daviddoji/pbdl-book
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

formatting updates

Browse Source
This commit is contained in:
NT
2021-05-16 11:12:16 +08:00
parent f533fd7a4f
commit 05d2783759
3 changed files with 14 additions and 7 deletions
Download Patch File Download Diff File Expand all files Collapse all files

1
overview-ns-forw.ipynb
View File

@@ -19,6 +19,7 @@
"\n", "\n",
"\n", "\n",
"Here $\\mathbf{g}$ collects the forcing terms. Below we'll use a simple buoyancy model. We'll solve this PDE on a closed domain with Dirchlet boundary conditions $\\mathbf{u}=0$ for the velocity, and Neumann boundaries $\\frac{\\partial p}{\\partial x}=0$ for pressure, on a domain $\\Omega$ with a physical size of $100 \\times 80$ units. \n", "Here $\\mathbf{g}$ collects the forcing terms. Below we'll use a simple buoyancy model. We'll solve this PDE on a closed domain with Dirchlet boundary conditions $\\mathbf{u}=0$ for the velocity, and Neumann boundaries $\\frac{\\partial p}{\\partial x}=0$ for pressure, on a domain $\\Omega$ with a physical size of $100 \\times 80$ units. \n",
"[[run in colab]](https://colab.research.google.com/github/tum-pbs/pbdl-book/blob/main/overview-ns-forw.ipynb)\n",
"\n" "\n"
] ]
}, },

17
physicalloss.md
View File

@@ -20,18 +20,21 @@ We can improve this setting by trying to bring the model equations (or parts the
into the training process. E.g., given a PDE for $\mathbf{u}(\mathbf{x},t)$ with a time evolution, into the training process. E.g., given a PDE for $\mathbf{u}(\mathbf{x},t)$ with a time evolution,
we can typically express it in terms of a function $\mathcal F$ of the derivatives we can typically express it in terms of a function $\mathcal F$ of the derivatives
of $\mathbf{u}$ via of $\mathbf{u}$ via
$
$$
\mathbf{u}_t = \mathcal F ( \mathbf{u}_{x}, \mathbf{u}_{xx}, ... \mathbf{u}_{xx...x} ) , \mathbf{u}_t = \mathcal F ( \mathbf{u}_{x}, \mathbf{u}_{xx}, ... \mathbf{u}_{xx...x} ) ,
$ $$
where the $_{\mathbf{x}}$ subscripts denote spatial derivatives with respect to one of the spatial dimensions where the $_{\mathbf{x}}$ subscripts denote spatial derivatives with respect to one of the spatial dimensions
of higher and higher order (this can of course also include derivatives with respect to different axes). of higher and higher order (this can of course also include derivatives with respect to different axes).
In this context we can employ DL by approximating the unknown $\mathbf{u}$ itself In this context we can employ DL by approximating the unknown $\mathbf{u}$ itself
with a NN, denoted by $\tilde{\mathbf{u}}$. If the approximation is accurate, the PDE with a NN, denoted by $\tilde{\mathbf{u}}$. If the approximation is accurate, the PDE
naturally should be satisfied, i.e., the residual $R$ should be equal to zero: naturally should be satisfied, i.e., the residual $R$ should be equal to zero:
$
R = \mathbf{u}_t - \mathcal F ( \mathbf{u}_{x}, \mathbf{u}_{xx}, ... \mathbf{u}_{xx...x} ) = 0 $$
$. R = \mathbf{u}_t - \mathcal F ( \mathbf{u}_{x}, \mathbf{u}_{xx}, ... \mathbf{u}_{xx...x} ) = 0 .
$$
This nicely integrates with the objective for training a neural network: similar to before This nicely integrates with the objective for training a neural network: similar to before
we can collect sample solutions we can collect sample solutions
@@ -42,7 +45,9 @@ get solutions with random offset or other undesirable components. Hence the supe
help to _pin down_ the solution in certain places. help to _pin down_ the solution in certain places.
Now our training objective becomes Now our training objective becomes
$\text{arg min}_{\theta} \ \alpha_0 \sum_i (f(x_i ; \theta)-y_i)^2 + \alpha_1 R(x_i) $, $$
\text{arg min}_{\theta} \ \alpha_0 \sum_i (f(x_i ; \theta)-y_i)^2 + \alpha_1 R(x_i) ,
$$ (physloss-training)
where $\alpha_{0,1}$ denote hyperparameters that scale the contribution of the supervised term and where $\alpha_{0,1}$ denote hyperparameters that scale the contribution of the supervised term and
the residual term, respectively. We could of course add additional residual terms with suitable scaling factors here. the residual term, respectively. We could of course add additional residual terms with suitable scaling factors here.

3
references.bib
View File

@@ -114,7 +114,7 @@
} }
@article{thuerey2020deepFlowPred, @article{thuerey2020dfp,
title={Deep learning methods for Reynolds-averaged Navier--Stokes simulations of airfoil flows}, title={Deep learning methods for Reynolds-averaged Navier--Stokes simulations of airfoil flows},
author={Thuerey, Nils and Weissenow, Konstantin and Prantl, Lukas and Hu, Xiangyu}, author={Thuerey, Nils and Weissenow, Konstantin and Prantl, Lukas and Hu, Xiangyu},
journal={AIAA Journal}, year={2020}, journal={AIAA Journal}, year={2020},
@@ -123,6 +123,7 @@
url={https://ge.in.tum.de/publications/2018-deep-flow-pred/}, url={https://ge.in.tum.de/publications/2018-deep-flow-pred/},
} }
@article{prantl2019rtliq, @article{prantl2019rtliq,
title ={{Generating Liquid Simulations with Deformation-Aware Neural Networks}}, title ={{Generating Liquid Simulations with Deformation-Aware Neural Networks}},
author={Lukas Prantl and Boris Bonev and Nils Thuerey}, author={Lukas Prantl and Boris Bonev and Nils Thuerey},