Neural network calculation method for nonlinear singular perturbation differential equation

Abstract

The invention discloses a neural network calculation method for a nonlinear singular perturbation differential equation. The method comprises the following steps: initializing hyper-parameters of truncated logarithmic normal distribution for training sample point generation; initializing the number of hidden layers in the neural network, the number of corresponding neurons in the ith hidden layer and an activation function; initializing weight matrix parameters and bias vector parameters of all hidden layers in the neural network; determining an optimization algorithm and a learning rate; sampling from the truncated logarithmic normal distribution to obtain an input sample point set; and constructing an approximate solution of a singular perturbation problem through a neural network, and substituting the approximate solution into the differential equation to obtain a loss function. According to the embodiment of the invention, aiming at the boundary layer characteristics of a nonlinear singular perturbation equation solution, a truncated logarithmic normal distribution sampling method is introduced to construct the sample points, so that the prior information of the singular perturbation problem is fully utilized, and compared with a traditional method, the solution time and precision are improved to different degrees.

Description

technical field [0001] The invention relates to the field of engineering, gas dynamics, chemical reaction, fluid mechanics and elastic mechanics singular perturbation model solving technical field, in particular to a neural network calculation method for nonlinear singular perturbation differential equations. Background technique [0002] At present, the asymptotic expansion method and the numerical method are two methods for studying nonlinear singular perturbed differential equations. Asymptotic expansion methods include matching asymptotic expansion method, boundary layer function method, stretching synthesis method, differential inequality method and multi-scale method, etc. Numerical solution methods include finite difference method, finite element method, variational method, rational spectrum collocation method, layer adaptive grid method and adaptive grid method, etc. [0003] However, the above algorithm requires a small step size calculation to achieve better accur...

AI Technical Summary

Problems solved by technology

[0005] The purpose of the present invention is to provide a neural network calculation method for nonlinear singular perturbation differential equations, aiming to solve problems in the fields of engineering, gas dynamics, chemical reactions, fluid mechanics and elastic mechanics in the prior art Problems with large computational workload, long solution time and low precision

Images