Deep Learning-Based Methods For Solving Partial Differential Equations

As one of the most critical tools to describe the objective laws of the physical world,partial differential equations(PDEs)are widely used in various scientific researches and engineering applications.In fact,it is difficult to obtain the analytical solution of the partial differential equation,so the approximate numerical solution is generally considered.In this paper,the conventional numerical methods for solving PDEs and also the deep learning method—physics informed neural network(PINN)are introduced first.The traditional numerical methods often suffer from the curse of dimensionality,and it is difficult to balance the calculation efficiency and solution accuracy with the use of these methods.Although the solver based on deep learning is mesh-free,it also loses some interpretability because of the " black box”properties of the full connected neural network.Based on the connection between the finite element basis function and ReLU neural network proposed by He et al.[1],in this paper meshless numerical methods are incorporated with deep learning to develop the RBF-PINN model with pure physical constraints.In the training process of this model,the radial basis function and the neural network complement each other.The introduction of the radial basis function makes the hidden layer structure partially interpretable.The parameter optimization system of deep learning gives the idea of self-adaptability in both the central nodes in the radial basis function and the corresponding bandwidth parameters,which further improves the generalization ability of the model.In order to explore the implications of kernel functions on the proposed models,a new class of squareplus kernel functions as well as generalized neural network kernels are presented for comparative analysis with Gaussian kernels.We make experiments on several different equations,including differential equations with Dirichlet boundary conditions and Neumann boundary conditions,and two-dimensional Poisson equations on regular or irregular domains.The effect of the number of radial basis center points and different kernel functions on the solution error of the RBF-PINN model are emphasized in this paper,and the adaptabilities of the central nodes and corresponding bandwidth parameters are also analyzed.The experimental results show that,compared with the PINN model,the RBF-PINN model has higher solution accuracy because of the sparse structure of the network,which can further reduce the computational complexity and improve the efficiency of the algorithm.In particular,for solving differential equations with Neumann boundary conditions,the PINN model based on the Hermite collocation method is developed,which can further reduce the derivative error of the approximate function in the boundary region.Under the framework given in this paper,the radial basis collocation method can be easily extended to other meshless models,such as the F-PINN model based on Fourier expansion,and it is tested in cases where the target functions to be approximated exhibit multi-scale behaviours.In order to better capture the time dependence in partial differential equations,the TD-RBFN algorithm with time discretization technique and the ST-RBFN algorithm for the entire space-time domain based on the RBF-PINN model are designed in this paper.Numerical experiments on linear advection equations,Burgers equations and Allen-Cahn equations are conducted.The results show that the TD-RBFN algorithm has higher solution accuracy,especially for describing the shock wave of the Burgers equations and the phase separation of the solutions of the Allen-Cahn equations.