Application of data driven algorithms on dynamical systems

U ovome radu prikazana je teorijska podloga Koopmanovog operatora. To je linearni operator koji nelinearne dinamičke sustave pretvara u linearne. Kompromis te transformacije je žrtvovanje konačnosti dimenzija sustava za beskonačno mnogo dimenzija. Tek naglim razvojem računala omogućeno je rješavanje ovakve problematike. Jedan od najkorištenijih algoritama ove problematike je DMD (eng. Dynamic Mode Decomposition) algoritam. To je podacima vođen algoritam što znači da za analizu sustava nije potrebno poznavati jednadžbe koje taj sustav opisuje, već se cijela analiza temelji na izmjerenim podacima u različitim vremenskim trenutcima. Za primjer analize, pomoću OpenFOAM simulacija, pripremljeni su podaci za nekoliko primjera iz dinamike fluida. Utvrđeno je da je moguće rekonstruirati sliku promatranih varijabli već za mali broj modova te da se pojedini sustav može opisati korištenjem nekoliko svojstvenih modova i svojstvenih vrijednosti sustava. Uz analize strujanja fluida dana je i analiza podataka dobivenih tijekom epidemije Sars-Cov 2 virusa. Predikcija kako će se razvijati epidemija je moguća uz pomno odabrane postavke analize kao što je veličina podataka za treniranje. Točnost same predikcije ovisi o promatranom broju dana. Na kraju rada dana su glavna razmatranja te prijedlozi za buduće implementacije ovakve metode.This paper presents the theoretical foundations of the Koopman operator. It is a linear operator that transforms nonlinear dynamical systems into linear systems. The trade-off of this transformation is the abandonment of the finiteness of the dimensions of the system in favor of an infinite number of dimensions. Only the rapid development of computers made it possible to solve such problems. One of the most commonly used algorithms for this problem is the DMD (Dynamic Mode Decomposition) algorithm. It is a data-driven algorithm, i.e., for the system analysis, it is not necessary to know the equations describing the system, but the whole analysis is based on the inclusion of measured data at different points in time. For testing purposes, data from several examples of fluid dynamics were processed using OpenFOAM simulations. It was found that it is possible to reconstruct the picture of the observed variables for a small number of modes and that a given system can be described with several eigenmodes and eigenvalues. In addition to the analyzes of the flows, an analysis of the data obtained during the Sars-Cov 2 virus epidemic was also performed. Prediction of the evolution of the epidemic is possible with carefully chosen analysis settings such as the size of the training data. The accuracy of the prediction itself depends on the number of days observed. At the end of the paper, the main considerations and suggestions for future implementations of this method are given