Computation of stabilization parameters by deep learning

• Název práce v češtině: Určení stabilizačních parametrů pomocí hlubokého učení
• Název v anglickém jazyce: Computation of stabilization parameters by deep learning
• Klíčová slova: konvektivně-dominantní úloha|metoda konečných prvků|stabilizovaná metoda|stabilizační parametr|hluboké učení|neuronová síť|PINN|numerická studie
• Klíčová slova anglicky: convection-dominated problem|finite element method|stabilized method|stabilization parameter|deep learning|neural network|PINN|numerical study
• Akademický rok vypsání: 2022/2023
• Typ práce: disertační práce
• Jazyk práce: angličtina
• Ústav: Katedra numerické matematiky (32-KNM)
• Vedoucí / školitel: doc. Mgr. Petr Knobloch, Dr., DSc.
• Řešitel: skrytý - zadáno a potvrzeno stud. odd.
• Datum přihlášení: 07.09.2023
• Datum zadání: 07.09.2023
• Datum potvrzení stud. oddělením: 03.10.2023

Zásady pro vypracování

The scalar steady-state convection-diffusion-reaction equation represents an important model problem for developing and analyzing numerical methods for many more complicated mathematical models. In the challenging case of small diffusion, a widely used class of numerical methods for the mentioned equation are stabilized finite element methods. These methods contain stabilization parameters which are difficult to define in an optimal way. The aim of the thesis is to develop techniques that compute stabilization parameters by applying deep learning. This will lead to a cheap numerical method for which time-consuming approaches like nonlinear algebraic flux correction will be used only for training the considered neural networks. One should study various types of neural networks and different training strategies, compare them by extensive numerical studies, and propose their extensions to other classes of problems.

Seznam odborné literatury

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