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）以求解各类偏微分方程。在所提出的RBF神经网络中，采用根据目标偏微分方程控制方程推导得到的物理启发核函数（PIKFs）作为激活函数，取代传统RBF函数。与众所周知的物理启发神经网络（PINNs）类似，本文提出的物理启发核函数神经网络（PIKFNNs）同样将目标偏微分方程的物理信息融入网络结构。区别在于，PINNs将该物理信息置于损失函数中，而PIKFNNs则将其融入激活函数。通过采用满足齐次、非齐次、瞬态等各类偏微分方程的物理启发核函数作为激活函数，网络训练仅需边界/初始数据。最后，通过高波数波动传播问题、无穷域问题、非齐次问题、长时演化问题、反问题、空间结构导数扩散模型等基准算例验证了所提PIKFNNs的可行性与准确性。