Our recent intensive study has found that physics-informed neural networks
(PINN) tend to be local approximators after training. This observation leads to
this novel physics-informed radial basis network (PIRBN), which can maintain
the local property throughout the entire training process. Compared to deep
neural networks, a PIRBN comprises of only one hidden layer and a radial basis
"activation" function. Under appropriate conditions, we demonstrated that the
training of PIRBNs using gradient descendent methods can converge to Gaussian
processes. Besides, we studied the training dynamics of PIRBN via the neural
tangent kernel (NTK) theory. In addition, comprehensive investigations
regarding the initialisation strategies of PIRBN were conducted. Based on
numerical examples, PIRBN has been demonstrated to be more effective and
efficient than PINN in solving PDEs with high-frequency features and ill-posed
computational domains. Moreover, the existing PINN numerical techniques, such
as adaptive learning, decomposition and different types of loss functions, are
applicable to PIRBN. The programs that can regenerate all numerical results can
be found at this https URL

作者提出了一种带有径向基函数的物理信息神经网络用于求解具有高频特征和不适定计算域的偏微分方程问题，该网络可通过梯度下降法收敛到高斯过程，并且比传统 PINN 方法更加高效。