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Numerical solution of the incompressible Navier-Stokes equation by a deep branching algorithm

Jiang Yu NguwiGuillaume PenentNicolas Privault
Dec 2022
摘要
We present an algorithm for the numerical solution of systems of fullynonlinear PDEs using stochastic coded branching trees. This approach coversfunctional nonlinearities involving gradient terms of arbitrary orders, and itrequires only a boundary condition over space at a given terminal time $T$instead of Dirichlet or Neumann boundary conditions at all times as in standardsolvers. Its implementation relies on Monte Carlo estimation, and uses neuralnetworks that perform a meshfree functional estimation on a space-time domain.The algorithm is applied to the numerical solution of the Navier-Stokesequation and is benchmarked to other implementations in the cases of theTaylor-Green vortex and Arnold-Beltrami-Childress flow.
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