MRF-PINN: A Multi-Receptive-Field convolutional physics-informed neural network for solving partial differential equations
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Physics-informed neural networks (PINN) can achieve lower development and solving cost than traditional partial differential equation (PDE) solvers in scenarios such as reconstructing the physics field and solving the inverse problem. Due to the advantages of parameter sharing, spatial feature extraction and low inference cost, convolutional neural networks (CNN) are increasingly used in PINN. To adapt convolutional PINN to different equations, researchers have to spend much time tuning critical hyperparameters. Furthermore, the effects of finite difference accuracy, model complexity, and mesh resolution on the prediction result of convolutional PINN are unclear. To fill the above research gaps, in this paper, (1) A Multi-Receptive-Field PINN (MRF-PINN) model is constructed to adapt different equation types and mesh resolutions without manual tuning.(2) The generality and advantages of the MRF-PINN are verified in three typical linear PDEs (elliptic, parabolic, hyperbolic) and nonlinear PDEs (Navier-Stokes equations). (3) The contribution of each receptive field to the final MRF-PINN result is analyzed, and the influence of finite difference accuracy, model complexity (channel number) and mesh resolution on the MRF-PINN result is tested. This paper shows that MRF-PINN can adapt to completely different equation types and mesh resolutions without any hyperparameter tuning. Further, the solving error is significantly decreased under high-order finite difference, large channel number, and high mesh resolution, which is expected to become a general convolutional PINN scheme.
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