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clarified JVPs

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diffphys.md
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@ -42,14 +42,13 @@ to simulate -- if this is missing we're in trouble. But luckily, we can
tap into existing collections of model equations and established methods tap into existing collections of model equations and established methods
for discretizing continuous models. for discretizing continuous models.
Let's assume we have a continuous formulation $\mathcal P^*(\mathbf{x}, \nu)$ of the physical quantity of Let's assume we have a continuous formulation $\mathcal P^*(\mathbf{u}, \nu)$ of the physical quantity of
interest $\mathbf{u}(\mathbf{x}, t): \mathbb R^d \times \mathbb R^+ \rightarrow \mathbb R^d$, interest $\mathbf{u}(\mathbf{u}, t): \mathbb R^d \times \mathbb R^+ \rightarrow \mathbb R^d$,
with model parameters $\nu$ (e.g., diffusion, viscosity, or conductivity constants). with model parameters $\nu$ (e.g., diffusion, viscosity, or conductivity constants).
The components of $\mathbf{u}$ will be denoted by a numbered subscript, i.e., The components of $\mathbf{u}$ will be denoted by a numbered subscript, i.e.,
$\mathbf{u} = (u_1,u_2,\dots,u_d)^T$. $\mathbf{u} = (u_1,u_2,\dots,u_d)^T$.
%and a corresponding discrete version that describes the evolution of this quantity over time: $\mathbf{u}_t = \mathcal P(\mathbf{x}, \mathbf{u}, t)$.
Typically, we are interested in the temporal evolution of such a system. Typically, we are interested in the temporal evolution of such a system.
Discretization yields a formulation $\mathcal P(\mathbf{x}, \nu)$ Discretization yields a formulation $\mathcal P(\mathbf{u}, \nu)$
that we re-arrange to compute a future state after a time step $\Delta t$. that we re-arrange to compute a future state after a time step $\Delta t$.
The state at $t+\Delta t$ is computed via sequence of The state at $t+\Delta t$ is computed via sequence of
operations $\mathcal P_1, \mathcal P_2 \dots \mathcal P_m$ such that operations $\mathcal P_1, \mathcal P_2 \dots \mathcal P_m$ such that
@ -63,7 +62,7 @@ $\partial \mathcal P_i / \partial \mathbf{u}$.
``` ```
Note that we typically don't need derivatives Note that we typically don't need derivatives
for all parameters of $\mathcal P(\mathbf{x}, \nu)$, e.g., for all parameters of $\mathcal P(\mathbf{u}, \nu)$, e.g.,
we omit $\nu$ in the following, assuming that this is a we omit $\nu$ in the following, assuming that this is a
given model parameter with which the NN should not interact. given model parameter with which the NN should not interact.
Naturally, it can vary within the solution manifold that we're interested in, Naturally, it can vary within the solution manifold that we're interested in,
@ -114,7 +113,7 @@ E.g., for two of them
$$ $$
\frac{ \partial (\mathcal P_1 \circ \mathcal P_2) }{ \partial \mathbf{u} } \Big|_{\mathbf{u}^n} \frac{ \partial (\mathcal P_1 \circ \mathcal P_2) }{ \partial \mathbf{u} } \Big|_{\mathbf{u}^n}
= =
\frac{ \partial \mathcal P_1 }{ \partial \mathbf{u} } \big|_{\mathcal P_2(\mathbf{u}^n)} \frac{ \partial \mathcal P_1 }{ \partial \mathcal P_2 } \big|_{\mathcal P_2(\mathbf{u}^n)}
\ \
\frac{ \partial \mathcal P_2 }{ \partial \mathbf{u} } \big|_{\mathbf{u}^n} \ , \frac{ \partial \mathcal P_2 }{ \partial \mathbf{u} } \big|_{\mathbf{u}^n} \ ,
$$ $$