The Construction of the Approximate Solution of the Chemical Reactor Problem Using the Feedforward Multilayer Neural Network

Abstract

A significant proportion of phenomena and processes in physical and technical systems is described by boundary value problems for ordinary differential equations. Methods of solving these problems are the subject of many works on mathematical modeling. In most works, the end result is a solution in the form of an array of numbers, which is not the best for further research. In the future, we move from the table of numbers to more suitable objects, for example, functions based on interpolation, graphs, etc. We believe that such an artificial division of the problem into two stages is inconvenient. We and some other researchers used the neural network approach to construct the solution directly as a function. This approach is based on finding an approximate solution in the form of an artificial neural network trained on the basis of minimizing some functional which formalizing the conditions of the problem. The disadvantage of this traditional neural network approach is the time-consuming procedure of neural network training. In this paper, we propose a new approach that allows users to build a multi-layer neural network solution without the use of time-consuming neural network training procedures based on that mentioned above functional. The method is based on the modification of classical formulas for the numerical solution of ordinary differential equations, which consists in their application to the interval of variable length. We demonstrated the efficiency of the method by the example of solving the problem of modeling processes in a chemical reactor.