Title: Deep neural networks for accelerating fluid-dynamics simulations

In this talk we discuss the use of deep neural networks for augmenting
classical finite element simulations in fluid-dynamics. 

First, we present the deep neural network multigrid method (DNN-MG)
[1,2] that can be considered as a multiscale scheme based on classical
finite elements for the coarse scale and a deep neural network for
predicting the fine scales. Embedded in a Newton-multigrid framework,
the neural network is only acting locally. This allows for very small
and efficient networks that require only a small amount of high
fidelity data for training. At the same time it shows very promising
generalization capabilities, since the network does not need any
information about the problem geometry. We present first numerical
results and discuss the approximation and stability properties of the
augmented finite element / neural network framework.

In detail we will discuss two aspects of the DNN-MG approach: can
we guarantee that the network augmented solution is able to preserve
important physical structures like mass conservation of the fluid
velocity field. And second, can we guarantee bounds on the
approximation quality of the neural network enhanced solution? 

If time permits we also give a quick overview on the efficient neural
network accelerated simulation of particulate flows with non-spherical
particles [3,4]. We have in mind configurations with many and small
particles, which cannot be resolved in detail in a classical
simulation. On the other hand, for general non-spherical particles,
like blood cells or polygonal objects, there exist no analytical laws
for predicting hydrodynamical coefficients like drag, lift or
torque. Based on resolved simulations we train neural networks for
establishing exactly such a mapping from the particle- and
flow-configuration to the resulting forces. We discuss different use
cases and demonstrate the accuracy and generalizability of this
approach.  



[1] D. Hartmann, C. Lessig, N. Margenberg and T. Richter.
"A neural network multigrid solver for the Navier-Stokes equations"
arxiv, 2021
https://arxiv.org/abs/2008.11520

[2] N. Margenberg, C. Lessig and T. Richter.
"Structure Preservation for the Deep Neural Network Multigrid Solver"
Electronic Transactions on Numerical Analysis, accepted 2021
http://numerik.uni-hd.de/~richter/pdf/https://arxiv.org/abs/2012.05290

[3] M. Minakowska, T. Richter and S. Sager.
"A finite element / neural network framework for modeling suspensions of non-spherical particles. Concepts and medical applications"
Vietnam Journal of Mathematics, Vol. 49, No. 1, p.207–235, 2021..
https://link.springer.com/article/10.1007/s10013-021-00477-9

[4] H. von Wahl, T. Richter.
"Using a deep neural network to predict the motion of under-resolved triangular rigid bodies in an incompressible flow"
2021, https://arxiv.org/abs/2102.11636