Metode strojnog učenja u predviđanju najniže svojstvene vrijednosti Laplaceovog operatora

master's thesis

• Author: Davor Penzar
• Mentor: Luka Grubišić (mentor)
• Committee member: Luka Grubišić (predsjednik povjerenstva)
• Committee member: Boris Širola (član povjerenstva)
• Committee member: Matija Bašić (član povjerenstva)
• Committee member: Mladen Vuković (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-02-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Proučavajući teorijski svojstvene vrijednosti i svojstvene funkcije Laplaceovog operatora uočena su obilježja kojima se njihovo računanje može pojednostaviti. Prvenstveno, one ne ovise o afinim izometričnim transformacijama skupova (osim što se domena svojstvenih funkcija, očito, izometrično transformira). Također, skaliranjem skupa svojstvene se vrijednosti skaliraju multiplikativnim inverzom kvadrata skalara, dok se svojstvene funkcije komponiraju zdesna sa skalarom. Nadalje, one neprekidno ovise o domeni i svojstvene su vrijednosti padajuće s obzirom na relaciju inkluzije. Iz svega se navedenog zaključeno je da se određene domene u svrhu numeričkog računanja svojstvenih vrijednosti i svojstvenih funkcija Laplaceovog operatora (to jest, najmanje svojstvene vrijednosti u slučaju ovog rada) mogu aproksimirati jednostavnijim, dovoljno sličnim i normaliziranim domenama. Kao jedan tip jednostavnijih domena koje mogu poprimiti razne oblike odabrani su poligoni. Zbog opaženih obilježja svojstvenih vrijednosti i svojstvenih funkcija Laplaceovog operatora, u svrhu njihova računanja modelom strojnog učenja bilo je poželjno poligone opisati karakterizacijama koje su dovoljno diskriminirajuće, ali i robusne na određene transformacije. U radu su ponuđene nude tri različite karakterizacije poligona, a u praktičnom nastavku rada proučavani su samo trokuti kao prototipi poligona. Za svaku od karakterizacija, nakon proučavanja literature i eksploratorne analize izrađenog skupa podataka, odabrana je jedna pogodna metoda strojnog učenja za početni problem. Tako su konstruirani dva modela linearne regresije, neuronska mreža i konvolucijska neuronska mreža. Nakon treniranja modela njihove su uspješnosti ispitane na testnim skupovima podataka i modeli su uspoređeni s obzirom na uspješnosti i brzine. Rezultati su u većini slučajeva bili obećavajući, ali generalizabilnost modela na složenije poligone nije potvrđena jer model koji se najjednostavnije može generalizirati---konvolucijska neuronska mreža---luči najlošije rezultate.

Abstract (english)

Studying the theory behind the eigenvalues and the eigenfunctions of Dirichlet Laplacian it was observed that they possess some characteristics that allow us to simplify calculations to find them. First of all, they are independent of affine isometric transformations of sets (except that the domain of the eigenfunctions is obviously isometrically transformed). Also, scaling a set scales its eigenvalues by the factor of the square of the multiplicative inverse of the scaling factor, and the resulting eigenfunctions are compositions of the original eigenfunctions on the left and the scaling factor on the right. Furthermore, they depend continuously on the domain and the eigenvalues are monotonically decreasing in regard to the order of inclusion of sets. Considering all this it was then concluded that some domains can be approximated by other, simpler yet similar enough, normalised domains to numerically find the eigenvalues and the eigenfunctions of Dirichelt Laplacian (i. e. the minimal eigenvalue, as is the case in this paper) on them. Polygons were then chosen as an example of simple domains with many different shapes. Because of the observed characteristics of the eigenvalues and the eigenfunctions, to predict them on polygons using a machine learning model, polygons should be characterised in a way that discriminates them but is also resistant to specific transformations. In this paper three distinct characterisations were proposed, and then in the following part of the paper, the practical part, only triangles were observed as they can be considered prototypes of polygons. After studying the sources and conducting exploratory analysis of the generated dataset, a suitable machine learning method was chosen for each of the characterisations to solve the original problem. Subsequently two linear regression models, a neural network, and a convolutional neural network were designed. After the training their success was measured on testing datasets and they were compared for their successfulness and time consumption. In most cases the results were promising, although generalisability of the models to more complex polygons was not confirmed since the model that could be generalised most easily—the convolutional neural network—yeilds the worst results.

• Keywords: Laplaceov operator; poligoni; strojno učenje
• Keywords (english): Dirichlet Laplacian; polygons; machine learning
• Language: croatian
• URN:NBN: urn:nbn:hr:217:509741
• Study programme: Title: Computer Science and Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra računarstva i matematike (magistar/magistra računarstva i matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2020-09-21 11:36:42