smaller fixes of notation

diffphys.md

@@ -89,7 +89,7 @@ $$ \begin{aligned}
 \end{aligned} $$
 %
 where, as above, $d$ denotes the number of components in $\mathbf{u}$. As $\mathcal P$ maps one value of
-$\mathbf{u}$ to another, the Jacobian is square and symmetric here. Of course this isn't necessarily the case
+$\mathbf{u}$ to another, the Jacobian is square here. Of course this isn't necessarily the case
 for general model equations, but non-square Jacobian matrices would not cause any problems for differentiable 
 simulations.
 
@@ -97,7 +97,7 @@ In practice, we rely on the _reverse mode_ differentiation that all modern DL
 frameworks provide, and focus on computing a matrix vector product of the Jacobian transpose
 with a vector $\mathbf{a}$, i.e. the expression: 
 $
-    ( \frac{\partial \mathcal P_i }{ \partial \mathbf{u} } )^T \mathbf{a}
+    \big( \frac{\partial \mathcal P_i }{ \partial \mathbf{u} } \big)^T \mathbf{a}
 $. 
 If we'd need to construct and store all full Jacobian matrices that we encounter during training, 
 this would cause huge memory overheads and unnecessarily slow down training.
@@ -117,7 +117,7 @@ $$
 $$
 
 which is just the vector valued version of the "classic" chain rule
-$f(g(x))' = f'(g(x)) g'(x)$, and directly extends for larger numbers of composited functions, i.e. $i>2$.
+$f\big(g(x)\big)' = f'\big(g(x)\big) g'(x)$, and directly extends for larger numbers of composited functions, i.e. $i>2$.
 
 Here, the derivatives for $\mathcal P_1$ and $\mathcal P_2$ are still Jacobian matrices, but knowing that 
 at the "end" of the chain we have our scalar loss (cf. {doc}`overview`), the right-most Jacobian will invariably

physicalloss-discuss.md

@@ -68,5 +68,7 @@ goals of the next sections.
 - Largely incompatible with _classical_ numerical methods.
 - Accuracy of derivatives relies on learned representation.
 
-Next, let's look at how we can leverage numerical methods to improve the DL accuracy and efficiency
+To address these issues,
+we'll next look at how we can leverage existing numerical methods to improve the DL process
 by making use of differentiable solvers.
+

physicalloss.md

@@ -44,7 +44,7 @@ therefore help to _pin down_ the solution in certain places.
 Now our training objective becomes
 
 $$
-\text{arg min}_{\theta} \ \alpha_0 \sum_i \big( f(x_i ; \theta)-y^*_i \big)^2 + \alpha_1 R(x_i) ,
+\text{arg min}_{\theta} \ \sum_i \alpha_0 \big( f(x_i ; \theta)-y^*_i \big)^2 + \alpha_1 R(x_i) ,
 $$ (physloss-training)
 
 where $\alpha_{0,1}$ denote hyperparameters that scale the contribution of the supervised term and 
@@ -100,7 +100,7 @@ Nicely enough, in this case we don't even need additional supervised samples, an
 An example implementation can be found in this [code repository](https://github.com/tum-pbs/CG-Solver-in-the-Loop).
 
 Overall, this variant 1 has a lot in common with _differentiable physics_ training (it's basically a subset). As we'll discuss differentiable physics in a lot more detail
-in {doc}`diffphys` and after, we'll focus on the direct NN representation (variant 2) from now on. 
+in {doc}`diffphys` and after, we'll focus on direct NN representations (variant 2) from now on. 
 
 ---
 
@@ -147,5 +147,6 @@ For higher order derivatives, such as $\frac{\partial^2 u}{\partial x^2}$, we ca
 The approach above gives us a method to include physical equations into DL learning as a soft constraint: the residual loss.
 Typically, this setup is suitable for _inverse problems_, where we have certain measurements or observations
 for which we want to find a PDE solution. Because of the high cost of the reconstruction (to be 
-demonstrated in the following), the solution manifold shouldn't be overly complex. E.g., it is not possible 
+demonstrated in the following), the solution manifold shouldn't be overly complex. E.g., it is typically not possible 
-to capture a wide range of solutions, such as with the previous supervised airfoil example, with such a physical residual loss.
+to capture a wide range of solutions, such as with the previous supervised airfoil example, by only using a physical residual loss.
+