SOLVING POISSON EQUATION WITH CONVOLUTIONAL NEURAL NETWORKS
Keywords:machine learning, Poisson equation, convolutional neural network
Context. The Poisson equation is the one of fundamental differential equations, which used to simulate complex physical processes, such as fluid motion, heat transfer problems, electrodynamics, etc. Existing methods for solving boundary value problems based on the Poisson equation require an increase in computational time to achieve high accuracy. The proposed method allows solving the boundary value problem with significant acceleration under the condition of acceptable loss of accuracy.
Objective. The aim of our work is to develop artificial neural network architecture for solving a boundary value problem based on the Poisson equation with arbitrary Dirichlet and Neumann boundary conditions.
Method. The method of solving boundary value problems based on the Poisson equation using convolutional neural network is proposed. The network architecture, structure of input and output data are developed. In addition, the method of training dataset generation is described.
Results. The performance of the developed artificial neural network is compared with the performance of the numerical finite difference method for solving the boundary value problem. The results showed an acceleration of the computational speed in x10–700 times depending on the number of sampling nodes.
Conclusions. The proposed method significantly accelerated speed of solving a boundary value problem based on the Poisson equation in comparison with the numerical method. In addition, the developed approach to the design of neural network architecture allows to improve the proposed method to achieve higher accuracy in modeling the process of pressure distribution in areas of arbitrary size.
Tu J. Computational Fluid Dynamics: A Practical Approach. Oxford: Butterworth-Heinemann, 2018, 495 p.
Wu C. Y., Ferng Y. M., Ciieng C. C., Liu C. C. Investigating the advantages and disadvantages of realistic approach and porous approach for closely packed pebbles in CFD simulation, Nuclear Engineering and Design, 2010, Vol. 240, pp. 1151–1159.
Simon H. D., Gropp W., Lusk E. Parallel Computational Fluid Dynamics: Implementations and Results (Scientific and Engineering Computation). Cambridge, The MIT Press, 1992, 362 p.
Runnels B., Agrawal V., Zhang W., Almgren A. Massively parallel finite difference elasticity using block-structured adaptive mesh refinement with a geometric multigrid solver, ArXiv e-prints, 2020, https://arxiv.org/abs/2001.04789v2, DOI: https://doi.org/10.1016/j.jcp.2020.110065
Raghu M., Schmidt E. A Survey of Deep Learning for Scientific Discovery, ArXiv e-prints, 2020, https://arxiv.org/abs/2003.11755v1.
Dissanayake M. W. M. G., Phan-Thien N. Neural-networkbased approximations for solving partial differential equations, Communications in Numerical Methods in Engineering, 1994, Vol. 10, No. 3, pp. 195–201.
Sirignano J., Spiliopoulos K. DGM: A deep learning algorithm for solving partial differential equations, Journal of Computational Physics, 2018, Vol. 375, pp. 1339–1364.
Lee H., Kang I. S. Neural algorithm for solving differential equations, Journal of Computational Physics, 1990, Vol. 91, No. 1, pp. 110–131.
Smaoui N., Al-Enezi S. Modelling the dynamics of nonlinear partial differential equations using neural networks, Journal of Computational and Applied Mathematics, 2004, Vol. 170, No. 1, pp. 27–58.
Kumar M. and Yadav N. Multilayer perceptrons and radial basis function neural network methods for the solution of differential equations: A survey, Computers & Mathematics with Applications, 2011, Vol. 62, №10, pp. 3796–3811.
Xiao X., Zhou Y., Wang H., and Yang X. A novel cnn-based poisson solver for fluid simulation, IEEE Transactions on Visualization and Computer Graphics, 2020, Vol. 26, No. 3, pp. 1454–1465. DOI: 10.1109/TVCG.2018.2873375
Tompson J., Schlachter K., Sprechmann P., and Perlin K. Accelerating eulerian fluid simulation with convolutional networks, ArXiv e-prints, 2017, https://arxiv.org/abs/1607.03597.
Lin T.-Y., Dollar P., Girshick R., He K., Hariharan B., Belongie S. Feature pyramid networks for object detection, ArXiv eprints, 2017, https://arxiv.org/abs/1612.03144v2
Olson L. N., Schroder J. B., PyAMG: Algebraic multigrid solvers in Python v4.0.0, 2018, https://github.com/pyamg
Pedregosa F. et al. Scikit-learn: machine learning in Python, Journal of Machine Learning Research, 2011, Vol. 12, pp. 2825–2830.
Duchi J., Hazan E., Singer Y. Adaptive subgradient methods for online learning and stochastic optimization, Journal of Machine Learning Research, 2011, Vol. 12, pp. 2121–2159.
How to Cite
Copyright (c) 2022 V. A. Kuzmych, M. A. Novotarskyi, O. B. Nesterenko
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Creative Commons Licensing Notifications in the Copyright Notices
The journal allows the authors to hold the copyright without restrictions and to retain publishing rights without restrictions.
The journal allows readers to read, download, copy, distribute, print, search, or link to the full texts of its articles.
The journal allows to reuse and remixing of its content, in accordance with a Creative Commons license СС BY -SA.
Authors who publish with this journal agree to the following terms:
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License CC BY-SA that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.