FAR EASTERN BRANCH OF THE RUSSIAN ACADEMY OF SCIENCES
INSTITUTE OF APPLIED MATHEMATICS KHABAROVSK DIVISION
Far Eastern Mathematical Journal
Inequalities for modulus of rational functions |
Parkhomenko D.A. |
2025, issue 1, P. 67-80 DOI: https://doi.org/10.47910/FEMJ202506 |
Abstract |
| The paper presents results on the solution of the one-dimensional Burgers equation and the two-dimensional Poisson equation followed by model inference and visualization. The solution has been obtained using performance lightweight neural networks in the developed agile project. The final architecture modularly combines a feed-forward neural network (FFNN) as an encoder and a Kolmogorov–Arnold neural network (KAN) as a decoder. |
Keywords: Physics-Informed Neural Networks, Deep Learning, Partial Differential Equations. |
Download the article (PDF-file) |
References |
| [1] Ngan et V.N. al., “A reduced-order model for segregated fluid-structure interaction solvers based on an ALE approach”, 2023, arXiv: 2305.13613. [2] Lu J., Peters E., Kuipers J.A.M., “Direct numerical simulation of coupled heat and mass transfer in fluid-solid systems”, Proceedings of the 12th International Conference on Computational Fluid Dynamics in the Oil & Gas, Metallurgical and Process Industries, SINTEF Academic Press, 2017. [3] Faroughi et S.A. al., “Physics-guided, physics-informed, and physics-encoded neural networks in scientific computing”, 2022, arXiv: 2211.07377. [4] Jin H., Zhang E., Espinosa H.D., “Recent advances and applications of machine learning in experimental solid mechanics: A review”, Applied Mechanics Reviews, 2023. [5] Hu et X. al., “Physics-guided deep neural networks for power flow analysis”, IEEE Transactions on Power Systems, 2020. [6] Hornik K., Stinchcombe M., White H., “Multilayer feedforward networks are universal approximators”, Neural networks, 1989. [7] Cuomo et S. al., “Scientific machine learning through physics–informed neural networks: Where we are and what’s next”, Journal of Scientific Computing, 2022. [8] Guasti Junior W., Santos I.P., “Solving differential equations using feedforward neural networks”, Computational Science and Its Applications – ICCSA 2021: 21st International Conference, Cagliari, Italy, September 13–16, 2021, Proceedings, Part IV 21, Springer, 2021. [9] Bonkile et M.P. al., “A systematic literature review of Burgers’ equation with recent advances”, Pramana, 2018. [10] Li Y., Mei F., “Deep learning-based method coupled with small sample learning for solving partial differential equations”, Multimedia Tools and Applications, 2021. [11] Wang Y., Zhong L., “NAS-PINN: neural architecture search-guided physics-informed neural network for solving PDEs”, Journal of Computational Physics, 2024. [12] Goodfellow I., Deep learning, MIT press, 2016. [13] Liu et Z. al., “Kan: Kolmogorov-arnold networks”, 2024, arXiv: 2404.19756. |