Comparative Analysis of Methods for Constructing Neural Network Models of Wastewater

Abstract

Modern mathematical modeling methods are considered - physically informed neural networks (PINNS) and analytical modification of classical numerical methods using the example of a system of differential equations modeling biological wastewater treatment processes. The optimal parameters of the models (the number of neurons and layers), as well as the hyperparameters of the learning method (the number of trial points, the penalty multiplier in the error functional, characterizing the accuracy of the problem solution) are being searched. The conducted research has demonstrated the effectiveness of using the studied neural network modeling methods to build adaptive mathematical models of real objects and processes in a functional form. The developed methods demonstrate a fairly high accuracy in comparison with numerical solutions of the initial system of differential equations obtained in the Mathematica environment. This ensures a significant reduction in computational costs for further adaptation of the constructed models, in case new information about the object being modeled becomes available. A feature of the proposed approach is the possibility of obtaining solutions in an analytical form, which expands the possibilities of their practical application, in particular, for embedded systems. The optimal number of neurons and the optimal weighting factor in the error functional formula are determined for different approaches. We study an approach with a constant number of neurons, the same or different for the concentration of nitrogen and bacteria, growing networks in which neurons are added one at a time. The possibility of increasing the accuracy of models by adding a term to the loss function determined by the discrepancy between the neural network model and the measurement data is also investigated. The considered methods can be applied without special difficulties to other problems of mathematical modeling of real objects and phenomena, the initial mathematical models of which are differential equations and systems.