A Deep-Learning-Based Method for Solving Nonlinear Singular Lane-Emden Type Equation

Abstract

The nonlinear Lane-Emden type equation can be used to describe many physical phenomena. To solve this type of equation, a method based on deep neural network is proposed. The output layer of this network has two layers, the last one of which scaling the outputs of their neighbors with an aim at coping with issues where the values of function to be approximated are much less or larger than the order of 1. The Lane-Emden equation and its initial conditions are employed to construct the loss function, and the problem of solving Lane-Emden equation is transformed into an optimization problem. A hybrid method combined Adam and L-BFGS-B methods is used to solve the optimization problem and consequently the Lane-Emden type equation is solved. To increase the accuracy, an adaptive strategy is incorporated into the training data sampling method. Especially, a strategy in coping with the issue about solving white-dwarf problem is proposed. Numerical experiments are conducted in which reference solutions including analytical solutions and numerical solutions given by Runge-Kutta method are used to verify the effectiveness of our proposed method. More importantly, the solutions over large domains are calculated by the use of the proposed method. The results show that the results given by our method are in good agreement with the reference solutions, and in cases where existent neural-network-based method fails, our method is able to produce convincible results.INDEX TERMS Deep learning, artificial neural network, Lane-Emden type equation, polytropic gas sphere model, isothermal gas sphere model, white-dwarf model.

Modelling Lane-Emden Type Equations Using Physics-Informed Neural Networks

“…In other words, a hybrid data loss function is used. Other options exist like solving an equivalent system of two first-order differential equations instead of a single second order one, as for a standard analytical approach (He et al, 2020). Nevertheless, as shown by our results, our method is efficient.…”

“…Some methods are analytical ones, numerical ones, or combinations of both. A rather complete summary of the various approaches can be found in He et al (2020). Classical numerical methods including the traditional Runge-Kutta methods have to circumvent the singularity problem, by using some transformation, Taylor-like expansion, or avoiding the origin.…”

Application of Lane-Emden differential equation numerical method in fair value analysis of financial accounting

“…By analysing the application of fair value in financial accounting, it can be found that the application of fair value in the process of financial accounting is based on the enterprise actual economic situation; Furthermore it helps enterprise decision-makers with financial report issued by the financial and accounting controls over a period of enterprise economics to take decisions, helps to make accurate projections for enterprise development and also the direction, strengthens the work level and ability of enterprise financial accounting and helps to realise the transformation of enterprise financial accounting to management accounting [7]. From the traditional way of enterprise financial accounting work, it will be very difficult for the decision makers to know the real financial information and the risks involved in real-time and moreover it will take certain time to realise that.…”