MRF-PINN：一个求解偏微分方程的多接收域卷积物理信息神经网络
MRF-PINN: A Multi-Receptive-Field convolutional physics-informed neural network for solving partial differential equations
摘要：
与传统的偏微分方程数值求解方法相比，物理信息神经网络（PINN）在物理场重构和反问题求解等方面表现出了极大的节省开发工作量和计算成本的优势.卷积神经网络（CNN）具有参数共享、空间特征提取和低推理代价等优点，在PINN中得到越来越多的应用。然而，仍然存在以下一些挑战。为了适应卷积PINN来解决不同的PDE，通常需要相当大的努力来调整关键的超参数。此外，有限差分精度和网格分辨率对卷积PINN预测能力的影响尚未解决。为了填补上述空白，本文提出了三个倡议：（1）多接收域PINN（2）在求解Navier-Stokes方程时，采用量纲平衡法估计损失权;（3）采用泰勒多项式填充边界附近的虚拟节点，实现高阶有限差分。通过求解三种典型的线性偏微分方程（椭圆型、抛物型、双曲型）和一系列非线性偏微分方程（Navier-Stokes偏微分方程），验证了该方法的通用性和优越性.本文表明，MRF-PINN可以适应完全不同的方程类型和网格分辨率，而无需任何超参数调整。维数平衡法节省了计算时间，提高了求解Navier-Stokes偏微分方程的收敛性。在高阶有限差分、大通道数和高网格分辨率条件下，求解误差显著减小，有望成为一种通用的卷积PINN算法。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
