MultiPINN: multi-head enriched physics-informed neural networks for differential equations solving

Recently, the physics-informed neural network (PINN) has attracted much attention in solving partial differential equations (PDEs). The success is due to the strong generalization ability of the neural network (NN), which is supported by the universal approximation theorem, and its mesh-free implementation. In this paper, we propose a multi-head NN enriched PINN (MultiPINN) for solving differential equations. The trial function is built based on the radial basis function (RBF)-interpolation, which makes NN training parameters partially interpre. The loss function is constructed by embedding the physics information of differential equations and boundary conditions. Then the parameters in MultiPINN are trained using the ADAM optimizer. A significant feature of MultiPINN is that it combines the traditional RBF interpolation method with machine learning (ML) techniques. The ML technique is employed to learn the basis feature enrichment that provides global information. The multi-head mechanism is used so that each node has multiple bases, which can improve the accuracy of the MultiPINN solution. Two ordinary differential equations and three partial differential equations, i.e. the convection equation, the Burgers equation, and the Poisson equation, are used in the numerical experiments. The experimental outcomes demonstrate that MultiPINN produces solutions consistent with both analytical solutions and solutions obtained through traditional numerical methods. Additionally, MultiPINN shows robustness and adaptability over the other NN-based methods in the implementations. © The Author(s), under exclusive licence to Springer-Verlag London Ltd., part of Springer Nature 2024.