Hluboké učení pro řešení diferenciálních rovnic

• Thesis title in Czech: Hluboké učení pro řešení diferenciálních rovnic
• Thesis title in English: Deep learning for the solution of differential equations
• Key words: Strojové učení|hluboké učení|diferenciální rovnice|metoda konečných prvků|neuronová síť
• English key words: Machine learning|deep learning|differential equations|finite element method|physics-informed neural network
• Academic year of topic announcement: 2022/2023
• Thesis type: Bachelor's thesis
• Thesis language: čeština
• Department: Department of Numerical Mathematics (32-KNM)
• Supervisor: Scott Congreve, Ph.D.
• Author: hidden - assigned and confirmed by the Study Dept.
• Date of registration: 15.12.2022
• Date of assignment: 22.12.2022
• Confirmed by Study dept. on: 23.05.2023
• Date and time of defence: 13.09.2023 09:00
• Date of electronic submission: 11.07.2023
• Date of submission of printed version: 24.07.2023
• Date of proceeded defence: 13.09.2023
• Opponents: doc. RNDr. Václav Kučera, Ph.D.

Guidelines

Traditionally the numerical solution of differential equations is performed using standard numerical
methods, such as the finite element method [1, 2]. In recent years, there has been research into
the application of machine/deep learning to solve differential equations; cf. [3] and the references
therein. In this thesis, we will study various machine learning techniques for the solution of
partial differential equations, implement code for performing these techniques in Python (or a
similar programming language), and compare the results to a traditional finite element solution.

References

[1] S. C. Brenner and L. R. Scott. The Mathematical Theory of Finite Element Methods. SpringerVerlag, 2008.

[2] V. Dolejší, P. Knobloch, V. Kučera and M. Vlasák. Finite element methods: Theory, applications
and implementations. Matfyzpress, Praha, 2013.

[3] L. Lu, X. Meng, Z. Mao and G. E. Karniadakis. DeepXDE: a deep learning library for solving
differential equations. SIAM Review, 63(1):208-228, 2021.
url: https://doi.org/10.1137/19M1274067

[4] C. F. Higham and D. J. Higham. Deep learning: an introduction for applied mathematicians.
SIAM Review, 61(4):860-891, 2019.
url: https://doi.org/10.1137/18M1165748

[5] M. Abadi, P. Barham, J. Chen, Z. Chen, A. Davis, J. Dean, M. Devin, S. Ghemawat, G.
Irving, M. Isard, M. Kudlur, J. Levenberg, R. Monga, S. Moore, D. G. Murray, B. Steiner,
P. Tucker, V. Vasudevan, P. Warden, M. Wicke, Y. Yu and X. Zheng. TensorFlow: a system
for large-scale machine learning. In Proceedings of the 12th USENIX Conference on Operating
Systems Design and Implementation, OSDI’16, pages 265-283, Savannah, GA, USA. USENIX
Association, 2016.

Preliminary scope of work

Cílem této práce je studovat aplikace strojového/hlubokého učení pro řešení diferenciálních rovnic
a porovnat je s tradičními numerickými metodami.

Preliminary scope of work in English

The goal of this thesis is to study the application of machine/deep learning to the solution
of differential equations, and compare to traditional numerical methods.