  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.
