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

TL;DR

This paper introduces PIRBN, a local approximating neural network for solving nonlinear PDEs, demonstrating improved efficiency over PINN, with convergence properties and compatibility with existing PINN techniques.

Contribution

The paper proposes PIRBN, a novel physics-informed neural network with a single hidden layer and radial basis activation, maintaining local properties and demonstrating convergence and effectiveness.

Findings

PIRBN outperforms PINN in high-frequency PDEs

PIRBN training converges to Gaussian processes under certain conditions

PIRBN is compatible with existing PINN techniques

Abstract

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.