corrected notation for burgers example

physicalloss-code.ipynb

@@ -39,11 +39,9 @@
     "where now $x_i$ denotes a space-time point $x_i=[p_i,t_i]$, the reference solutions are $y^*_i = y^*(x_i)$, and the index $i$ indicates different sampling points for our data set.\n",
     "Both $f$ and $y^*$ represent the solution of $u$ at different locations in space and time, and\n",
     "as we're dealing with a 1D velocity, $f, y^*: \\mathbb{R}^2 \\rightarrow \\mathbb{R}$.\n",
-    "In this example, $y^*$ denotes a reference $u$ for $\\mathcal{P}$ being Burgers equation, which $f$ should approximate as closely as possible at a chosen space-time point $x_i=[p_i,t_i]$. \n",
+    "In this example, $y^*$ denotes a reference $u$ for $\\mathcal{P}$ being Burgers equation, which $f$ should approximate as closely as possible at all chosen space-time points $x_i=[p_i,t_i]$. \n",
     "\n",
-    "\n",
-    "\n",
-    "The residual function $R$ above collects additional evaluations of $f(;\\theta)$ and its derivatives to formulate the residual for $\\mathcal{P}$. This approach -- using derivatives of a neural network to compute a PDE residual -- is typically called a _physics-informed_ approach, yielding a _physics-informed neural network_ (PINN) {cite}`raissi2019pinn` to represent a solution for the inverse reconstruction problem.\n",
+    "While the first term above is the \"supervised\" data term, the second one denotes the residual function $R$. It collects additional evaluations of $f(;\\theta)$ and its derivatives to formulate the residual for $\\mathcal{P}$. This approach -- using derivatives of a neural network to compute a PDE residual -- is typically called a _physics-informed_ approach, yielding a _physics-informed neural network_ (PINN) {cite}`raissi2019pinn` to represent a solution for the inverse reconstruction problem.\n",
     "\n",
     "Thus, in the formulation above, $R$ should simply converge to zero above. We've omitted scaling factors in the objective function for simplicity. Note that, effectively, we're only dealing with individual point samples of a single solution $u$ for $\\mathcal{P}$ here.\n",
     "\n"