Construction of a multilayer solution of the Duffing equation based on analytical modification of numerical methods

Abstract

Context and relevance. The purpose of the article is to study a new approach to constructing approximate solutions of parametric differential equations. The new approach combines the analytical modification of iterative formulas of conventional numerical methods with physically informed neural networks. The new method allows us to build a hierarchy of approximate solutions with high accuracy with a small neural network size. For problems with parameters, the method allows you to build parametric models with solutions in the form of functions. For these models, the task parameters are among the input variables. Methods and materials. The operation of the method is illustrated by the example of the Duffing equation. An analytical modification of the Heun's method is used to construct an approximate parametric solution. Our approach allows us to build mathematical models of three levels. As models of the first level, we investigate the method of analytical modification with the selection of the optimal parameter. As second-level models, we build models in which the parameter is a neural network, which is input to the task parameter. These models are somewhat more complex, but also more accurate. The models of the third level are considered models in which the parameter of the Heun's method is a neural network, the input of which is a task parameter and an independent variable. Results. The result is a hierarchy of adaptive models of varying complexity and accuracy, which makes it easy to adapt these models to the observational data of the object under study. Conclusions. The proposed method for constructing a multilayer neural network solution for parametric ordinary differential equations based on an analytical modification of numerical methods can be applied to for other differential equations, systems of differential equations, partial differential equations, et al.