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

updated latex code for pdf output

Browse Source
This commit is contained in:
NT
2021-03-26 21:50:46 +08:00
parent 4576bc09f3
commit 55a602e814
1 changed files with 21 additions and 12 deletions
Download Patch File Download Diff File Expand all files Collapse all files

33
diffphys-code-sol.ipynb
View File

@@ -57,32 +57,41 @@
"$\\newcommand{\\vr}[1]{\\mathbf{r}_{#1}} \\vr{t+n}$. \n", "$\\newcommand{\\vr}[1]{\\mathbf{r}_{#1}} \\vr{t+n}$. \n",
"This is what we will address with an NN in the following.\n", "This is what we will address with an NN in the following.\n",
"\n", "\n",
"We'll use an $L^2$-norm in the following to quantify the deviations, i.e., an error function \n", "We'll use an $L^2$-norm in the following to quantify the deviations, i.e., \n",
"$\n", "an error function $\\newcommand{\\loss}{e} \n",
"\\newcommand{\\loss}{e} \n",
"\\newcommand{\\corr}{\\mathcal{C}} \n", "\\newcommand{\\corr}{\\mathcal{C}} \n",
"\\newcommand{\\vc}[1]{\\mathbf{s}_{#1}} \n", "\\newcommand{\\vc}[1]{\\mathbf{s}_{#1}} \n",
"\\newcommand{\\vr}[1]{\\mathbf{r}_{#1}} \n", "\\newcommand{\\vr}[1]{\\mathbf{r}_{#1}} \n",
"\\loss (\\vc{t},\\project \\vr{t})=\\Vert\\vc{t}-\\project \\vr{t}\\Vert_2$. \n", "\\loss (\\vc{t},\\project \\vr{t})=\\Vert\\vc{t}-\\project \\vr{t}\\Vert_2$. \n",
"Our learning goal is to train at a correction operator $\\corr ( \\vc{} )$ such that \n", "Our learning goal is to train at a correction operator \n",
"a solution to which the correction is applied has a lower error than the original unmodified (source) solution: \n", "$\\newcommand{\\corr}{\\mathcal{C}} \n",
"$\\loss ( \\pdec( \\corr (\\project \\vr{t_0}) ) , \\project \\vr{t_1}) < \\loss ( \\pdec( \\project \\vr{t_0} ), \\project \\vr{t_1})$. \n", "\\corr ( \\vc{} )$ such that \n",
"a solution to which the correction is applied has a lower error than the original unmodified (source) \n",
"solution: $\\newcommand{\\loss}{e} \n",
"\\newcommand{\\pdec}{\\pde_{s}}\n",
"\\newcommand{\\corr}{\\mathcal{C}} \n",
"\\newcommand{\\project}{\\mathcal{T}} \n",
"\\newcommand{\\vr}[1]{\\mathbf{r}_{#1}} \n",
"\\loss ( \\pdec( \\corr (\\project \\vr{t_0}) ) , \\project \\vr{t_1}) < \\loss ( \\pdec( \\project \\vr{t_0} ), \\project \\vr{t_1})$. \n",
"\n", "\n",
"The correction function \n", "The correction function \n",
"$\\newcommand{\\vcN}{\\mathbf{s}} \\newcommand{\\corr}{\\mathcal{C}} \\corr (\\vcN | \\theta)$ \n", "$\\newcommand{\\vcN}{\\mathbf{s}} \\newcommand{\\corr}{\\mathcal{C}} \\corr (\\vcN | \\theta)$ \n",
"is represented as a deep neural network with weights $\\theta$\n", "is represented as a deep neural network with weights $\\theta$\n",
"and receives the state $\\vcN$ to infer an additive correction field with the same dimension.\n", "and receives the state $\\mathbf{s}$ to infer an additive correction field with the same dimension.\n",
"To distinguish the original states $\\vcN$ from the corrected ones, we'll denote the latter with an added tilde\n", "To distinguish the original states $\\mathbf{s}$ from the corrected ones, we'll denote the latter with an added tilde $\\tilde{\\mathbf{s}}$.\n",
"$\\newcommand{\\vctN}{\\tilde{\\mathbf{s}}} \\vctN$.\n",
"The overall learning goal now becomes\n", "The overall learning goal now becomes\n",
"\n", "\n",
"$\n", "$\n",
"\\newcommand{\\pdec}{\\pde_{s}}\n",
"\\newcommand{\\corr}{\\mathcal{C}} \n",
"\\newcommand{\\project}{\\mathcal{T}} \n",
"\\newcommand{\\vr}[1]{\\mathbf{r}_{#1}} \n",
"\\text{argmin}_\\theta | ( \\pdec \\corr )^n ( \\project \\vr{t} ) - \\project \\vr{t}|^2\n", "\\text{argmin}_\\theta | ( \\pdec \\corr )^n ( \\project \\vr{t} ) - \\project \\vr{t}|^2\n",
"$\n", "$\n",
"\n", "\n",
"A crucial bit here that's easy to overlook is that the correction depends on the modified states, i.e.\n", "A crucial bit here that's easy to overlook is that the correction depends on the modified states, i.e.\n",
"it is a function of\n", "it is a function of\n",
"$\\newcommand{\\vctN}{\\tilde{\\mathbf{s}}} \\vctN$, so we have \n", "$\\tilde{\\mathbf{s}}$, so we have \n",
"$\\newcommand{\\vctN}{\\tilde{\\mathbf{s}}} \\newcommand{\\corr}{\\mathcal{C}} \\corr (\\vctN | \\theta)$.\n", "$\\newcommand{\\vctN}{\\tilde{\\mathbf{s}}} \\newcommand{\\corr}{\\mathcal{C}} \\corr (\\vctN | \\theta)$.\n",
"These states actually evolve over time when training. They don't exist beforehand.\n", "These states actually evolve over time when training. They don't exist beforehand.\n",
"\n", "\n",
@@ -203,9 +212,9 @@
"source": [ "source": [
"## Simulation Setup\n", "## Simulation Setup\n",
"\n", "\n",
"Now we can set up the _source_ simulation $\\newcommand{\\pdec}{\\pde_{s}} \\pdec$. \n", "Now we can set up the _source_ simulation $\\mathcal{P}_{s}$. \n",
"Note that we won't deal with \n", "Note that we won't deal with \n",
"$\\newcommand{\\pder}{\\pde_{r}} \\pder$\n", "$\\mathcal{P}_{r}$\n",
"below: the downsampled reference data is contained in the training data set. It was generated with a four times finer discretization. Below we're focusing on the interaction of the source solver and the NN. \n", "below: the downsampled reference data is contained in the training data set. It was generated with a four times finer discretization. Below we're focusing on the interaction of the source solver and the NN. \n",
"\n", "\n",
"This code block and the next ones will define lots of functions, that will be used later on for training.\n", "This code block and the next ones will define lots of functions, that will be used later on for training.\n",