The Application Of Deep Neural Network Based On Physical Mechanism In Numerically Solving The Nonlinear Degasperis-Procesi Equation

The Degasperis Procesi(DP)equation is an important class of nonlinear partial differential equation with high-order derivatives,which can have both peaked solution and discontinuous shock solution.The low regularity of the solution is the main challenge for numerical simulation.The traditional numerical methods suffer to numerical oscillations and require post-processing techniques.In recent years,deep neural networks have been widely used in numerically solving partial differential equations.This paper aims to construct physics-informed neural network surrogate model to provide efficient algorithms for solving DP equation and at the meantime to identify the parameters in DP equation with observed data.We first review the multilayer feedforward neural networks and the basic ideas using physics-informed neural network to solve partial differential equations.The network construction strategies,parameter optimization methods for neural networks and loss functions are considered.Then a deep neural network surrogate is established based on the hyperbolic elliptic system of the DP equation.The loss function is then constructed via the initial boundary data and the equation.We use Adam and L-BFGS optimization methods to train the neural network to get the optional network parameter.We demonstrate the effectiveness of the algorithm through several numerical experiments,namely:(1)Smooth soliton solutions.(2)Single peakon and anti-peakon traveling solutions.(3)Two peakons interaction and two anti-peakons interaction.(4)Shock peakon solution.(5)Peakon and anti-peakon interaction.Finally,we use PINN to solve the parameter inversion problem of the DP equation,which expect to infer the equation that is consistent with the observed data.We validate the effectiveness of the PINN surrogate via noise data and noise-free data.The simulation results show that the physics-informed neural network does not need spatio-temporal meshing or postprocessing to eliminate numerical oscillations when solving the forward and inverse problems of the DP equation.Thus it provides an efficient method for numerically solving the DP equation.