An Improved Physics-Informed Neural Network For Biharmonic Equation

Biharmonic equation is a typical fourth-order partial differential equation(PDE),which is widely used in physics and engineering fields such as elastic deformation,fluid flow and electromagnetic wave propagation.The traditional numerical methods for solving biharmonic equation mainly include finite element method,finite difference method,boundary element method,etc.In order to ensure the accuracy and validity of numerical results,traditional numerical methods often require partitioning high-quality grids and using large-scale linear/nonlinear solvers.However,as the problems become more complex,the heavy meshing and time-consuming solving process greatly limit the application efficiency of traditional numerical methods for solving PDE.In recent years,deep learning methods represented by physical-informed neural network(PINN)have been widely used to solve a variety of PDE,and have become a hot research method in the field of scientific and engineering computing.PINN fully utilizes the powerful function approximation ability of the neural network,embedding the characterize physical information PDE and initial and boundary conditions that into the loss function of the neural network.Through training the model parameters of the neural network by minimizing the loss function,and finally obtaining the mapping model between the input spatial-temporal coordinates and the output function to be evaluated.PINN has the property of meshless method,which does not need to divide the grid during the whole solving process,and the trained model can predict the function value of any point in the calculation computational domain,which greatly improves the efficiency of numerical methods.In this paper,PINN is used to solve linear and nonlinear biharmonic equations,and two improved PINN methods are developed.Firstly,the basic principles of fully connected feedforward deep neural network(DNN)and automatic differential(AD)technique are introduced,and corresponding loss functions are constructed for biharmonic equation with clamped boundary condition or simply supported boundary condition.The numerical results show that PINN is feasible to solve the biharmonic equation,but the calculation accuracy and efficiency need to be improved.To this end,this paper develops physics-informed neural network with auxiliary function(AF-PINN)and gradient auxiliary physics-informed neural network without penalty parameters(GA-PINN).The idea of AF-PINN is to introduce an auxiliary function to split the biharmonic equation into two Poisson equations,which can avoid calculating high-order derivatives.Numerical experiments show that compared with PINN,AF-PINN has greatly improved both calculation accuracy and efficiency.The idea of GA-PINN is to divides the biharmonic equation into a third-order or second-order differential equation system by introducing multiple auxiliary functions,and then constructs a neural network composite function that satisfies different boundary conditions during the training process.Compared with PINN and AF-PINN,GA-PINN automatically satisfies boundary conditions at all stages of training.The numerical results prove that GA-PINN not only improves calculation accuracy but also accelerates the learning speed of neural network.However,GA-PINN is only applicable to regular domain,and it is a future development direction to apply this method to irregular domain and other PDE.