MRF-PINN: a multi-receptive-field convolutional physics-informed neural network for solving partial differential equations

Compared with conventional numerical approaches to solving partial differential equations (PDEs), physics-informed neural networks (PINN) have manifested the capability to save development effort and computational cost, especially in scenarios of reconstructing physical fields and solving inverse problems. Considering the advantages of parameter sharing, spatial feature extraction, and low inference cost, convolutional neural networks (CNN) tend to be integrated into PINN. To pursue a convolutional PINN framework for solving PDEs with low solving errors, low-costing tuning, and high convergence rate, we propose three initiatives in this paper: (1) a multiple receptive field convolutional PINN (MRF-PINN) is proposed; (2) a Taylor polynomial is used to pad the virtual nodes near the boundaries for implementing high-order finite difference; (3) a dimensional balance method is developed to estimate loss weights, especially for solving nonlinear PDEs. The proposed MRF-PINN with high-order finite difference and the dimensional balance method is validated for solving three typical linear PDEs (elliptic, parabolic, hyperbolic) and a series of nonlinear PDEs (Navier–Stokes equations). Limited by the inherent approximation error and the challenge of non-convex optimisation, MRF-PINN has not yet fully matched to conventional FEM/FVM in terms of solving accuracy and computational cost. Nevertheless, MRF-PINN significantly improves the solving accuracy and convergence of the existing PINN methods, such as MLP-PINN and UNet-PINN. MRF-PINN is promising to become a unified and efficient approach for solving both forward and inverse PDE problems, which are challenging for FEM/FVM methods to address. It can be demonstrated that MRF-PINN has the potential to serve as a branch of the convolutional PINN, thereby enhancing the utilization of Graphics Processing Units (GPUs) hardware and Artificial Intelligence (AI) software in the field of computational mechanics.