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Volume 34, Issue 2
Numerical Solution of the Incompressible Navier-Stokes Equation by a Deep Branching Algorithm

Jiang Yu Nguwi, Guillaume Penent & Nicolas Privault

DOI: 10.4208/cicp.OA-2022-0140

Commun. Comput. Phys., 34 (2023), pp. 261-289.

Published online: 2023-09

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  • Abstract

We present an algorithm for the numerical solution of systems of fully nonlinear PDEs using stochastic coded branching trees. This approach covers functional nonlinearities involving gradient terms of arbitrary orders, and it requires only a boundary condition over space at a given terminal time $T$ instead of Dirichlet or Neumann boundary conditions at all times as in standard solvers. Its implementation relies on Monte Carlo estimation, and uses neural networks that perform a meshfree functional estimation on a space-time domain. The algorithm is applied to the numerical solution of the Navier-Stokes equation and is benchmarked to other implementations in the cases of the Taylor-Green vortex and Arnold-Beltrami-Childress flow.

  • Keywords

Fully nonlinear PDEs, systems of PDEs, Navier-Stokes equations, Monte Carlo method, deep neural network, branching process, random tree.

  • AMS Subject Headings

35G20, 76M35, 76D05, 60H30, 60J85, 65C05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-34-261, author = {Nguwi , Jiang YuPenent , Guillaume and Privault , Nicolas}, title = {Numerical Solution of the Incompressible Navier-Stokes Equation by a Deep Branching Algorithm}, journal = {Communications in Computational Physics}, year = {2023}, volume = {34}, number = {2}, pages = {261--289}, abstract = {

We present an algorithm for the numerical solution of systems of fully nonlinear PDEs using stochastic coded branching trees. This approach covers functional nonlinearities involving gradient terms of arbitrary orders, and it requires only a boundary condition over space at a given terminal time $T$ instead of Dirichlet or Neumann boundary conditions at all times as in standard solvers. Its implementation relies on Monte Carlo estimation, and uses neural networks that perform a meshfree functional estimation on a space-time domain. The algorithm is applied to the numerical solution of the Navier-Stokes equation and is benchmarked to other implementations in the cases of the Taylor-Green vortex and Arnold-Beltrami-Childress flow.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2022-0140}, url = {http://global-sci.org/intro/article_detail/cicp/21969.html} }
TY - JOUR T1 - Numerical Solution of the Incompressible Navier-Stokes Equation by a Deep Branching Algorithm AU - Nguwi , Jiang Yu AU - Penent , Guillaume AU - Privault , Nicolas JO - Communications in Computational Physics VL - 2 SP - 261 EP - 289 PY - 2023 DA - 2023/09 SN - 34 DO - http://doi.org/10.4208/cicp.OA-2022-0140 UR - https://global-sci.org/intro/article_detail/cicp/21969.html KW - Fully nonlinear PDEs, systems of PDEs, Navier-Stokes equations, Monte Carlo method, deep neural network, branching process, random tree. AB -

We present an algorithm for the numerical solution of systems of fully nonlinear PDEs using stochastic coded branching trees. This approach covers functional nonlinearities involving gradient terms of arbitrary orders, and it requires only a boundary condition over space at a given terminal time $T$ instead of Dirichlet or Neumann boundary conditions at all times as in standard solvers. Its implementation relies on Monte Carlo estimation, and uses neural networks that perform a meshfree functional estimation on a space-time domain. The algorithm is applied to the numerical solution of the Navier-Stokes equation and is benchmarked to other implementations in the cases of the Taylor-Green vortex and Arnold-Beltrami-Childress flow.

Nguwi , Jiang YuPenent , Guillaume and Privault , Nicolas. (2023). Numerical Solution of the Incompressible Navier-Stokes Equation by a Deep Branching Algorithm. Communications in Computational Physics. 34 (2). 261-289. doi:10.4208/cicp.OA-2022-0140
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