This paper proposes a hybrid PINN + LSTM framework for the high-precision solution of malware propagation dynamics in wireless sensor networks. A seven-compartment SVEHLQR model is developed to capture this complex transmission process. To overcome the limitations of standard physics-informed neural networks (PINNs) in long-term prediction, including gradient vanishing and error accumulation, we integrate LSTM’s temporal memory capability into the PINN architecture. Comprehensive comparisons are conducted among the proposed PINN + LSTM, standard PINN, and Fourier PINN, using the fourth-order Runge–Kutta method as the benchmark. Experimental results demonstrate that PINN + LSTM significantly outperforms both baseline methods, achieving an average relative error of 3.88×10−3 compared to 7.20×10−2 for PINN and 2.81×10−1 for Fourier PINN, representing a 94.6% accuracy improvement over PINN. These results validate that incorporating LSTM’s recursive memory mechanism enables the accurate and efficient solution of complex time-dependent dynamical systems. Additionally, the model’s robustness is verified under 1%, 5%, and 10% Gaussian noise. PINN + LSTM maintains extremely low relative errors, not exceeding 0.0049, and outperforms PINN and Fourier PINN significantly, confirming its strong noise immunity and stable dynamics learning ability in realistic environments.
Zhang et al. (Thu,) studied this question.