Linearized Learning Methods with Multiscale Deep Neural Networks for Stationary Navier-Stokes Equations with Oscillatory Solutions

TL;DR

This paper introduces linearized learning methods combined with multiscale deep neural networks to efficiently and accurately solve stationary Navier-Stokes equations with oscillatory solutions, accelerating training convergence.

Contribution

The paper develops and integrates four linearization schemes into multiscale neural network training for stationary Navier-Stokes equations, improving convergence speed and solution accuracy.

Findings

Fast and accurate training of multiscale neural networks achieved.

Effective handling of highly oscillating stationary flows demonstrated.

Linearized schemes enhance convergence in complex domain problems.

Abstract

In this paper, we present linearized learning methods to accelerate the convergence of training for stationary nonlinear Navier-Stokes equations. To solve the stationary nonlinear Navier-Stokes (NS) equation, we integrate the procedure of linearization of the nonlinear convection term in the NS equation into the training process of multi-scale deep neural network approximation of the NS solution. Four forms of linearizations are considered. After a benchmark problem, we solve the highly oscillating stationary flows utilizing the proposed linearized learning with multi-scale neural network for complex domains. The results show that multiscale deep neural network combining with the linearized schemes can be trained fast and accurately.