WaveletInception-BiLSTM Network

• The paper introduces a deep learning architecture that integrates a learnable wavelet packet transform, 1D Inception modules, and a bidirectional LSTM for per-beam railpad and ballast stiffness estimation.
• It efficiently processes non-stationary, multi-scale drive-by vibration data by fusing temporal features with operational conditions like speed via an LSTM-based fusion stage.
• Empirical results show reduced overall MAPE and faster convergence compared to baselines, balancing high prediction accuracy with computational efficiency.

Searching arXiv for the specified paper and closely related work to ground the article. WaveletInception-BiLSTM is a deep learning framework for drive-by vibration-based infrastructure health monitoring that is designed to infer local, component-level health states from vehicle-borne vibration response signals without hand-crafted preprocessing and with robustness to variations in operational conditions, especially speed. Introduced for railway track stiffness estimation from axle-box acceleration (ABA) signals, it combines a Learnable Wavelet Packet Transform (LWPT) stem for early spectral decomposition, 1D Inception modules for multi-scale temporal feature extraction, an LSTM-based fusion stage for integrating operational conditions, and a bidirectional LSTM (BiLSTM) estimator for beam-level prediction of railpad stiffness $k_p$ and ballast stiffness $k_b$ (Samani et al., 17 Jul 2025).

1. Problem setting and monitoring objective

The network addresses a central problem in drive-by vibration-based monitoring: estimating local structural condition from vibration signals measured on a moving vehicle rather than from fixed, instrumented infrastructure. In the reported case study, the target is beam-level or sleeper-level stiffness estimation over a segment of $N=10$ sleepers, using drive-by ABA signals acquired at different measurement speeds: $35$, $50$, $55$, and $65$ km/h (Samani et al., 17 Jul 2025).

The motivating difficulty is that drive-by vibration data are both non-stationary and multi-scale. According to the paper, stiffness-related phenomena appear in characteristic frequency bands while also evolving over time and along the alignment. Existing CNN/LSTM approaches are described as often requiring fixed-size inputs and spectrogram resizing, and as typically ignoring operational factors such as speed, even though speed modulates excitation levels and spatial sampling. WaveletInception-BiLSTM is proposed specifically to handle variable speeds, variable sequence lengths, minimal preprocessing, and beam-level localization within a single end-to-end architecture (Samani et al., 17 Jul 2025).

The monitoring objective is formulated as per-beam regression rather than global classification. Each beam is assigned two real-valued outputs, $\hat{k}_p$ and $\hat{k}_b$, so that the method performs component-level assessment rather than only segment-level condition recognition. This design is tied to the paper’s emphasis on high-resolution, localized, and fully automated monitoring (Samani et al., 17 Jul 2025).

2. End-to-end architecture

The pipeline operates directly on raw time-domain vibration signals from drive-by measurements. Inputs are variable-length ABA sequences; for batching, signals are zero-padded to the maximum length within each batch. The architecture then applies four major stages: an LWPT stem, 1D Inception feature extraction, LSTM-based feature fusion with operational conditions, and a BiLSTM estimator head for beam-level prediction (Samani et al., 17 Jul 2025).

A compact summary of the architecture is given below.

Component	Function	Reported details
LWPT stem	Early spectral decomposition	$L$-level wavelet packet tree; downsampling by $k_b$0 at each level
1D Inception blocks	Multi-scale temporal feature extraction	Parallel temporal convolutions; outputs concatenated
LSTM fusion	Joint modeling of vibration features and operational conditions	Speed embedding is resampled and concatenated channel-wise
Beam-aligned selection	Mapping temporal features to structural components	$k_b$1 uniformly spaced time steps, with $k_b$2
BiLSTM estimator	Per-beam health estimation	Predicts $k_b$3 and $k_b$4 via dense layers at each time step

After the LWPT and Inception stages, the model resamples an embedding of operational conditions so that it matches the temporal dimension of the vibration-derived feature sequence. These fused features are processed by an LSTM, and then $k_b$5 feature vectors are selected at uniformly spaced time steps along the downsampled temporal axis. The paper states that this establishes a one-to-one mapping between features and beams. A BiLSTM then processes the resulting sequence and predicts railpad and ballast stiffness at each beam (Samani et al., 17 Jul 2025).

The paper does not report explicit tensor shapes for intermediate layers. It states only that LWPT downsamples by a factor of two per level and that the temporal length after the LWPT/Inception stages determines the sequence length seen by the LSTM. This omission is important for implementation fidelity because several architectural hyperparameters are described as tuned but not enumerated (Samani et al., 17 Jul 2025).

3. WaveletInception feature extractor

The defining feature extractor consists of a learnable wavelet front-end followed by 1D Inception modules. The LWPT stem applies an $k_b$6-level wavelet packet tree to the time-domain signal. Filters are initialized from discrete wavelet transform (DWT) bases, specifically Haar and Daubechies db4, and are subsequently learned by backpropagation. The paper emphasizes orthogonality and perfect reconstruction as the basis for downsampling without information loss while preserving high-frequency detail (Samani et al., 17 Jul 2025).

The DWT filtering and downsampling relation reported in the paper is

$k_b$7

For clarity, the standard wavelet-packet forward relations included in the source material are

$k_b$8

where $k_b$9 and $N=10$0 denote learnable low-pass and high-pass filters, and downsampling by $N=10$1 occurs at each level. The paper does not present these recurrence equations explicitly, but it does describe the WPT conceptually and states that the filter coefficients are trainable (Samani et al., 17 Jul 2025).

The subsequent 1D Inception blocks operate only along the temporal axis. They use parallel convolutional branches with different kernel sizes to realize multi-scale receptive fields, and their outputs are concatenated. The paper does not disclose the exact kernel sizes, whether bottlenecks are used, activation functions, stride or dilation settings, or the number of branches. Those choices are treated as hyperparameters determined during validation. The stated rationale is that multi-scale temporal filters complement LWPT by extracting higher-level abstract patterns from multi-resolution wavelet outputs, which is appropriate for vibration signatures that span multiple time-frequency scales (Samani et al., 17 Jul 2025).

Feature-extractor ablation results indicate that the learnable wavelet stem matters. The reported overall MAPE is $N=10$2 for db4-WaveletInception and $N=10$3 for Haar-WaveletInception, while non-learnable Haar/db4 stems yield overall errors in the range $N=10$4–$N=10$5. The same comparison reports $N=10$6 for Laplace-Inception, $N=10$7 for Inception-only, $N=10$8 for LSTM, and $N=10$9 for VGG, all paired with the proposed fusion and estimator design (Samani et al., 17 Jul 2025).

4. Temporal fusion, beam alignment, and bidirectional estimation

Operational-condition fusion is a core architectural choice rather than an auxiliary input appended only at the end. In the case study, measurement speed is encoded through an embedding layer, resampled to match the temporal dimension of the vibration-derived feature sequence, and concatenated channel-wise before an LSTM layer. The source material describes this stage as learning interrelated temporal dependencies among modes of information, including the relation between measurement speed, signal length, and excitation level, as well as dependencies within each modality (Samani et al., 17 Jul 2025).

For completeness, the standard LSTM gate equations included in the source material are

$35$0

$35$1

$35$2

The paper does not print these equations explicitly and does not specify the number of LSTM units or dropout rate; these are described as validation-tuned hyperparameters (Samani et al., 17 Jul 2025).

The beam-aligned selection stage transforms the fused temporal sequence into a structural sequence. From the LSTM outputs, $35$3 feature vectors are chosen at uniformly spaced time steps, with $35$4 equal to the number of beams or sleepers in the segment; in the case study, $35$5. This selection creates the sequence consumed by the BiLSTM estimator, which predicts $35$6 and $35$7 at each beam via fully connected layers (Samani et al., 17 Jul 2025).

The rationale for the BiLSTM head is explicitly physical. The paper notes that neighboring beams are interdependent, and the average ABA power at one sleeper changes when adjacent sleepers lose stiffness. A BiLSTM therefore captures both forward and backward dependencies across the beam sequence. This suggests that the estimator is intended not only to model temporal order in the machine-learning sense but also to exploit spatial coupling induced by the moving vehicle and the track structure (Samani et al., 17 Jul 2025).

5. Task formulation, data, and optimization

The supervised task is per-beam regression of railpad stiffness $35$8 and ballast stiffness $35$9 over segments of ten sleepers. The reported loss is mean squared error,

$50$0

Optimization uses Adam together with a learning-rate scheduler having factor $50$1 and patience $50$2, keyed to validation loss. No additional regularization terms are reported (Samani et al., 17 Jul 2025).

The training set comprises $50$3 simulated ABA records generated from a finite-element vehicle–track model from Shen et al., across four speeds and four stiffness scenarios: uniform, drop in $50$4 sleeper, drop in $50$5 sleepers, and transition zones. Two parameters vary per sleeper, namely railpad stiffness $50$6 and ballast stiffness $50$7, within ranges denoted $50$8 in the paper’s parameter table. Additive Gaussian noise at $50$9 noise-to-signal ratio is introduced to simulate measurement noise. The data split is $55$0 for train/validation/test (Samani et al., 17 Jul 2025).

Hyperparameter tuning is performed through Bayesian optimization using Weights & Biases, with $55$1 configurations per model. Reported training times correspond to $55$2 epochs. Batch size, initial learning rate, and hardware are not reported. The model operates directly on time-domain signals with zero-padding per batch, and the paper states that no resampling or spectrogram generation is required. It also reports no explicit speed normalization or data augmentation (Samani et al., 17 Jul 2025).

A practical consequence of these design choices is that variability in sequence length is handled structurally rather than through handcrafted preprocessing. The source material presents this as one of the main differences from prior fixed-size spectrogram-based pipelines (Samani et al., 17 Jul 2025).

6. Empirical performance and computational profile

On the test set, the proposed model is reported to outperform the compared methods in overall MAPE for stiffness estimation. The published comparison is as follows (Samani et al., 17 Jul 2025).

Model	$55$3 / $55$4 test MAPE	Overall MAPE
WI-BiLSTM	$55$5 / $55$6	$55$7
WI-LSTM	$55$8 / $55$9	$65$0
GoogLeNet	$65$1 / $65$2	$65$3
Laplace-Inception	$65$4 / $65$5	$65$6
LSTM-BiLSTM	$65$7 / $65$8	$65$9
VGG	$\hat{k}_p$0 / $\hat{k}_p$1	$\hat{k}_p$2

Scenario-wise results for WI-BiLSTM show that the uniform case is the easiest and the single-sleeper drop case is the hardest. The reported overall MAPE values are $\hat{k}_p$3 for scenario I (uniform), $\hat{k}_p$4 for scenario II (drop in $\hat{k}_p$5 sleeper), $\hat{k}_p$6 for scenario III (drop in $\hat{k}_p$7 sleepers), and $\hat{k}_p$8 for scenario IV (transition zones). The corresponding RMSE values in MN/m are, respectively: scenario I, $\hat{k}_p$9 for $\hat{k}_b$0 and $\hat{k}_b$1 for $\hat{k}_b$2; scenario II, $\hat{k}_b$3 and $\hat{k}_b$4; scenario III, $\hat{k}_b$5 and $\hat{k}_b$6; scenario IV, $\hat{k}_b$7 and $\hat{k}_b$8 (Samani et al., 17 Jul 2025).

The paper also reports that validation loss curves show WI-BiLSTM converging fastest and to the lowest loss among all models. Qualitative prediction plots are described as showing close alignment between estimated and ground-truth values for both $\hat{k}_b$9 and $L$0 across all scenarios. This suggests that the gain is not restricted to average test error but extends to optimization behavior and per-scenario consistency (Samani et al., 17 Jul 2025).

The computational comparison is notable because the best-performing model is not the largest.

Model	Parameters	Training / inference time
WI-BiLSTM	$L$1M	$L$2 min / $L$3 min per record
WI-LSTM	$L$4M	$L$5 min / $L$6
GoogLeNet	$L$7M	$L$8 min / $L$9
Laplace-Inception	$k_b$00M	$k_b$01 min / $k_b$02
LSTM-BiLSTM	$k_b$03M	$k_b$04 min / $k_b$05
VGG	$k_b$06M	$k_b$07 min / $k_b$08

The paper characterizes the proposed models as striking a strong balance between accuracy and efficiency, attributing the favorable inference profile to LWPT downsampling and 1D convolutions. It further states that inference time is an order of magnitude lower than that of many baselines (Samani et al., 17 Jul 2025).

7. Interpretation, limitations, and future directions

The source material provides a physical interpretation of each major architectural choice. The LWPT filters are described as learning to emphasize wavelet packets whose frequency bands are sensitive to stiffness-related resonances and damping while retaining the essential structure of Haar or db4 initialization. Speed is treated as a causal operational variable affecting both excitation level and apparent spatial frequency; by injecting speed during feature extraction rather than only at the output stage, the model avoids ad hoc resampling or spectrogram hyperparameters. The BiLSTM estimator is interpreted as encoding the physical coupling along the track because local stiffness reductions alter neighboring ABA power (Samani et al., 17 Jul 2025).

Several limitations are stated explicitly. First, the results are based on simulated vehicle–track ABA data, so generalization to diverse real-world networks, vehicle types, and fastening systems remains to be validated. Second, robustness to noise is currently achieved through learning rather than through explicit denoising, and the paper notes that denoising could improve field performance. Third, exact values for several important hyperparameters are withheld, including the number of LWPT levels, the Inception kernel sizes and branches, and the LSTM/BiLSTM widths and layer counts. This means that the method is conceptually specified but not exhaustively parameterized for exact reproduction from the paper alone (Samani et al., 17 Jul 2025).

The future directions proposed in the paper include residual or skip connections, transfer learning, alternative temporal encoders such as GRU or attention, and incorporation of broader operational factors including rail profile and alignment. A plausible implication is that the architecture is intended as a generalizable template for sequential structural inference from mobile sensing, but the paper’s validated scope remains railway stiffness estimation under simulated drive-by conditions (Samani et al., 17 Jul 2025).