High-efficient machine learning projection method for incompressible Navier-Stokes equations

TL;DR

This paper introduces a machine learning projection framework for incompressible Navier-Stokes equations that uses generated data instead of traditional datasets, achieving high efficiency and accuracy.

Contribution

It proposes novel ML-based projection methods with data generated from physical laws, eliminating the need for high-fidelity training datasets and significantly improving computational speed.

Findings

Poisson-NN trained with generated data achieves high accuracy.

WTCNN-MG method accelerates computations by over 7 times.

Physical law-based data improves high-frequency approximation.

Abstract

This study proposes a high-efficient machine learning (ML) projection method using forward-generated data for incompressible Navier-Stokes equations. A Poisson neural network (Poisson-NN) embedded method and a wavelet transform convolutional neural network multigrid (WTCNN-MG) method are proposed, integrated into the projection method framework in patchwork and overall differentiable manners with MG method, respectively. The solution of the pressure Poisson equation split from the Navier-Stokes equations is first generated either following a random field (e.g. Gaussian random field, GRF, computational complexity O(NlogN), N is the number of spatial points) or physical laws (e.g. a kind of spectra, computational complexity O(NM), M is the number of modes), then the source terms, boundary conditions and initial conditions are constructed via balance of equations, avoiding the difficulties of obtaining high-fidelity training datasets. The feasibility of generated data for training Poisson-NN and WTCNN as well as the acceleration performances of the Poisson-NN embedded method and WTCNN-MG method are validated. The results indicate that even without any DNS data, the generated data can train these two models with excellent generalization and accuracy. The data following physical laws can significantly improve the high-frequency approximation, convergence rate, generalization and accuracy than that generated following GRF. The ML projection method offers significant improvements in computational efficiency. Particularly, the Poisson-NN embedded method achieves an average speed-up of 5.83 times over the traditional MG method, while the WTCNN-MG method offers an even greater average speed-up of 7.03 times, demonstrating impressive acceleration performance.