Structure Preservation for the Deep Neural Network Multigrid Solver

  title={Structure Preservation for the Deep Neural Network Multigrid Solver},
  author={Nils Margenberg and Christian Lessig and Thomas Richter},
The simulation of partial differential equations is a central subject of numerical analysis and an indispensable tool in science, engineering and related fields. Existing approaches, such as finite elements, provide (highly) efficient tools but deep neural network-based techniques emerged in the last few years as an alternative with very promising results. We investigate the combination of both approaches for the approximation of the Navier-Stokes equations and to what extent structural… 

Figures and Tables from this paper

Deep neural networks for geometric multigrid methods
The results demonstrate that larger networks are able to capture the flow behavior better while requiring only little additional training time, and can even reduce the computation time compared to a classical multigrid simulation through a faster convergence of the nonlinear solve that is required at every time step.
Distributed multigrid neural solvers on megavoxel domains
A scalable framework is presented that integrates two distinct advances that significantly reduces the time to solve and opens up the possibility of fast and scalable training of neural PDE solvers on heterogeneous clusters.
A priori and a posteriori error estimates for the Deep Ritz method applied to the Laplace and Stokes problem
An a posteriori error estimator for neural network approximations of partial differential equations obtained with Physics Informed Neural Networks is developed, destined to serve as a stopping criterion that guarantees the accuracy of the solution independently of the design of the neural network training.
Deep learning for surrogate modeling of two-dimensional mantle convection
Deep learning techniques can produce reliable parameterized surrogates (i.e. surrogates that predict state variables such as temperature based only on parameters) of the underlying partial differential equations of mantle convection, using a dataset of 10, 525 two-dimensional simulations of the thermal evolution of the mantle of a Mars-like planet.
Deep learning for surrogate modelling of 2D mantle convection
Using a dataset of 10, 525 two-dimensional simulations of the thermal evolution of the mantle of a Mars-like planet, it is shown that deep learning techniques can produce reliable parameterized surrogates of the underlying partial differential equations.


Learning data-driven discretizations for partial differential equations
Data-driven discretization is proposed, a method for learning optimized approximations to PDEs based on actual solutions to the known underlying equations that uses neural networks to estimate spatial derivatives.
A finite element pressure gradient stabilization¶for the Stokes equations based on local projections
A variant of the classical weighted least-squares stabilization for the Stokes equations has improved accuracy properties, especially near boundaries, and is based on local projections of the residual terms which are used in order to achieve consistency of the method.
DeepXDE: A Deep Learning Library for Solving Differential Equations
An overview of physics-informed neural networks (PINNs), which embed a PDE into the loss of the neural network using automatic differentiation, and a new residual-based adaptive refinement (RAR) method to improve the training efficiency of PINNs.
Neural Time-Dependent Partial Differential Equation
This work proposes a sequence deep learning framework called Neural-PDE, which allows to automatically learn governing rules of any time-dependent PDE system from existing data by using a bidirectional LSTM encoder, and predict the next n time steps data.
Fourier Neural Operator for Parametric Partial Differential Equations
This work forms a new neural operator by parameterizing the integral kernel directly in Fourier space, allowing for an expressive and efficient architecture and shows state-of-the-art performance compared to existing neural network methodologies.
Solving high-dimensional partial differential equations using deep learning
A deep learning-based approach that can handle general high-dimensional parabolic PDEs using backward stochastic differential equations and the gradient of the unknown solution is approximated by neural networks, very much in the spirit of deep reinforcement learning with the gradient acting as the policy function.
Stationary Flow Predictions Using Convolutional Neural Networks
An alternative approach for computing flow predictions using Convolutional Neural Networks (CNNs) is described; in particular, a classical CNN as well as the U-Net architecture are used.