minor typo fixes

intro-teaser.ipynb

@@ -15,7 +15,7 @@
    "source": [
     "Let's start with a very reduced example that highlights some of the key capabilities of physics-based learning approaches. Let's assume our physical model is a very simple equation: a parabola along the positive x-axis.\n",
     "\n",
-    "Despite being very simple, there are two solutions for every point along x, i.e. we have two modes, one above the other one below the x-axis, as shown on the left below. If we don't take care a conventional learning approach will give us an approximation like the red one shown in the middle, which is completely off. With an improved learning setup, ideally, by using a discretized numerical solver, we can at least accurately represent one of the modes of the solution (shown in green on the right).\n",
+    "Despite being very simple, there are two solutions for every point along x, i.e. we have two modes, one above the other one below the x-axis, as shown on the left below. If we don't take care, a conventional learning approach will give us an approximation like the red one shown in the middle, which is completely off. With an improved learning setup, ideally, by using a discretized numerical solver, we can at least accurately represent one of the modes of the solution (shown in green on the right).\n",
     "\n",
     "```{figure} resources/intro-teaser-side-by-side.png\n",
     "---\n",
@@ -198,7 +198,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": "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 truncated
+      "image/png": "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 truncated
       "text/plain": [
        "<Figure size 432x288 with 1 Axes>"
       ]
@@ -274,7 +274,7 @@
    "id": "conscious-budapest",
    "metadata": {},
    "source": [
-    "The loss function is the crucial point for training: we directly incorporate the function f into the loss. In this simple case, the `loss_dp` function simply computes the square of the prediction `y_pred`. \n",
+    "The loss function is the crucial point for training: we directly incorporate the function $f$ into the loss. In this simple case, the `loss_dp` function simply computes the square of the prediction `y_pred`. \n",
     "\n",
     "Later on, a lot more could happen here: we could evaluate finite-difference stencils on the predicted solution, or compute a whole implicit time-integration step of a solver. Here we have a simple _mean-squared error_ term of the form $|y_{\\text{pred}}^2 - y_{\\text{true}}|^2$, which we are minimizing during training. It's not necessary to make it so simple: the more knowledge and numerical methods we can incorporate, the better we can guide the training process."
    ]
@@ -339,7 +339,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": "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 truncated
+      "image/png": "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 truncated
       "text/plain": [
        "<Figure size 432x288 with 1 Axes>"
       ]
@@ -375,7 +375,7 @@
     "\n",
     "- We're still only getting one side of the curve! This is to be expected because we're representing the solutions with a deterministic function. Hence, we can only represent a single mode. Interestingly, whether it's the top or bottom mode is determined by the random initialization of the weights in $f$ - run the example a couple of times to see this effect in action. To capture multiple modes we'd need to extend the NN to capture the full distribution of the outputs and parametrize it with additional dimensions.\n",
     "\n",
-    "- The region with $x$ near zero is typically still off in this example. The network essentially learns a linear approximation of one half of the parabola here. This is partially caused by the weak neural network: it is very small and shallow. In addition, the evenly spread of sample points along the x-axis bias the NN towards the larger y values. These contribute more to the loss, and hence the network invests most of its resources to reduce the error in this region.\n"
+    "- The region with $x$ near zero is typically still off in this example. The network essentially learns a linear approximation of one half of the parabola here. This is partially caused by the weak neural network: it is very small and shallow. In addition, the evenly spread of sample points along the x-axis bias the NN towards the larger $y$ values. These contribute more to the loss, and hence the network invests most of its resources to reduce the error in this region.\n"
    ]
   },
   {
@@ -389,7 +389,7 @@
     "\n",
     "Good and obvious examples are bifurcations in fluid flow. Smoke rising above a candle will start out straight, and then, due to tiny perturbations in its motion, start oscillating in a random direction. The images below illustrate this case via _numerical perturbations_: the perfectly symmetric setup will start turning left or right, depending on how the approximation errors build up. Averaging the two modes would give an unphysical, straight flow similar to the parabola example above.\n",
     "\n",
-	"Similarly, we have different modes in many numerical solutions, and typically it's important to recover them, rather than averaging them out. Hence, we'll show how to leverage training via _differentiable physics_ in the following chapters for more practical and complex cases.\n",
+    "Similarly, we have different modes in many numerical solutions, and typically it's important to recover them, rather than averaging them out. Hence, we'll show how to leverage training via _differentiable physics_ in the following chapters for more practical and complex cases.\n",
     "\n",
     "```{figure} resources/intro-fluid-bifurcation.jpg\n",
     "---\n",

overview.md

@@ -119,7 +119,7 @@ where a solver targets cases from a certain well-defined problem
 domain repeatedly, it can for instance make a lot of sense to once invest 
 significant resources to train
 a neural network that supports the repeated solves. Based on the
-domain-specific specialization of this network, such a hybrid
+domain-specific specialization of this network, such a hybrid solver
 could vastly outperform traditional, generic solvers. And despite
 the many open questions, first publications have demonstrated
 that this goal is not overly far away {cite}`um2020sol,kochkov2021`.