smaller cleanup

diffphys-code-burgers.ipynb

@@ -182,6 +182,14 @@
     "Now we have our simulation graph in TF, we can use TF to give us a gradient for the initial state for the loss. All we need to do is run `tf.gradients(loss, [state_in.velocity.data]`, which will give us a \n",
     "\n",
     "Thus now we have \"search direction\" for each velocity variable. Based on a linear approximation, the gradient tells us how to change each of them to increase the loss function (gradients _always_ point \"upwards\"). In the following code block, we're additionally saving all these gradients in a list called `grads`, such that we can visualize them later on. (Normally, we could discard each gradient after performing an update step.)\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Optimization \n",
     "\n",
     "Based on the gradient, we can now take a step in the opposite direction to bring the loss down (instead of increasing it). Below we're using a learning rate `LR=5` for this step. Afterwards, we're re-evaluating the loss for the updated state to check how we did. "
    ]

diffphys-control.ipynb

@@ -41,7 +41,7 @@
     "$\n",
     "\\newcommand{\\pde}{\\mathcal{P}}\n",
     "\\newcommand{\\net}{\\mathrm{CFE}}\n",
-    "\\mathbf{u}_{n},d_{n} = \\pdec(\\net(\\pdec(\\net(\\cdots \\pdec(\\net( \\mathbf{u}_0,d_0  ))\\cdots)))) = (\\pdec\\net)^n ( \\mathbf{u}_0,d_0 ) .\n",
+    "\\mathbf{u}_{n},d_{n} = \\pde(\\net(~\\pde(\\net(\\cdots \\pde(\\net( \\mathbf{u}_0,d_0  ))\\cdots)))) = (\\pde\\net)^n ( \\mathbf{u}_0,d_0 ) .\n",
     "$\n",
     "\n",
     "minimizes the loss above. The $\\mathrm{OP}$ network is a predictor that determines the action of the $\\mathrm{CFE}$ network given the target $d^*$, i.e., $\\mathrm{OP}(\\mathbf{u},d,d^*)=d_{OP}$,\n",

intro.md

@@ -87,6 +87,7 @@ See also... Test link: {doc}`supervised`
 - PINNs: often need weighting of added loss terms for different parts
 - DP intro, check transpose of Jacobians in equations
 - DP control, show targets at bottom?
+- finish pictures...
 
 
 ## TODOs , Planned content