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.
翻译:我们提出了一个算法,用于利用随机编码分枝树对完全非线性PDE系统进行数字解析。这个算法涵盖涉及任意命令梯度条件的非线性功能性非线性,它只要求在特定终端时间空间的边界条件(T$)而不是标准解答器中的Drichlet或Neumann边界条件(Neumann边界条件),其实施取决于蒙特卡洛估计,并使用神经网络,对时空域进行无线功能估计。这个算法适用于纳维-斯托克斯方程式的数字解算,并在泰勒-绿色涡旋体和Arnold-Beltrami-Ceress流动中以其他执行方式为基准。