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WaveletKernelNet: An Interpretable Deep Neural Network for Industrial Intelligent Diagnosis
Tianfu Li, Zhibin Zhao, Chuang Sun, Li Cheng, Xuefeng Chen, Ruqiang Yan, Robert X. Gao
TL;DR
CNNs are effective for PHM but provide limited physical interpretation of their architecture and decisions. The paper introduces WaveletKernelNet, replacing the first convolution with a continuous wavelet convolution layer whose scale and translation parameters are learned from raw vibration signals. Across three laboratory datasets, WKN shows interpretable first-layer outputs alongside higher accuracy, fewer parameters, and faster convergence than standard CNNs.
Problem
CNNs have demonstrated effectiveness in PHM, but the physical meaning of their architectures and decisions has rarely been studied.
Method
WaveletKernelNet replaces the first standard CNN convolution with a CWConv layer that learns wavelet scale and translation parameters from raw signals.
Results
Across three laboratory fault datasets, WKN variants achieve higher classification accuracy than standard CNNs, with interpretable outputs, fewer parameters, and faster convergence.
Takeaways & Limitations
The CWConv layer provides physically meaningful and interpretable first-layer features while enhancing the reported diagnostic performance of CNN-based models.
Abstract
from arXiv · showhide
Convolutional neural network (CNN), with ability of feature learning and nonlinear mapping, has demonstrated its effectiveness in prognostics and health management (PHM). However, explanation on the physical meaning of a CNN architecture has rarely been studied. In this paper, a novel wavelet driven deep neural network termed as WaveletKernelNet (WKN) is presented, where a continuous wavelet convolutional (CWConv) layer is designed to replace the first convolutional layer of the standard CNN. This enables the first CWConv layer to discover more meaningful filters. Furthermore, only the scale parameter and translation parameter are directly learned from raw data at this CWConv layer. This provides a very effective way to obtain a customized filter bank, specifically tuned for extracting defect-related impact component embedded in the vibration signal. In addition, three experimental verification using data from laboratory environment are carried out to verify effectiveness of the proposed method for mechanical fault diagnosis. The results show the importance of the designed CWConv layer and the output of CWConv layer is interpretable. Besides, it is found that WKN has fewer parameters, higher fault classification accuracy and faster convergence speed than standard CNN.
I. INTRODUCTION
PHM and CNNs support mechanical fault diagnosis, but conventional approaches depend on handcrafted features while CNN decisions and learned filters often lack physical interpretability. WaveletKernelNet addresses these issues by replacing the first CNN convolution with a continuous wavelet convolution layer whose learned parameters produce physically meaningful signal filters.
- Traditional PHM pipelines rely on manually selected features, making extraction time-consuming and potentially degrading classification when features are inadequate.
- CNNs learn features directly from monitoring data and have achieved successful applications in mechanical fault diagnosis.
- CNNs remain black boxes in mechanical fault diagnosis, limiting insight into their decisions and potentially challenging decision credibility.
- For one-dimensional vibration signals, existing interpretable networks lack a specific kernel design for extracting fault-related impulsive components.
- WaveletKernelNet replaces the first convolution with a continuous wavelet convolution layer to discover filters with physical meaning.
- The CWConv layer learns only scale and translation parameters from raw signals, producing a customized wavelet filter bank for signal-feature extraction.
II. THEORETICAL FOUNDATION
The theoretical foundation describes standard CNN feature extraction, activation, pooling, classification, and optimization operations. These components establish the conventional convolutional pipeline that the proposed wavelet-based first layer is intended to replace.
- A CNN combines an input layer, hidden feature-extraction layers, and an output layer, with hidden layers typically containing convolution, pooling, activation, and fully connected operations.
- In a standard convolutional layer, randomly initialized kernels convolve with the input signal to produce feature-map values before nonlinear activation.
- Pooling downsamples activated feature maps to reduce model parameters while retaining their main features.
- After stacked convolution and pooling layers, flattened feature maps are classified or regressed by a fully connected layer.
- CNN classification commonly uses softmax probabilities and cross-entropy loss, with backpropagation and stochastic gradient descent updating weights.
B. Continuous Wavelet Transform
The continuous wavelet transform analyzes signals by matching them with translated and scaled wavelet bases. Its two parameters provide time-scale representations suited to non-stationary signals.
- The continuous wavelet transform calculates similarity coefficients between portions of a signal and a selected wavelet basis.
- The wavelet dictionary is generated by dilating the scale parameter and translating the mother wavelet with the translation parameter.
- Scale is inversely proportional to frequency: lower scales yield more detailed representations, whereas higher scales stretch the wavelet and reduce detail.
- Unlike full-band Fourier analysis, wavelet transformation provides multi-scale analysis of non-stationary signals through its scale and translation parameters.
- Changing the scale parameter and repeating the analysis decomposes the signal across multiple scales.
- Wavelet dictionaries project a signal into two-dimensional time and scale dimensions after traversing the analyzed scales.
C. The Continuous Wavelet Convolutional Layer
The CWConv layer replaces the first standard convolution with wavelet kernels whose scale and translation parameters are learned, making first-layer feature maps physically interpretable for vibration-signal diagnosis.
- The first standard CNN convolution is replaced by a CWConv layer because conventional FIR filtering may not accurately extract impact components or produce human-readable features.
- CWConv performs convolution with predefined wavelet functions, while learning only the scale and translation parameters for each wavelet filter.The output h is defined as the convolution of the input x with the wavelet function ψ_u,s(t).
- Wavelet filters are updated during back-propagation through their scale and translation parameters, rather than by directly learning every filter-vector element.
- The WKN diagnosis pipeline normalizes and windows sensor vibration signals, feeds them to a CNN backbone with its first layer replaced by CWConv, and trains the resulting model.
IV. EXPERIMENTAL VERIFICATION
Experiments on bearing, helical gear, and aeroengine bevel gear datasets compare WKN with standard CNN models to evaluate its mechanical fault diagnosis performance.
- Three datasets—bearing fault, helical gear fault, and aeroengine bevel gear fault—were used to evaluate WKN and compare it with standard CNN models.
A. Bearing Fault Dataset
The bearing experiment evaluates seven fault classes using tapered roller bearing vibration data and compares several CWConv wavelets, SinCNN, and standard CNN baselines.
- The bearing dataset contains seven labels from fault-free bearings and six prefabricated fault locations in 3,0205 tapered roller bearings.
- Vibration data were collected at 1,200 r/min with a 96 kHz sampling rate, recording 2 minutes for each fault type.
- Sliding-window segmentation produced 1,000-point samples, with 7,474 training samples and 1,869 testing samples per failure mode.
- Mexhat, Morlet, and Laplace wavelets formed three CWConv variants, while a 16-point filter and Adam optimizer were used across comparative experiments averaged over 20 runs.
- All three WaveletKernelNets achieved higher classification accuracy and smaller training loss than standard CNN, while LaplaceWaveletNet had the highest mean accuracy and fastest convergence.
B. Helical Gear Fault Dataset
The helical gear dataset represents seven fault and health classes, and WKN variants outperform traditional CNN and SinCNN in accuracy and convergence behavior.
- Dataset: The dataset covers tooth surface wear, tooth root crack, and normal-state conditions in a 7-class classification task.Tooth surface wear has two failure modes, while tooth root crack has four.
- Dataset: 6,748 training samples and 1,687 testing samples were used for the 7-class classification task.Each failure mode contributed 964 training samples.
- Results: All three WKNs achieved higher mean classification accuracy and smaller training loss than traditional CNN on the helical gear dataset.The comparison results are reported in Table II and Fig. 9.
- Results: LaplaceWaveletNet achieved the highest mean classification accuracy and fastest convergence among the three WKNs, while SinCNN was slowest to converge.The broader comparison reports that SinCNN also had the lowest classification accuracy.
C. Aeroengine Bevel Gear Fault Dataset
The aeroengine bevel gear experiments classify four operating-state categories across multiple working conditions. WKN outperforms traditional CNN and SinCNN in each condition, with LaplaceWaveletNet achieving the highest diagnostic accuracy.
- Dataset: Vibration signals were collected from aeroengine bevel gears using acceleration sensors fixed on the accessory casing.The test device and sensor layout are shown in Fig. 10.
- Dataset: Each working condition includes broken teeth, tooth surface wear, overhaul after returning to the factory, and no-fault states.Each fault mode under the same condition has 800 training samples and 200 testing samples.
- Evaluation: The experiment evaluates classification results across working conditions, with PCA visualizations reported for LaplaceWaveletNet.The results are listed in Table III and visualized in Fig. 11.
- Results: WKN outperforms traditional CNN and SinCNN in each working condition.The comparison is based on the reported diagnostic results for the aeroengine bevel gear dataset.
- Results: LaplaceWaveletNet achieves the highest diagnostic accuracy and matches aeroengine bevel-gear vibration components well.The paper directly links its highest accuracy to matching the vibration components.
V. MODEL INTERPRETABILITY
The paper evaluates WKN interpretability through feature-map and waveform visualizations, together with an inner-product-matching perspective.
- Interpretability: WKN interpretability is examined through CWConv feature-map visualization, waveform visualization, and inner product matching.These analyses target the behavior and interpretability of the CWConv layer.
A. The Feature Map of CWConv Layer and Its Waveform
The experiments compare feature maps and learned waveforms from CWConv variants, traditional CNN, and a sin-function filter using a trained network and one signal sample.
- Experimental setup: A signal sample is passed through a trained network to compare feature maps and kernel waveforms across five first-layer designs.The designs include Mexhat, Morlet, Laplace, traditional convolution, and sin-function filters.
- Feature maps: MexFM and MorFM disperse energy across frequency bands, whereas LapFM, TCFM, and SFM concentrate energy in the signal’s impact portion.The feature-map comparison is shown in Fig. 13.
- Feature maps: LapFM indicates the impact position and energy more clearly than the other compared feature maps.The paper states that this explains LaplaceWaveletNet’s best performance across the three datasets.
- Waveforms: Traditional CNN filters are not always well-defined and can appear noisy, while WKN retains a fixed waveform.The waveform comparison is reported in Fig. 14.
- Waveforms: WKN continuously learns scale and time-shift parameters, giving its filters multi-scale and multiresolution analysis capabilities.The resulting filters can match the input waveform and extract signal characteristics effectively.
B. Inner Product Matching
Inner-product matching explains why wavelet-based first-layer filters can better capture fault-related vibration patterns. The CWConv design also reduces learned parameters relative to conventional convolution, while experiments report improved diagnostic performance and convergence.
- Inner-product matching: LaplaceWaveletNet performs best across all three datasets, whereas SinCNN performs worst.The difference is explained through inner-product matching between filters and signal components.
- Inner-product matching: Bearing and gear vibration signals primarily contain periodic impulses that Laplace wavelets are designed to extract.Sin functions target stationary periodic signals and therefore match the impact component less effectively.
- Parameter efficiency: CWConv learns F x 2 parameters instead of the conventional convolutional layer’s F x L parameters.Each CWConv filter is specified by a scale parameter and a translation parameter, reducing parameter count and accelerating convergence.
- Model implications: The proposed WKN improves interpretability and fault-diagnosis accuracy while converging faster than standard CNN models.Its first wavelet convolution layer is intended to discover filters with physical meaning.