.822, precision 0.866, recall 0.800) and efficiency, demonstrating its potential for real-time PD detection in smart grids.