Physics-Based Neural Network Methods for Solving Parameterized Singular Perturbation Problem

Tatiana Lazovskaya,Dmitry Tarkhov,Galina Malykhina

doi:10.3390/computation9090097

Tatiana Lazovskaya, Dmitry Tarkhov + Show 1 more

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https://doi.org/10.3390/computation9090097

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Abstract

This work is devoted to the description and comparative study of some methods of mathematical modeling. We consider methods that can be applied for building cyber-physical systems and digital twins. These application areas add to the usual accuracy requirements for a model the need to be adaptable to new data and the small computational complexity allows it to be used in embedded systems. First, we regard the finite element method as one of the “pure” physics-based modeling methods and the general neural network approach as a variant of machine learning modeling with physics-based regularization (or physics-informed neural networks) and their combination. A physics-based network architecture model class has been developed by us on the basis of a modification of classical numerical methods for solving ordinary differential equations. The model problem has a parameter at some values for which the phenomenon of stiffness is observed. We consider a fixed parameter value problem statement and a case when a parameter is one of the input variables. Thus, we obtain a solution for a set of parameter values. The resulting model allows predicting the behavior of an object when its parameters change and identifying its parameters based on observational data.

Highlights

IntroductionAs a part of the operating of these systems, the received information (data) is synchronized with a corresponding physical or information object
The development of technologies that make it possible to obtain data from environments, observed objects, and processes and process them with sufficient speed and efficiency leads to the widespread implementation of cyber-physical systems (CPSs) [1]
The issue of building digital twin (DT) and the simultaneous development of artificial intelligence technologies have led to the emergence of so-called “pure” data-driven modeling “black boxes”, in which a model is created on the basis of deep learning using big data

Summary

Introduction

As a part of the operating of these systems, the received information (data) is synchronized with a corresponding physical or information object The result of such synchronization is a digital twin (DT) of an object, i.e., a model of a complex system. For some complex industrial and technological processes, the acquisition of a large amount of data itself can be accompanied by high expenditure and, sometimes, a lack of measurement possibilities. In these cases, the data can be utilized to verify a model built by classical modeling methods based on the physics of an object, including numerical methods for solving corresponding problems of mathematical physics

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