DNN-MG: A Hybrid Neural Network/Finite Element Method with Applications to 3D Simulations of the Navier-Stokes Equations
arXiv:2307.14837 · doi:10.1016/j.cma.2023.116692
Abstract
We extend and analyze the deep neural network multigrid solver (DNN-MG) for the Navier-Stokes equations in three dimensions. The idea of the method is to augment a finite element simulation on coarse grids with fine scale information obtained using deep neural networks. The neural network operates locally on small patches of grid elements. The local approach proves to be highly efficient, since the network can be kept (relatively) small and since it can be applied in parallel on all grid patches. However, the main advantage of the local approach is the inherent generalizability of the method. Since the network only processes data of small sub-areas, it never ``sees'' the global problem and thus does not learn false biases. We describe the method with a focus on the interplay between the finite element method and deep neural networks. Further, we demonstrate with numerical examples the excellent efficiency of the hybrid approach, which allows us to achieve very high accuracy with a coarse grid and thus reduce the computation time by orders of magnitude.
arXiv comment40 pages, 24 figures, 11 tables. Submitted to Computer Methods in Applied Mechanics and Engineering
References in corpus (21)
- Decoupled Weight Decay Regularization
- DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
- Artificial Neural Networks for Solving Ordinary and Partial Differential Equations
- FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators
- Transformers for Modeling Physical Systems
- GraphCast: Learning skillful medium-range global weather forecasting
- Hybrid FEM-NN models: Combining artificial neural networks with the finite element method
- Pangu-Weather: A 3D High-Resolution Model for Fast and Accurate Global Weather Forecast
- MgNet: A Unified Framework of Multigrid and Convolutional Neural Network
- Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems
- A neural network multigrid solver for the Navier-Stokes equations
- A least-squares Galerkin gradient recovery method for fully nonlinear elliptic equations
- Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problems
- Accelerating high order discontinuous Galerkin solvers using neural networks: 3D compressible Navier-Stokes equations
- A priori and a posteriori error estimates for the Deep Ritz method applied to the Laplace and Stokes problem
- Neural Control of Discrete Weak Formulations: Galerkin, Least-Squares and Minimal-Residual Methods with Quasi-Optimal Weights
- Structure Preservation for the Deep Neural Network Multigrid Solver
- Optimal Dirichlet Boundary Control by Fourier Neural Operators Applied to Nonlinear Optics
- A hybrid finite element/neural network solver and its application to the Poisson problem
- Deep neural networks for geometric multigrid methods
- Error analysis for hybrid finite element/neural network discretizations