Introduction to Differentiable Physics
=======================
As a next step towards a tighter and more generic combination of deep learning
methods and physical simulations we will target incorporating _differentiable
numerical simulations_ into the learning process. In the following, we'll shorten
these "differentiable numerical simulations of physical systems" to just "differentiable physics" (DP).
The central goal of these methods is to use existing numerical solvers, and equip
them with functionality to compute gradients with respect to their inputs.
Once this is realized for all operators of a simulation, we can leverage
the autodiff functionality of DL frameworks with backpropagation to let gradient
information flow from a simulator into an NN and vice versa. This has numerous
advantages such as improved learning feedback and generalization, as we'll outline below.
In contrast to physics-informed loss functions, it also enables handling more complex
solution manifolds instead of single inverse problems.
E.g., instead of using deep learning
to solve single inverse problems as in the previous chapter,
differentiable physics can be used to train NNs that learn to solve
larger classes of inverse problems very efficiently.
```{figure} resources/diffphys-shortened.jpg
---
height: 220px
name: diffphys-short-overview
---
Training with differentiable physics means that a chain of differentiable operators
provide directions in the form of gradients to steer the learning process.
```
## Differentiable operators
With the DP direction we build on existing numerical solvers. I.e.,
the approach is strongly relying on the algorithms developed in the larger field
of computational methods for a vast range of physical effects in our world.
To start with, we need a continuous formulation as model for the physical effect that we'd like
to simulate -- if this is missing we're in trouble. But luckily, we can
tap into existing collections of model equations and established methods
for discretizing continuous models.
Let's assume we have a continuous formulation $\mathcal P^*(\mathbf{x}, \nu)$ of the physical quantity of
interest $\mathbf{u}(\mathbf{x}, t): \mathbb R^d \times \mathbb R^+ \rightarrow \mathbb R^d$,
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.,
$\mathbf{u} = (u_1,u_2,\dots,u_d)^T$.
%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.
Discretization yields a formulation $\mathcal P(\mathbf{x}, \nu)$
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
operations $\mathcal P_1, \mathcal P_2 \dots \mathcal P_m$ such that
$\mathbf{u}(t+\Delta t) = \mathcal P_m \circ \dots \mathcal P_2 \circ \mathcal P_1 ( \mathbf{u}(t),\nu )$,
where $\circ$ denotes function decomposition, i.e. $f(g(x)) = f \circ g(x)$.
```{note}
In order to integrate this solver into a DL process, we need to ensure that every operator
$\mathcal P_i$ provides a gradient w.r.t. its inputs, i.e. in the example above
$\partial \mathcal P_i / \partial \mathbf{u}$.
```
Note that we typically don't need derivatives
for all parameters of $\mathcal P(\mathbf{x}, \nu)$, e.g.,
we omit $\nu$ in the following, assuming that this is a
given model parameter with which the NN should not interact.
Naturally, it can vary within the solution manifold that we're interested in,
but $\nu$ will not be the output of an NN representation. If this is the case, we can omit
providing $\partial \mathcal P_i / \partial \nu$ in our solver. However, the following learning process
naturally transfers to including $\nu$ as a degree of freedom.
## Jacobians
As $\mathbf{u}$ is typically a vector-valued function, $\partial \mathcal P_i / \partial \mathbf{u}$ denotes
a Jacobian matrix $J$ rather than a single value:
%
$$ \begin{aligned}
\frac{ \partial \mathcal P_i }{ \partial \mathbf{u} } =
\begin{bmatrix}
\partial \mathcal P_{i,1} / \partial u_{1}
& \ \cdots \ &
\partial \mathcal P_{i,1} / \partial u_{d}
\\
\vdots & \ & \
\\
\partial \mathcal P_{i,d} / \partial u_{1}
& \ \cdots \ &
\partial \mathcal P_{i,d} / \partial u_{d}
\end{bmatrix}
\end{aligned} $$
%
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 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.
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:
$
\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.
Instead, for backpropagation, we can provide faster operations that compute products
with the Jacobian transpose because we always have a scalar loss function at the end of the chain.
Given the formulation above, we need to resolve the derivatives
of the chain of function compositions of the $\mathcal P_i$ at some current state $\mathbf{u}^n$ via the chain rule.
E.g., for two of them
$$
\frac{ \partial (\mathcal P_1 \circ \mathcal P_2) }{ \partial \mathbf{u} } \Big|_{\mathbf{u}^n}
=
\frac{ \partial \mathcal P_1 }{ \partial \mathbf{u} } \big|_{\mathcal P_2(\mathbf{u}^n)}
\
\frac{ \partial \mathcal P_2 }{ \partial \mathbf{u} } \big|_{\mathbf{u}^n} \ ,
$$
which is just the vector valued version of the "classic" chain rule
$f\big(g(x)\big)' = f'\big(g(x)\big) g'(x)$, 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
be a matrix with 1 column, i.e. a vector. During reverse mode, we start with this vector, and compute
the multiplications with the left Jacobians, $\frac{ \partial \mathcal P_1 }{ \partial \mathbf{u} }$ above,
one by one.
For the details of forward and reverse mode differentiation, please check out external materials such
as this [nice survey by Baydin et al.](https://arxiv.org/pdf/1502.05767.pdf).
## Learning via DP operators
Thus, once the operators of our simulator support computations of the Jacobian-vector
products, we can integrate them into DL pipelines just like you would include a regular fully-connected layer
or a ReLU activation.
At this point, the following (very valid) question arises: "_Most physics solvers can be broken down into a
sequence of vector and matrix operations. All state-of-the-art DL frameworks support these, so why don't we just
use these operators to realize our physics solver?_"
It's true that this would theoretically be possible. The problem here is that each of the vector and matrix
operations in tensorflow and pytorch is computed individually, and internally needs to store the current
state of the forward evaluation for backpropagation (the "$g(x)$" above). For a typical
simulation, however, we're not overly interested in every single intermediate result our solver produces.
Typically, we're more concerned with significant updates such as the step from $\mathbf{u}(t)$ to $\mathbf{u}(t+\Delta t)$.
Thus, in practice it is a very good idea to break down the solving process into a sequence
of meaningful but _monolithic_ operators. This not only saves a lot of work by preventing the calculation
of unnecessary intermediate results, it also allows us to choose the best possible numerical methods
to compute the updates (and derivatives) for these operators.
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 $O(n)$ runtime of multigrid solvers for matrix inversion.
The flip-side 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.
Thus, if we want to employ DL for model equations that we don't have a proper grasp of, it might not be a good
idea to directly go for learning via a DP approach. However, if we don't really understand our model, we probably
should go back to studying it a bit more anyway...
Also, in practice we should be _greedy_ with the derivative operators, and only
provide those which are relevant for the learning task. E.g., if our network
never produces the parameter $\nu$ in the example above, and it doesn't appear in our
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
---
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 $\theta$.
```
---
## A practical example
As a simple example let's consider the advection of a passive scalar density $d(\mathbf{x},t)$ in
a velocity field $\mathbf{u}$ as physical model $\mathcal P^*$:
$$
\frac{\partial d}{\partial{t}} + \mathbf{u} \cdot \nabla d = 0
$$
Instead of using this formulation as a residual equation right away (as in v2 of {doc}`physicalloss`),
we can discretize it with our favorite mesh and discretization scheme,
to obtain a formulation that updates the state of our system over time. This is a standard
procedure for a _forward_ solve.
To simplify things, we assume here that $\mathbf{u}$ is only a function in space,
i.e. constant over time. We'll bring back the time evolution of $\mathbf{u}$ later on.
%
Let's denote this re-formulation as $\mathcal P$. It maps a state of $d(t)$ into a
new state at an evolved time, i.e.:
$$
d(t+\Delta t) = \mathcal P ( ~ d(t), \mathbf{u}, t+\Delta t)
$$
As a simple example of an inverse problem and learning task, let's consider the problem of
finding a velocity field $\mathbf{u}$.
This velocity should transform a given initial scalar density state $d^{~0}$ at time $t^0$
into a state that's evolved by $\mathcal P$ to a later "end" time $t^e$
with a certain shape or configuration $d^{\text{target}}$.
Informally, we'd like to find a flow that deforms $d^{~0}$ through the PDE model into a target state.
The simplest way to express this goal is via an $L^2$ loss between the two states. So we want
to minimize the loss function $L=|d(t^e) - d^{\text{target}}|^2$.
Note that as described here this inverse problem is a pure optimization task: there's no NN involved,
and our goal is to obtain $\mathbf{u}$. We do not want to apply this velocity to other, unseen _test data_,
as would be custom in a real learning task.
The final state of our marker density $d(t^e)$ is fully determined by the evolution
from $\mathcal P$ via $\mathbf{u}$, which gives the following minimization problem:
$$
\text{arg min}_{~\mathbf{u}} | \mathcal P ( d^{~0}, \mathbf{u}, t^e) - d^{\text{target}}|^2
$$
We'd now like to find the minimizer for this objective by
_gradient descent_ (GD), where the
gradient is determined by the differentiable physics approach described earlier in this chapter.
Once things are working with GD, we can relatively easily switch to better optimizers or bring
an NN into the picture, hence it's always a good starting point.
To make things easier to read below, we'll omit the transpose of the Jacobians in the following.
Unfortunately, the Jacobian is defined this way, but we actually never need the un-transposed one.
Keep in mind that in practice we're dealing with tranposed Jacobians $\big( \frac{ \partial a }{ \partial b} \big)^T$
that are "abbreviated" by $\frac{ \partial a }{ \partial b}$.
As the discretized velocity field $\mathbf{u}$ contains all our degrees of freedom,
all we need to do is to update the velocity by an amount
$\Delta \mathbf{u} = \partial L / \partial \mathbf{u}$,
which is decomposed into
$\Delta \mathbf{u} =
\frac{ \partial d }{ \partial \mathbf{u}}
\frac{ \partial L }{ \partial d} $.
The $\frac{ \partial L }{ \partial d}$ component is typically simple enough: we'll get
$$
\frac{ \partial L }{ \partial d}
= \frac{ \partial | \mathcal P ( d^{~0}, \mathbf{u}, t^e) - d^{\text{target}}|^2 }{ \partial d }
= 2 \big( d(t^e)-d^{\text{target}} \big).
$$
If $d$ is represented as a vector, e.g., for one entry per cell of a mesh,
$\frac{ \partial L }{ \partial d}$ will likewise be a column vector of equivalent size.
This stems from the fact that $L$ is always a scalar loss function, and so the Jacobian
matrix will have a dimension of 1 along the $L$ dimension.
Intuitively, this vector will simply contain the differences between $d$ at the end time
in comparison to the target densities $d^{\text{target}}$.
The evolution of $d$ itself is given by our discretized physical model $\mathcal P$,
and we use $\mathcal P$ and $d$ interchangeably.
Hence, the more interesting component is the Jacobian
$\partial d / \partial \mathbf{u} = \partial \mathcal P / \partial \mathbf{u}$ to
compute the full $\Delta \mathbf{u} =
\frac{ \partial d }{ \partial \mathbf{u}}
\frac{ \partial L }{ \partial d}$.
We luckily don't need $\partial d / \partial \mathbf{u}$ as a full
matrix, but instead only multiplied by $\frac{ \partial L }{ \partial d}$.
So what is the actual Jacobian for $d$? To compute it we first need
to finalize our PDE model $\mathcal P$, such that we get an expression which we can derive.
In the next section we'll choose a specific advection scheme and a discretization
so that we can be more specific.
%the vector obtained from the derivative of our scalar loss function $L$.
%the $L^2$ loss $L= |d(t^e) - d^{\text{target}}|^2$, thus
%So what are the actual Jacobians here? The one for $L$ is simple enough, we simply get a column vector with entries of the form $2(d_i(t^e) - d^{\text{target}}_i)$ for one component $i$.
%$\partial \mathcal P / \partial \mathbf{u}$ is more interesting: here we'll get derivatives of the chosen advection operator w.r.t. each component of the velocities.
%...to obtain an explicit update of the form $d(t+\Delta t) = A d(t)$, where the matrix $A$ represents the discretized advection step of size $\Delta t$ for $\mathbf{u}$. ... we'll get a matrix that essentially encodes linear interpolation coefficients for positions $\mathbf{x} + \Delta t \mathbf{u}$. For a grid of size $d_x \times d_y$ we'd have a
### Introducing a specific advection scheme
In the following we'll make use of a simple [first order upwinding scheme](https://en.wikipedia.org/wiki/Upwind_scheme)
on a Cartesian grid in 1D, with marker density $d_i$ and velocity $u_i$ for cell $i$.
We omit the $(t)$ for quantities at time $t$ for brevity, i.e., $d_i(t)$ is written as $d_i$ below.
From above, we'll use our _physical model_ that updates the marker density
$d_i(t+\Delta t) = \mathcal P ( d_i(t), \mathbf{u}(t), t + \Delta t)$, which
gives the following:
$$ \begin{aligned}
& d_i(t+\Delta t) = d_i - \Delta t \big[ u_i^+ (d_{i+1} - d_{i}) + u_i^- (d_{i} - d_{i-1}) \big] \text{ with } \\
& u_i^+ = \text{min}(u_i / \Delta x,0) \\
& u_i^- = \text{max}(u_i / \Delta x,0)
\end{aligned} $$
```{figure} resources/diffphys-advect1d.jpg
---
height: 150px
name: advection-upwind
---
1st-order upwinding uses a simple one-sided finite-difference stencil that takes into account the direction of the flow
```
Thus, for a negative $u_i$, we're using $u_i^+$ to look in the opposite direction of the velocity, i.e., _backward_ in terms of the motion. $u_i^-$ will be zero in this case. For positive $u_i$ it's vice versa, and we'll get a zero'ed $u_i^+$, and a backward difference stencil via $u_i^-$.
To pick the former case, for a negative $u_i$ we get
$$
\mathcal P ( d_i(t), \mathbf{u}(t), t + \Delta t) = (1 + \frac{u_i \Delta t }{ \Delta x}) d_i - \frac{u_i \Delta t }{ \Delta x} d_{i+1}
$$ (eq:advection)
and hence $\partial \mathcal P / \partial u_i$ gives
$\frac{\Delta t }{ \Delta x} d_i - \frac{\Delta t }{ \Delta x} d_{i+1}$. Intuitively,
the change of the velocity $u_i$ depends on the spatial derivatives of the densities.
Due to the first order upwinding, we only include two neighbors (higher order methods would depend on
additional entries of $d$)
In practice this step is equivalent to evaluating a transposed matrix multiplication.
If we rewrite the calculation above as
$ \mathcal P ( d_i(t), \mathbf{u}(t), t + \Delta t) = A \mathbf{u}$,
then $\big( \partial \mathcal P / \partial \mathbf{u} \big)^T = A^T$.
However, in many practical cases, a matrix free implementation of this multiplication might
be preferable to actually constructing $A$.
Another derivative that we can consider for the advection scheme is that w.r.t. the previous
density state, i.e. $d_i(t)$, which is $d_i$ in the shortened notation.
$\partial \mathcal P / \partial d_i$ for cell $i$ from above gives $1 + \frac{u_i \Delta t }{ \Delta x}$. However, for the full gradient we'd need to add the potential contributions from cells $i+1$ and $i-1$, depending on the sign of their velocities. This derivative will come into play in the next section.
### Time evolution
So far we've only dealt with a single update step of
$d$ from time $t$ to $t+\Delta t$, but we could of course have an arbitrary number of such
steps. After all, above we stated the goal to advance the initial marker state $d(t^0)$ to
the target state at time $t^e$, which could encompass a long interval of time.
In the expression above for $d_i(t+\Delta t)$, each of the $d_i(t)$ in turn depends
on the velocity and density states at time $t-\Delta t$, i.e., $d_i(t-\Delta t)$. Thus we have to trace back
the influence of our loss $L$ all the way back to how $\mathbf{u}$ influences the initial marker
state. This can involve a large number of evaluations of our advection scheme via $\mathcal P$.
This sounds challenging at first:
e.g., one could try to insert equation {eq}`eq:advection` at time $t-\Delta t$
into equation {eq}`eq:advection` at time $t$ and repeat this process recursively until
we have a single expression relating $d^{~0}$ to the targets. However, thanks
to the linear nature of the Jacobians, we treat each advection step, i.e.,
each invocation of our PDE $\mathcal P$ as a seperate, modular
operation. And each of these invocations follows the procedure described
in the previous section.
Given the machinery above, the backtrace is fairly simple to realize:
for each of the advection steps
in $\mathcal P$ we compute a Jacobian product with the _incoming_ vector of derivatives
from the loss $L$ or a previous advection step. We repeat this until we have traced the chain from the
loss with $d^{\text{target}}$ all the way back to $d^{~0}$.
Theoretically, the velocity $\mathbf{u}$ could be a function of time like $d$, in which
case we'd get a gradient $\Delta \mathbf{u}(t)$ for every time step $t$. However, to simplify things
below, let's we assume we have field that is constant in time, i.e., we're
reusing the same velocities $\mathbf{u}$ for every advection via $\mathcal P$. Now, each time step
will give us a contribution to $\Delta \mathbf{u}$ which we accumulate for all steps.
$$ \begin{aligned}
\Delta \mathbf{u} =&
\frac{ \partial d(t^e) }{ \partial \mathbf{u} }
\frac{ \partial L }{ \partial d(t^e) } \\
& + \
\frac{ \partial d(t^e - \Delta t) }{ \partial \mathbf{u}}
\frac{ \partial d(t^e) }{ \partial d(t^e - \Delta t) }
\frac{ \partial L }{ \partial d(t^e)}
\\
&
+ \ \cdots \ \\
&
+ \ \Big( \frac{ \partial d(t^0) }{ \partial \mathbf{u}} \cdots
\frac{ \partial d(t^e - \Delta t) }{ \partial d(t^e - 2 \Delta t) }
\frac{ \partial d(t^e) }{ \partial d(t^e - \Delta t) }
\frac{ \partial L }{ \partial d(t^e)} \Big)
\end{aligned} $$
Here the last term above contains the full backtrace of the marker density to time $t^0$.
The terms of this sum look unwieldy
at first, but looking closely, each line simply adds an additional Jacobian for one time step on the left hand side.
This follows from the chain rule, as shown in the two-operator case above.
So the terms of the sum contain a lot of similar Jacobians, and in practice can be computed efficiently
by backtracing through the sequence of computational steps that resulted from the forward evaluation of our PDE.
(Note that, as mentioned above, we've omitted the tranpose of the Jacobians here.)
This structure also makes clear that the process is very similar to the regular training
process of an NN: the evaluations of these Jacobian vector products from nested function calls
is exactly what a deep learning framework does for training an NN (we just have weights $\theta$ instead
of a velocity field there). And hence all we need to do in practice is to provide a custom
function the Jacobian vector product for $\mathcal P$.
---
## Implicit gradient calculations
As a slightly more complex example let's consider Poisson's equation $\nabla^2 a = b$, where
$a$ is the quantity of interest, and $b$ is given.
This is a very fundamental elliptic PDE that is important for
a variety of physical problems, from electrostatics to gravitational fields. It also arises
in the context of fluids, where $a$ takes the role of a scalar pressure field in the fluid, and
the right hand side $b$ is given by the divergence of the fluid velocity $\mathbf{u}$.
For fluids, we typically have
$\mathbf{u}^{n} = \mathbf{u} - \nabla p$, with
$\nabla^2 p = \nabla \cdot \mathbf{u}$. Here, $\mathbf{u}^{n}$ denotes the _new_, divergence-free
velocity field. This step is typically crucial to enforce the hard-constraint $\nabla \cdot \mathbf{u}=0$,
and also goes under the name of _Chorin Projection_, or _Helmholtz decomposition_.
It is a direct consequence of the fundamental theorem of vector calculus.
If we now introduce an NN that modifies $\mathbf{u}$ in a solver, we inevitably have to
backpropagate through the Poisson solve. I.e., we need a gradient for $\mathbf{u}^{n}$, which in this
notation takes the form $\partial \mathbf{u}^{n} / \partial \mathbf{u}$.
In combination, we aim for computing $\mathbf{u}^{n} = \mathbf{u} - \nabla \left( (\nabla^2)^{-1} \nabla \cdot \mathbf{u} \right)$. The outer gradient (from $\nabla p$) and the inner divergence ($\nabla \cdot \mathbf{u}$) are both linear operators, and their gradients are simple to compute. The main difficulty lies in obtaining the
matrix inverse $(\nabla^2)^{-1}$ from Poisson's equation (we'll keep it a bit simpler here, but it's often time-dependent, and non-linear).
In practice, the matrix vector product for $(\nabla^2)^{-1} b$ with $b=\nabla \cdot \mathbf{u}$ is not explicitly computed via matrix operations, but approximated with a (potentially matrix-free) iterative solver. E.g., conjugate gradient (CG) methods are a very popular choice here. Thus, we theoretically could treat this iterative solver as a function $\mathcal{S}$,
with $p = \mathcal{S}(\nabla \cdot \mathbf{u})$.
It's worth noting that matrix inversion is a non-linear process, despite the matrix itself being linear. As solvers like CG are also based on matrix and vector operations, we could decompose $\mathcal{S}$ into a sequence of simpler operations over the course of all solver iterations as $\mathcal{S}(x) = \mathcal{S}_n( \mathcal{S}_{n-1}(...\mathcal{S}_{1}(x)))$, and backpropagate through each of them. This is certainly possible, but not a good idea: it can introduce numerical problems, and will be very slow.
As mentioned above, by default DL frameworks store the internal states for every differentiable operator like the $\mathcal{S}_i()$ in this example, and hence we'd organize and keep a potentially huge number of intermediate states in memory. These states are completely uninteresting for our original PDE, though. They're just intermediate states of the CG solver.
If we take a step back and look at $p = (\nabla^2)^{-1} b$, it's gradient $\partial p / \partial b$
is just $((\nabla^2)^{-1})^T$. And in this case, $(\nabla^2)$ is a symmetric matrix, and so $((\nabla^2)^{-1})^T=(\nabla^2)^{-1}$. This is the identical inverse matrix that we encountered in the original equation above, and hence we re-use our unmodified iterative solver to compute the gradient. We don't need to take it apart and slow it down by storing intermediate states. However, the iterative solver computes the matrix-vector-products for $(\nabla^2)^{-1} b$. So what is $b$ during backpropagation? In an optimization setting we'll always have our loss function $L$ at the end of the forward chain. The backpropagation step will then give a gradient for the output, let's assume it is $\partial L/\partial p$ here, which needs to be propagated to the earlier operations of the forward pass. Thus, we simply invoke our iterative solve during the backward pass to compute $\partial p / \partial b = \mathcal{S}(\partial L/\partial p)$. And assuming that we've chosen a good solver as $\mathcal{S}$ for the forward pass, we get exactly the same performance and accuracy in the backwards pass.
If you're interested in a code example, the [differentiate-pressure example]( https://github.com/tum-pbs/PhiFlow/blob/master/demos/differentiate_pressure.py) of phiflow uses exactly this process for an optimization through a pressure projection step: a flow field that is constrained on the right side, is optimized for the content on the left, such that it matches the target on the right after a pressure projection step.
The main take-away here is: it is important _not to blindly backpropagate_ through the forward computation, but to think about which steps of the analytic equations for the forward pass to compute gradients for. In cases like the above, we can often find improved analytic expressions for the gradients, which we then approximate numerically.
```{admonition} Implicit Function Theorem & Time
:class: tip
**IFT**:
The process above essentially yields an _implicit derivative_. Instead of explicitly deriving all forward steps, we've relied on the [implicit function theorem](https://en.wikipedia.org/wiki/Implicit_function_theorem) to compute the derivative.
**Time**: we _can_ actually consider the steps of an iterative solver as a virtual "time",
and backpropagate through these steps. In line with other DP approaches, this enables an NN to _interact_ with an iterative solver. An example is to learn initial guesses of CG solvers from {cite}`um2020sol`.
[Details and code can be found here.](https://github.com/tum-pbs/CG-Solver-in-the-Loop)
```
## Summary of differentiable physics so far
To summarize, using differentiable physical simulations
gives us a tool to include physical equations with a chosen discretization into DL.
In contrast to the residual constraints of the previous chapter,
this makes it possible to let NNs seamlessly interact with physical solvers.
We'd previously fully discard our physical model and solver
once the NN is trained: in the example from {doc}`physicalloss-code`
the NN gives us the solution directly, bypassing any solver or model equation.
The DP approach substantially differs from the physics-informed NNs (v2) from {doc}`physicalloss`,
it has more in common with the controlled discretizations (v1). They are essentially a subset, or partial
application of DP training.
However in contrast to both residual approaches, DP makes it possible to train an NN alongside
a numerical solver, and thus we can make use of the physical model (as represented by
the solver) later on at inference time. This allows us to move beyond solving single
inverse problems, and yields NNs that quite robustly generalize to new inputs.
Let's revisit the example problem from {doc}`physicalloss-code` in the context of DPs.