Research On Finite Difference And Neural Network Methods For Two Types Of Partial Differential Equations

The fourth-order stochastic fractional damped wave equation and delayed partial differential equations are two kinds of special partial differential equations,the former arise from the research of elasto-plastic-microstructures models,the later are required tools when modelling phenomena with hereditary characteristics in a wide variety of scientific and technical fields,such as in population ecology,control theory,viscoelasticity,materials with thermal memory,etc.In this work,we solve these two types of differential equation by finite difference method and Delay-physics-informed neural networks,respectively.The main research contents are as follows:Firstly,we proposes an energy dissipative finite difference method for solving a fourth-order nonlinear wave equation driven by space fractional damping and multiplicative noise.We use a ’fractional centered derivative’ approach to approximate the Riesz fractional derivative in damping term,and develop a three-level implicit Crank-Nicolson scheme for the temporal-spatial approximation.Subsequently,we discussed the expected value of the discrete energy and proved that the proposed method attain the convergence orders O(Δt2+h2)under the expected sense.Finally,a numerical experiment is given to verify the efficiency of the scheme and confirm the correctness of theoretical results,and it also shows that the fractional damping and noise term can influence the global behavior of this evolution system.Secondly,we propose a new framework of physics-informed neural networks(called Delay-PINN)to approach ordinary/partial differential equations with proportional delay,subtractive delay and time variable delay.We transform the well-posed system into an optimization problem by setting delayed dataset for the delay term.By making full use of the modern Auto-Differentiation tools,we can find the optimal parameters that enabled the neural network fits the solution well.Four numerical results illustrate the efficiency and the robustness of this method.Moreover,we explained our method can be extended to other explicit delay problems naturally,such as delayed initial/boundary conditions and mixed delay problems.Finally,we summarize the mathematical principles,error sources,advantages and disadvantages of these two methods.We illustrated the complementarity of these two kinds of methods from two aspects,and we listed suitable numerical schemes for common special types of partial differential equations,which provides reference for researchers to choose numerical algorithms in different application scenarios.