论文标题
MRF-PINN：一种多触觉场卷积物理学的神经网络，用于求解部分微分方程
MRF-PINN: A Multi-Receptive-Field convolutional physics-informed neural network for solving partial differential equations
论文作者
论文摘要
与求解偏微分方程（PDE）的常规数值方法相比，物理知识的神经网络（PINN）表现出了节省开发工作和计算成本的能力，尤其是在重建物理领域并解决反相反问题的情况下。考虑到参数共享，空间特征提取和低推理成本的优势，卷积神经网络（CNN）越来越多地用于PINN中。但是，仍然存在一些挑战如下。为了适应卷积Pinn解决不同的PDE，通常需要大量的努力来调整关键的超参数。此外，未解决有限差异准确性以及网格分辨率对卷积PINN预测性的影响。为了填补上面的空白，我们在本文中提出了三个计划：（1）建立了多率的场pinn（MRF-PINN）模型，以在无需手动调整的情况下在各种网格分辨率上求解不同类型的PDE； （2）尺寸平衡方法用于估计求解Navier-Stokes方程时的损耗； （3）Taylor多项式用于填充边界附近的虚拟节点，以实现高阶有限差。对拟议的MRF-PINN进行了测试，以求解三个典型的线性PDE（椭圆形，抛物线，双曲线）和一系列非线性PDE（Navier-Stokes PDES），以证明其一般性和优越性。本文表明，MRF-PINN可以适应完全不同的方程式类型和网格分辨率，而无需进行任何高参数调整。尺寸平衡方法节省了计算时间，并改善了解决Navier-Stokes PDE的融合。此外，在高阶有限差，较大的通道数和高网格分辨率下，解决误差显着下降，这预计将是一般卷积PINN方案。
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.
