Domain Decomposition with Neural Network Interface Approximations for time-harmonic Maxwell’s equations with different wave numbers
DOI:
https://doi.org/10.17268/sel.mat.2023.01.01Keywords:
Time-Harmonic Maxwell’s Equations, Machine Learning, Feedforward Neural Network, Domain Decomposition MethodAbstract
In this work, we consider the time-harmonic Maxwell’s equations and their numerical solution with a domain decomposition method. As an innovative feature, we propose a feedforward neural network-enhanced approximation of the interface conditions between the subdomains. The advantage is that the interface condition can be updated without recomputing the Maxwell system at each step. The main part consists of a detailed description of the construction of the neural network for domain decomposition and the training process. To substantiate this proof of concept, we investigate a few subdomains in some numerical experiments with low frequencies. Therein the new approach is compared to a classical domain decomposition method. Moreover, we highlight current challenges of training and testing with different wave numbers and we provide information on the behaviour of the neural-network, such as convergence of the loss function, and different activation functions.
References
Shi L, Babushkin I, Husakou A, Melchert O, Frank B, Yi J, et al. Femtosecond field-driven on-chip unidirectional electronic currents in nonadiabatic tunneling regime. Laser & Photonics Reviews. 2021;(15).
Melchert O, Kinnewig S, Dencker F, Perevoznik D, Willms S, Babushkin I, et al. Soliton compression and supercontinuum spectra in nonlinear diamond photonics. 2022, arXiv preprint arXiv:221100492. Available from: arXiv:2211.00492.
Monk P. Finite element methods for Maxwell’s equations. Oxford Science Publications; 2003.
Demkowicz L. Computing with hp-adaptive finite elements. Volume 1 One and Two Dimensional Elliptic and Maxwell Problems. Chapman and Hall/CRC; 2006.
Langer U, Pauly D, Repin S, editors. Maxwell’s Equations: Analysis and Numerics. De Gruyter; 2019. Available from: https://www.degruyter.com/document/doi/10.1515/9783110543612/html.
Rodriguez AA, Bertolazzi E, Valli A. In: Langer U, Pauly D, Repin S, editors. 1. The curl–div system: theory and finite element approximation. Berlin, Boston: De Gruyter; 2019. p. 1-44. Available from: https://doi.org/10.1515/9783110543612-001 [cited 2023-01-18].
Nédélec JC. Mixed finite elements in R3. Numerische Mathematik. 1980 Sep; 35(3):315-41. Available from: https: //doi.org/10.1007/BF01396415.
Nédélec JC. A new family of mixed finite elements in R3. Numerische Mathematik. 1986 Jan;50(1):57-81. Available from: https://doi.org/10.1007/BF01389668.
Kinnewig S, Wick T, Beuchler S. Resolving the sign conflict problem for hanging nodes on hp-hexahedral Nédélec elements; 2023. In preparation.
Carstensen C, Demkowicz L, Gopalakrishnan J. Breaking spaces and forms for the DPG method and applications including Maxwell equations. Computers & Mathematics with Applications. 2016; 72(3):494-522. Available from: https://
www.sciencedirect.com/science/article/pii/S0898122116302620.
Nicaise S, Tomezyk J. In: Langer U, Pauly D, Repin S, editors. 9. The time-harmonic Maxwell equations with impedance boundary conditions in polyhedral domains. Berlin, Boston: De Gruyter; 2019. p. 285-340. Available from: https://doi.org/10.1515/9783110543612-009 [cited 2023-01-18].
Hiptmair R. Multigrid method for Maxwell’s equations. SIAM Journal on Numerical Analysis. 1998;36(1):204-25.
Henneking S, Demkowicz L. A numerical study of the pollution error and DPG adaptivity for long waveguide simulations. Computers & Mathematics with Applications. 2021;95:85-100.
Faustmann M, Melenk JM, Parvizi M. H-matrix approximability of inverses of FEM matrices for the time-harmonic Maxwell equations. Advances in Computational Mathematics. 2022;48(5).
Dolean V, Gander Mj, Gerardo-Giorda L. Optimized Schwarz Methods for Maxwell’s Equations. SIAM Journal on Scientific Computing. 2009;31(3):2193-213. Available from: https://epubs.siam.org/doi/abs/10.1137/080728536.
Schöberl J. A Posteriori Error Estimates for Maxwell Equations. Mathematics of Computation. 2008;77(262):633-49. Available from: https://www.jstor.org/stable/40234527.
Bürg M. A residual-based a posteriori error estimator for the hp-finite element method for Maxwell’s equations. Applied Numerical Mathematics. 2012 08; 62:922–940.
Beuchler S, Kinnewig S, Wick T. In: Brenner SC, Chung ETS, Klawonn A, Kwok F, Xu J, Zou J, editors. Parallel domain decomposition solvers for the time harmonic Maxwell equations. vol. 145 of Lecture Notes in Computational Science and Engineering. Springer; 2023. p. 615-22.
Toselli A,Widlund O. Domain decomposition methods - algorithms and theory. Volume 34 of Springer Series in Computational Mathematics. Berlin, Heidelberg: Springer; 2005.
El Bouajaji M, Thierry B, Antoine X, Geuzaine C. A quasi-optimal domain decomposition algorithm for the timeharmonic Maxwell’s equations. Journal of Computational Physics. 2015;294:28-57. Available from: https://www.sciencedirect.com/science/article/pii/S0021999115001965.
Arndt D, BangerthW, Davydov D, Heister T, Heltai L, Kronbichler M, et al. The deal.II finite element library: Design, features, and insights. Computers & Mathematics with Applications. 2020. Available from: http://www.sciencedirect.com/science/article/pii/S0898122120300894.
Arndt D, Bangerth W, Feder M, Fehling M, Gassm¨oller R, Heister T, et al. The deal.II library, Version 9.4. Journal of Numerical Mathematics. 2022;30(3):231-46. Available from: https://dealii.org/deal94-preprint.pdf.
Bishop CM. Pattern recognition and machine learning. Springer; 2006.
Higham CF, Higham DJ. Deep Learning: An introduction for applied mathematicians. SIAM review. 2019;61(4):860-91.
Kinnewig S, Kolditz L, Roth J, Wick T. Numerical methods for algorithmic systems and neural networks. Hannover : Institutionelles Repositorium der Leibniz Universität Hannover, Lecture Notes. Institut f¨ur Angewandte Mathematik, Leibniz Universität Hannover; 2022.
Paszke A, Gross S, Massa F, Lerer A, Bradbury J, Chanan G, et al. PyTorch: An imperative style, high-performance deep learning library. In: Advances in Neural Information Processing Systems 32. Curran Associates, Inc.; 2019. p. 8024-35. Available from: http://papers.neurips.cc/paper/9015-pytorch-an-imperative-style-high-performance-deep-learning-library.pdf.
Knoke T, Kinnewig S, Wick T, Beuchler S. Neural network interface condition approximation in a domain decomposition method applied to Maxwell’s equations; 2022. In review.
Girault V, Raviart PA. Finite element methods for Navier-Stokes equations. vol. 5 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin; 1986. Theory and algorithms. Available from: https://doi.org/10.1007/978-3-642-61623-5.
Zaglmayr S. High order finite element methods for electromagnetic field computation. Johannes Kepler University Linz; 2006.
Copony S. Dynamisches Verhalten in neuronalen Netzen. Universität Hamburg; 2007. Available from: https://www.math.uni-hamburg.de/home/gunesch/Vorlesung/SoSe2007/Sem_DS_ODE/Vortrag/Copony_Vortrag.pdf.
Kriesel D. A Brief Introduction to Neural Networks. http://www.dkriesel.com/en/science/neural networks; 2005.
Ben-David S, Shalev-Shwartz S. Understanding machine learning : from theory to algorithms. Cambridge: Cambridge UniversityPress; 2014.
Nielson A. Neural networks and Deep Learning. Determination Press; 2015.
Knoke T, Wick T. Solving differential equations via artificial neural networks: Findings and failures in a model problem. Examples and Counterexamples. 2021;1:100035. Available from: https://www.sciencedirect.com/science/ article/pii/S2666657X21000197.
Ellacott SW. Aspects of the numerical analysis of neural networks. Acta Numerica. 1994;3:145–202.
Kingma DP, Ba J. Adam: A Method for Stochastic Optimization. In: Bengio Y, LeCun Y, editors. 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings;. Available from: http://arxiv.org/abs/1412.6980.
Ernst OG, Gander MJ. In: Why it is Difficult to Solve Helmholtz Problems with Classical Iterative Methods. Lecture Notes in Computational Science and Engineering. Springer; 2012. p. 325-63. Available from: https://doi.org/10.1007/978-3-642-22061-6_10.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Selecciones Matemáticas
This work is licensed under a Creative Commons Attribution 4.0 International License.
The authors who publish in this journal accept the following conditions:
1. The authors retain the copyright and assign to the journal the right of the first publication, with the work registered with the Creative Commons Attribution License,Atribución 4.0 Internacional (CC BY 4.0) which allows third parties to use what is published whenever they mention the authorship of the work And to the first publication in this magazine.
2. Authors may make other independent and additional contractual arrangements for non-exclusive distribution of the version of the article published in this journal (eg, include it in an institutional repository or publish it in a book) provided they clearly state that The paper was first published in this journal.
3. Authors are encouraged to publish their work on the Internet (for example, on institutional or personal pages) before and during the review and publication process, as it can lead to productive exchanges and to a greater and more rapid dissemination Of the published work.
