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[{"key": "dc.contributor.advisor", "value": "Repin, Sergey", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.advisor", "value": "P\u00f6l\u00f6nen, Ilkka", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.advisor", "value": "Julin, Vesa", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Halonen, Vilho", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2021-11-19T08:00:49Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2021-11-19T08:00:49Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2021", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/78718", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa verrataan analyyttisien menetelmien ja koneoppimismallien toimivuutta ep\u00e4tarkkuudesta johtuvien virheiden kontrolloinnissa. Tarkasteltavana matemaattisena esimerkkiongelmana k\u00e4ytet\u00e4\u00e4n lineaarista variaatio-ongelmaa. Tuloksena havaitaan, ett\u00e4 neuroverkot toimivat hyvin ja ovat k\u00e4yt\u00e4nt\u00f6\u00f6n mahdollisesti soveltuva keino tehd\u00e4 virhearviointia.\nMonille osittaisdifferentiaaliyht\u00e4l\u00f6ille on johdettu analyyttisia kontrollointikeinoja viime vuosikymmenien aikana (katso [1], [2]). Ensimm\u00e4iset luvut k\u00e4yt\u00e4mme analyyttisien virhearvioiden todistamiseen tunnettujen analyysin ty\u00f6kalujen avulla tarkasteltavalle variaatio-ongelmalle. Virhearvioita testataan numeerisesti ja huomataan, ett\u00e4 vaikka analyyttiset rajat ovat varmoja ja halpoja laskennallisesti, ne ovat monesti toivottua ep\u00e4tarkempia.\nTutkielman toisessa osiossa luodaan koneoppimismalleja, joilla pyrit\u00e4\u00e4n arvioimaan tarkalleen ep\u00e4tarkkuuden aiheuttamaa virhett\u00e4. Valittu koneoppimismalli on neuroverkko. Mallien kouluttamiseen k\u00e4ytetty data luodaan itse numeerisilla menetelmill\u00e4.\nViimeisess\u00e4 luvussa verrataan analyyttisien metodien ja luotujen neuroverkkojen toimivuutta. Vertailussa k\u00e4ytet\u00e4\u00e4n koulutusdatasta eroavaa generoitua dataa jolle lasketaan analyyttiset rajat, numeeriset approksimaatiot ja neuroverkkojen tulokset. Havaitaan, ett\u00e4 neuroverkot suoriutuvat teht\u00e4v\u00e4st\u00e4 niin hyvin, ett\u00e4 voidaan sanoa niiden olevan kilpailullisia analyyttisien metodien kanssa. Jos vastaavia koneoppimismalleja pystyt\u00e4\u00e4n luomaan vaikeammille moniulotteisille ongelmille, t\u00e4m\u00e4 menetelm\u00e4 voi osoittautua varsin hy\u00f6dylliseksi simuloinnissa ja insin\u00f6\u00f6rity\u00f6ss\u00e4.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.abstract", "value": "In this thesis we compare the performance of analytical methods and neural networks trained with numerically produced data in controlling uncertainty errors of a linear variational problem. We find that neural networks perform well and are feasible to use in practical computations in place of analytical control methods.\nAnalytical methods for controlling uncertainty errors have been derived for various differential problems (see [1], [2]) in recent decades. The first chapters are devoted to deriving by known methods analytical error bounds for the linear variational problem which we will study. These error bounds are numerically tested and we find that the bounds while they are guaranteed and cheap to compute are not always as sharp as an engineer might hope.\nThe second part of this thesis consists of creating machine learning models with the goal of approximating the exact error caused by uncertainty in our mathematical model. The chosen type of machine learning model is a deep neural network. The training data used for training the models is generated by numerical computations.\nIn the final chapter we compare the performance of the analytical methods and machine learning models and we conclude that neural networks can be competitive in this task. If such models are made and found to work for more complicated nonlinear PDE\u2019s this method could prove very useful in computer simulations and engineering.", "language": "en", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2021-11-19T08:00:49Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2021-11-19T08:00:49Z (GMT). No. of bitstreams: 0\n Previous issue date: 2021", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "63", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "eng", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.title", "value": "Accuracy analysis of uncertain variational problems with analytical and machine learning methods", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202111195729", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", 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