Environment
---
action $a_{t}$ state $s_{t}$
---
reward $r_{t}$
Figure 13: Reinforcement learning in which an agent interacts with an
environment with actions $a_{t}$, states $s_{t}$, and rewards $r_{t}$. Figure
adapted from [587].

###  6.1 Applications

Deep reinforcement learning is mainly used for inverse problems (see [8] for a
review within fluid mechanics), where the PDE solver is treated as a black
box, and assumed to not be differentiable.

The most prominent application are control problems. One example is
discovering swimming strategies for fish – with the goal of efficiently
minimizing the distance to a leader fish [588, 589]. The environment is given
by the Navier Stokes equation. Another example is balancing rigid bodies with
fluid jets while using as little force as possible [590]. Similarly, [591]
control jets in order to reduce the drag around a cylinder. Reducing the drag
around a cylinder is also achieved by controlling small rotating cylinders in
the wake of the flow [592]. A more complex example is controlling unmanned
aerial vehicles [593]. The control schemes are learned by interacting with
simulations and, subsequently, applied in experiments.

Further applications in connection with inverse problems are learning filters
to perturb flows in order to match target flows [594]. Also, constitutive laws
can be identified. The individual arithmetic manipulations within a
constitutive law can be represented as graphs. An agent constructs the graph
in order to best match simulation and measurement [595], which yields an
interpretable law.

Topology optimization has also been tackled by reinforcement learning.
Specifically, the ability to predict only binary states (material or no
material) is desirable – instead of intermediate states, as in solid isotropic
material with penalization [596, 597]. This has been shown with binary truss
structures, modeled with graphs in order to minimize the total structural
volume under stress constraints. In [598], an agent removes trusses from
existing structures, and trusses are added in [599]. Similarly, [600] removes
finite elements in solid structures to modify the topology. Instead, [601]
pursues design diversity. Here a NN surrogate model predicts near optimal
structures from reference designs. The agent then learns to generate reference
designs as input, such that the corresponding optimal structures are as
diverse as possible.

Also, high-dimensional PDEs have been solved with reinforcement learning [602,
603]. This is achieved by recasting the PDE into stochastic control problems,
thereby solving these with reinforcement learning.

Finally, adaptive mesh refinement algorithms have been learned by
reinforcement learning [604]. An agent decides whether an element is to be
refined based on the current state, i.e., the mesh and solution. The reward is
subsequently defined in terms of the error reduction, which is computed with a
ground truth solution. The trained agent can thus be applied to adaptive mesh
refinement to previously unseen simulations.

####  6.1.1 Extensions

Each interaction with the environment requires solving the differential
equation, which, due to the many interactions, makes reinforcement learning
expensive. The learning can be accelerated through some basic modifications.
The learning can be perfectly parallelized by using multiple environments
simultaneously [605], or by using multiple agents within the same environment
[606]. Another idea is to construct a surrogate model of the environment and
thereby exploit model-based approaches [607, 608, 609, 610]. The general
procedure consists of three steps:

  * •

model learning: learn surrogate of environment,

  * •

behavior learning: learn policy or value function,

  * •

environment interaction: apply learned policy and collect data.

Most approaches construct the surrogate with data-driven modeling (Section
2.1), but physics-informed approaches have been proposed as well [607, 609]
(Section 3.2).

##  7 Conclusion

In order to structure the state-of-the-art, an overview of the most prominent
deep learning methods employed in computational mechanics was presented. Six
main categories were identified: simulation substitution, simulation
enhancement, discretizations as NNs, generative approaches, and deep
reinforcement learning.

Despite the variety and abundance of the literature, few approaches are
competitive in comparison to classical methods. With only few exceptions,
current research is still in its early stages, with a focus on showcasing
possibilities without focusing too much attention on accuracy and efficiency.
Future research must, nevertheless, shift its focus to incorporate more in-
depth investigations into the performance of the developed methods – including
thorough and meaningful comparisons to classical methods. This is in agreement
with the recent review article on deep learning in topology optimization [5],
where critical and fair assessments are requested. This includes the
determination of generalization capabilities, greater transparency by
including, e.g., worst case performances to illustrate reliability, and
computation times without disregard of the training time.

In line with this, and to the best of our knowledge, we provide a final
overview outlining the potentials and limitations of the discussed methods.

  * •

Simulation substitution has potential for surrogate modeling of parameterized
models that need to be evaluated many times. This is however, currently only
realizable for small parameter spaces, due to the amount of data required.
Complex problems can still be solved if they are first reduced to a low-
dimensional space through model order reduction techniques. Physics-informed
learning further reduces the amount of required data and improves the
generalization capabilities. However, enforcing physics through penalty terms
increases the computational effort, where the solutions still do not
necessarily satisfy the corresponding physical laws. Instead, enforcing
physics by construction, which guarantees the enforced physics, seems more
favorable.

  * •

Simulation enhancement is currently one of the most useful approaches. It is
in particular beneficial for tasks where classical methods show difficulties.
An excellent example for this is the formulation of constitutive laws, which
are inherently phenomenological and thereby well-suited to be identified from
data using tools such as deep learning. In addition, simulation enhancement,
makes it possible to draw on insights gained from classical methods developed
since the inception of computational mechanics. Furthermore, it is currently
more realistic to learn smaller components of the simulation chain with NNs
rather than the entire model. These components should ideally be expensive and
have limited requirements regarding the accuracy. Lastly, it is also easier to
assess whether a method enhanced by deep learning outperforms the classical
method, as direct and fair comparisons are readily possible.

  * •

An important research direction is to employ discretizations as NNs, as this
offers the potential to discover NNs tailored to computational mechanics
tasks, such as CNNs for computer vision and RNNs and transformers for natural
language processing. Their main benefit comes from exploiting the
computational benefits of tools and hardware that were created for the wider
community of deep learning – such as NN libraries and GPUs.

  * •

Generative approaches have been shown to be highly versatile in applications
of computational mechanics since the accuracy of a specific instance under
investigation is less of a concern here. They have been used for realistic
data generation to train other machine learning models, incorporate vague
constraints based on data within optimization frameworks, and detect
anomalies.

  * •

Deep reinforcement learning has already shown encouraging results – for
example in controlling unmanned vehicles in complex physics environments. It
is mainly applicable for problems where efficient differentiable physics
solvers are unavailable, which is why it is popular in control problems for
turbulence. In the presence of differentiable solvers, gradient-based methods
are however still the state-of-the-art [418] and thus preferred.

## Acknowledgements

The authors gratefully acknowledge the funding through the joint research
project Geothermal-Alliance Bavaria (GAB) by the Bavarian State Ministry of
Science and the Arts (StMWK) as well as the Georg Nemetschek Institut (GNI)
under the project DeepMonitor.

## Declarations

Conflict of interest No potential conflict of interest was reported by the
authors.

##  Appendix A Deep Reinforcement Learning

In reinforcement learning, the environment is commonly modeled as a Markov
Decision Process (MDP). This mathematical model is defined by a set of all
possible states $S$, actions $A$, and associated rewards $R$. Furthermore, the
probability of getting to the next state $s_{t+1}$ from the previous $s_{t}$
with action $a_{t}$ is given by $\mathbb{P}(s_{t+1}|s_{t},a_{t})$. Thus, the
environment is not necessarily deterministic. One key aspect of a Markov
Decision Process is the Markov property, stating that future states depend
solely on the current state and action, and not the history of states and
actions.

The goal of a reinforcement learning algorithm is to determine a policy
$\pi(s,a)$ which dictates the next action $a_{t}$ in order to maximize the
cumulative reward $R_{\Sigma}$. The cumulative reward $R_{\Sigma}$ is
discounted by a discount factor $\gamma^{t}$ in order to give more importance
to immediate rewards.

$R_{\Sigma}=\sum_{t=0}^{\infty}\gamma^{t}r_{t}$ (81)

The quality of a policy $\pi(s,a)$ can be assessed by a state-value function
$V_{\pi}(s)$, defined as the expected future reward given the current state
$s$ and following the policy $\pi$. Similarly, an action-value function
$Q_{\pi}(s)$ determines the expected future reward given the current state $s$
and action $a$ and then following the policy $\pi$. The expected value along a
policy $\pi$ is denoted as $\mathbb{E}_{\pi}$.

$\displaystyle V_{\pi}(s)=\mathbb{E}_{\pi}\bigl{[}R_{\Sigma}(t)|s\bigr{]}$
(82) $\displaystyle
Q_{\pi}(s,a)=\mathbb{E}_{\pi}\bigl{[}R_{\Sigma}(t)|s,a\bigr{]}$ (83)

The optimal value and quality function correspondingly follow the optimal
policy:

$\displaystyle V(s)=\max_{\pi}V_{\pi}(s),$ (85) $\displaystyle
Q(s,a)=\max_{\pi}Q_{\pi}(s).$ (86)

The approaches can be subdivided into model-based and model-free. Model-based
methods incorporate a model of the environment. In the most general sense, a
probabilistic environment, this entitles the probability distribution of the
next state $\mathbb{P}(s_{t+1}|s_{t},a_{t})$ and of the next reward
$\mathbb{R}(r_{t+1}|s_{t+1},s_{t},a_{t})$. The model of the environment can be
cheaply sampled to improve the policy $\pi$ with model-free reinforcement
learning techniques [611, 612, 613, 614] discussed in the sequel (Sections A.1
and A.2). However, if the model is differentiable, the gradient of the reward
can directly be used to update the policy [615, 616, 617, 618, 619, 620]. This
is identical to the optimization through differentiable physics solvers
discussed in Section 3.3.3. Model-free reinforcement learning techniques can
be used to enhance the optimization.

A further distinction is made between policy-based [621, 622, 623, 624, 625]
and value-based [626, 627, 628] approaches. Policy-based methods, such as deep
policy networks [21] (Section A.1), directly optimize the policy. By contrast,
value-based methods, such as deep Q-learning [628] (Section A.2) learn the
value function from which the optimal policy is selected. Actor-critic
methods, such as proximal policy optimization [629] combine the ideas with an
actor that performs a policy and a critic that judges its quality. Both can be
modeled by NNs.

###  A.1 Deep Policy Networks

In deep policy networks, the policy, i.e., the mapping of states to actions,
is modeled by a NN $\hat{a}=\pi(s;\boldsymbol{\theta})$. The quality of the NN
is assessed by the expected cumulative reward $R_{\Sigma}$, formulated in
terms of the action-value function $Q(s,a)$.

$C=R_{\Sigma}=\mathbb{E}\bigl{[}Q(s,a)\bigr{]}\\\ $ (87)

Its gradient (see [21, 622, 624] for a derivation), given as:

$\nabla_{\boldsymbol{\theta}}R_{\Sigma}=\mathbb{E}\bigl{[}Q(s,a)\nabla_{\boldsymbol{\theta}}\log\bigl{(}\pi(s,a;\boldsymbol{\theta})\bigr{)}\bigr{]}$
(88)

can be applied within a gradient ascent scheme to learn the optimal policy.

###  A.2 Deep Q-Learning

Deep Q-learning identifies the optimal action-value function $Q(s,a)$ from
which the optimal policy is extracted. Q-Learning relies on the Bellman
optimality criterion [630, 631]. By separating the reward $r_{0}$ at the first
step, the recursion formula of the optimal state-value function, i.e., the
Bellman optimality criterion, can be established:

$\displaystyle V(s)$
$\displaystyle=\max_{\pi}\mathbb{E}_{\pi}\Bigl{[}\sum_{t=0}^{\infty}\gamma^{t}r_{t}|s_{0}=s\Bigr{]}$
(89)
$\displaystyle=\max_{\pi}\mathbb{E}_{\pi}\Bigl{[}r_{0}+\sum_{t=1}^{\infty}\gamma^{t}r_{t}|s_{1}=s^{\prime}\Bigr{]}$
(90) $\displaystyle=\max_{\pi}\mathbb{E}_{\pi}\bigl{[}r_{0}+\gamma
V(s^{\prime})\bigr{]}.$ (91)

Here, $s^{\prime}$ represents the next state after $s$. This can be done
analogously for the action-value function.

$Q(s,a)=\max_{\pi}\mathbb{E}_{\pi}\bigl{[}r_{0}+\gamma
Q(s^{\prime},a^{\prime})\bigr{]}$ (92)

The recursion enables an update formula, referred to as temporal difference
(TD) learning [632, 633]. Specifically, the current estimate $Q^{(m)}$ at
state $s_{t}$ is compared to the more accurate estimate at the next state
$s_{t+1}$ using the obtained reward $r_{t}$, referred to as the TD target
estimate. The difference is the TD error, which in combination with a learning
rate $\alpha$ is used to update the function $Q^{(m)}$:

$Q^{(m+1)}(s_{t},a_{t})=Q^{(m)}(s_{t},a_{t})+\alpha\overbrace{\bigl{(}\underbrace{r_{t}+\gamma\max_{a}Q(s_{t+1},a)}_{\textrm{TD
target estimate}}-\underbrace{Q^{(m)}(s_{t},a_{t})}_{\textrm{model
prediction}}\bigr{)}}^{\textrm{TD error}}.$ (93)

Here, the TD target estimate only looks one step ahead – and is therefore
referred to as TD(0). The generalization is called TD(N). In the limit
$N\rightarrow\infty$, the method is equivalent to Monte Carlo learning, where
all steps are performed and a true target is obtained.

Deep Q-learning introduces a NN for the action-value function
$Q(s,a;\boldsymbol{\theta})$. Its quality is assessed with a loss composed of
the mean squared error of the TD error.

$C=\mathbb{E}\Bigl{[}\frac{1}{2}\bigl{(}r_{t}+\gamma\max_{a}Q(s_{t+1},a;\boldsymbol{\theta})-Q(s_{t},a_{t};\boldsymbol{\theta})\bigr{)}^{2}\Bigr{]}$
(94)

Lastly, the optimal policy $\pi(s)$ maximizing the action-value function
$Q(s,a;\boldsymbol{\theta})$ is extracted:

$\pi(s)=\underset{a}{\arg\max}\textrm{ }Q(s,a;\boldsymbol{\theta})$ (95)

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