Shock wave pressure modeling using long short-term memory network based on variational mode decomposition processing

Shock wave pressure sensor acquisition systems exhibit both high-frequency and low-frequency dynamic characteristics, while traditional transfer-function-based modeling and compensation methods were unable to achieve accurate full-band modeling, thereby limiting further improvement in compensation accuracy and reconstructed signal fidelity under complex dynamic conditions. To address this limitation, a fusion modeling method integrating the sparrow search algorithm (SSA), variational mode decomposition (VMD), and a long short-term memory (LSTM) network was developed to improve the dynamic characteristic modeling accuracy of shock wave pressure acquisition systems. In this method, SSA was employed to globally optimize the mode number and penalty factor of VMD using a comprehensive fitness function that combined sample entropy and the Pearson correlation coefficient, which enhanced the adaptability of decomposition for nonstationary response signals contaminated by oscillations and noise. With the optimized parameters, VMD decomposed the sensor response signal into multiple intrinsic modal components; the frequency-domain characteristics of each component were then analyzed, and correlation coefficients together with jump durations were calculated and compared according to the spectral distribution characteristics of shock waves to identify the signal types contained in each mode. Based on this identification, the high-frequency oscillatory modes and noise modes were discarded, thereby achieving the reconstruction of the effective shock wave signal. A sinusoidal signal generator was used to obtain pressure acquisition waveforms over 0.1–10 Hz, amplitudes were converted into decibels to form the low-frequency magnitude–frequency characteristic curve, and a frequency-domain rational function fitting procedure was applied to model the low-frequency transfer function. Using the transfer function, low-frequency dynamic compensation was performed on the reconstructed signal, and the compensated low-frequency signal was combined with the original sensor response signal to construct an input-output dataset that simultaneously preserved compensated dynamic information and original response characteristics. Based on this dataset, SSA was further used to search key LSTM hyperparameters, including the number of hidden units, the maximum training epochs, and the initial learning rate, and an LSTM network was trained to model the nonlinear, time-dependent, and memory-dependent behavior of the acquisition system, enabling fusion modeling of high- and low-frequency dynamic characteristics within a unified framework. Simulation analyses and live explosion test results demonstrated that, compared with the traditional inverse filtering compensation method, the proposed method reduced the mean absolute percentage error (MAPE) between the compensated signal and the reference pressure curve by approximately 75% and decreased oscillation residuals by about 38%, meeting the accuracy requirements for input pressure signals; compared with a single LSTM-based modeling approach, the VMD-LSTM fusion modeling method reduced the overall modeling error to 13%, indicating improved accuracy and robustness. These results show that the SSA-optimized VMD, transfer-function-based low-frequency compensation, and SSA-tuned LSTM fusion modeling jointly provide an effective full-band modeling route, and the proposed framework now offers a robust solution for accurate dynamic characteristic modeling and compensation in shock wave pressure sensor acquisition systems.