  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 the physics field and solving the
inverse problem. Considering the advantages of parameter sharing, spatial
feature extraction and low inference cost, convolutional neural networks (CNN)
are increasingly used in PINN. However, some challenges still remain as
follows. To adapt convolutional PINN to solve different PDEs, considerable
effort is usually needed for tuning critical hyperparameters. Furthermore, the
effects of the finite difference accuracy, and the mesh resolution on the
predictivity of convolutional PINN are not settled. To fill the gaps above, we
propose three initiatives in this paper: (1) A Multi-Receptive-Field PINN
(MRF-PINN) model is established to solve different types of PDEs on various
mesh resolutions without manual tuning; (2) The dimensional balance method is
used to estimate the loss weights when solving Navier-Stokes equations; (3) The
Taylor polynomial is used to pad the virtual nodes near the boundaries for
implementing high-order finite difference. The proposed MRF-PINN is tested for
solving three typical linear PDEs (elliptic, parabolic, hyperbolic) and a
series of nonlinear PDEs (Navier-Stokes PDEs) to demonstrate its generality and
superiority. This paper shows that MRF-PINN can adapt to completely different
equation types and mesh resolutions without any hyperparameter tuning. The
dimensional balance method saves computational time and improves the
convergence for solving Navier-Stokes PDEs. Further, the solving error is
significantly decreased under high-order finite difference, large channel
number, and high mesh resolution, which is expected to be a general
convolutional PINN scheme.
