corrected notation for burgers example

overview-equations.md

@@ -73,7 +73,7 @@ The corresponding fields are either d-dimensional vector fields, for instance $\
 or scalar $\mathbf{p}: \mathbb{R}^d \times \mathbb{R}^{+} \rightarrow \mathbb{R}$.
 The components of a vector are typically denoted by $x,y,z$ subscripts, i.e.,
 $\mathbf{v} = (v_x, v_y, v_z)^T$ for $d=3$, while
-positions are denoted by $\mathbf{x} \in \Omega$.
+positions are denoted by $\mathbf{p} \in \Omega$.
 
 To obtain unique solutions for $\mathcal P^*$ we need to specify suitable
 initial conditions, typically for all quantities of interest at $t=0$,

physicalloss-code.ipynb

@@ -33,12 +33,15 @@
     "In terms of the $x,y^*$ notation from {doc}`overview-equations` and the previous section, this reconstruction problem means we are solving\n",
     "\n",
     "$$\n",
-    "\\text{arg min}_{\\theta} \\sum_i ( f(x_i ; \\theta)-y^*_i )^2 + R(x_i) ,\n",
+    "\\text{arg min}_{\\theta} \\sum_i \\big( f(x_i ; \\theta)-y^*_i \\big)^2 + R(x_i) ,\n",
     "$$\n",
     "\n",
-    "where $x$ and $y^*$ are solutions of $u$ at different locations in space and time. As we're dealing with a 1D velocity, $x,y^* \\in \\mathbb{R}$.\n",
-    "They both represent two-dimensional solutions\n",
-    "$x(p_i,t_i)$ and $y^*(p_i,t_i)$ for a spatial coordinate $p_i$ and a time $t_i$, where the index $i$ sums over a set of chosen $p_i,t_i$ locations at which we evaluate the PDE and the approximated solutions. Thus $y^*$ denotes a reference $u$ for $\\mathcal{P}$ being Burgers equation, which $x$ should approximate as closely as possible. Thus our neural network representation of $x$ will receive $p,t$ as input to produce a velocity solution at the specified position.\n",
+    "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",
+    "\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",
     "\n",
@@ -914,4 +917,4 @@
  },
  "nbformat": 4,
  "nbformat_minor": 1
-}
+}