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 https://github.com/JinshuaiBai/PIRBN.

翻译：我们近期的深入研究发现，物理信息神经网络（PINN）在训练后往往呈现出局部逼近特性。这一发现催生了本文提出的新型物理信息径向基网络（PIRBN），该网络能够在整个训练过程中保持局部性质。与深度神经网络相比，PIRBN仅包含一个隐藏层和一个径向基"激活"函数。在适当条件下，我们证明了使用梯度下降法训练PIRBN可收敛至高斯过程。此外，我们通过神经正切核（NTK）理论研究了PIRBN的训练动力学特性，并对PIRBN的初始化策略进行了全面探究。基于数值算例，PIRBN在求解具有高频特征及不适定计算域的偏微分方程时，被证实比PINN更具高效性与有效性。值得注意的是，现有的PINN数值技术（如自适应学习、分解策略及不同形式的损失函数）均适用于PIRBN。可复现所有数值结果的程序代码详见 https://github.com/JinshuaiBai/PIRBN。