DiscretizationNet: A Machine-Learning based solver for Navier-Stokes Equations using Finite Volume Discretization

TL;DR

DiscretizationNet is a novel ML-based solver for Navier-Stokes equations that integrates finite volume discretization and iterative algorithms to improve accuracy, stability, and convergence in complex fluid dynamics problems.

Contribution

This work introduces DiscretizationNet, combining PDE discretization schemes with CNN-based models and iterative training to efficiently solve 3D Navier-Stokes equations.

Findings

Successfully solves 3D Navier-Stokes cases like lid-driven cavity and flow past a cylinder.

Achieves faster convergence and improved stability over traditional methods.

Demonstrates the effectiveness of ML in complex fluid dynamics simulations.

Abstract

Over the last few decades, existing Partial Differential Equation (PDE) solvers have demonstrated a tremendous success in solving complex, non-linear PDEs. Although accurate, these PDE solvers are computationally costly. With the advances in Machine Learning (ML) technologies, there has been a significant increase in the research of using ML to solve PDEs. The goal of this work is to develop an ML-based PDE solver, that couples important characteristics of existing PDE solvers with ML technologies. The two solver characteristics that have been adopted in this work are: 1) the use of discretization-based schemes to approximate spatio-temporal partial derivatives and 2) the use of iterative algorithms to solve linearized PDEs in their discrete form. In the presence of highly non-linear, coupled PDE solutions, these strategies can be very important in achieving good accuracy, better stability and faster convergence. Our ML-solver, DiscretizationNet, employs a generative CNN-based encoder-decoder model with PDE variables as both input and output features. During training, the discretization schemes are implemented inside the computational graph to enable faster GPU computation of PDE residuals, which are used to update network weights that result into converged solutions. A novel iterative capability is implemented during the network training to improve the stability and convergence of the ML-solver. The ML-Solver is demonstrated to solve the steady, incompressible Navier-Stokes equations in 3-D for several cases such as, lid-driven cavity, flow past a cylinder and conjugate heat transfer.