Application Of Deep Learning In Solving Partial Differential Equations For Engineering Problems

When solving engineering problems,the unknowns are usually functions of multiple variables in time and space,and the basic approaches are to solve the partial differential equations representing the problems.However,in most cases the analytical solutions of partial differential equations are difficult to obtain,and the numerical solutions may prevail.To obtain numerical solutions,the finite element method,finite difference method,and finite volume method are among the most popular numerical methods.The core idea of these methods is to discretize the area into independent grid units,approximate the unknown quantities on each unit,transform partial differential equations into algebraic equations,and finally solve such engineering problems.However,these traditional methods still have some shortcomings.For example,when facing complex engineering problems,if there is a high-dimensional partial differential equations,using traditional numerical methods may lead to the "curse of dimensionality" and increase the computational cost.Different from the traditional numerical algorithms,this research attempts to apply deep learning methods into the solution of engineering problems.The Physics Informed Neural Network(PINN)is employed as a welcomed deep learning method.PINN’s basic idea is to use neural networks to approximate the solution of differential equations.In this approach,the engineering problems are represented by a set of control equations,boundary conditions,and initial conditions.First,the loss function is defined by the degree of matching between the simulation results of the neural network and the actual state.Then algorithms such as gradient descent are used to minimize the loss function,and optimize the parameters of the neural network in the process.After the training results converge,the final approximate solution of the equation can be obtained.In this research,the PINN method is applied in the one-dimensional Burgers equation and the two-dimensional Cook Panel problem.The former is a partial differential equation involving the time domain,while the latter is a large scale,multi-equation elasticity problem.The physical information describing their current engineering state is embedded into the neural network for training,and the corresponding numerical solutions are obtained.In these two examples,the PINN has obtained good simulation results,which shows the development potential of this method.In addition,this research also proposes a way to optimize large-scale engineering models.Finally,the training results of PINN are compared with the results of traditional numerical solutions,and a summarization is made on the advantages and disadvantages of this new method in dealing with engineering problems.