smaller updates to figures and captions

diffphys-discuss.md

@@ -49,7 +49,7 @@ In this case we compute a (potentially very long) sequence of PDE solver steps i
 
 ```{figure} resources/diffphys-multistep.jpg
 ---
-height: 220px
+height: 180px
 name: diffphys-mulitstep
 ---
 Time stepping with interleaved DP and NN operations for $k$ solver iterations.
@@ -59,7 +59,7 @@ Note that this picture (and the ones before) have assumed an _additive_ influenc
 
 DP setups with many time steps can be difficult to train: the gradients need to backpropagate through the full chain of PDE solver evaluations and NN evaluations. Typically, each of them represents a non-linear and complex function. Hence for larger numbers of steps, the vanishing and exploding gradient problem can make training difficult (see {doc}`diffphys-code-sol` for some practical tipps how to alleviate this).
 
-## Alternatives - Noise
+## Alternatives: Noise
 
 It is worth mentioning here that other works have proposed perturbing the inputs and 
 the iterations at training time with noise {cite}`sanchez2020learning` (somewhat similar to

diffphys.md

@@ -147,7 +147,7 @@ to compute the updates (and derivatives) for these operators.
 %in practice break down into larger, monolithic components
 E.g., as this process is very similar to adjoint method optimizations, we can re-use many of the techniques
 that were developed in this field, or leverage established numerical methods. E.g., 
-we could leverage the $O(n)$ complexity of multigrid solvers for matrix inversion.
+we could leverage the $O(n)$ runtime of multigrid solvers for matrix inversion.
 
 The flipside of this approach is, that it requires some understanding of the problem at hand, 
 and of the numerical methods. Also, a given solver might not provide gradient calculations out of the box.
@@ -161,13 +161,14 @@ never produces the parameter $\nu$ in the example above, and it doesn't appear i
 loss formulation, we will never encounter a $\partial/\partial \nu$ derivative
 in our backpropagation step.
 
+The following figure summarizes the DP-based learning approach, and illustrates the sequence of operations that are typically processed within a single PDE solve. As many of the operations are non-linear in practice, this often leads to a challenging learning task for the NN:
 
 ```{figure} resources/diffphys-overview.jpg
 ---
 height: 220px
 name: diffphys-full-overview
 ---
-TODO , details...
+DP learning with a PDE solver that consists of $m$ individual operators $\mathcal P_i$. The gradient travels backward through all $m$ operators before influencing the network weights $\theta$.
 ```