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5
diffphys.md
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@ -68,7 +68,7 @@ naturally transfers to including $\nu$ as a degree of freedom.
As $\mathbf{u}$ is typically a vector-valued function, $\partial \mathcal P_i / \partial \mathbf{u}$ denotes As $\mathbf{u}$ is typically a vector-valued function, $\partial \mathcal P_i / \partial \mathbf{u}$ denotes
a Jacobian matrix $J$ rather than a single value: a Jacobian matrix $J$ rather than a single value:
% test %
$$ \begin{aligned} $$ \begin{aligned}
\frac{ \partial \mathcal P_i }{ \partial \mathbf{u} } = \frac{ \partial \mathcal P_i }{ \partial \mathbf{u} } =
\begin{bmatrix} \begin{bmatrix}
@ -83,6 +83,7 @@ $$ \begin{aligned}
\partial \mathcal P_{i,d} / \partial u_{d} \partial \mathcal P_{i,d} / \partial u_{d}
\end{bmatrix} \end{bmatrix}
\end{aligned} $$ \end{aligned} $$
%
where, as above, $d$ denotes the number of components in $\mathbf{u}$. As $\mathcal P$ maps one value of where, as above, $d$ denotes the number of components in $\mathbf{u}$. As $\mathcal P$ maps one value of
$\mathbf{u}$ to another, the jacobian is square and symmetric here. Of course this isn't necessarily the case $\mathbf{u}$ to another, the jacobian is square and symmetric here. Of course this isn't necessarily the case
for general model equations, but non-square Jacobian matrices would not cause any problems for differentiable for general model equations, but non-square Jacobian matrices would not cause any problems for differentiable
@ -139,12 +140,10 @@ state of the forward evaluation for backpropagation (the "$g(x)$" above). For a
simulation, however, we're not overly interested in every single intermediate result our solver produces. simulation, however, we're not overly interested in every single intermediate result our solver produces.
Typically, we're more concerned with significant updates such as the step from $\mathbf{u}(t)$ to $\mathbf{u}(t+\Delta t)$. Typically, we're more concerned with significant updates such as the step from $\mathbf{u}(t)$ to $\mathbf{u}(t+\Delta t)$.
%provide discretized simulator of physical phenomenon as differentiable operator.
Thus, in practice it is a very good idea to break down the solving process into a sequence Thus, in practice it is a very good idea to break down the solving process into a sequence
of meaningful but _monolithic_ operators. This not only saves a lot of work by preventing the calculation of meaningful but _monolithic_ operators. This not only saves a lot of work by preventing the calculation
of unnecessary intermediate results, it also allows us to choose the best possible numerical methods of unnecessary intermediate results, it also allows us to choose the best possible numerical methods
to compute the updates (and derivatives) for these operators. to compute the updates (and derivatives) for these operators.
%in practice break down into larger, monolithic components
E.g., as this process is very similar to adjoint method optimizations, we can re-use many of the techniques E.g., as this process is very similar to adjoint method optimizations, we can re-use many of the techniques
that were developed in this field, or leverage established numerical methods. E.g., that were developed in this field, or leverage established numerical methods. E.g.,
we could leverage the $O(n)$ runtime of multigrid solvers for matrix inversion. we could leverage the $O(n)$ runtime of multigrid solvers for matrix inversion.

3
others-intro.md
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@ -10,11 +10,10 @@ implementations for each of them.
More specifically, we will look at: More specifically, we will look at:
* Time series predictions, i.e., using to DL predict the evolution of a physical system. * Model reduction and time series predictions, i.e., using to DL predict the evolution of a physical system in a latent space.
This typically replaces a numerical solver, and we can make use of special techniques from the DL area that target time series. This typically replaces a numerical solver, and we can make use of special techniques from the DL area that target time series.
* Generative models are likewise an own topic in DL, and here especially generative adversarial networks were shown to be powerful tools. They also represent a highly interesting training approach involving to separate NNs. * Generative models are likewise an own topic in DL, and here especially generative adversarial networks were shown to be powerful tools. They also represent a highly interesting training approach involving to separate NNs.
{cite}`xie2018tempoGan`
* Meshless methods and unstructured meshes are an important topic for classical simulations. Here, we'll look at a specific Lagrangian method that employs learning in the context of dynamic, particle-based representations. * Meshless methods and unstructured meshes are an important topic for classical simulations. Here, we'll look at a specific Lagrangian method that employs learning in the context of dynamic, particle-based representations.
{cite}`prantl2019tranquil` {cite}`prantl2019tranquil`

8
overview-equations.md
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@ -11,7 +11,9 @@ models, and there are lots of great introductions to deep learning.
Hence, we'll keep it short: Hence, we'll keep it short:
our goal is to approximate an unknown function our goal is to approximate an unknown function
$f^*(x) = y^*$ , $$
f^*(x) = y^* ,
$$ (learn-base)
where $y^*$ denotes reference or "ground truth" solutions. where $y^*$ denotes reference or "ground truth" solutions.
$f^*(x)$ should be approximated with an NN representation $f(x;\theta)$. We typically determine $f$ $f^*(x)$ should be approximated with an NN representation $f(x;\theta)$. We typically determine $f$
@ -20,7 +22,9 @@ of the NN.
This gives a minimization problem to find $f(x;\theta)$ such that $e$ is minimized. This gives a minimization problem to find $f(x;\theta)$ such that $e$ is minimized.
In the simplest case, we can use an $L^2$ error, giving In the simplest case, we can use an $L^2$ error, giving
$\text{arg min}_{\theta} | f(x;\theta) - y^* |_2^2$ $$
\text{arg min}_{\theta} | f(x;\theta) - y^* |_2^2
$$ (learn-l2)
We typically optimize, i.e. _train_, We typically optimize, i.e. _train_,
with some variant of a stochastic gradient descent (SGD) optimizer. with some variant of a stochastic gradient descent (SGD) optimizer.

149
references.bib
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@ -14,6 +14,24 @@
@article{chu2021physgan,
author = {Chu, Mengyu and Thuerey, Nils},
title ={{Learning Meaningful Controls for Fluids}},
journal = ACM_TOG,
volume = {40(4)},
year = {2021},
publisher = {ACM},
}
% url={https://ge.in.tum.de/publications/tempogan/},
@article{franz2021globtrans,
author = {Franz, Erik and Solenthaler, Barbara and Thuerey, Nils},
title ={{Global Transport for Fluid Reconstruction with Learned Self-Supervision}},
journal = CVPR,
year = {2021},
url={https://ge.in.tum.de/publications/},
}
@article{um2020sol, @article{um2020sol,
title={Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers}, title={Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers},
author={Um, Kiwon and Brand, Robert and Holl, Philipp and Fei, Raymond Thuerey, Nils}, author={Um, Kiwon and Brand, Robert and Holl, Philipp and Fei, Raymond Thuerey, Nils},
@ -50,12 +68,11 @@
url={https://ge.in.tum.de/publications/2019-tecogan-chu/}, url={https://ge.in.tum.de/publications/2019-tecogan-chu/},
} }
@inproceedings{weiss2020ssc, @inproceedings{weiss2020ssc,
title={Correspondence-Free Material Reconstruction using Sparse Surface Constraints}, title={Correspondence-Free Material Reconstruction using Sparse Surface Constraints},
author={Weiss, Sebastian and Maier, Robert and Cremers, Daniel and Westermann, Rudiger and Thuerey, Nils}, author={Weiss, Sebastian and Maier, Robert and Cremers, Daniel and Westermann, Rudiger and Thuerey, Nils},
booktitle=CVPR, booktitle=CVPR,
pages={4686--4695}, pages={4686--4695},
year={2020}, year={2020},
url={https://ge.in.tum.de/publications/2020-cvpr-weiss/}, url={https://ge.in.tum.de/publications/2020-cvpr-weiss/},
} }
@ -250,68 +267,68 @@
} }
@article{thuerey2017interpolations, @article{thuerey2017interpolations,
title ={{Interpolations of Smoke and Liquid Simulations}}, title ={{Interpolations of Smoke and Liquid Simulations}},
author={Thuerey, Nils}, author={Thuerey, Nils},
journal=ACM_TOG, journal=ACM_TOG,
volume={36}, volume={36},
number={1}, number={1},
pages={3}, pages={3},
year={2017}, year={2017},
publisher={ACM} publisher={ACM}
} }
@article{canabal2016dispersion, @article{canabal2016dispersion,
title ={{Dispersion kernels for water wave simulation}}, title ={{Dispersion kernels for water wave simulation}},
author={Canabal, Jos{\'e} A and Miraut, David and Thuerey, Nils and Kim, Theodore and Portilla, Javier and Otaduy, Miguel A}, author={Canabal, Jos{\'e} A and Miraut, David and Thuerey, Nils and Kim, Theodore and Portilla, Javier and Otaduy, Miguel A},
journal=ACM_TOG, journal=ACM_TOG,
volume={35}, volume={35},
number={6}, number={6},
pages={202}, pages={202},
year={2016}, year={2016},
publisher={ACM} publisher={ACM}
} }
@article{jones2016example, @article{jones2016example,
title ={{Example-based plastic deformation of rigid bodies}}, title ={{Example-based plastic deformation of rigid bodies}},
author={Jones, Ben and Thuerey, Nils and Shinar, Tamar and Bargteil, Adam W}, author={Jones, Ben and Thuerey, Nils and Shinar, Tamar and Bargteil, Adam W},
journal=ACM_TOG, journal=ACM_TOG,
volume={35}, volume={35},
number={4}, number={4},
pages={34}, pages={34},
year={2016}, year={2016},
publisher={ACM} publisher={ACM}
} }
@inproceedings{ferstl2016narrow, @inproceedings{ferstl2016narrow,
title ={{Narrow band FLIP for liquid simulations}}, title ={{Narrow band FLIP for liquid simulations}},
author={Ferstl, Florian and Ando, Ryoichi and Wojtan, Chris and Westermann, R{\"u}diger and Thuerey, Nils}, author={Ferstl, Florian and Ando, Ryoichi and Wojtan, Chris and Westermann, R{\"u}diger and Thuerey, Nils},
booktitle =EG_CGF, booktitle =EG_CGF,
volume={35(2)}, volume={35(2)},
pages={225--232}, pages={225--232},
year={2016}, year={2016},
organization={Wiley Online Library} organization={Wiley Online Library}
} }
@article{monszpart2016smash, @article{monszpart2016smash,
title ={{SMASH: physics-guided reconstruction of collisions from videos}}, title ={{SMASH: physics-guided reconstruction of collisions from videos}},
author={Monszpart, Aron and Thuerey, Nils and Mitra, Niloy J}, author={Monszpart, Aron and Thuerey, Nils and Mitra, Niloy J},
journal=ACM_TOG, journal=ACM_TOG,
volume={35}, volume={35},
number={6}, number={6},
pages={199}, pages={199},
year={2016}, year={2016},
publisher={ACM} publisher={ACM}
} }
@article{mercier2015surface, @article{mercier2015surface,
title ={{Surface turbulence for particle-based liquid simulations}}, title ={{Surface turbulence for particle-based liquid simulations}},
author={Mercier, Olivier and Beauchemin, Cynthia and Thuerey, Nils and Kim, Theodore and Nowrouzezahrai, Derek}, author={Mercier, Olivier and Beauchemin, Cynthia and Thuerey, Nils and Kim, Theodore and Nowrouzezahrai, Derek},
journal=ACM_TOG, journal=ACM_TOG,
volume={34}, volume={34},
number={6}, number={6},
pages={202}, pages={202},
year={2015}, year={2015},
publisher={ACM} publisher={ACM}
} }
@ARTICLE{ Ando:2015:streamfunc, @ARTICLE{ Ando:2015:streamfunc,
@ -696,25 +713,6 @@
} }
@ARTICLE{ Thuerey:2007b:phd,
AUTHOR = {N. Thuerey},
TITLE = {{Physically based Animation of Free Surface Flows with the Lattice Boltzmann Method}},
YEAR = {2007},
JOURNAL = {PhD thesis},
PUBLISHER = {Dept. of Computer Science 10, University of Erlangen-Nuremberg},
VOLUME = {ISBN 978-3-89963-519-5}
}
@ARTICLE{ Thuerey:2006:drdobbs,
AUTHOR = {N. Thuerey},
TITLE = {{Fluid Simulation with Blender}},
YEAR = {2006},
JOURNAL = {Dr. Dobbs Journal},
PUBLISHER = {CMP Media}
}
@ARTICLE{ Iglberger:2005:movNanoPart, @ARTICLE{ Iglberger:2005:movNanoPart,
AUTHOR = {Iglberger and N. Thuerey and U. Ruede and H. Schmid and W. Peukert}, AUTHOR = {Iglberger and N. Thuerey and U. Ruede and H. Schmid and W. Peukert},
TITLE = {{Simulation of moving Nano-Particles with the Lattice Boltzmann Method in 3D}}, TITLE = {{Simulation of moving Nano-Particles with the Lattice Boltzmann Method in 3D}},
@ -751,16 +749,6 @@
} }
@ARTICLE{ Thuerey:2003:lbmMetallschaum,
AUTHOR = {N. Thuerey and U. Ruede and C. Koerner},
TITLE = {{Simulation von Metallschaum mittels der Lattice-Boltzmann Methode}},
YEAR = {2003},
JOURNAL = {Konwihr Quartl},
PUBLISHER = {KONWIHR},
VOLUME = {35}
}
@ARTICLE{ Thuerey:2003:singlePhaseFsLbm, @ARTICLE{ Thuerey:2003:singlePhaseFsLbm,
AUTHOR = {N. Thuerey}, AUTHOR = {N. Thuerey},
TITLE = {{A Lattice Boltzmann method for single-phase free surface flows in 3D}}, TITLE = {{A Lattice Boltzmann method for single-phase free surface flows in 3D}},
@ -781,7 +769,6 @@
% ----------------- external -------------------- % ----------------- external --------------------

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