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9
physicalloss.md
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@@ -39,9 +39,10 @@ $$
This nicely integrates with the objective for training a neural network: we can train for This nicely integrates with the objective for training a neural network: we can train for
minimizing this residual in combination with direct loss terms. minimizing this residual in combination with direct loss terms.
Similar to before, we can make use of sample solutions Similar to before, we can use pre-computed solutions
$[x_0,y_0], ...[x_n,y_n]$ for $\mathbf{u}$ with $\mathbf{u}(\mathbf{x})=y$. $[x_0,y_0], ...[x_n,y_n]$ for $\mathbf{u}$ with $\mathbf{u}(\mathbf{x})=y$ as constraints
This is typically important, as most practical PDEs we encounter do not have unique solutions in addition to the residual terms.
This is typically important, as most practical PDEs do not have unique solutions
unless initial and boundary conditions are specified. Hence, if we only consider $R$ we might unless initial and boundary conditions are specified. Hence, if we only consider $R$ we might
get solutions with random offset or other undesirable components. The supervised sample points get solutions with random offset or other undesirable components. The supervised sample points
therefore help to _pin down_ the solution in certain places. therefore help to _pin down_ the solution in certain places.
@@ -71,7 +72,7 @@ In order to compute the residuals at training time, it would be possible to stor
the unknowns of $\mathbf{u}$ on a computational mesh, e.g., a grid, and discretize the equations of the unknowns of $\mathbf{u}$ on a computational mesh, e.g., a grid, and discretize the equations of
$R$ there. This has a fairly long "tradition" in DL, and was proposed by Tompson et al. {cite}`tompson2017` early on. $R$ there. This has a fairly long "tradition" in DL, and was proposed by Tompson et al. {cite}`tompson2017` early on.
A popular variant of employing physical soft-constraints {cite}`raissi2018hiddenphys` A popular variant of employing physical soft-constraints {cite}`raissi2019pinn`
instead uses fully connected NNs to represent $\mathbf{u}$. This has some interesting pros and cons that we'll outline in the following, and we will also focus on it in the following code examples and comparisons. instead uses fully connected NNs to represent $\mathbf{u}$. This has some interesting pros and cons that we'll outline in the following, and we will also focus on it in the following code examples and comparisons.
The central idea here is that the aforementioned general function $f$ that we're after in our learning problems The central idea here is that the aforementioned general function $f$ that we're after in our learning problems

13
references.bib
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@@ -812,16 +812,17 @@
pages = {3424--3433} pages = {3424--3433}
} }
@article{raissi2018hiddenphys, @article{raissi2019pinn,
title={Hidden physics models: Machine learning of nonlinear partial differential equations}, title={Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations},
author={Raissi, Maziar and Karniadakis, George Em}, author={Raissi, Maziar and Perdikaris, Paris and Karniadakis, George},
journal={Journal of Computational Physics}, journal={Journal of Computational Physics},
volume={357}, volume={378},
pages={125--141}, pages={686--707},
year={2018}, year={2019},
publisher={Elsevier} publisher={Elsevier}
} }
@book{stocker2014climate, @book{stocker2014climate,
title={Climate change 2013: the physical science basis: Working Group I contribution to the Fifth assessment report of the Intergovernmental Panel on Climate Change}, title={Climate change 2013: the physical science basis: Working Group I contribution to the Fifth assessment report of the Intergovernmental Panel on Climate Change},
author={Stocker, Thomas}, author={Stocker, Thomas},