A Method Of Deep Learning For Solving Partial Differential Equations

In this paper,a numerical method for partial differential equations based on deep learning is presented,it solves nonlinear elliptic equations,linear elastic equations and fourth-order biharmonic equations.We use a deep network to represent the numerical solutions of partial differential equations.In the two different structures of no residual connection and residual connection,we trace the construction process of numerical so-lution to derive the more accurate expression of partial derivative of function.Then we get the energy functional through the variational principle,and then add the constraints of the boundary term to construct the loss function of partial differential equation.Fur-thermore,we use the more accurate expression of partial derivative of function in spe-cific network structure instead of the difference method to express the derivative,which makes the model use fewer parameters and the loss function more accurate.Finally,we give some numerical example.