# A physics-informed GAN Framework based on Model-free Data-Driven
Computational Mechanics

Kerem Ciftci<EMAIL_ADDRESS>Klaus Hackl<EMAIL_ADDRESS>Institute of
Mechanics of Materials, Ruhr University Bochum, Universitätsstrasse 150, 44801
Bochum, Germany.

###### Abstract

Model-free data-driven computational mechanics, first proposed by Kirchdoerfer
and Ortiz, replace phenomenological models with numerical simulations based on
sample data sets in strain-stress space. In this study, we integrate this
paradigm within physics-informed generative adversarial networks (GANs). We
enhance the conventional physics-informed neural network framework by
implementing the principles of data-driven computational mechanics into GANs.
Specifically, the generator is informed by physical constraints, while the
discriminator utilizes the closest strain-stress data to discern the
authenticity of the generator’s output. This combined approach presents a new
formalism to harness data-driven mechanics and deep learning to simulate and
predict mechanical behaviors.

###### keywords:

Model-free Data-Driven , Generative Adversarial Networks , Data-Driven
Computing , Physics-informed Neural Networks

††journal: Computer Methods in Applied Mechanics and Engineering

##  1 Introduction

The simulation of boundary value problems typically contains two equations:
conservation and constitutive laws. While conservation laws are derived from
universal principles, constitutive laws are usually obtained by fitting model
parameters to given strain-stress data [1]. Nevertheless, material modeling
can be ill-posed and adds uncertainties to the solutions, particularly in
highly complex systems. The model-free data-driven method, introduced by
Kirchdoerfer and Ortiz [2], bypasses the step of material modeling,
incorporating experimental data directly into the numerical simulations of
boundary-value problems.
The data-driven scheme bypasses the empirical material modeling step by
computing the closest point in the material data set consistent with the
problem’s compatibility and equilibrium condition. Consequently, it provides
an alternative formulation of the classical initial-boundary-value problem
based on nearest-neighbor clustering.
The approach has been fine-tuned for diverse applications: from non-linear
elasticity [2, 3, 4, 5, 6] to dynamics [7] and finite strain [8]. It’s also
been adapted for material data identification [9], non-local mechanics [10],
electro-mechanical problems [11], homogenization schemes [12], and model-
driven coupling [13]. Ibañez et al. [14, 15] refined the approach using a
manifold learning method that maps data into a lower-dimensional space to use
the locally linear embeddings. Eggersmann et al. [16] presented a second-order
data-driven approach that uses tensor voting [17] to obtain point-wise tangent
space, enabling the search for additional states close to the original data.
For inelastic boundary value problems, Eggersmann et al. [18] include local
histories in the data set to investigate materials with memory. Karapiperis et
al. [19] have also suggested a variation of the scheme, considering multiscale
modeling. In addition, we recently developed a paradigm incorporating the
tangent space into the distance-minimizing data-driven formulation and
classifies the underlying data structure into subsets according to various
material behaviors [20]. The framework features a parametrization of the
material history and an optimal sampling of the mechanical system’s state
space.
The paradigm’s dependence on the nearest-neighbor clustering of data points
proposes research areas in machine-learning methods, particularly Artificial
Neural Networks (ANNs), that are known to approximate any continuous function
for appropriate network parameters [21, 22]. The flexibility and quality of
neural networks led to success in a wide range of applications, e.g., image
recognition [23], language processing [24], or generative modeling [25, 26].
An extension to neural networks is physics-informed deep learning,
successfully used in solving physical-related problems such as fluid mechanics
[27, 28], aerodynamics [29, 30], shell structures [31] or material science
[32, 33]. Physics-Informed Neural Networks (PINNs) can be trained to fulfill
training data and learn optimal solutions for allocated physics-governing
equations by specifying appropriate loss functions [34, 35]. The physics-based
loss competes against a data-based loss, which is needed to provide
fundamental knowledge of the system. Thus, partial differential equations act
as additional constraints during network training, resulting in a multi-
objective optimization problem. Optimizing data and physics give physics-
informed neural networks flexibility in solving forward and inverse problems
[30, 28, 29, 36, 37, 38]. The trade-off between the individual losses can be
influenced using hyper-parameter [39, 40, 41]. For example, adaptive
activation functions [42, 43], or manually weighted losses [44], can improve
the quality of the neural network for specific problems. Another approach to
overcome the local convergence issue due to global approximation is the usage
of adaptive training strategies and domain decomposition [45]. This
investigation combines the model-free data-driven approach with Generative
Adversarial networks (GANs). In machine learning, GANs have emerged as a
powerful tool consisting of two neural networks – the generator, which creates
data, and the discriminator, which evaluates the authenticity of the generated
data. Through their adversarial game, GANs are adept at generating high-
fidelity data, often indistinguishable from actual data [26]. An extension is
the integration of physics-informed neural networks with the GAN structure.
For instance, the pursuit of robust uncertainty quantification within the
framework of PINNs has led to recent methodologies. The PIG-GAN framework [46]
harnesses the capabilities of a physics-informed generator to address
adversarial uncertainty. On the other hand, the PID-GAN approach [47] uses a
physics-informed discriminator, carving out a distinct avenue to achieve
reliable uncertainty quantification while maintaining fidelity to the
governing physics. Another stride in this direction is the DeqGAN, which
offers a unique perspective on PINNs by learning the loss function via
generative adversarial networks. This methodology provides a robust avenue for
solving the challenges traditionally associated with defining appropriate loss
functions for PINNs [48]. In our approach, the generator is a physics-informed
neural network, and the discriminator employs the closest strain-stress data
to evaluate the authenticity of the generator’s results. This synergized
methodology matches model-free data-driven computational mechanics and deep
learning principles, aspiring to more accurately simulate and predict
intricate mechanical behaviors.

Section 2 provides a general setting by introducing the definitions and
derivation of the distance-minimizing data-driven computing method based on
[4]. Section 3 introduces the framework of artificial neural networks and
generative adversarial networks. In addition, we propose using a physics-
informed GAN to solve the distance-minimizing data-driven problem. Section 4
exhibits the performance of the proposed method using a numerical example
involving a non-linear elastic in-plane boundary value problem. Finally,
Section 5 summarizes the results and suggests future research subjects.

##  2 Model-free Data-driven setting

The following will summarize the classical data-driven computational mechanics
method for the reader’s convenience based on the definitions and formulations
in [4]. We consider an elastic body $\Omega\subset\mathbb{R}^{d}$ whose
internal states are defined by displacement field
$\mbox{\boldmath$u$}:\Omega\to\mathbb{R}^{d}$ and the compatibility and
equilibrium conditions

$\displaystyle\mbox{\boldmath$\varepsilon$}(\mathbb{x})-\nabla^{\textrm{sym}}\mbox{\boldmath$u$}(\mathbb{x})=\mathbb{0},$
$\displaystyle\text{in }\Omega,$ (1)
$\displaystyle\nabla\cdot\mbox{\boldmath$\sigma$}(\mathbb{x})-\mbox{\boldmath$f$}(\mathbb{x})=\mathbb{0},$
$\displaystyle\text{in }\Omega,$

and boundary conditions

$\displaystyle\mbox{\boldmath$u$}(\mathbb{x})=\mbox{\boldmath$g$}(\mathbb{x}),$
$\displaystyle\text{on }\Gamma_{D},$ (2)
$\displaystyle\mbox{\boldmath$\sigma$}(\mathbb{x})\cdot\mbox{\boldmath$n$}(\mathbb{x})=\mbox{\boldmath$t$}(\mathbb{x}),$
$\displaystyle\text{on }\Gamma_{N},$

where
$\mbox{\boldmath$\varepsilon$}:\Omega\to\mathbb{R}^{{d}\times{d}}_{\textrm{sym}}$
is the strain field and
$\mbox{\boldmath$\sigma$}:\Omega\to\mathbb{R}^{{d}\times{d}}_{\textrm{sym}}$
is the stress field. The boundary $\Gamma$ of the domain $\Omega$ is defined
by the Dirichlet ($\Gamma_{D}$) and Neumann ($\Gamma_{N}$) with
$\Gamma=\Gamma_{D}\cup\Gamma_{N}$ and $\Gamma_{D}\cap\Gamma_{N}=\emptyset$. In
addition, $\mbox{\boldmath$f$}:\Omega\to\mathbb{R}^{d}$ is the body force, and
$\mbox{\boldmath$g$},\mbox{\boldmath$t$},\mbox{\boldmath$n$}:\Gamma\to\mathbb{R}^{d}$
define the boundary displacement, applied traction and outer normal,
respectively.
We define
$Z_{\textrm{loc}}\subset\mathbb{R}^{{d}\times{d}}_{\textrm{sym}}\times\mathbb{R}^{{d}\times{d}}_{\textrm{sym}}$
as the local phase space consisting of pairs
$\mathbb{z}(\mathbb{x})=(\mbox{\boldmath$\varepsilon$}(\mathbb{x}),\mbox{\boldmath$\sigma$}(\mathbb{x}))$
describing the local state of the system at material point $\mathbb{x}$. The
global phase space $Z$ is defined as the collection of the state functions,
i.e.

$\displaystyle Z=\\{\mathbb{z}\,:\,\mathbb{z}\in Z_{\textrm{loc}}\\}.$ (3)

The data-driven distance-minimization problem, introduced by [2], reads

$\displaystyle\mathop{\mathrm{arg\,min}}_{\mathbb{z}\in\mathcal{C},\hat{\mathbb{z}}\in\mathcal{D}}d(\mathbb{z},\hat{\mathbb{z}}),$
(4)

where $\mathcal{C}\subset Z$ denotes the constraint set defined by

$\mathcal{C}:=\Big{\\{}\mathbb{z}\in
Z:\eqref{eq:constraint}\;\text{and}\;\eqref{eq:bc_constraint}\Big{\\}};$ (5)

containing all states fulfilling compatibility and equilibrium. The set
$\mathcal{D}\subset Z$ consists of a finite number of experimental
measurements achieved from small-scale simulations and is defined by

$\mathcal{D}=\\{\mathbb{z}\in
Z\,:\,\mathbb{z}(\mathbb{x})\in\mathcal{D}_{\textrm{loc}}\\}\quad\text{with}\quad\mathcal{D}_{\textrm{loc}}=\\{(\mbox{\boldmath$\varepsilon$}_{i},\mbox{\boldmath$\sigma$}_{i})\\}_{i=1}^{n_{e}},$
(6)

where $n_{e}\in\mathbb{N}$ is the number of local data points associated with
the material point. The distance $d:Z\times Z\to\mathbb{R}$ is defined by

$\displaystyle
d(\mathbb{z},\hat{\mathbb{z}}):=\|\mathbb{z}-\hat{\mathbb{z}}\|,$ (7)

metricized by the norm

$\|\mathbb{z}\|^{2}:=\int\limits_{\Omega}\left(\frac{1}{2}\mbox{\boldmath$C$}\mbox{\boldmath$\varepsilon$}\cdot\mbox{\boldmath$\varepsilon$}+\frac{1}{2}\mbox{\boldmath$C$}^{-1}\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$\sigma$}\right)\mathbb{d}\mathbb{x},$
(8)

where $\mbox{\boldmath$C$}\in\mathbb{R}^{{d}\times{d}}_{\textrm{sym}}$ is a
symmetric positive definite matrix typically being of the type of elastic
stiffness. Thus, the data-driven paradigm aims to find the closest point
$\mathbb{z}$ in the constraint set $\mathcal{C}$ to $\hat{\mathbb{z}}$ in the
material data set $\mathcal{D}$.

Challenges such as data availability, noise, inconsistency, and high
dimensionality frequently arise in the data-driven paradigm. Traditional
analytical and computational methods may need to be adjusted when addressing
these issues. Consequently, the incorporation of machine learning,
particularly methods like generative adversarial networks coupled with
physics-informed generators, is considered. This integration is aimed at
effectively handling the complexities of data-driven datasets, ensuring the
outcomes remain consistent with domain-specific knowledge. The following
sections will present a detailed discussion on the principles of artificial
neural networks and physics-informed neural networks, illustrating the
approach of physics-informed generative adversarial networks to solve the
data-driven boundary value problem (4).

##  3 Generative adversarial networks with physics-informed generators for
model-free data-driven problems

This section delves into the application of Generative Adversarial Networks
(GANs) equipped with Physics-Informed Generators for addressing the model-free
data-driven problem. A GAN involves a competitive dynamic between two neural
networks, forming a zero-sum game: one network’s success implies the other’s
setback. To harness GANs for resolving the data-driven boundary value problem
depicted in (4), Section 3.1 initiates with a concise overview of Artificial
Neural Networks (ANNs) and explains physics-informed neural networks (PINNs).
Section 3.2 lays out the foundational principles of GANs, and in Section 3.3,
we pivot to the novel approach of leveraging GANs augmented with PINNs to
solve the data-driven boundary value problem.

###  3.1 Physics-informed neural networks

Based on the universal function approximation theorem [49], an artificial
neural network is a parametrized, non-linear function composition that can
approximate arbitrary Borel measurable functions. This section introduces the
basic concept based on the definitions and formulations in [45]. For this
purpose, we introduce a densely connected feed-forward neural network, denoted
by the map $\mathcal{N}:\mathbb{R}^{d_{x}}\times[0,T]\to\mathbb{R}^{d_{y}}$,
which is defined by a composition of $n_{L}\in\mathbb{N}$ non-linear
functions:

$\displaystyle\mathcal{N}:\mathbb{R}^{d_{x}}\times[0,T]$
$\displaystyle\to\mathbb{R}^{d_{y}}$ (9) $\displaystyle(\mathbb{x},t)$
$\displaystyle\mapsto\mathcal{N}(\mathbb{x},t)=\mathbb{y}^{(\ell)}\circ\ldots\circ\mathbb{y}^{(0)}=\mathbb{y},$
(10)

for $\ell=1,\ldots,n_{L}$, where $\mathbb{x}$ denotes the spatial part of the
input vector of dimension $d_{x}\in\mathbb{N}$ at time $t\in[0,T]$ with $T>0$
and $\mathbb{y}$ denotes the output vector of dimension $d_{y}\in\mathbb{N}$.
In this context, $\mathbb{y}^{(0)}$ and $\mathbb{y}^{(n_{L})}$ are called the
input and output layer, such that

$\displaystyle\mathbb{y}^{(0)}=(\mathbb{x},t),\qquad\mathbb{y}^{(n_{L})}=\mathcal{N}(\mathbb{x},t).$
(11)

The functions $\mathbb{y}^{(\ell)}$ are called hidden layers and define a
$\ell-$fold composition, mapping the input $(\mathbb{x},t)$ to the output
$\mathbb{y}$ by

$\displaystyle\mathbb{y}^{(\ell)}=\\{\mathbb{y}^{(\ell)}_{\eta},\,\eta=1,\ldots,\eta_{u}\\},\text{
with
}\mathbb{y}^{(\ell)}_{\eta}=\mathrm{act}^{(\ell)}\left(\mathbb{W}^{(\ell)}_{\eta}\mathbb{y}^{(\ell-1)}+\mathbb{b}_{\eta}^{(\ell)}\right).$
(12)

We call $\mathbb{y}^{(\ell)}_{\eta}$ the $\eta^{\text{th}}$ neural unit of the
$\ell^{\text{th}}$ layer $\mathbb{y}^{(\ell)}$, where $\eta_{u}\in\mathbb{N}$
is the total number of neural units per layer. $\mathbb{W}_{\eta}^{(\ell)}$
and $\mathbb{b}_{\eta}^{(\ell)}$ denote the weight matrix and bias vector of
the $\eta^{\text{th}}$ neural unit in the $\ell^{\text{th}}$ layer
$\mathbb{y}^{(\ell)}$. Furthermore
$\mathrm{act}^{(\ell)}(\cdot):\mathbb{R}\to\mathbb{R}$ is a non-linear
activation function. All weights and biases of all layers
$\mathbb{y}^{(\ell)}$ are assembled in

$\displaystyle\bm{\theta}=\Big{\\{}\left(\mathbb{W}_{\eta}^{(\ell)},\mathbb{b}_{\eta}^{(\ell)}\right);\,\ell=1,\ldots,n_{L},\,\eta=1,\ldots,\eta_{u}\Big{\\}},$
(13)

including all parameters of the neural network $\mathcal{N}(\mathbb{x},t)$. As
a result, the notation $\mathcal{N}(\mathbb{x},t;\bm{\theta})$ highlights the
dependence of a neural network’s output on the input and the current
realization of the weights and biases. Figure 1 illustrates the network’s
topology, a combination of layers, neural units, and activation functions.

Figure 1: Schematic representation of the neural network architecture
$\mathcal{N}(\mathbb{x},t;\bm{\theta})$, starting with multi-input parameters
$\mathbb{x}$ and $t$ at input layer $\mathbb{y}^{(0)}$, progressing through
sequential hidden layer $\mathbb{y}^{(1)},\ldots,\mathbb{y}^{(n_{L}-1)}$. The
architecture concludes with an output layer $\mathbb{y}^{(n_{L})}$, producing
the final output $\mathbb{y}$.

The main idea of solving boundary value problems with an artificial neural
network is the reformulation to an optimization problem [50, 35, 51], where
the residual of the differential equations is to be minimized. To solve the
differential equation (1) and (2), a suitable topology for the artificial
neural network and, consequently, the physics-informed neural networks
described in Section 3.1 has to be chosen. Since (1) is stationary, we can
reduce the artificial neural network to $\mathcal{N}(\mathbb{x};\bm{\theta})$.
Thus, we can define neural networks as an ansatz for the displacement and
stress field i.e.

$\displaystyle\mbox{\boldmath$u$}(\mathbb{x},t)\approx\mathcal{N}_{u}(\mathbb{x};\bm{\theta}_{u}),$
(14)
$\displaystyle\mbox{\boldmath$\sigma$}(\mathbb{x},t)\approx\mathcal{N}_{\sigma}(\mathbb{x};\bm{\theta}_{\sigma}),$
(15)

with trainable network parameters
$\bm{\theta}:=\\{\bm{\theta}_{u},\bm{\theta}_{\sigma}\\}$. Notably, there is
no separate network for the strain tensor. The strain tensor is deduced using
the kinematics and differentiation applied to the displacement network, i.e.
$\mbox{\boldmath$\varepsilon$}=\nabla^{\textrm{sym}}\mathcal{N}_{u}(\mathbb{x};\bm{\theta}_{u})$.
The architecture of this artificial neural network is visualized in Fig.2.

Figure 2: Schematic representation of the neural network’s topology,
illustrating the progression from input through multiple hidden layers,
resulting in displacement components $u_{i}$ and stress components
$\sigma_{ij}$.

Using the neural network ansatz we can rewrite the physics (1) and (2) as

$\displaystyle R_{\Omega}$
$\displaystyle=\nabla\cdot\mathcal{N}_{\sigma}(\mathbb{x},\bm{\theta}_{\sigma})-\mbox{\boldmath$f$}(\mathbb{x}),$
$\displaystyle\text{in }\Omega,$ (16) $\displaystyle R_{\Gamma_{D}}$
$\displaystyle=\mathcal{N}_{u}(\mathbb{x},\bm{\theta}_{u})-\mbox{\boldmath$g$}(\mathbb{x}),$
$\displaystyle\text{on }\Gamma_{D},$ $\displaystyle R_{\Gamma_{N}}$
$\displaystyle=\mathcal{N}_{\sigma}(\mathbb{x},\bm{\theta}_{\sigma})\cdot\mbox{\boldmath$n$}(\mathbb{x})-\mbox{\boldmath$t$}(\mathbb{x}),$
$\displaystyle\text{on }\Gamma_{N},$

where $R_{\textrm{PDE}}$ penalizes the residual of the equilibrium equation,
and the equations $R_{\Gamma_{D}}$ and $R_{\Gamma_{N}}$ describe the
discrepancy of the Dirichlet and Neumann boundary conditions. Notice that if
$\mathcal{N}_{u}$ and $\mathcal{N}_{\sigma}$ is a solution to the original
boundary value problem, they minimize the differential equation-based
residuals.
The parameters $\bm{\theta}$ of the networks can be found by incorporating the
physics-induced residuals into the training process of a neural network as
components of the loss function. For this, we use a collocation method
discretizing the domain $\Omega$ and the boundary
$\Gamma:=\Gamma_{D}\cup\Gamma_{N}$ into sets of sample points $S_{\Omega}$ and
$S_{\Gamma}$ with cardinalities $|S_{\Omega}|$ and $|S_{\Gamma}|$. Then, an
optimization problem to find the optimal parameters $\bm{\theta}^{\star}$,
also called training, is defined as

$\displaystyle\bm{\theta}^{\star}$
$\displaystyle=\mathop{\mathrm{arg\,min}}\limits_{\bm{\theta}}L_{\mathcal{C}}$
(17)

with
$L_{\mathcal{C}}:=L_{\Omega}(\mathbb{x},\bm{\theta})+L_{\Gamma}(\mathbb{x},\bm{\theta})$
given by the local losses

$\displaystyle L_{\Omega}$
$\displaystyle=\frac{1}{|S_{\Omega}|}\sum_{\mathbb{x}\in
S_{\Omega}}\|R_{\Omega}(\mathbb{x};\bm{\theta})\|^{2}_{2},$ (18)
$\displaystyle L_{\Gamma}$
$\displaystyle=\frac{1}{|S_{\Gamma}|}\left(\sum_{\mathbb{x}\in
S_{\Gamma_{D}}}\|R_{\Gamma_{D}}(\mathbb{x};\bm{\theta})\|^{2}_{2},+\sum_{\mathbb{x}\in
S_{\Gamma_{N}}}\|R_{\Gamma_{N}}(\mathbb{x};\bm{\theta})\|^{2}_{2}\right).$
(19)

The expressions penalize the residual of the governing equations and the
discrepancy of the Dirichlet and Neumann boundary conditions, respectively.
Notice that in the three-dimensional setting, one defines the neural networks
as tuples, i.e.

$\displaystyle\mathcal{N}_{u}(\mathbb{x};\bm{\theta}_{u})$
$\displaystyle=\\{\mathcal{N}_{u_{i}}(\mathbb{x};\bm{\theta}_{u_{i}})\,|\,i=1,2\\},$
(20) $\displaystyle\mathcal{N}_{\sigma}(\mathbb{x};\bm{\theta}_{\sigma})$
$\displaystyle=\\{\mathcal{N}_{\sigma_{ij}}(\mathbb{x};\bm{\theta}_{\sigma_{ij}})\,|\,i,j=1,2,3\text{
and }ij=ji\\},$ (21)

including the three components $u_{i}$ of displacement $u$ and six stress
components $\sigma_{ij}$, where $ij=ji$ ensures the symmetry of the stress
tensor $\sigma$. Fig. 3 illustrates the complete network’s structure.

Figure 3: Schematic representation of the physics-informed neural network’s
topology, illustrating the progression from input through multiple hidden
layers, resulting in displacement components $u_{i}$ and stress components
$\sigma_{ij}$. These outputs undergo automatic differentiation to compute a
physics-based loss function $L_{\mathcal{C}}$ . The loss is minimized using an
optimizer to refine the network parameters $\theta$.

While the PINN framework provides a straightforward method to solve physical-
enhanced problems, it has challenges. Notably, there have been instances where
the optimization yields solutions with unexpected or non-physical behaviors
even when carefully tailored to encapsulate the physics [31]. Additionally,
the current PINN formulation must minimize the difference between the
network’s outputs and the available strain-stress data $\mathcal{D}$ due to
the nature of the data-driven distance minimization problem (4). If we
integrate the distance as an additional loss into the global loss, the whole
problem becomes a nested optimization, leading to training challenges. The
neural network could optimize in an undesired direction during each training
epoch. If the approximated strain-stress point is not accurate, the
corresponding data point might be suboptimal concerning the optimization
algorithm, further complicating the learning process. To address these
challenges, we consider the integration of PINNs with generative adversarial
networks. GANs are proficient at generating outputs with the same properties
as actual data, providing a potential approach to generating realistic strain-
stress solutions. Their flexibility ensures adaptability across diverse data
types suited for various physical conditions. Moreover, the inherent
capability of GANs to discern and capitalize on intricate patterns may lead to
a more robust representation of underlying physics. Additionally, with
conditional GANs, generating outputs based on specific conditions becomes
feasible, allowing for more targeted solutions. The combined PINN-GAN approach
seeks to ensure physical consistency and alignment with observed data,
leveraging the strengths of both methodologies. For clarity, we will provide a
brief overview of GAN theory in the following.

###  3.2 Intermezzo to generative adversarial networks

Introduced by Goodfellow et al. [25], generative adversarial networks
illustrate a novel approach to generating data using neural architectures.
These networks comprise two distinctive neural entities: the generator ($G$)
and the discriminator ($D$). The underlying goal of a GAN is to generate data
instances that emulate the properties of actual data. The generation is
achieved by setting the two networks against each other in a competitive game,
often described as a dual-player minimax game.

Taking reference from the definitions provided in (9), we define the real data
space as $\mathbb{D}_{\text{real}}\subset\mathbb{R}^{d_{y}},$ where $d_{y}$ is
the dimension of the space, i.e., $d_{y}=\dim(\mathbb{D}_{\text{real}}).$ The
main objective of GANs is to produce synthetic data denoted as
$\mathbf{y}_{\mathrm{syn}}$, residing in the same space as our real data
$\mathbf{y}_{\text{real}}$. The generator can be defined as a function
$G:\mathbb{R}^{d_{x}}\to\mathbb{R}^{d_{y}},$ which transforms a random noise
vector $\mathbf{x}$ into synthetic data $\mathbf{y}_{\mathrm{syn}}$. In
contrast, the discriminator operates as a function
$D:\mathbb{R}^{d_{y}}\to\mathbb{R},$ that provides a measure of authenticity
for a given data sample. Mathematically, these networks can be illustrated as:

$\displaystyle\begin{split}G:\mathbb{R}^{d_{x}}&\to\mathbb{R}^{d_{y}}\\\
\mathbf{x}&\mapsto\mathcal{N}_{G}(\mathbf{x};\bm{\theta}_{G}),\end{split}\qquad\text{and}\qquad\begin{split}D:\mathbb{R}^{d_{y}}&\to[0,1]\\\
\mathbf{y}&\mapsto\mathcal{N}_{D}(\mathbf{y};\bm{\theta}_{D}).\end{split}$
(22)

Here, $\mathcal{N}_{G}(\mathbf{x};\bm{\theta}_{G})$ and
$\mathcal{N}_{D}(\mathbf{y};\bm{\theta}_{D})$ describe the neural networks
with their corresponding trainable parameters $\bm{\theta}_{G}$ and
$\bm{\theta}_{D}$. The adversarial game between the generator and the
discriminator during training can be encapsulated in the following objective

$\displaystyle L(G,D)=\mathbb{E}_{\mathbf{y}\sim p_{\text{data}}}[\ln
D(\mathbf{y})]+\mathbb{E}_{\mathbf{x}\sim
p_{\mathbf{x}}}[\ln(1-D(G(\mathbf{x})))],$ (23)

leading to the optimization:

$\displaystyle\min_{G}\max_{D}L(G,D),$ (24)

where $\mathbb{E}$ represents a random variable’s expectation or expected
value. It provides a weighted average of a function concerning its probability
distribution. Specifically,

$\displaystyle\mathbb{E}_{\mathbf{y}\sim p_{\text{data}}}[\ln D(\mathbf{y})]$
(25)

represents the average logarithmic score assigned by the discriminator to
actual data samples drawn from the distribution $p_{\text{data}}$. On the
other hand, the expression

$\displaystyle\mathbb{E}_{\mathbf{x}\sim
p_{\mathbf{x}}}[\ln(1-D(G(\mathbf{x})))]$ (26)

reflects the average logarithmic score the discriminator accords to the
synthetic or generated data, which is created from a random noise vector
$\mathbf{x}$ following the noise distribution $p_{\mathbf{x}}$.
The competition between the two networks is straightforward: the generator $G$
aims to produce data that the discriminator $D$ cannot distinguish from
accurate data. In contrast, the discriminator tries to better distinguish real
data from fake data produced by $G$. The probability distributions
$p_{\text{data}}$ and $p_{\mathbf{x}}$ depict the actual data and noise
distributions, respectively. The terms in the objective function essentially
capture the average confidence levels of the discriminator in judging the
authenticity of both original and fake data samples. The procedure of the
GAN’s interplay between the generator and the discriminator is illustrated in
Fig. 4.

Figure 4:  Schematic representation of a generative adversarial network (GAN)
showcasing the interaction between the generator producing data from random
input and the discriminator evaluating the authenticity of both real and
generated data.

Traditional GANs deploy a sigmoid activation function for the discriminator’s
final layer, ensuring its outputs fall within [0,1]. The GANs can suffer from
issues like mode collapse (where the generator generates limited varieties of
samples), vanishing gradients, and general training instability. To address
some of these challenges, the Wasserstein GAN (WGAN) [52] changes the
objective function to leverage the Wasserstein distance [53]. The WGAN
objective can be described as:

$\displaystyle L_{WGAN}(G,D)=\mathbb{E}_{\mathbf{y}\sim
p_{\text{data}}}[D(\mathbf{y})]-\mathbb{E}_{\mathbf{x}\sim
p_{\mathbf{x}}}[D(G(\mathbf{x}))],$ (27)

leading to the following optimization:

$\displaystyle\min_{G}\max_{D}L_{WGAN}(G,D).$ (28)

WGANs are known to provide more stable and consistent training dynamics [52].
Building on the WGAN, the Wasserstein GAN with Gradient Penalty (WGAN-GP)
introduced a regularization term to ensure that the discriminator’s gradients
remain bounded [54]. This gradient penalty aims to enforce the Lipschitz
continuity condition, which addresses the vanishing gradient problem. The
gradient penalty is defined as:

$\displaystyle\text{GP}=\mathbb{E}\left[\left(\lVert\nabla_{\tilde{\mathbf{y}}}D(\tilde{\mathbf{y}})\rVert_{2}-1\right)^{2}\right],$
(29)

where
$\tilde{\mathbf{y}}=\delta\mathbf{y}_{\text{real}}+(1-\delta)\mathbf{y}_{\mathrm{syn}}$
and $\delta$ is sampled from a uniform distribution in $[0,1]$. The
optimization for WGAN-GP thus becomes

$\displaystyle\min_{G}\max_{D}L_{WGAN}(G,D)+\omega\cdot\text{GP},$ (30)

where $\omega\in\mathbb{R}_{+}$ is a hyperparameter determining the weight of
the gradient penalty in the overall objective [55].

###  3.3 Physics-informed GANs for data-driven mechanics problems

In the classical data-driven computational mechanics paradigm Section 2, the
objective is to find the closest point $\mathbb{z}$ in the constraint set
$\mathcal{C}$ to $\hat{\mathbb{z}}$ in the material dataset $\mathcal{D}$, as
formalized in equation (4). This context motivates our modified GAN approach
for data-driven mechanics problems. To utilize GANs for solving differential
equations in a data-driven mechanics setting, we propose a novel approach
wherein the generator in the GAN architecture is identified as a physics-
informed neural network (PINN). In this paradigm, while the generator outputs
plausible solutions adhering to the underlying physics, the discriminator is
trained to distinguish between the generator’s predictions and actual strain-
stress data.
In the conventional GAN setup from Section 3.2, the generator $G$ maps the
input vector $\mathbf{x}$ into synthetic data, $\mathbf{y}_{\mathrm{syn}}$.
Instead of treating $\mathbf{x}$ as a random noise vector, it represents the
collocation points $\mathbb{x}$ in the domain $S_{\Omega}$. Thus, the
generator is formalized as a mapping
$G:S_{\Omega}\to(\mathcal{N}_{u},\mathcal{N}_{\sigma})$, where
$\mathcal{N}_{u}$ and $\mathcal{N}_{\sigma}$ represents the neural network
approximation for the displacement and stress field, respectively. Therefore,
the generator can be defined as:

$\displaystyle
G(\mathbb{x},\bm{\theta}_{G}):=(\mathcal{N}_{u}(\mathbb{x};\bm{\theta}_{u}),\mathcal{N}_{\sigma}(\mathbb{x};\bm{\theta}_{\sigma}))$
(31)

where $\bm{\theta}_{G}:=(\bm{\theta}_{u},\bm{\theta}_{\sigma})$ denotes the
trainable parameters of the generator network. Building upon the physics-
informed aspect, we differentiate $\mathcal{N}_{u}$ and employ the kinematics
equation to obtain the strain $\varepsilon$. Given
$\mbox{\boldmath$\varepsilon$}=\nabla^{\textrm{sym}}\mathcal{N}_{u}$, the
generator’s output evolves from merely the neural network predictions
$\mathcal{N}_{u}$ and $\mathcal{N}_{\sigma}$ to the strain-stress pair
$\mathbb{z}:=(\mbox{\boldmath$\varepsilon$},\mathcal{N}_{\sigma})$.

Once we obtain the strain-stress output from the generator, to stay consistent
with the data-driven mechanics’ paradigm, we compute the strain-stress data
points $\hat{\mathbb{z}}\in\mathcal{D}$ closest to the output $\mathbb{z}$,
which corresponds to:

$\hat{\mathbb{z}}=\mathop{\mathrm{arg\,min}}_{\hat{\mathbb{z}}\in\mathcal{D}}d(\mathbb{z},\hat{\mathbb{z}}),$
(32)

with distance (7). We then use $\mathbb{z}$ and $\hat{\mathbb{z}}$ as
synthetic and real data for the discriminator’s training. For the
discriminator $D(\mathbb{y},\bm{\theta}_{D})$, we establish the mapping
$D:\mathbb{R}^{2d}\to[0,1]$, aligning with the conventional GAN framework. To
accommodate strain-stress pairs as inputs for the discriminator, we convert a
pair into a $2d$-vector $\mathbb{y}$ by applying Voigt-Notation to both the
strain and stress, then merging them into a single vector. Given strain-stress
data $\hat{\mathbb{z}}\in\mathcal{D}$, it assesses the data’s authenticity,
furnishing scores to guide the generator’s training. With the generator now
representing a PINN, the adversarial loss in equation (23) has to integrate
the physics-informed loss $L_{\mathcal{C}}$, derived from the residuals of the
governing differential equations:

$L(G,D)=\mathbb{E}_{\hat{\mathbb{z}}\sim p_{\mathcal{D}}}[\ln
D(\hat{\mathbb{z}})]+\mathbb{E}_{\mathbb{x}\sim
p_{S_{\Omega}}}[\ln(1-D(G(\mathbb{x})))+L_{\mathcal{C}}].$ (33)

The collaborative training between the discriminator and the physics-informed
generator ensures that the latter learns to craft data that confounds the
discriminator and aligns closely with intrinsic physics. Fig. 5 illustrated
the physics-enhanced GAN approach for the data-driven mechanics problem.
Regarding Wasserstein GANs and their gradient penalty variants, their
objectives concerning the physics-informed generator must be modified. For
instance, with the Wasserstein GAN objective, the loss function becomes:

$L_{\textrm{WGAN}}(G,D)=\mathbb{E}_{\hat{\mathbb{z}}\sim
p_{\mathcal{D}}}[D(\hat{\mathbb{z}})]-\mathbb{E}_{\mathbb{x}\sim
p_{S_{\Omega}}}[D(G(\mathbb{x}))+L_{\mathcal{C}}],$ (34)

Moreover, for the WGAN-GP, the combined objective is:

$L(G,D)=L_{\textrm{WGAN}}+\omega\cdot\text{GP}.$ (35)

Figure 5:  Schematic representation of a physics-informed generative
adversarial network (GAN) incorporating collocation points and strain-stress
data for physical point generation and discrimination.

By incorporating GANs with physics-informed principles, the models produce
data that adheres to the statistics of observed datasets and the underlying
differential equations. This integration addresses the nested optimization
issue commonly found in the PINN-based data-driven mechanics. With the
capability of GANs to generate outputs mirroring accurate data, the solutions
are both statistically relevant and in line with physical principles. Using
GANs simplifies the optimization process, making the training more stable and
less prone to errors from inaccurate strain-stress approximations. However, it
is worth noting that the loss values obtained while training a traditional GAN
are often unreliable. In many studies, qualitative and quantitative evaluation
methods are employed to assess the performance of the GAN. Qualitative
evaluations, while offering a quick visual validation, can be subjective.
Typically, they involve human observers who evaluate the realism of a
generated sample. The overall presumption has been that if the generated
sample appears realistic, the GAN’s training is deemed successful, regardless
of potential fluctuations in loss values. Nevertheless, such evaluations can
be biased and do not always represent the complete performance spectrum of the
GAN. For instance, the generated samples might still appear high quality even
in scenarios where mode collapse occurs. Considering these challenges,
especially in the context of our work where the goal is not generating images
but accurately representing strain-stress states, we decided on WGAN + GP.
Unlike traditional GANs, the loss of WGANs has a convergence point. Ideally,
this point is reached when the generator is so adept at producing samples that
no Lipschitz continuous discriminator can differentiate between real and
generated samples. This characteristic of WGAN provides a more stable and
consistent evaluation metric, ensuring that the generated strain-stress states
are physically accurate. The effectiveness of this method will be showcased in
a two-dimensional numerical example.

##  4 Numerical benchmark of a non-Linear elastic plate with hole

This section illustrates the application of GANs to the data-driven computing
paradigm [2] in a typical benchmark, considering stress analysis of non-linear
elastic material. We discuss the problem setup and test environments and give
a proper definition of the geometry and boundary conditions and the material
parameters for data generation. We limit the simulation to noiseless synthetic
data sets, which consist of strain-stress points created numerically using a
material model rather than obtained by actual experimental measurements.
However, experimental data is often noisy and contains outliers. This issue
can be addressed with noise reduction algorithms such as tensor voting [56],
Kalman filtering [57], and deep learning-based techniques.

In this benchmark, we investigate a $2d$ in-plain plate with a hole subject to
a distributed force. The geometry, boundary conditions, and displacements are
chosen according to a similar test presented in [16] and illustrated in Fig.
6.

#### Geometry:

The system is defined by
$\Omega=\big{[}-\frac{\ell}{2},\frac{\ell}{2}\big{]}^{2}\setminus B_{r}(0)$,
where $B_{r}$ refers to the open ball of radius $r=\frac{\ell}{4}$ centered at
the origin $(0,0)$. The side lengths of the plate are equal to
$\ell=2\textrm{m}$. Due to the symmetry of geometry, only one-quarter of the
system is simulated, cf. Fig 6.

Figure 6: Illustration of a square plate subjected to external forces,
alongside its top-right quadrant representing the symmetry section with
specified boundary conditions and force distribution $t(y)$ applied.

Displacements are fixed at the quarter plate’s left surface $x=0$ in
$x$-direction and at the bottom surface $y=0$ in $y$-direction. The
corresponding conditions read as follows:

$\displaystyle\begin{cases}u_{x}=0,&\text{if }x=0;\\\ u_{y}=0,&\text{if
}y=0;\\\ \end{cases}$ (36)

where $\sigma_{x}$ is the stress and $u_{x}$ and $u_{y}$ are the displacements
in $x$ and $y$-directions, respectively. In addition, we define boundary
conditions for the stress, especially for $x=\frac{\ell}{2}$ the plate is
subjected to a distributed force $t(y)=200\cos(\frac{\pi y}{2})$ in
$x$-direction. The boundary conditions for the stress components read

$\displaystyle\begin{cases}\sigma_{xx}=t(y),&\text{if }x=1;\\\
\sigma_{yy}=0,&\text{if }y=1;\\\ \sigma_{xy}=0,&\text{if
}(x,y)\in\partial\Omega.\end{cases}$ (37)

Notice that numerical methods based on the weak form of a boundary value
problem innately satisfy shear-free boundary conditions on free boundaries.
However, our PINN approach utilizes the strong form of the boundary value
problem, so it is crucial to impose the zero stress boundary conditions
directly [45].

To train the network, we utilize $128^{2}$ quasi-random points produced using
the Sobol sequence [58]. For testing, we generate $256^{2}$ domain points
using a uniform random distribution.

#### Material parameters:

The boundary value problem considers the non-linear elastic material behavior
of [16] defined by

$\displaystyle\mbox{\boldmath$\sigma$}=\lambda
g(\mathrm{tr}(\mbox{\boldmath$\varepsilon$}))\bm{I}+\mu\mbox{\boldmath$\varepsilon$}+\mbox{\boldmath$C$}\mbox{\boldmath$\varepsilon$},$
(38)

with $g(x)=((|x|+a)^{p}-a^{p})\mathrm{sgn}(x)$ and $a,p\in\mathbb{R}$. The
applied material parameters are Young’s modulus $E$, Poisson’s ratio $\nu$,
and orthotropic elasticity tensor for plane strain given by

$\displaystyle\mbox{\boldmath$C$}=\begin{pmatrix}C_{11}&&2\nu(\bar{\lambda}+G_{\perp})&&0\\\
2\nu(\bar{\lambda}+G_{\perp})&&\bar{\lambda}+2G_{\perp}&&0\\\
0&&0&&G_{\parallel}\end{pmatrix},$ (39)

where $\lambda=\frac{E\nu}{(1+\nu)(1-2\nu)}$ and $\mu=\frac{E}{2(1+\nu)}$ are
the well known Lamé constants and $C_{11}=4.6875E$, $G_{\perp}=0.3E$,
$G_{\parallel}=0.2E$ and $\bar{\lambda}=\frac{2\nu^{2}+1}{15-20\nu^{2}}E$ are
additional material parameters. The exact parameter values used for the
reference solution and synthetic data are given in Table 1.

$E\,[\text{MPa}]$ | $\nu\,[-]$ | $a\,[-]$ | $p\,[-]$
---|---|---|---
$1\times 10^{4}$ | $0.3$ | $0.001$ | $0.005$
Table 1: Material parameters

#### Synthetic data:

In order to simulate actual experimental measurements, we generate data
artificially using the non-linear material model (38) based on the given
material parameters. We investigate normal data distributions of $100^{3}$
strain-stress data points with a fixed random seed. The data is created by a
zero-mean normal distribution with a standard deviation of $0.005$ in all
strain dimensions.

#### WGAN parameter:

For the adversarial network, the model consists of the generator and the
discriminator setup. The generator

$\displaystyle
G(\mathbb{x},\bm{\theta}_{G})=\\{\mathcal{N}_{u}(\mathbb{x},\bm{\theta}_{u}),\mathcal{N}_{\sigma}(\mathbb{x},\bm{\theta}_{\sigma})\\}$
(40)

with $\mathbb{x}=(x,y)$,
$\bm{\theta}_{G}=(\bm{\theta}_{u},\bm{\theta}_{\sigma})$ and

$\displaystyle\mathcal{N}_{u}(\mathbb{x},\bm{\theta}_{u})$
$\displaystyle=\\{\mathcal{N}_{u_{i}}(\mathbb{x},\theta_{u_{i}})\,|\,i=x,y\\},$
(41) $\displaystyle\mathcal{N}_{\sigma}(\mathbb{x},\bm{\theta}_{\sigma})$
$\displaystyle=\\{\mathcal{N}_{\sigma_{ij}}(\mathbb{x},\theta_{\sigma_{ij}})\,|\,i,j=x,y\\},$
(42)

is constructed with a series of fully connected layers. The architecture
utilizes $4$ hidden layers, each with $64$ neurons. The activation function
used across these layers is the Swish function, defined as

$\displaystyle\textrm{Hardswish}(x)=\begin{cases}0&\text{if }x\leq-3,\\\
x&\text{if }x\geq 3,\\\ \frac{x^{2}+3x}{6}&\text{otherwise}.\end{cases}$ (43)

In addition, to optimize the network training, we hard enforce the boundary
conditions from (36) and (37), such that the output of the generator is given
by

$\displaystyle\begin{aligned}
\mathcal{N}_{u_{x}}(\mathbb{x};\theta_{u_{x}})&=x\cdot\hat{\mathcal{N}}_{u_{x}}(\mathbb{x};\theta_{u_{x}}),\\\
\mathcal{N}_{u_{y}}(\mathbb{x};\theta_{u_{y}})&=y\cdot\hat{\mathcal{N}}_{u_{x}}(\mathbb{x};\theta_{u_{y}}),\\\
\mathcal{N}_{\sigma_{xx}}(\mathbb{x};\theta_{\sigma_{xx}})&=(1-x)\cdot\hat{\mathcal{N}}_{\sigma_{xx}}(\mathbb{x};\theta_{\sigma_{xx}}),\\\
\mathcal{N}_{\sigma_{yy}}(\mathbb{x};\theta_{\sigma_{yy}})&=(1-y)\cdot\hat{\mathcal{N}}_{\sigma_{yy}}(\mathbb{x};\theta_{\sigma_{yy}}),\\\
\mathcal{N}_{\sigma_{xy}}(\mathbb{x};\theta_{\sigma_{xy}})&=xy(x^{2}+y^{2}-0.25)\cdot\hat{\mathcal{N}}_{\sigma_{xy}}(\mathbb{x};\theta_{\sigma_{xy}}),\end{aligned}$
(44)

with $\mathbb{x}=(x,y)$ and
$\bm{\theta}=(\theta_{u_{i}},\theta_{\sigma_{ij}})$ being the tuple of all
trainable network parameters regarding the displacement and stress component.
In order to obtain the strains and optimize the loss function, the spatial
derivatives are obtained by automatic differentiation.
On the other hand, the discriminator $D(\mathbb{y},\bm{\theta}_{D})$ comprises
a network architecture of $3$ hidden layers, each with $16$ neurons, which
uses the LeakyReLU activation function defined as

$\displaystyle\textrm{LeakyReLU}(x)=\begin{cases}x&\text{if }x\geq 0,\\\
\alpha x&\text{if }x<0,\\\ \end{cases}$ (45)

with a slope of $\alpha=0.2$ for negative values. Regarding optimization, both
the generator and the discriminator use the ADAM optimizer with a learning
rate of $0.02$. The beta values for the moment estimates are set as
$(0.5,0.999)$. A learning rate scheduler is employed with a maximum learning
rate of $0.02$. It is set to adjust the rate over a total of $200$ steps for
both the Generator and Discriminator.

#### Result:

The WGAN+GP frameworks were utilized in our numerical evaluations to
investigate their effectiveness in computing non-linear elastic materials
through a data-driven approach. Figure 7 depicts the distribution of strain-
stress achieved after $200$ training epochs. Due to the utilization of batch
processing during training, the number of training steps exceeded this epoch
count. The findings provide a profound understanding of the training quality
and effectiveness.
We investigated the loss values during training for a clearer perspective on
model behavior. The Wasserstein-enhanced architectures showcased robustness
and consistency during training. Fig. 8 displays the losses for the
discriminator and generator of the model across the epochs. Notably, shallow
loss values for either the generator or the discriminator can be
counterproductive. It generally indicates that one network is dominating the
other, leading to a stagnation in the training process. Ideally, there should
be a balance where both networks challenge each other, encouraging continuous
improvement. Despite their effectiveness, traditional adversarial networks
present non-interpretable loss values, making it challenging to discern
training quality. The WGAN+GP approach offers direct insights into the quality
of data generation, making it more user-friendly in solution analysis. In
addition, we plot the minimum distance between the generated states and the
data set $\mathcal{D}$ in Fig. 9. Given the data-driven approach, the learning
process trains with a finite set of data points. Consequently, the losses do
not converge to zero but to a positive lower bound. In our case, both losses
decrease over training, representing this convergence. In data-driven
mechanics, the approach displayed a commendable ability to simulate stress-
strain distributions. WGAN, with improved loss interpretability and smoother
training, stands out as the preferable choice for intricate computational
mechanics tasks.

Figure 7: Visualization of displacement and stress distribution after $200$
training epochs offering insights into the material’s behavior under the
applied loads and conditions (36) and (37). From top-left to bottom-right:
$u_{x}$ showcases a gradient, indicating a maximum displacement at
$(x,y)=(1,0)$; $u_{y}$ reveals a displacement trend with negative values
highlighted in $x=0$; $\sigma_{x}$ shows a maximum stress magnitude at $y=0$;
$\sigma_{y}$ displays a similar gradient; and $\sigma_{xy}$ captures a
pronounced shear stress distribution inside the plate. Figure 8: Comparative
visualization of the discriminator and generator loss metrics over training
iterations for a WGAN+GP model, showcasing the dynamic interplay and
convergence patterns. The shaded area shows the maximum range of loss for
individual training batches. Figure 9: Visualization of the distance metrics
over $200$ epochs.The distance of the generated $\mathbb{z}$ to the dataset
$\mathcal{D}$ illustrates how closely the model-generated outputs match the
dataset over training iterations.

##  5 Conclusion

The model-free data-driven method, developed by Kirchdoerfer and Ortiz, uses
experimental data directly in simulations, bypassing the entire material
modeling step. The paradigm uses nearest-neighbor clustering to reformulate
boundary value problems. The approach has been diversified for many
applications. Challenges such as data availability, noise, inconsistency, and
high dimensionality frequently arise in the data-driven paradigm. Traditional
analytical and computational methods may need to be adjusted when addressing
these issues.
Consequently, the incorporation of machine learning methods is considered,
especially physics-informed neural networks. In solving boundary value
problems with ANNs, the idea is to transform it into an optimization problem.
The residual of the differential equations is minimized, and the neural
network approximates the displacement and stress field. However, there are
challenges with PINNs. There have been instances where the optimization yields
solutions with unexpected or non-physical behaviors even when carefully
tailored to encapsulate the physics. If we integrate the distance as an
additional loss into the global loss, the whole problem becomes a nested
optimization, leading to training challenges. In addition, approximated
strain-stress fields can correspond to suboptimal data points influencing the
direction and rate of the convergence.
To address these challenges, we consider the integration of PINNs with
generative adversarial networks. GANs are proficient at generating outputs
with the same properties as actual data, providing a potential approach to
generating realistic strain-stress solutions. Their flexibility ensures
adaptability across diverse data types suited for various physical conditions.
Moreover, the inherent capability of GANs to distinguish and capitalize on
intricate patterns may lead to a more robust representation of underlying
physics. The combined PINN-GAN approach seeks to ensure physical consistency
and alignment with observed data, leveraging the strengths of both
methodologies.
This research introduced an approach to WGANs + GP tailored for data-driven
mechanics problems. The generator is identified as a PINN, ensuring that
generated outputs conform to underlying physical principles. Instead of random
noise, the generator utilizes collocation points from the domain and maps them
to neural network approximations of strain and stress fields. The
discriminator is then trained using the generated and the closest actual
strain-stress data. By integrating WGANs with physics-informed principles, the
model outputs adhere to observed dataset statistics and differential
equations. This results in improved optimization, more stable training, and
accurate, physically consistent solutions. In this regard, we investigated a
non-linear elastic plate with a hole benchmark. The results indicate that our
proposed method provides reasonable outcomes. Furthermore, we observed robust
and consistent training of the networks and noted the convergence of the data-
driven solution as data size increased.
As we advance our research, we aim to delve deeper into other convergence
criteria for the GAN or WGAN. We plan to explore metrics such as the Inception
Score [59], Frechet Inception Distance [60], and perceptual similarity
measures [61] to provide a broader assessment of the generated outputs. These
metrics will help to analyze the quality of the generated material states.
Another area of interest is using the discriminator in the GAN framework for
material identification. The discriminator’s ability to distinguish between
actual and generated outputs can be used to identify different material
states. This approach could offer a novelty to classify materials, and we want
to explore this further.
In addition, we plan to extend our method to more complex and varied material
properties. We also consider integrating advanced machine learning techniques
to improve prediction accuracy, especially when dealing with sparse datasets.
We are considering hybrid network architectures that combine convolutional and
regression layers. The traditional image-based GAN structure inspires this
design. By adding these layers, we hope to combine the advantages of image-
based GANs with our current data-focused method.

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