论文标题
深层分支算法的不可压缩Navier-Stokes方程的数值解
Numerical solution of the incompressible Navier-Stokes equation by a deep branching algorithm
论文作者
论文摘要
我们提出了一种使用随机编码的分支树的完全非线性PDES系统的数值解的算法。这种方法涵盖了涉及任意订单的梯度术语的功能非线性，并且在给定终端时间$ t $而不是dirichlet或neumann边界条件的情况下，它仅需要在空间上的边界条件，就像标准求解器中一样。它的实现依赖于蒙特卡洛的估计，并使用对时空域进行无网格功能估计的神经网络。该算法应用于Navier-Stokes方程的数值解决方案，并在Taylor-Green Vortex和Arnold-Beltrami-Childress流动的情况下对其他实现进行了标准。
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.
