A neural network multigrid solver for the Navier-Stokes equations

Abstract

We present the deep neural network multigrid solver (DNN-MG) that we develop for the instationary Navier-Stokes equations. DNN-MG improves computational efficiency using a judicious combination of a geometric multigrid solver and a recurrent neural network with memory. DNN-MG uses the multi-grid method to classically solve on coarse levels while the neural network corrects interpolated solutions on fine ones, thus avoiding the increasingly expensive computations that would have to be performed there. This results in a reduction in computation time through DNN-MG's highly compact neural network. The compactness results from its design for local patches and the available coarse multigrid solutions that provides a "guide" for the corrections. A compact neural network with a small number of parameters also reduces training time and data. Furthermore, the network's locality facilitates generalizability and allows one to use DNN-MG trained on one mesh domain also on different ones. We demonstrate the efficacy of DNN-MG for variations of the 2D laminar flow around an obstacle. For these, our method significantly improves the solutions as well as lift and drag functionals while requiring only about half the computation time of a full multigrid solution. We also show that DNN-MG trained for the configuration with one obstacle can be generalized to other time dependent problems that can be solved efficiently using a geometric multigrid method.

32 pages, 15 figures

References in corpus (5)

• PyTorch: An Imperative Style, High-Performance Deep Learning Library
• Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
• Physics-informed Autoencoders for Lyapunov-stable Fluid Flow Prediction
• Symplectic Recurrent Neural Networks
• A Newton multigrid framework for optimal control of fluid-structure interactions

Cited by in corpus (11)

• Solving the Discretised Neutron Diffusion Equations using Neural Networks
• A priori and a posteriori error estimates for the Deep Ritz method applied to the Laplace and Stokes problem
• DNN-MG: A Hybrid Neural Network/Finite Element Method with Applications to 3D Simulations of the Navier-Stokes Equations
• Elvet -- a neural network-based differential equation and variational problem solver
• Structure Preservation for the Deep Neural Network Multigrid Solver
• Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation
• Preconditioned FEM-based Neural Networks for Solving Incompressible Fluid Flows and Related Inverse Problems
• A hybrid finite element/neural network solver and its application to the Poisson problem
• A Model-Constrained Tangent Slope Learning Approach for Dynamical Systems
• Deep neural networks for geometric multigrid methods
• A robust and stable hybrid neural network/finite element method for 2D flows that generalizes to different geometries