# Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers

@article{Um2020SolverintheLoopLF, title={Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers}, author={Kiwon Um and Yun Fei and Philipp Holl and Robert Brand and Nils Th{\"u}rey}, journal={ArXiv}, year={2020}, volume={abs/2007.00016} }

Finding accurate solutions to partial differential equations (PDEs) is a crucial task in all scientific and engineering disciplines. It has recently been shown that machine learning methods can improve the solution accuracy by correcting for effects not captured by the discretized PDE. We target the problem of reducing numerical errors of iterative PDE solvers and compare different learning approaches for finding complex correction functions. We find that previously used learning approaches are…

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## References

SHOWING 1-10 OF 81 REFERENCES

### Learning Neural PDE Solvers with Convergence Guarantees

- Computer ScienceICLR
- 2019

This work proposes an approach to learn a fast iterative solver tailored to a specific domain by learning to modify the updates of an existing solver using a deep neural network, and achieves 2-3 times speedup compared to state-of-the-art solvers.

### Learning data-driven discretizations for partial differential equations

- Computer ScienceProceedings of the National Academy of Sciences
- 2019

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.

### Learning to Control PDEs with Differentiable Physics

- Computer ScienceICLR
- 2020

It is shown that by using a differentiable PDE solver in conjunction with a novel predictor-corrector scheme, this work can train neural networks to understand and control complex nonlinear physical systems over long time frames.

### Accelerating Eulerian Fluid Simulation With Convolutional Networks

- Computer ScienceICML
- 2017

This work proposes a data-driven approach that leverages the approximation power of deep-learning with the precision of standard solvers to obtain fast and highly realistic simulations of the Navier-Stokes equations.

### PDE-Net: Learning PDEs from Data

- Computer ScienceICML
- 2018

Numerical experiments show that the PDE-Net has the potential to uncover the hidden PDE of the observed dynamics, and predict the dynamical behavior for a relatively long time, even in a noisy environment.

### End-to-End Differentiable Physics for Learning and Control

- Computer ScienceNeurIPS
- 2018

This paper demonstrates how to perform backpropagation analytically through a physical simulator defined via a linear complementarity problem, and highlights the system's ability to learn physical parameters from data, efficiently match and simulate observed visual behavior, and readily enable control via gradient-based planning methods.

### DGM: A deep learning algorithm for solving partial differential equations

- Computer ScienceJ. Comput. Phys.
- 2018

### Learning to Simulate Complex Physics with Graph Networks

- Computer ScienceICML
- 2020

A machine learning framework and model implementation that can learn to simulate a wide variety of challenging physical domains, involving fluids, rigid solids, and deformable materials interacting with one another, and holds promise for solving a wide range of complex forward and inverse problems.

### A Machine Learning Strategy to Assist Turbulence Model Development

- Computer Science
- 2015

This work investigates the feasibility of a new data-driven approach to turbulence model development by attempting to reproduce, through a machine learning methodology, the results obtained with the well-established Spalart-Allmaras model.

### Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Data

- Computer ScienceArXiv
- 2018

HFM is a physics informed deep learning framework capable of encoding an important class of physical laws governing fluid motions, namely the Navier-Stokes equations, and can be used in physical and biomedical problems to extract valuable quantitative information for which direct measurements may not be possible.