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

TL;DR

This paper introduces a novel deep branching algorithm that uses stochastic trees and neural networks to solve fully nonlinear PDEs like the Navier-Stokes equations, requiring only terminal boundary conditions and demonstrating competitive results.

Contribution

The paper presents a new meshfree, neural network-based stochastic algorithm for solving nonlinear PDEs with minimal boundary condition requirements, applied to fluid dynamics.

Findings

Successfully applied to Navier-Stokes equations

Benchmarked against standard flow problems

Demonstrates meshfree and boundary condition flexibility

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.