Rješavanje običnih diferencijalnih jednadžbi korištenjem neuronske mreže

master's thesis

• Title (english): Solving ordinary differential equations by use of neural network
• Author: Ivan Badrov
• Mentor: Mladen Jurak (mentor)
• Committee member: Mladen Jurak (predsjednik povjerenstva)
• Committee member: Hrvoje Planinić (član povjerenstva)
• Committee member: Dražen Adamović (član povjerenstva)
• Committee member: Bojan Basrak (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2024-09-26, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu predstavljena je i implementirana nova vrsta neuronskih mreža koja je u posljednje vrijeme pobudila interes mnogih znanstvenika. Riječ je o fizikom informiranim neuronskim mrežama koje poprilično uspješno koristeći moderne tehnike, transformiraju dobro poznati problem rješavanja sustava diferencijalnih jednadžbi. U uvodnom poglavlju predstavlja se potrebna teorijska podloga za razumijevanje problema koji se adresira, a to su diferencijalne jednadžbe i njihovi sustavi. Zatim, nakon... More upoznavanja klasičnih numeričkih metoda i njihovih ograničenja, dolazimo do osnova neuronskih mreža. Konačno, na red stižu i najavljene fizikom informirane mreže: njihova funkcionalnost i upoznavanje s ulogom, odnosno položajem u trenutnom znanstvenom i industrijskom okruženju. Za kraj bavimo se implementacijom metode i analizom rezultata za odabrane sustave. Glavni cilj rada bila je realizacija vektoriziranog algoritma kroz čiju konstrukciju je potrebno spoznati ograničena i prednosti, te osvijetliti smjer i potencijal razvoja u budućnosti. Glavni rezultati ukazuju na optimizam oko komplementarnosti ovog tipa neuronske mreže i numeričkih metoda koje su sazrjele i preživjele zub vremena. Less

Abstract (english)

In this paper, a new type of neural network that has recently attracted the interest of many scientists is presented and implemented. These are Physics-Informed Neural Networks (PINNs), which quite successfully transform and deal with the well-known problem of solving systems of differential equations using modern techniques. The introductory chapter presents the necessary theoretical background for understanding the addressed problem, which includes differential equations and their systems.... More After introducing classical numerical methods and their limitations, the basics of neural networks are covered. Finally, the announced Physics-Informed Neural Networks are discussed: their functionality and their role or position in the current scientific and industrial landscape. The paper concludes with the implementation of the method and an analysis of the results for selected systems. The main goal of the work was the realization of a vectorized algorithm, through whose construction it was necessary to understand its limitations and advantages, and to shed light on the direction and potential for future development. The main results indicate optimism about the complementarity of this type of neural network with numerical methods that have matured and withstood the test of time. Less

• Keywords: neuronska mreža; fizikom informirane neuronske mreže; problem rješavanja sustava diferencijalnih jednadžbi
• Keywords (english): neural network; Physics-Informed Neural Networks; PINN; problem of solving systems of differential equations
• Language: croatian
• URN:NBN: urn:nbn:hr:217:853010
• Study programme: Title: Financial and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar matematike (sveučilišni magistar matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2025-02-05 12:54:52