# Structure Preservation for the Deep Neural Network Multigrid Solver

@article{Margenberg2021StructurePF, title={Structure Preservation for the Deep Neural Network Multigrid Solver}, author={Nils Margenberg and Christian Lessig and Thomas Richter}, journal={ArXiv}, year={2021}, volume={abs/2012.05290} }

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…

## 7 Citations

Deep neural networks for geometric multigrid methods

- Computer ScienceArXiv
- 2021

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.

A neural network multigrid solver for the Navier-Stokes equations

- Computer ScienceJ. Comput. Phys.
- 2022

Distributed multigrid neural solvers on megavoxel domains

- Computer ScienceSC
- 2021

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

- Computer Science, Mathematics
- 2021

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

- Computer Science, PhysicsPhysical Review Fluids
- 2021

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

- Computer Science, PhysicsArXiv
- 2021

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.

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