An efficient framework for solving forward and inverse problems of nonlinear partial differential equations via enhanced physics-informed neural network based on adaptive learning

Abstract

In recent years, the advancement of deep learning has led to the utilization of related technologies to enhance the efficiency and accuracy of scientific computing. Physics-Informed Neural Networks (PINNs) are a type of deep learning method applied to scientific computing, widely used to solve various partial differential equations (PDEs), demonstrating tremendous potential. This study improved upon original PINNs and applied them to forward and inverse problems in the nonlinear science field. In addition to incorporating the constraints of PDEs, the improved PINNs added constraints on gradient information, which further enhanced the physical constraints. Moreover, an adaptive learning method was used to update the weight coefficients of the loss function and dynamically adjust the weight proportion of each constraint term. In the experiment, the improved PINNs were used to numerically simulate localized waves and two-dimensional lid-driven cavity flow described by partial differential equations. Meanwhile, we critically evaluate the accuracy of the prediction results. Furthermore, the improved PINNs were utilized to solve the inverse problems of nonlinear PDEs, where the results showed that even with noisy data, the unknown parameters could be discovered satisfactorily. The study results indicated that the improved PINNs were significantly superior to original PINNs, with shorter training time, increased accuracy in prediction results, and greater potential for application.

MetaPINNs: Predicting soliton and rogue wave of nonlinear PDEs via the improved physics-informed neural networks based on meta-learned optimization

“…[15][16][17][18] For example, Chen et al proposed a variable coefficient physics-informed neural network (VC-PINN), [19] which specializes in dealing with the forward and inverse problems of variable coefficient partial differential equations. Cao et al adopted an adaptive learning method to update the weight coefficients of the loss function for gradientenhanced PINNs and dynamically adjusted the weight proportion of each constraint term, [20] and numerically simulated localized waves and complex fluid dynamics phenomena, which achieved better prediction results than the traditional PINNs, and the improved method also performed well for the inverse problem under noise interference.…”