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updated latex code for pdf output

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NT 2021-03-26 21:50:46 +08:00
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diffphys-code-sol.ipynb
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@ -57,32 +57,41 @@
"$\\newcommand{\\vr}[1]{\\mathbf{r}_{#1}} \\vr{t+n}$. \n",
"This is what we will address with an NN in the following.\n",
"\n",
"We'll use an $L^2$-norm in the following to quantify the deviations, i.e., an error function \n",
"$\n",
"\\newcommand{\\loss}{e} \n",
"We'll use an $L^2$-norm in the following to quantify the deviations, i.e., \n",
"an error function $\\newcommand{\\loss}{e} \n",
"\\newcommand{\\corr}{\\mathcal{C}} \n",
"\\newcommand{\\vc}[1]{\\mathbf{s}_{#1}} \n",
"\\newcommand{\\vr}[1]{\\mathbf{r}_{#1}} \n",
"\\loss (\\vc{t},\\project \\vr{t})=\\Vert\\vc{t}-\\project \\vr{t}\\Vert_2$. \n",
"Our learning goal is to train at a correction operator $\\corr ( \\vc{} )$ such that \n",
"a solution to which the correction is applied has a lower error than the original unmodified (source) solution: \n",
"$\\loss ( \\pdec( \\corr (\\project \\vr{t_0}) ) , \\project \\vr{t_1}) < \\loss ( \\pdec( \\project \\vr{t_0} ), \\project \\vr{t_1})$. \n",
"Our learning goal is to train at a correction operator \n",
"$\\newcommand{\\corr}{\\mathcal{C}} \n",
"\\corr ( \\vc{} )$ 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",
"The correction function \n",
"$\\newcommand{\\vcN}{\\mathbf{s}} \\newcommand{\\corr}{\\mathcal{C}} \\corr (\\vcN | \\theta)$ \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",
"To distinguish the original states $\\vcN$ from the corrected ones, we'll denote the latter with an added tilde\n",
"$\\newcommand{\\vctN}{\\tilde{\\mathbf{s}}} \\vctN$.\n",
"and receives the state $\\mathbf{s}$ to infer an additive correction field with the same dimension.\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",
"The overall learning goal now becomes\n",
"\n",
"$\n",
"\\newcommand{\\pdec}{\\pde_{s}}\n",
"\\newcommand{\\corr}{\\mathcal{C}} \n",
"\\newcommand{\\project}{\\mathcal{T}} \n",
"\\newcommand{\\vr}[1]{\\mathbf{r}_{#1}} \n",
"\\text{argmin}_\\theta | ( \\pdec \\corr )^n ( \\project \\vr{t} ) - \\project \\vr{t}|^2\n",
"$\n",
"\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",
"$\\newcommand{\\vctN}{\\tilde{\\mathbf{s}}} \\vctN$, so we have \n",
"$\\tilde{\\mathbf{s}}$, so we have \n",
"$\\newcommand{\\vctN}{\\tilde{\\mathbf{s}}} \\newcommand{\\corr}{\\mathcal{C}} \\corr (\\vctN | \\theta)$.\n",
"These states actually evolve over time when training. They don't exist beforehand.\n",
"\n",
@ -203,9 +212,9 @@
"source": [
"## Simulation Setup\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",
"$\\newcommand{\\pder}{\\pde_{r}} \\pder$\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",
"\n",
"This code block and the next ones will define lots of functions, that will be used later on for training.\n",