The improved PINNs for solving partial differential equations with nonlinear couple systems

For partial differential equations, traditional methods are characterized by high accuracy, yet they face difficulties in handling complex high-dimensional. Although physics-informed neural networks (PINNs) can effectively solve high-dimensional problems, their accuracy still needs to be improved. The Fourier neural operator (FNO) has an advantage in capturing the global characteristics of physical systems, but it relies on a large amount of training data. To address these limitations, this paper proposes a hybrid method, FNO-PINN, which integrates the FNO with a PINN. This method devises a hybrid architecture consisting of an enhanced FNO, high precision difference calculation, multi-objective physical constraints, and post processing mechanisms. Meanwhile, a dynamic weight adjustment strategy and an iterative auxiliary data loss for different time-scale variables are proposed to effectively alleviate the imbalance problem during the training process. The numerical experimental results of three examples show that compared with the traditional PINNs method, the proposed FNO-PINN method has a faster convergence speed and smaller error, and it demonstrates superior performance in handling coupled systems with multiple time scales.