cs.LGApr 8, 2026
SavePD-SOVNet: A Physics-Driven Second-Order Vibration Operator Network for Estimating Wheel Polygonal Roughness from Axle-Box Vibrations
Organizations: School of Ocean Engineering, Harbin Institute of Technology, West Wenhua Road, Weihai, 264209, Shandong, China · Technical Center, Bogie Development Department, CRRC Qingdao Sifang Locomotive and Rolling Stock Co., Ltd, Jinhong East Road, Qingdao, 266111, Shandong, China
Abstract
Quantitative estimation of wheel polygonal roughness from axle-box vibration signals is a challenging yet practically relevant problem for rail-vehicle condition monitoring. Existing studies have largely focused on detection, identification, or severity classification, while continuous regression of multi-order roughness spectra remains less explored, especially under real operational data and unseen-wheel conditions. To address this problem, this paper presents PD-SOVNet, a physics-guided gray-box framework that combines shared second-order vibration kernels, a MIMO coupling module, an adaptive physical correction branch, and a Mamba-based temporal branch for estimating the 1st--40th-order wheel roughness spectrum from axle-box vibrations. The proposed design embeds modal-response priors into the model while retaining data-driven flexibility for sample-dependent correction and residual temporal dynamics. Experiments on three real-world datasets, including operational data and real fault data, show that the proposed method provides competitive prediction accuracy and relatively stable cross-wheel performance under the current data protocol, with its most noticeable advantage observed on the more challenging Dataset III. Noise injection experiments further indicate that the Mamba temporal branch helps mitigate performance degradation under perturbed inputs. These results suggest that structured physical priors can be beneficial for stabilizing roughness regression in practical rail-vehicle monitoring scenarios, although further validation under broader operating conditions and stricter comparison protocols is still needed.
Figures & tables
Figure 1: Schematic comparison of the regression behavior of a purely data-driven network and a physics-driven second-order vibration operator network under the current unseen-wheel evaluation setting. The method on the left mainly relies on input-dependent dynamic weighting mechanisms to learn the mapping relationship and may therefore depend more heavily on the training distribution when the response–label mapping varies across wheel groups. The method on the right explicitly embeds physical priors, such as gain, natural frequency, and damping, into the model through shared second-order vibration operator kernels, with the aim of making prediction behavior more stable under the current data protocol.
Figure 2: Schematic illustration of the data acquisition and preprocessing pipeline. The upper part shows the sources of the input data, namely axle-box vibration and speed signals collected during vehicle operation; the lower part shows the source of the label data, namely wheel out-of-roundness profiles measured by an underfloor wheel lathe. After speed filtering, angle-domain resampling, and order extraction, the input samples and the 1–40-order roughness labels for supervised regression are finally constructed.
Figure 3: Illustration of the raw time-domain vibration and speed data. The left axis represents vibration acceleration, and the right axis represents operating speed. It can be seen that although the speed is relatively stable in some high-speed intervals, the vibration response still changes significantly. Therefore, speed filtering is first required before subsequent modeling.
Figure 4: Examples of angle-domain resampled samples. Panel A shows the angle-domain vibration, the corresponding order spectrum, and the roughness labels of Wheel No.1 under similar speed conditions; Panel B shows the angle-domain vibration, the corresponding order spectrum, and the roughness labels of Wheel No.2 under similar speed conditions. The yellow dashed boxes indicate regions with relatively obvious differences in the order domain, showing that even under similar speed conditions, the vibration patterns and order responses may still differ significantly across wheels. Note: the vertical-axis values are masked due to data-sensitivity requirements from the data provider.
Figure 5: Overall architecture of PD-SOVNet. (A) Shared second-order vibration kernels. (B) MIMO coupling with modal mixing, weighting, and summation. (C) Physics-driven branch with adaptive physical correction. (D) Overall workflow of the proposed framework.
Figure 6: Schematic illustration of the vehicle–bogie–wheelset–rail system. Panels A–C show the multi-level coupled vibration relationships among the carbody, bogie frame, wheelset, and rail. Panel D shows the sensor placement in the axle-box region, indicating that the measured vibration signal is essentially a composite response of wheel–rail excitation after transmission through multiple structural levels.
Figure 7: Mamba structure and selective SSM.
Figure 8: Field collection of wheel roughness and axle-box vibration data. (A) Wheel roughness data acquisition process. (B) Data acquisition equipment. (C) Measurement location of axle-box vibration and on-site sensor installation.
| Model | Params (M) | Avg. latency (ms/sample) | Throughput (samples/s) | Peak memory (MiB) |
| BiLSTM | 1.71 | 0.027 | 36758 | 100.5 |
| CNN | 1.44 | 0.033 | 46734 | 94.0 |
| FRF | 2.32 | 0.030 | 33630 | 40.5 |
| Transformer | 8.48 | 0.074 | 13655 | 121.4 |
| Mamba | 3.07 | 0.092 | 10925 | 128.8 |
| Ours | 0.55 | 0.140 | 7184 | 203.0 |
Table 1: Computational characteristics of the compared models under the same measurement pipeline with a unified batch size of 64. The table reports parameter count, average per-sample inference latency, throughput, and peak memory allocation. These metrics are provided to complement the accuracy-oriented evaluation and to illustrate deployment-related trade-offs, rather than to serve as the primary basis for model ranking.GPU(3080ti)
Figure 9: Comparison results of six methods on the three datasets under the low-training-error stability analysis. (A) shows examples of the validation roughness spectra on Dataset I, Dataset II, and Dataset III, together with the curves of MAE (lower is better) and (higher is better) during training for each method. The validation roughness spectra are used to show how different methods fit the target roughness distribution over orders, while the MAE and curves reflect the error convergence and fitting trends during training. (B) shows the absolute error distributions over 40 orders on the three datasets, which are used to compare the error level and stability of different methods in order-wise prediction. This analysis is used mainly to examine performance stability when the models are trained to a lower training-error range under the unseen-wheel split.
Figure 10: Comparison results of six methods on the three datasets under the training-state sensitivity analysis. (A) shows examples of the validation roughness spectra on Dataset I, Dataset II, and Dataset III, together with the curves of MAE (lower is better) and (higher is better) during training for each method. The validation roughness spectra are used to show how different methods fit the target roughness distribution over orders, while the MAE and curves reflect the error convergence and fitting trends during training. (B) shows the absolute error distributions over 40 orders on the three datasets, which are used to compare the error level and stability of different methods in order-wise prediction. In this analysis, the purely data-driven baselines are regularized, while FRF and Ours are additionally shown in a further-optimized training state.
Figure 11: Summary comparison of the final results of different methods on the three datasets under the two evaluation analyses. (A)–(C) correspond to Dataset I, Dataset II, and Dataset III, respectively. In each subfigure, the bar chart represents the validation MAE, and the line plot represents the validation ; the horizontal axis corresponds to the further-optimized training state and the regularized training state. Here, the term “overfitting” in the figure is used only to denote the lower-training-error state after further optimization, rather than as a formal diagnosis of statistical overfitting for all methods.
Figure 12: Supplementary training-process results of FRF and Ours on the three datasets under the training-state sensitivity analysis. (A) shows the MAE curves over training epochs; (B) shows the curves over training epochs. This figure is mainly used to supplement the training and validation results of FRF and Ours on Dataset I, Dataset II, and Dataset III in the previous main comparison experiment.
Figure 13: Integrated comparison of the final results of six methods on the three datasets under two training states. (A)–(C) correspond to Dataset I, Dataset II, and Dataset III, respectively. The bar chart represents the validation MAE, and the line plot represents the validation ; the horizontal axis corresponds to the further-optimized training state and the regular training state. Here, the term “overfitting” in the figure is used only to denote the lower-training-error state after further optimization, rather than as a formal diagnosis of statistical overfitting for all methods.
| Physical branch | Time-domain Mamba | Dataset I | Dataset II | Dataset III | |||
| MAE | MAE | MAE | |||||
| 6.156 | 0.456 | 6.242 | 0.435 | 5.781 | 0.555 | ||
| 5.132 | 0.645 | 5.223 | 0.621 | 4.895 | 0.692 | ||
| 4.992 | 0.663 | 5.086 | 0.658 | 4.674 | 0.726 | ||
Table 2: Ablation results of the physical branch and time-domain Mamba
| MIMO coupling | Adaptive correction | Dataset I | Dataset II | Dataset III | |||
| MAE | MAE | MAE | |||||
| 5.332 | 0.619 | 5.384 | 0.618 | 5.031 | 0.650 | ||
| 5.263 | 0.625 | 5.313 | 0.621 | 4.983 | 0.672 | ||
| 5.052 | 0.643 | 5.141 | 0.639 | 4.759 | 0.689 | ||
| 4.992 | 0.663 | 5.086 | 0.658 | 4.674 | 0.726 | ||
Table 3: Ablation results of MIMO coupling and adaptive physical correction.
Figure 14: Comparison results of the training-set noise injection experiments. (A)–(C) correspond to Dataset I, Dataset II, and Dataset III, respectively. Each subfigure shows the error variations of the six methods under four noise conditions ( dB, dB, dB, and dB), where the bar chart denotes the mean error and the error bars indicate the fluctuation range. All models are evaluated using the weights saved when the training MAE first decreased to approximately , so as to support an aligned comparison under a common training reference. This figure is used to compare the stability and noise robustness of different methods under perturbed inputs.
Figure 15: Comparison of the noise ablation results of the temporal branch. (A)–(C) show the MAE performance of the proposed model on Dataset I, Dataset II, and Dataset III, respectively, when the temporal branch is implemented using Linear and Mamba under four levels of input noise perturbation ( dB, dB, dB, and dB). The bar chart denotes the mean error under different noise conditions, the error bars represent the corresponding fluctuation range, and the numerical labels indicate the specific MAE values under each noise level. The dashed lines are used to assist in visualizing the trend of error variation as the noise increases. It can be seen that under all three datasets and all four noise conditions, the model error with the Mamba temporal branch is consistently lower than that with the Linear branch, indicating that stronger temporal modeling capability can effectively alleviate the sensitivity of the physics-driven architecture to additional input perturbations near the deployment weights, thereby improving overall stability.
Figure 16: Illustration of time-series anomaly detection models for future active data acquisition and abnormal-sample screening. The figure shows three candidate models, including DeepSVDDTS, BiMamba, and Anomaly Transformer. The purpose of introducing this figure in the Discussion section is not to rank their structures, but to illustrate that anomaly detection can be used as an auxiliary module in the data closed loop to identify high-value samples that are inconsistent with the current training distribution, and to provide a reference for subsequent supplementary sampling and label matching.
Figure 17: Visualization analysis of DeepSVDDTS-style anomaly detection results on training samples and held-out samples. (A) shows the sample distribution in the embedding space for observing the overall shift between the training distribution and the held-out samples. (B) shows the convergence process of the training loss. (C) gives the ROC curve and the corresponding AUC result. (D) shows the boxplot of anomaly scores for the two types of samples. (E) gives the prediction confusion matrix based on threshold partitioning. (F) shows the anomaly-score distribution and the corresponding threshold position. This figure is used to illustrate that, for the “training distribution consistency detection” task considered in this work, the anomaly scores already exhibit certain statistical differences between the training samples and the held-out samples, thus providing a feasible screening signal for subsequent active data supplementation.
Figure 18: Comparison results of anomaly scores between training samples and validation samples for Anomaly Transformer and BiMamba. (A)–(C) correspond to Anomaly Transformer, including the anomaly-score distribution, statistical boxplots of the anomaly scores, and the relationship between time-domain scores and order-domain scores; (D)–(F) correspond to BiMamba, including the same three types of visualizations. This figure is used to compare the suitability of different time-series anomaly detection models for the task of “training distribution consistency detection,” and to illustrate that the differences in the mean, variance, and joint distribution of anomaly scores between training samples and validation samples can serve as reference signals for subsequent active sample screening and data supplementation.
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