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Our recent intensive study has found that physics-informed neural networks(PINN) tend to be local approximators after training. This observation leads tothis novel physics-informed radial basis network (PIRBN), which can maintainthe local property throughout the entire training process. Compared to deepneural networks, a PIRBN comprises of only one hidden layer and a radial basis"activation" function. Under appropriate conditions, we demonstrated that thetraining of PIRBNs using gradient descendent methods can converge to Gaussianprocesses. Besides, we studied the training dynamics of PIRBN via the neuraltangent kernel (NTK) theory. In addition, comprehensive investigationsregarding the initialisation strategies of PIRBN were conducted. Based onnumerical examples, PIRBN has been demonstrated to be more effective andefficient than PINN in solving PDEs with high-frequency features and ill-posedcomputational domains. Moreover, the existing PINN numerical techniques, suchas adaptive learning, decomposition and different types of loss functions, areapplicable to PIRBN. The programs that can regenerate all numerical results canbe found at https://github.com/JinshuaiBai/PIRBN.

Physics-informed radial basis network (PIRBN): A local approximating neural network for solving nonlinear PDEs