SELF-REGRESSIVE PHYSICS-INFORMED NEURAL NETWORK BASED ON RUNGE-KUTTA METHOD FOR SOLVING PARTIAL DIFFERENTIAL EQUATIONS

Physics-informed neural networks discrete-time model (PINN-RK) is a product of combining deep learning techniques with Runge-Kutta method, which has excellent stability and high accuracy in solving partial differential equations. As an emerging computational tool, PINN-RK has been widely applied in solving various complex problems in scientific and engineering fields. However, due to the limitations of the Runge-Kutta algorithm itself, the PINN-RK model can only achieve single-step time prediction and has low computational efficiency. To achieve multi-step time prediction and improve the computational efficiency of the model, this paper proposes a novel self-regressive physics-informed neural networks model based on the Runge-Kutta method (SR-PINN-RK). The SR-PINN-RK model builds upon the PINN-RK model by incorporating a self-regressive time-advancing mechanism which allows the SR-PINN-RK model to learn the temporal dynamics of the partial differential equation more effectively, resulting in improved training performance and accuracy. In SR-PINN-RK model, except for the label data at the initial time given by the user, all other training labels are provided by the neural network model itself. The new PINN-RK model is a significant improvement over the PINN-RK model, with a much smaller number of training parameters and a significant boost in computational efficiency. This makes SR-PINN-RK model much faster and easier to train, while still maintaining the same level of accuracy. The SR-PINN-RK model uses a self-regressive mechanism to dynamically update the label data, which allows it to successfully achieve multi-step time prediction of partial differential equation solutions. This represents a remarkable improvement compared to the PINN-RK model, which is limited to single-step predictions. In order to verify the accuracy and computational efficiency of the SR-PINN-RK model, the Allen-Cahn equation and Burgers equation are solved using the SR-PINN-RK model and the predicted results are compared with the benchmark solutions in the literature. It is shown that the predicted solution of the SR-PINN-RK model is highly consistent with the benchmark solution qualitatively without significant differences. When it comes to the quantitative analysis, it can be observed that the maximum relative error in solving the Allen-Cahn and Burgers equations are both below 0.009, thereby exhibiting a remarkable level of precision in solving partial differential equations.