International Scientific Journal


Data-driven approaches have achieved remarkable success in different applications, however, their use in solving PDE has only recently emerged. Herein, we present the potential fluid method, which uses existing data to nest physical meanings into mathematical iterative processes. Potential fluid method is suitable for PDE, such as CFD problems, including Burgers’ equation. Potential fluid method can iteratively determine the steady-state space distribution of PDE. For mathematical reasons, we compare the potential fluid method with the finite difference method and give a detailed explanation.
PAPER REVISED: 2021-05-23
PAPER ACCEPTED: 2021-05-24
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2022, VOLUME 26, ISSUE Issue 2, PAGES [1165 - 1174]
  1. A. Krizhevsky, I. Sutskever, G.E. Hinton, ImageNet Classification with Deep Convolutional Neural Networks, in: ImageNet Classif. with Deep Convolutional Neural Networks, 2012: pp. 1097-1105.
  2. M. Raissi, P. Perdikaris, G.E. Karniadakis, Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations, (2017).
  3. M. Raissi, P. Perdikaris, G.E. Karniadakis, Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations, (2017).
  4. N. Lawrence, Probabilistic Non-linear Principal Component Analysis with Gaussian Process Latent Variable Models, 2005.
  5. C.M. Bishop, M.E. Tipping, Latent variable models and data visualisation, in: Stat. Neural Networks, 1997.
  6. S. Depeweg, J.M. Hernández-Lobato, F. Doshi-Velez, S. Udluft, Uncertainty Decomposition in Bayesian Neural Networks with Latent Variables, (2017).
  7. A.D. Beck, D.G. Flad, C.-D. Munz, Deep Neural Networks for Data-Driven Turbulence Models, (2018).
  8. J. Ling, A. Kurzawski, J. Templeton, Reynolds averaged turbulence modelling using deep neural networks with embedded invariance, J. Fluid Mech. 807 (2016) 155-166.
  9. J. Viquerat, J. Rabault, A. Kuhnle, H. Ghraieb, A. Larcher, E. Hachem, Direct shape optimization through deep reinforcement learning, (2019). (accessed November 2, 2019).
  10. X. Yan, J. Zhu, M. Kuang, X. Wang, Aerodynamic shape optimization using a novel optimizer based on machine learning techniques, Aerosp. Sci. Technol. (2019).
  11. S. Cai, S. Zhou, C. Xu, Q. Gao, Dense motion estimation of particle images via a convolutional neural network, Exp. Fluids. (2019).
  12. L.M. Abadie, J.M. Chamorro, Finite difference methods, in: Lect. Notes Energy, 2013.
  13. G.A. Watson, Numerical Analytics, 1977.

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence