A systematic comparative analysis of modeling methods and variable-dependent weight optimization strategy for PINNs

Dengue fever is a mosquito-borne infectious disease that poses a significant threat to public health and safety. It is crucial to establish the most appropriate mathematical model and estimate the most reliable parameters to simulate the epidemic and quantify control measures. Therefore, an ODE model, a FDE model and a discrete model are constructed. Three parameter estimation methods (PINNs/fPINNs/Euler-PINNs) are applied to three models (ODE/FDE/discrete), respectively, with data in Singapore. Specifically, a variable-dependent weight optimization strategy (VDWOS) for the inverse problem is newly proposed in this paper, which is compared with the traditional weight optimization strategy (TWOS). For theoretical analysis, there are essential differences: when the endemic equilibrium of the ODE/FDE model is globally asymptotically stable, the discrete model may exhibit a flip bifurcation, resulting in a disruption of stability. For numerical simulation, by comparing TWOS and VDWOS for three models, VDWOS yields more reliable parameters and time-varying parameter β(t) without specific function assumptions. Furthermore, an LSTM network is applied to predict β(t) obtained by VDWOS, and the prediction solution is highly consistent with the test data. Meanwhile, through sensitivity analysis, the model-based numerical reconstruction solution exhibits significant responsiveness, which neural networks cannot achieve. However, the ODE model and the discrete model exhibit a high degree of consistency in their specific numerical conclusions, while the FDE model yields significantly different results. This study provides valuable insights into the crucial issue of model selection in the fields of infectious disease prediction and early warning, pest control, and tumor treatment.