This paper proposed a novel radial basis function neural network (RBFNN) to solve various partial differential equations (PDEs). In the proposed RBF neural networks, the physics-informed kernel functions (PIKFs), which are derived according to the governing equations of the considered PDEs, are used to be the activation functions instead of the traditional RBFs. Similar to the well-known physics-informed neural networks (PINNs), the proposed physics-informed kernel function neural networks (PIKFNNs) also include the physical information of the considered PDEs in the neural network. The difference is that the PINNs put this physical information in the loss function, and the proposed PIKFNNs put the physical information of the considered governing equations in the activation functions. By using the derived physics-informed kernel functions satisfying the considered governing equations of homogeneous, nonhomogeneous, transient PDEs as the activation functions, only the boundary/initial data are required to train the neural network. Finally, the feasibility and accuracy of the proposed PIKFNNs are validated by several benchmark examples referred to high-wavenumber wave propagation problem, infinite domain problem, nonhomogeneous problem, long-time evolution problem, inverse problem, spatial structural derivative diffusion model, and so on.

翻译：本文提出了一种新型径向基函数神经网络（RBFNN）用于求解各类偏微分方程。在所提出的RBF神经网络中，采用根据待解偏微分方程控制方程推导出的物理信息核函数作为激活函数，取代了传统的径向基函数。与知名的物理信息神经网络类似，所提出的物理信息核函数神经网络同样将待解偏微分方程的物理信息纳入神经网络中。两者的区别在于：物理信息神经网络将物理信息置于损失函数中，而本文提出的物理信息核函数神经网络则将待解控制方程的物理信息嵌入激活函数中。通过采用满足齐次、非齐次及瞬态偏微分方程控制方程的物理信息核函数作为激活函数，仅需边界/初始数据即可完成神经网络训练。最后，通过高波数波传播问题、无限域问题、非齐次问题、长时间演化问题、反问题、空间结构导数扩散模型等多个基准算例验证了所提出的物理信息核函数神经网络的可行性与准确性。