SHM

    structural health monitoring
DIC

    digital image correlation
ESPI

    electronic speckle pattern interferometry
PDE

    partial differential equation
MCMC

    Markov chain Monte Carlo
PDF

    probability density function
NLS

    nonlinear least-squares
ANN

    artificial neural network
FFNN

    feed-forward neural network
PINN

    physics-informed neural network
FE

    finite element
FEM

    finite element method
BC

    boundary condition
NLS-FEM

    nonlinear least-squares finite element method
VFM

    virtual fields method
RE

    relative error
MAE

    mean absolute error
$\text{rL}^{2}$

    relative $\text{rL}^{2}$ norm
RD

    relative deviation
ARE

    absolute relative error
SEM

    standard error of the mean
GPU

    graphics processing unit

[1]David Anton

1]Institute for Computational Modeling in Civil Engineering, Technische
Universität Braunschweig, Pockelsstraße 3, Braunschweig, 38106, Germany

2]Institute of Applied Mechanics, Clausthal University of Technology, Adolph-
Roemer-Straße 2A, Clausthal-Zellerfeld, 38678, Germany

3]Institute for Acoustics and Dynamics, Technische Universität Braunschweig,
Langer Kamp 19, Braunschweig, 38106, Germany

4]Computational Mechanics Group, Eidgenössische Technische Hochschule Zürich,
Tannenstrasse 3, Zürich, 8092, Switzerland

# Deterministic and statistical calibration of constitutive models from full-
field data with parametric physics-informed neural networks

<EMAIL_ADDRESS>Jendrik-Alexander Tröger  jendrik-
<EMAIL_ADDRESS>Henning Wessels  h.wessels@tu-
braunschweig.de Ulrich Römer<EMAIL_ADDRESS>Alexander Henkes
<EMAIL_ADDRESS>Stefan Hartmann<EMAIL_ADDRESS>[  [  [  [

###### Abstract

The calibration of constitutive models from full-field data has recently
gained increasing interest due to improvements in full-field measurement
capabilities. In addition to the experimental characterization of novel
materials, continuous structural health monitoring is another application that
is of great interest. However, monitoring is usually associated with severe
time constraints, difficult to meet with standard numerical approaches.
Therefore, parametric physics-informed neural networks for constitutive model
calibration from full-field displacement data are investigated. In an offline
stage, a parametric PINN can be trained to learn a parameterized solution of
the underlying partial differential equation. In the subsequent online stage,
the parametric PINN then acts as a surrogate for the parameters-to-state map
in calibration. We test the proposed approach for the deterministic least-
squares calibration of a linear elastic as well as a hyperelastic constitutive
model from noisy synthetic displacement data. We further carry out Markov
chain Monte Carlo-based Bayesian inference to quantify the uncertainty. A
proper statistical evaluation of the results underlines the high accuracy of
the deterministic calibration and that the estimated uncertainty is valid.
Finally, we consider experimental data and show that the results are in good
agreement with a Finite Element Method-based calibration. Due to the fast
evaluation of PINNs, calibration can be performed in near real-time. This
advantage is particularly evident in many-query applications such as Markov
chain Monte Carlo-based Bayesian inference.

###### keywords:

model calibration, parametric physics-informed neural networks, uncertainty
quantification, solid mechanics

##  1 Introduction

The calibration of constitutive models is a major research field in
computational as well as experimental solid mechanics and has a wide range of
applications in practice. The interest in appropriate methods for constitutive
model calibration recently increased further with the improvement of full-
field measurement capabilities and the associated increase in available full-
field displacement data. Probably the most obvious application in the context
of experimental solid mechanics is the characterization of novel materials
from experimental data. Another application that is gaining increasing
interest is continuous structural health monitoring (SHM) [1, 2]. Material
parameters directly reflect the resistance to external impacts and indicate
damage and material degradation and thus provide crucial information for the
assessment of existing structures. Since in SHM stress data is typically not
accessible, the material parameters of interest must be identified from
displacement or strain data, measured by, e.g., digital image correlation
(DIC) [3] or electronic speckle pattern interferometry (ESPI) [4],
respectively.

The connection between constitutive model parameters and the measured full-
field data is then established by the inverse solution of the parametric
mechanical model. Traditionally, this inverse problem is solved by numerical
methods, such as the nonlinear least-squares finite element method (NLS-FEM),
see, for instance, [5, 6], or the virtual fields method (VFM) [7, 8]. While
both NLS-FEM and VFM are well established in experimental mechanics, their
application in SHM is oftentimes prohibitive since their computational costs
do not meet the severe time constraints in online applications. Thus, there is
great interest in methods that are suitable for repeated calibration in the
laboratory or in online applications.

Recently, it has been shown that physics-informed neural networks (PINNs) [9]
are particularly suited for solving inverse problems. PINN are a framework for
solving forward and inverse problems involving nonlinear partial differential
equations from the field of physics-informed machine learning [10]. The idea
behind this method goes back to the 1990s [11, 12], but it became applicable
only recently due to developments in automatic differentiation [13], software
frameworks, such as TensorFlow [14] and PyTorch [15], and more powerful
hardware. The main advantages of PINNs are a straightforward inclusion of
training data and their use as a continuous ansatz function. Thanks to the
latter, all quantities can be computed directly on the sensor locations,
bypassing the need for interpolation as, e.g., in finite element method
(FEM)-based calibration approaches.

In general, most numerical methods for calibrating constitutive models from
full-field data can be classified into reduced and all-at-once approaches, see
[16] for a recent review. Therein, an unifying framework for model calibration
in computational solid mechanics has been developed. The reduced approach
assumes that a parameters-to-state map exists, which is provided, e.g., by a
PINN or a finite element (FE) simulation. In contrast, in the all-at-once
approach, the state and the model parameters are inferred simultaneously. For
PINNs as well as other numerical methods, it is possible to formulate the
calibration problem both in the reduced as well as in the all-at-once setting.

In the literature, most contributions focusing on parameter identification
with PINNs are associated with the all-at-once approach. Such formulations are
also referred to as inverse PINNs. In [17, 18, 19, 20, 21, 22], inverse PINNs
have been applied to parameter identification from full-field displacement
data. However, many of the assumptions made therein do not match the
conditions of real-world applications. This mainly concerns the magnitude and
quality of the measured displacements. Some references, such as [20], even
consider the availability of full-field stress data for identification, which
in practice must be considered as unknown. In earlier work, some of the
authors have further developed inverse PINNs towards parameter identification
in a realistic regime [23], both concerning the magnitude of the material
parameters as well as the noise level of the displacement data. Nevertheless,
a severe restriction of inverse PINNs remains. In principle, they must be
trained from scratch each time new measurements become available. This
involves high computational costs and is a significant disadvantage when it
comes to repeated online calibration or standardized material tests, where the
setup basically remains the same.

In this contribution, we therefore focus on PINNs in a reduced approach. In an
offline stage, the PINN is trained to learn a parameterized solution of the
underlying parametric partial differential equation within a predefined range
of material parameters. For this purpose, the material parameters are
considered as additional inputs to the PINN, such that the predicted
displacement no longer depends on the spatial position only, but also on the
material parameters. To speed up the training process and to make it more
robust, we suggest to include some data in the training process. This data may
be generated by high-fidelity FE simulations. In the subsequent online stage,
the pre-trained PINN then acts as a surrogate for the parameters-to-state map
in calibration. This special variant of PINNs, known as parametric PINNs, have
already been deployed for thermal analysis of a laser powder bed fusion
process [24], magnetostatic problems [25], or for the optimization of an
airfoil geometry [26]. To the best of our knowledge, parametric PINNs have not
yet been used for the calibration of constitutive models in solid mechanics
using real-world experimental data. Building up on our results reported in
[16], we statistically evaluate the accuracy of the parametric PINNs for the
calibration of constitutive models from noisy synthetic full-field data,
extend the study to hyperelastic materials and consider experimental data.

We demonstrate that the parametric PINN approach enables an accurate and
efficient model calibration and uncertainty quantification of the inferred
material parameters in online applications, even though up to
$\mathcal{O}(10^{4})$ forward model evaluations are required. To illustrate
this, we first consider the constitutive models for both small strain linear
elasticity and finite strain hyperelasticity and perform a re-identification
of the material parameters from noisy synthetic displacement data. In the
deterministic setting, a nonlinear least-squares (NLS) problem is solved. A
statistical evaluation of the results shows that the point estimates obtained
by solving the NLS problem deviate only marginally from the true material
parameters. We further treat the material parameters as random variables,
conduct Bayesian statistical inference and quantify the uncertainty in the
estimated material parameters. The posterior distribution of the material
parameters is determined by carrying out a Markov chain Monte Carlo (MCMC)
analysis. In order to validate the quantified uncertainty from a frequentist
point of view, we perform a coverage test. The results for the statistical
calibration show that the estimated uncertainties are also valid. In addition
to the synthetic data, we calibrate the constitutive model for small strain
linear elasticity using experimental full-field displacement data obtained
from a tensile test. We demonstrate that the calibration with a parametric
PINN shows good results compared to using FEM for both the deterministic as
well as the statistical setting.

In summary, the advantages of using parametric PINNs as surrogates of the
parameters-to-state map in the context of constitutive model calibration are:

  * •

Parametric PINNs allow for a near real-time calibration. Once a PINN has been
trained in the offline stage, the evaluation of the parameters-to-state map in
the online stage is very cheap. This is a clear advantage, especially when
used in many-query approaches such as the deterministic NLS approach or the
statistical MCMC analysis.

  * •

Parametric PINNs are continuous ansatz functions. No interpolation between the
sensor locations and the numerical discretization is required for calibration.

  * •

Data can be easily integrated to speed up training. Data is not necessary for
training, but can speed up the training process and make it more robust. As
with projection-based reduced order modeling approaches [27, 28], such data
may arise from snapshots of a high-fidelity FE model.

To support the advantages mentioned above and to increase the acceptance of
parametric PINNs in the context of constitutive model calibration, the present
study aims towards the following key contributions:

  * •

We use parametric PINNs for uncertainty quantification. The parametric PINN is
used as surrogate of the parameters-to-state map within a MCMC analysis and
provides us with the posterior probability density of the parameters of
interest.

  * •

We perform a statistical evaluation of the numerical results. To validate the
estimated uncertainty in the Bayesian statistical setting from a frequentist
point of view, we perform a coverage test.

  * •

We consider real-world experimental displacement data. We calibrate a linear
elastic constitutive model using experimental data measured in a tensile test.

To the best of the authors knowledge, the above mentioned contributions in
connection with parametric PINNs have not yet been considered in the
literature.

The code for our numerical tests including the data generation, the training
and validation of parametric PINNs as well as the calibration methods is
implemented in the Python programming language. The PINN implementation is
mainly based on the PyTorch framework [15]. The code for the FE data
generation is built on top of the FEniCSx project [29]. Our research code is
open source and available both on GitHub and Zenodo [30]. In addition, we also
published the experimental data set on Zenodo [31].

The remainder of this paper is structured as follows: In Section 2, the
balance of linear momentum and the considered constitutive models are
recapitulated. We then provide a brief introduction to artificial neural
networks and parametric PINNs in Section 3. In this section, we also elaborate
on normalization steps necessary for real-world applications. In Section 4,
the calibration problem both in the deterministic as well as the Bayesian
statistical setting are formulated. Subsequently, in Section 5 and Section 6,
we provide the results for our numerical tests including both synthetic and
experimental full-field data, respectively. Finally, we conclude our
investigations with a critical discussion and point out possible directions of
future work in Section 7.

##  2 Solid mechanics preliminaries

The relationship between the measured displacements of a body and the material
parameters is defined by the framework of solid mechanics. In the following,
we briefly recapitulate the balance of linear momentum and elaborate on the
constitutive models for both small strain linear elasticity and finite strain
hyperelasticity. For a more in-depth introduction to solid mechanics, the
reader is referred to standard text books, e.g., [32, 33].

###  2.1 Fundamental equations

The displacement of a material point $\mathbf{X}\in\mathcal{B}_{\textrm{R}}$
in the undeformed reference configuration $\mathcal{B}_{\textrm{R}}$ (denoted
by subscript ${}_{\textrm{R}}$) is defined by

$\mathbf{u}(\mathbf{X},t)=\mathbf{x}-\mathbf{X}=\boldsymbol{\chi}_{\textrm{R}}(\mathbf{X},t)-\mathbf{X},$
(1)

where the vector $\mathbf{x}\in\mathcal{B}$ corresponds to the position of a
material point in the deformed configuration $\mathcal{B}$ at time $t$ and
$\boldsymbol{\chi}_{\textrm{R}}(\mathbf{X},t)$ represents the motion. In the
following, the explicit time dependence is omitted for brevity. Furthermore,
both the undeformed reference configuration $\mathcal{B}_{\textrm{R}}$ and the
deformed configuration $\mathcal{B}$ are modeled as a subset of the physical
Euclidean space $\mathbb{E}^{3}$ with orthonormal basis vectors. Then,
$\mathbb{E}^{3}$ can be identified with the common three-dimensional vector
space $\mathbb{R}^{3}$. More information on the geometrical treatment of
continuum mechanics can be found in [34, 35].

In the reference configuration $\mathcal{B}_{\textrm{R}}$, the balance of
linear momentum in its strong form and in static equilibrium states

$\operatorname{Div}\boldsymbol{\mathsf{P}}(\mathbf{X};\boldsymbol{\kappa})+\rho_{\textrm{R}}(\mathbf{X})\,\mathbf{b}(\mathbf{X})=\mathbf{0},\;\mathbf{X}\in\mathcal{B}_{\textrm{R}}.$
(2)

Here, $\operatorname{Div}$ denotes the divergence operator with respect to the
coordinates $\mathbf{X}$ and $\boldsymbol{\mathsf{P}}$ represents the first
Piola-Kirchhoff stress tensor. The density in the reference configuration is
denoted by $\rho_{\textrm{R}}$ and $\mathbf{b}$ are accelerations caused, for
instance, by gravity. Equation 2 needs to be satisfied for all points
$\mathbf{X}$ inside the domain $\mathcal{B}_{\textrm{R}}$. The stress depends
on the displacement $\mathbf{u}$ via the strains and is parameterized by a set
of material parameters $\boldsymbol{\kappa}{\,\in\mathbb{R}}^{n_{\kappa}}$.
The semicolon indicates parameterization of $\boldsymbol{\mathsf{P}}$ in
$\boldsymbol{\kappa}$.

The PDE 2 is complemented by a set of Dirichlet and Neumann boundary
conditions

$\displaystyle\mathbf{u}(\mathbf{X})$
$\displaystyle=\bar{\mathbf{u}},\;\mathbf{X}\in\Gamma_{\textrm{R}}^{\textrm{D}},$
(3a)
$\displaystyle\boldsymbol{\mathsf{P}}(\mathbf{X};\boldsymbol{\kappa})\cdot\boldsymbol{\mathsf{n}}_{\textrm{R}}(\mathbf{X})$
$\displaystyle=\bar{\mathbf{t}},\;\mathbf{X}\in\Gamma_{\textrm{R}}^{\textrm{N}},$
(3b)

with $\Gamma_{\textrm{R}}^{\textrm{D}}$ and $\Gamma_{\textrm{R}}^{\textrm{N}}$
denoting the complementary surfaces of the boundary
$\Gamma_{\textrm{R}}\subset\overline{\mathcal{B}_{\textrm{R}}}$, where
$\overline{\mathcal{B}_{\textrm{R}}}$ denotes the closure of
$\mathcal{B}_{\textrm{R}}$, with
$\Gamma_{\textrm{R}}^{\textrm{D}}\,\cup\,\Gamma_{\textrm{R}}^{\textrm{N}}=\Gamma_{\textrm{R}}$.
Furthermore, $\bar{\mathbf{u}}$ and $\bar{\mathbf{t}}$ are the prescribed
displacements and tractions on the boundaries, respectively, and
$\boldsymbol{\mathsf{n}}_{\textrm{R}}$ is the normal vector on the outer
surface of the reference configuration.

The system of equations arising from 2–3 is closed by the kinematics and a
constitutive model describing the stress state as a function of strain,
parameterized in the material parameters $\boldsymbol{\kappa}$. In the
following, we briefly recall the kinematics and constitutive equations for
linear elasticity and hyperelasticity.

###  2.2 Linear elasticity

For linear, isotropic elasticity and small strains, the constitutive model
states

$\boldsymbol{\mathsf{\sigma}}(\boldsymbol{\mathsf{\epsilon}};\boldsymbol{\kappa})=K\,\text{tr$\left(\boldsymbol{\mathsf{\epsilon}}\right)$}\boldsymbol{\mathsf{I}}+2G{\boldsymbol{\mathsf{\epsilon}}_{\textrm{D}}},$
(4)

where $\boldsymbol{\mathsf{\sigma}}$ is the Cauchy stress tensor,
$\boldsymbol{\mathsf{I}}$ is the second-order identity tensor and tr is the
trace operator. Note that in the linear elastic theory, it is assumed that
$\boldsymbol{\mathsf{P}}\approx\boldsymbol{\mathsf{\sigma}}$ in 2–3. The
linear strain tensor $\boldsymbol{\mathsf{\epsilon}}$ is defined as

$\boldsymbol{\mathsf{\epsilon}}=\frac{1}{2}\Bigl{(}\operatorname{Grad}{\mathbf{u}(\mathbf{X})}+(\operatorname{Grad}{\mathbf{u}(\mathbf{X})})^{\top}\Bigr{)},$
(5)

where the gradient $\operatorname{Grad}$ is defined with respect to the
coordinates $\mathbf{X}$. Here, $\mathbf{x}\approx\mathbf{X}$ is assumed.
Furthermore,
${\boldsymbol{\mathsf{\epsilon}}_{\textrm{D}}}=\boldsymbol{\mathsf{\epsilon}}-\text{tr$\left(\boldsymbol{\mathsf{\epsilon}}\right)$}/3\boldsymbol{\mathsf{I}}$
denotes the deviatoric part of $\boldsymbol{\mathsf{\epsilon}}$. The
constitutive model is parameterized in material parameters
$\boldsymbol{\kappa}=\\{K,G\\}^{\top}$ composed of the bulk modulus $K$ and
the shear modulus $G$.

###  2.3 Hyperelasticity

In the following, we consider finite strains and compressible, isotropic
hyperelastic materials. The first Piola-Kirchhoff stress tensor can be derived
from a strain energy density function $\psi_{\textrm{R}}$ expressed in terms
of the tensor-valued measure $\boldsymbol{\mathsf{C}}$ by

$\boldsymbol{\mathsf{P}}(\boldsymbol{\mathsf{F}};\boldsymbol{\kappa})=2\boldsymbol{\mathsf{F}}\frac{\partial\psi_{\textrm{R}}(\boldsymbol{\mathsf{C}};\boldsymbol{\kappa})}{\partial\boldsymbol{\mathsf{C}}}.$
(6)

The deformation gradient $\boldsymbol{\mathsf{F}}$ and the right Cauchy-Green
tensor $\boldsymbol{\mathsf{C}}$ are defined as

$\boldsymbol{\mathsf{F}}=\frac{\partial\boldsymbol{\chi}_{\textrm{R}}(\mathbf{X},t)}{\partial\mathbf{X}}=\operatorname{Grad}{\mathbf{u}(\mathbf{X})}+\boldsymbol{\mathsf{I}},\quad\boldsymbol{\mathsf{C}}=\boldsymbol{\mathsf{F}}^{\top}\boldsymbol{\mathsf{F}},$
(7)

where $\boldsymbol{\mathsf{I}}$ is again the second-order identity tensor.

The strain energy density function $\psi_{\textrm{R}}$ can be additively
decomposed into a volumetric and an isochoric part
$\psi_{\textrm{R}}^{\textrm{vol}}$ and $\psi_{\textrm{R}}^{\textrm{iso}}$,
respectively:

$\psi_{\textrm{R}}(\boldsymbol{\mathsf{C}};\boldsymbol{\kappa})=\psi_{\textrm{R}}^{\textrm{vol}}(\mathrm{J};\boldsymbol{\kappa})+\psi_{\textrm{R}}^{\textrm{iso}}(\bar{\boldsymbol{\mathsf{C}}};\boldsymbol{\kappa}).$
(8)

Here, $\mathrm{J}=\operatorname{det}(\boldsymbol{\mathsf{F}})$ is the
determinant of the deformation gradient and
$\bar{\boldsymbol{\mathsf{C}}}=\mathrm{J}^{-2/3}\boldsymbol{\mathsf{C}}$ is
the isochoric right Cauchy-Green tensor. There are many concurrent approaches
to model the volumetric part $\psi_{\textrm{R}}^{\textrm{vol}}$. A common
approach frequently stated in standard text books [32, 33] is to consider

$\psi_{\textrm{R}}^{\textrm{vol}}(\mathrm{J};\boldsymbol{\kappa})=\frac{K}{4}(\mathrm{J}^{2}-1-2\operatorname{ln}\mathrm{J}),$
(9)

where $K$ is again the bulk modulus. For the isochoric part
$\psi_{\textrm{R}}^{\textrm{iso}}$, a Neo-Hookean-type ansatz

$\psi_{\textrm{R}}^{\textrm{iso}}(\bar{\boldsymbol{\mathsf{C}}};\boldsymbol{\kappa})=\frac{G}{2}(\mathrm{I}_{\bar{\boldsymbol{\mathsf{C}}}}-3),$
(10)

with the first invariant
$\mathrm{I}_{\bar{\boldsymbol{\mathsf{C}}}}=\operatorname{tr}(\bar{\boldsymbol{\mathsf{C}}})$
is chosen, where $G$ defines the shear modulus in the small strain limit.

The relation between $K$ and $G$ might lead to a non-physical behavior for
large compressive and tensile states, see, for a discussion, [36, 37]. Thus,
both the relation between $K$ and $G$ as well as the amount of the deformation
has to be considered carefully. Again, as in the case of linear elasticity,
the material parameters $\boldsymbol{\kappa}$ can be summarized as
$\boldsymbol{\kappa}=\\{K,G\\}^{\top}$.

##  3 Parametric physics-informed neural networks

PINNs are a deep learning framework for solving forward and inverse problems
involving PDEs, in which ANNs act as a global ansatz function to the PDE
solution [9]. An extension of the ANN with additional inputs makes it even
possible to learn parameterized forward solutions of PDEs. We first review the
basics of ANNs. Subsequently, we lay emphasize on the key characteristic of
parametric PINNs which is the formulation of the loss function. We further
elaborate on necessary normalization steps for an application of the proposed
parametric PINN formulation in a real-world setting.

###  3.1 Artificial neural networks

ANN are parameterized, nonlinear function compositions which serve as an
approximation for an input-output mapping. There are several different
formulations of this mapping, such as convolutional and recurrent neural
networks. In the following, however, we restrict ourselves to fully-connected
feed-forward neural networks. For a more in-depth introduction to ANNs, the
reader is referred to standard text books, e.g., [38].

We consider a fully-connected FFNN $f_{\textrm{N}}$ composed of
${n_{\textrm{{L}}}}+1$ layers $\mathbf{h}^{(l)}$ that defines a mapping from
an input space $\mathbb{R}^{N}$ to an output space $\mathbb{R}^{M}$ in the
general form

$\displaystyle f_{\textrm{N}}:\mathbb{R}^{N}$
$\displaystyle\to\mathbb{R}^{M},$ (11) $\displaystyle\hat{\mathbf{x}}$
$\displaystyle\mapsto
f_{\textrm{N}}(\hat{\mathbf{x}})=\mathbf{h}^{({n_{\textrm{{L}}}})}\circ\mathbf{h}^{({n_{\textrm{{L}}}}-1)}\circ\ldots\circ\mathbf{h}^{(0)}=\hat{\mathbf{y}},$

where $\hat{\mathbf{x}}{\,\in\mathbb{R}}^{N}$ denotes the input vector,
$\hat{\mathbf{y}}{\,\in\mathbb{R}}^{M}$ the output vector and $\circ$ the
composition operator, such that $(f\circ g)(x)=f(g(x))$). Accordingly, the
first layer $\mathbf{h}^{(0)}$ and the last layer
$\mathbf{h}^{({n_{\textrm{{L}}}})}$ are the input and the output layer,
respectively, and defined as

$\mathbf{h}^{(0)}=\hat{\mathbf{x}}{\,\in\mathbb{R}}^{N},\;\;\mathbf{h}^{({n_{\textrm{{L}}}})}=\hat{\mathbf{y}}{\,\in\mathbb{R}}^{M}.$
(12)

The ${n_{\textrm{{L}}}}-1$ layers between the input and the output layer are
usually called hidden layers. The vector-valued output of the hidden layers
and the output layer are defined as

$\mathbf{h}^{(l)}=\phi^{(l)}\Bigl{(}\mathbf{W}^{(l)}\mathbf{h}^{(l-1)}+\mathbf{b}^{(l)}\Bigr{)}=\phi^{(l)}\Bigl{(}\mathbf{z}^{(l)}\Bigr{)},\;\;l=\\{1,\ldots,{n_{\textrm{{L}}}}\\}.$
(13)

Here, $\mathbf{z}^{(l)}$ denotes the result of an affine transformation of the
output vector of the downstream layer $\mathbf{h}^{(l-1)}$ controlled by the
matrix $\mathbf{W}^{(l)}$ and the bias vector $\mathbf{b}^{(l)}$. The output
of the hidden layers is computed by applying a nonlinear activation function
$\phi^{(l)}$ on top of the affine transformation $\mathbf{z}^{(l)}$. In the
output layer $\mathbf{h}^{({n_{\textrm{{L}}}})}$, the identity function is
used as activation, such that

$\mathbf{h}^{({n_{\textrm{{L}}}})}=\phi^{({n_{\textrm{{L}}}})}\Bigl{(}\mathbf{W}^{({n_{\textrm{{L}}}})}\mathbf{h}^{({n_{\textrm{{L}}}}-1)}+\mathbf{b}^{({n_{\textrm{{L}}}})}\Bigr{)}=\boldsymbol{\mathsf{I}}\,\mathbf{z}^{({n_{\textrm{{L}}}})}=\mathbf{z}^{({n_{\textrm{{L}}}})},$
(14)

where $\boldsymbol{\mathsf{I}}$ is the identity matrix of size
${n_{\textrm{{n}}}}^{({n_{\textrm{{L}}}})}\times{n_{\textrm{{n}}}}^{({n_{\textrm{{L}}}})}$
and ${n_{\textrm{{n}}}}^{({n_{\textrm{{L}}}})}$ is the size of the vector
$\mathbf{z}^{({n_{\textrm{{L}}}})}$ which is equivalent to the number of
neurons in this layer. In this contribution, we use the hyperbolic tangent as
activation functions in the hidden layers.

The weight matrices $\mathbf{W}^{(l)}$ and bias vectors $\mathbf{b}^{(l)}$
comprise the trainable parameters of the layers
$l=\\{1,\ldots,{n_{\textrm{{L}}}}\\}$. All parameters of the FFNN can be
combined in a single parameter vector $\boldsymbol{\theta}$ with

$\boldsymbol{\theta}=\Bigl{\\{}\mathbf{W}^{(l)},\mathbf{b}^{(l)}\Bigr{\\}}_{1\leq
l\leq{n_{\textrm{{L}}}}}.$ (15)

Taking the trainable parameters $\boldsymbol{\theta}$ into account, in the
following, the FFNN defined by 11, 12, 13, 14 and 15 is denoted by
$f_{\textrm{N}}(\hat{\mathbf{x}};\boldsymbol{\theta})$. This notation
highlights that the FFNN output $\hat{\mathbf{y}}$ does not only depend on the
input $\hat{\mathbf{x}}$ but is also parameterized in the current realization
of $\boldsymbol{\theta}$.

An appropriate point estimate of the parameters $\boldsymbol{\theta}$ can be
found by solving an optimization problem, often referred to as training of the
ANN. The objective of the optimization problem is to minimize a loss function
that provides a measure for the deviation of the ANN from the hidden input-
output mapping. According to the universal function approximation theorem, any
Borel measurable function can be approximated by an ANN with enough parameters
with only mild assumptions on the activation function [39, 40, 41]. However,
it should be noted that the issue of finding the optimal parameters of the ANN
is still an open question and highly problem dependent.

###  3.2 Parametric physics-informed neural network formulation

Parametric PINNs are an extension of standard PINNs for learning parameterized
forward solutions involving parametric PDEs. A parameterized ansatz is used to
approximate the solution which is realized by an ANN with additional inputs
besides the spatial coordinates. In the following, we apply parametric PINNs
for solving the model 2–3 parameterized in the material parameters
$\boldsymbol{\kappa}$. We start by defining our ansatz function for the
displacement field and the resulting discretized model. Subsequently, we
formulate the loss function and elaborate on the training process.

First, we approximate the displacement field by the parametric ansatz

$\mathbf{u}(\mathbf{X},\boldsymbol{\kappa})\approx\mathcal{U}(\mathbf{X},\boldsymbol{\kappa};\boldsymbol{\theta}),$
(16)

which acts as a function approximator to the solution of 2–3. Here,
$\mathcal{U}$ is a modified FFNN $f_{\textrm{N}}$, whereby the modifications
are explained later on. It should be noted that both the spatial coordinates
$\mathbf{X}$ and the material parameters $\boldsymbol{\kappa}$ are inputs to
the ansatz $\mathcal{U}$. The FFNN is parameterized by the weights and biases
$\boldsymbol{\theta}$ as defined in 15. Furthermore, in this work, we consider
the calibration from full-field displacement data as a two-dimensional problem
and thus $\mathcal{U}:\mathbb{R}^{2+n_{\boldsymbol{\kappa}}}\to\mathbb{R}^{2}$
where $n_{\boldsymbol{\kappa}}$ is the number of material parameters.

In particular, we use an ansatz for the displacement field that differs from a
standard FFNN as follows: We choose an ansatz function that strictly fulfills
the Dirichlet boundary conditions 3a by construction, which is referred to as
hard boundary conditions. Alternatively, the Dirichlet boundary conditions can
be imposed by a separate loss term. This approach is referred to as soft
boundary conditions. With the application of the hard boundary condition
according to [42], the FFNN $f_{\textrm{N}}$ modifies to

$\tilde{\mathcal{U}}_{\textrm{hbc}}(\mathbf{X},\boldsymbol{\kappa};\boldsymbol{\theta})=\mathbf{G}(\mathbf{X})+\mathbf{D}(\mathbf{X})\otimes
f_{\textrm{N}}(\bar{\mathbf{X}};\boldsymbol{\theta}),$ (17)

where $\tilde{\mathcal{U}}_{\textrm{hbc}}$ denotes an intermediate step in the
derivation of the parameterized ansatz $\mathcal{U}$. Moreover, $\mathbf{G}$
is a smooth extension of the boundary data and $\mathbf{D}$ is a smooth
distance function giving the distance of
$\mathbf{X}\in\mathcal{B}_{\textrm{R}}$ to the boundary
$\Gamma_{\textrm{R}}^{\textrm{D}}$. The vector
$\bar{\mathbf{X}}=\\{\mathbf{X}^{\top},\boldsymbol{\kappa}^{\top}\\}^{\top}$
is the summarized FFNN input vector. When selecting the distance function, it
is important to ensure that $\mathbf{D}$ vanishes on the boundary
$\Gamma_{\textrm{R}}^{\textrm{D}}$. It should be noted that $\mathbf{G}$ and
$\mathbf{D}$ are vector valued functions of the same dimension as the ansatz
output and that $\otimes$ in 17 denotes the element-wise Hadamard
multiplication operator, such that
$[\mathbf{a}\otimes\mathbf{b}]_{i}=a_{i}\cdot b_{i}$ for two vectors
$\mathbf{a},\mathbf{b}{\,\in\mathbb{R}}^{n}$. In this contribution, we use a
normalized linear distance function defined as

$\mathbf{D}(\mathbf{X})=(\mathbf{X}-\mathbf{X}_{\textrm{bc}})\oslash({\mathbf{X}_{\textrm{max}}}-{\mathbf{X}_{\textrm{min}}}),$
(18)

where ${\mathbf{X}_{\textrm{min}}}$ and ${\mathbf{X}_{\textrm{max}}}$ are
vectors containing the minimum and maximum coordinates for each dimension
within $\mathcal{B}_{\textrm{R}}$, respectively. In addition,
$\mathbf{X}_{\textrm{bc}}$ is a vector that contains the position of the
Dirichlet boundary condition in the respective dimension. The element-wise
Hadamard division operator $\oslash$ is defined as
$[\mathbf{a}\oslash\mathbf{b}]_{i}=a_{i}/b_{i}$ for two vectors
$\mathbf{a},\mathbf{b}{\,\in\mathbb{R}}^{n}$. Note that the distance function
defined in 18 assumes that there is only one Dirichlet boundary condition in
each dimension and that the Dirichlet boundaries are parallel to the Cartesian
coordinate system. In general, however, hard boundary conditions can also be
applied to complex geometries, as shown in [42].

Furthermore, we normalize the inputs and outputs of the ansatz because it is
well known that this accelerates the convergence of the training of ANNs.
According to [43], the mean value of each input feature should be close to
zero. Since we assume that the input is evenly distributed over the input
domain, we normalize the input by the following linear transformation

$\mathbf{N}_{f_{\textrm{N}}^{\textrm{in}}}(\bar{\mathbf{X}})=2(\bar{\mathbf{X}}-{\bar{\mathbf{X}}_{\textrm{min}}})\oslash({\bar{\mathbf{X}}_{\textrm{max}}}-{\bar{\mathbf{X}}_{\textrm{min}}})-\boldsymbol{1},$
(19)

which maps the entries of the real input vector $\bar{\mathbf{X}}$ to the
range $[-1,1]$. Here, ${\bar{\mathbf{X}}_{\textrm{min}}}$ and
${\bar{\mathbf{X}}_{\textrm{max}}}$ are vectors containing the minimum and
maximum input features, respectively, and $\boldsymbol{1}$ is a vector of
ones. In addition, we normalize the ansatz outputs. Depending on the problem,
the scales of the displacements can vary significantly in the different
dimensions, as, e.g., in uniaxial tensile tests. At the same time, error
metrics like the mean squared error are scale-sensitive. To give the
displacement field approximation the same relative importance in all
dimensions during training, we enforce the ansatz outputs to be also in the
range $[-1,1]$. Therefore, we renormalize the output in a last step by another
linear transformation

$\mathbf{N}^{-1}_{f_{\textrm{N}}^{\textrm{out}}}\Bigl{(}\tilde{\mathcal{U}}_{\textrm{n}}(\mathbf{X},\boldsymbol{\kappa};\boldsymbol{\theta})\Bigr{)}=\frac{1}{2}\Bigl{(}\tilde{\mathcal{U}}_{\textrm{n}}(\mathbf{X},\boldsymbol{\kappa};\boldsymbol{\theta})+\mathbf{1}\Bigr{)}\otimes({\mathbf{u}_{\textrm{max}}}-{\mathbf{u}_{\textrm{min}}})+{\mathbf{u}_{\textrm{min}}},$
(20)

where $\tilde{\mathcal{U}}_{\textrm{n}}$ is the intermediate normalized ansatz
with its outputs enforced to be in the range $[-1,1]$. The vectors
${\mathbf{u}_{\textrm{min}}}$ and ${\mathbf{u}_{\textrm{max}}}$ contain the
minimum and maximum expected displacements of the material body resulting from
the range of material parameters $\boldsymbol{\kappa}$ under consideration,
respectively. The intermediate normalized ansatz is defined as

$\tilde{\mathcal{U}}_{\textrm{n}}(\mathbf{X},\boldsymbol{\kappa};\boldsymbol{\theta})=\mathbf{N}_{f_{\textrm{N}}^{\textrm{out}}}\Bigl{(}\mathbf{G}(\mathbf{X})\Bigr{)}+\mathbf{D}(\mathbf{X})\otimes
f_{\textrm{N}}\Bigl{(}\mathbf{N}_{f_{\textrm{N}}^{\textrm{in}}}(\bar{\mathbf{X}});\boldsymbol{\theta}\Bigr{)}.$
(21)

In order to guarantee that the renormalized ansatz output $\mathcal{U}$ still
strictly fulfills the Dirichlet boundary conditions, the boundary extension
$\mathbf{G}$ in 21 must also be normalized by the inverse of 20 which is given
by

$\mathbf{N}_{f_{\textrm{N}}^{\textrm{out}}}\Bigl{(}\mathbf{G}(\mathbf{X})\Bigr{)}=2\Bigl{(}\mathbf{G}(\mathbf{X})-{\mathbf{u}_{\textrm{min}}}\Bigr{)}\oslash({\mathbf{u}_{\textrm{max}}}-{\mathbf{u}_{\textrm{min}}})-\boldsymbol{1}.$
(22)

Note that the input $\mathbf{X}$ to $\mathbf{D}(\mathbf{X})$ in 21 is also
normalized by definition 18.

Applying the normalization and renormalization steps from equations 18, 19,
20, 21 and 22 to the modified ansatz $\tilde{\mathcal{U}}_{\textrm{hbc}}$ from
equation 17, we finally obtain the ansatz

$\begin{split}\mathcal{U}(\mathbf{X},\boldsymbol{\kappa};\boldsymbol{\theta})&=\mathbf{N}^{-1}_{f_{\textrm{N}}^{\textrm{out}}}\Bigl{(}\tilde{\mathcal{U}}_{\textrm{n}}(\mathbf{X},\boldsymbol{\kappa};\boldsymbol{\theta})\Bigr{)}\\\
&=\mathbf{N}^{-1}_{f_{\textrm{N}}^{\textrm{out}}}\biggl{(}\mathbf{N}_{f_{\textrm{N}}^{\textrm{out}}}\Bigl{(}\mathbf{G}(\mathbf{X})\Bigr{)}+\mathbf{D}(\mathbf{X})\otimes
f_{\textrm{N}}\Bigl{(}\mathbf{N}_{f_{\textrm{N}}^{\textrm{in}}}(\bar{\mathbf{X}});\boldsymbol{\theta}\Bigr{)}\biggr{)}.\end{split}$
(23)

The normalization steps aim to condition the optimization problem that arises
during PINN training. While the required minimum and maximum input values are
given from the training data, the required minimum and maximum output values
can, e.g., be extracted from given experimental or simulation data or be
estimated based on prior knowledge such as boundary conditions. It is
important to emphasize that at any time during training and prediction, only
the non-normalized, extended inputs $\bar{\mathbf{X}}$ are fed into the
ansatz. Likewise, the ansatz always outputs non-normalized displacements. This
also means that the physics is not violated when the outputs are derived with
respect to the inputs during training.

For the following steps, we reformulate the governing equations introduced in
Section 2 as a function of the displacement state vector
$\mathbf{u}^{\textrm{s}}$ and the material parameters $\boldsymbol{\kappa}$
and define the discretized model

$\mathbf{F}(\mathbf{u}^{\textrm{s}},\boldsymbol{\kappa})=\begin{Bmatrix}[l]\mathbf{F}_{\textrm{C}}(\mathbf{u}^{\textrm{s}}_{\textrm{C}},\boldsymbol{\kappa})\\\
\mathbf{F}_{\textrm{D}}(\mathbf{u}^{\textrm{s}}_{\textrm{D}},\boldsymbol{\kappa})\\\
\mathbf{F}_{\textrm{N}}(\mathbf{u}^{\textrm{s}}_{\textrm{N}},\boldsymbol{\kappa})\\\
\end{Bmatrix}=\begin{Bmatrix}[r]\operatorname{Div}\boldsymbol{\mathsf{P}}(\mathbf{u}^{\textrm{s}}_{\textrm{C}};\boldsymbol{\kappa})+\rho_{\textrm{R}}\mathbf{b}\\\
\mathbf{u}^{\textrm{s}}_{\textrm{D}}-\bar{\mathbf{u}}_{\textrm{F}}\\\
\boldsymbol{\mathsf{P}}(\mathbf{u}^{\textrm{s}}_{\textrm{N}};\boldsymbol{\kappa})\cdot\boldsymbol{\mathsf{n}}_{\textrm{R}}-\bar{\mathbf{t}}_{\textrm{F}}\\\
\end{Bmatrix}=\mathbf{0}.$ (24)

A statically and kinematically admissible displacement field must fulfill
$\mathbf{F}$ everywhere in $\mathcal{B}_{\textrm{R}}$. With PINNs, however,
the displacement is only evaluated at discrete points, represented in the
state vector
$\mathbf{u}^{\textrm{s}}{\,\in\mathbb{R}}^{2({n_{\textrm{{C}}}}+{n_{\textrm{{D}}}}+{n_{\textrm{{N}}}})}$.
The latter comprises the displacement state vectors
$\mathbf{u}^{\textrm{s}}_{\textrm{C}}{\,\in\mathbb{R}}^{2{n_{\textrm{{C}}}}}$,
$\mathbf{u}^{\textrm{s}}_{\textrm{D}}{\,\in\mathbb{R}}^{2{n_{\textrm{{D}}}}}$
and
$\mathbf{u}^{\textrm{s}}_{\textrm{N}}{\,\in\mathbb{R}}^{2{n_{\textrm{{N}}}}}$,
where ${n_{\textrm{{C}}}}$, ${n_{\textrm{{D}}}}$ and ${n_{\textrm{{N}}}}$ are
the number of evaluation points inside the domain $\mathcal{B}_{\textrm{R}}$
and on the Dirichlet and Neumann boundaries $\Gamma_{\textrm{R}}^{\textrm{D}}$
and $\Gamma_{\textrm{R}}^{\textrm{N}}$, respectively. Accordingly,
$\mathbf{F}{\,\in\mathbb{R}}^{2({n_{\textrm{{C}}}}+{n_{\textrm{{D}}}}+{n_{\textrm{{N}}}})}$
comprises $\mathbf{F}_{\textrm{C}}{\,\in\mathbb{R}}^{2{n_{\textrm{{C}}}}}$,
$\mathbf{F}_{\textrm{D}}{\,\in\mathbb{R}}^{2{n_{\textrm{{D}}}}}$ and
$\mathbf{F}_{\textrm{N}}{\,\in\mathbb{R}}^{2{n_{\textrm{{N}}}}}$. Furthermore,
$\bar{\mathbf{u}}_{\textrm{F}}{\,\in\mathbb{R}}^{2{n_{\textrm{{D}}}}}$ and
$\bar{\mathbf{t}}_{\textrm{F}}{\,\in\mathbb{R}}^{2{n_{\textrm{{N}}}}}$ are the
vectors with the prescribed displacements and tractions, respectively. The
implementation of the discrete model 24 for solving the forward problem using
PINNs is introduced in the following.

Second, we define the loss function. The loss function encoding the physics in
the model 24 and enhanced by data is defined as

$F^{\textrm{L}}(\boldsymbol{\theta};\mathbf{T})=\lambda_{\textrm{C}}F^{\textrm{L}}_{\textrm{C}}(\boldsymbol{\theta};\mathbf{T}_{\textrm{C}})+\lambda_{\textrm{N}}F^{\textrm{L}}_{\textrm{N}}(\boldsymbol{\theta};\mathbf{T}_{\textrm{N}})+\lambda_{\textrm{d}}F^{\textrm{L}}_{\textrm{d}}(\boldsymbol{\theta};\mathbf{T}_{\textrm{d}}).$
(25)

The loss terms $F^{\textrm{L}}_{\textrm{C}}$, $F^{\textrm{L}}_{\textrm{N}}$
and $F^{\textrm{L}}_{\textrm{d}}$ penalize the mean squared error of the
approximation $\mathcal{U}$ defined in 23 with respect to the PDE, the Neumann
boundary condition and the data, respectively, and are defined as

$\displaystyle
F^{\textrm{L}}_{\textrm{C}}(\boldsymbol{\theta};\mathbf{T}_{\textrm{C}})$
$\displaystyle=\frac{1}{2{n_{\textrm{{C}}}}}\sum_{i=1}^{{n_{\textrm{{C}}}}}\left\lVert\mathbf{F}_{\textrm{C}}\Bigl{(}{\mathbf{u}^{\textrm{s}}_{\textrm{C}}}^{(i)},\boldsymbol{\kappa}^{(i)}\Bigr{)}\right\rVert^{2}$
(26a)
$\displaystyle=\frac{1}{2{n_{\textrm{{C}}}}}\sum_{i=1}^{{n_{\textrm{{C}}}}}\left\lVert\operatorname{Div}\boldsymbol{\mathsf{P}}\Bigl{(}\mathcal{U}(\mathbf{X}^{(i)},\boldsymbol{\kappa}^{(i)};\boldsymbol{\theta});\boldsymbol{\kappa}^{(i)}\Bigr{)}+\rho_{\textrm{R}}\Bigl{(}\mathbf{X}^{(i)}\Bigr{)}\,\mathbf{b}\Bigl{(}\mathbf{X}^{(i)}\Bigr{)}\right\rVert^{2},$
$\displaystyle
F^{\textrm{L}}_{\textrm{N}}(\boldsymbol{\theta};\mathbf{T}_{\textrm{N}})$
$\displaystyle=\frac{1}{2{n_{\textrm{{N}}}}}\sum_{k=1}^{{n_{\textrm{{N}}}}}\left\lVert\mathbf{F}_{\textrm{N}}\Bigl{(}{\mathbf{u}^{\textrm{s}}_{\textrm{N}}}^{(k)},\boldsymbol{\kappa}^{(k)}\Bigr{)}\right\rVert^{2}$
(26b)
$\displaystyle=\frac{1}{2{n_{\textrm{{N}}}}}\sum_{k=1}^{{n_{\textrm{{N}}}}}\left\lVert\boldsymbol{\mathsf{P}}\Bigl{(}\mathcal{U}(\mathbf{X}^{(k)},\boldsymbol{\kappa}^{(k)};\boldsymbol{\theta});\boldsymbol{\kappa}^{(k)}\Bigr{)}\cdot\boldsymbol{\mathsf{n}}_{\textrm{R}}\Bigl{(}\mathbf{X}^{(k)}\Bigr{)}-\bar{\mathbf{t}}_{\textrm{F}}^{(k)}\right\rVert^{2},$
$\displaystyle
F^{\textrm{L}}_{\textrm{d}}(\boldsymbol{\theta};\mathbf{T}_{\textrm{d}})$
$\displaystyle=\frac{1}{2{n_{\textrm{{d}}}}}\sum_{l=1}^{{n_{\textrm{{d}}}}}\left\lVert\mathcal{U}(\mathbf{X}^{(l)},\boldsymbol{\kappa}^{(l)};\boldsymbol{\theta})-\bar{\mathbf{u}}_{\textrm{d}}^{(l)}\right\rVert^{2},$
(26c)

where $\left\lVert\bullet\right\rVert^{2}$ denotes the squared
$\text{L}^{2}$-norm. The training data $\mathbf{T}$ consists of three sets
$\mathbf{T}_{\textrm{C}}$, $\mathbf{T}_{\textrm{N}}$ and
$\mathbf{T}_{\textrm{d}}$:

  1. (i)

$\mathbf{T}_{\textrm{C}}$ is referred to as a set of ${n_{\textrm{{C}}}}$
collocation points
$\left\\{\mathbf{X}^{(i)},\boldsymbol{\kappa}^{(i)}\right\\}_{i=1}^{{n_{\textrm{{C}}}}}$
sampled from the domain $\mathcal{B}_{\textrm{R}}$.

  2. (ii)

$\mathbf{T}_{\textrm{N}}$ consists of ${n_{\textrm{{N}}}}$ collocation points
$\left\\{\mathbf{X}^{(k)},\boldsymbol{\kappa}^{(k)},\bar{\mathbf{t}}_{\textrm{F}}^{(k)}\right\\}_{k=1}^{{n_{\textrm{{N}}}}}$
on the Neumann boundary $\Gamma_{\textrm{R}}^{\textrm{N}}$ with the prescribed
tractions $\bar{\mathbf{t}}_{\textrm{F}}^{(k)}$.

  3. (iii)

$\mathbf{T}_{\textrm{d}}$ contains ${n_{\textrm{{d}}}}$ points
$\left\\{\mathbf{X}^{(l)},\boldsymbol{\kappa}^{(l)},\bar{\mathbf{u}}_{\textrm{d}}^{(l)}\right\\}_{l=1}^{{n_{\textrm{{d}}}}}$
where the displacements $\bar{\mathbf{u}}_{\textrm{d}}^{(l)}$ can be obtained
from, e.g., FE simulations.

The individual loss terms in 25 can additionally be weighted by
$\lambda_{\textrm{C}}$, $\lambda_{\textrm{N}}$ and $\lambda_{\textrm{d}}$ to
balance them. The weight factors may also be adapted during training, see, for
instance, [44]. In order to calculate the partial derivatives required to
evaluate the loss terms 26a–26b, the displacement field in the constitutive
models 4 and 6 is approximated by the ansatz 16. The derivatives of the ansatz
outputs with respect to the inputs are calculated using automatic
differentiation [13]. If required, the loss function 25 may be complemented by
further, problem specific loss terms, such as symmetry boundary conditions.

It should be noted that the loss function 25 does not contain a separate loss
term for the Dirichlet boundary condition since we use a hard boundary
condition for this, see 23. Provided that the stress is also considered as an
output of the ANN in addition to the displacement, the Neumann boundary
condition can in principle also be replaced by a hard boundary condition. In
this work, we do not use hard Neumann boundary conditions, as we achieved high
accuracy without them and do not observe any problems with the weak imposition
of the Neumann boundary conditions.

Third, we optimize the ANN parameters $\boldsymbol{\theta}$. The optimization
problem for finding an appropriate point estimate for the ANN parameters
$\boldsymbol{\theta}^{*}$ is defined as

$\boldsymbol{\theta}^{*}=\operatorname*{arg\,min}_{\boldsymbol{\theta}}F^{\textrm{L}}(\boldsymbol{\theta};\mathbf{T}),$
(27)

and is usually carried out using gradient-based optimization algorithms, such
as ADAM [45] or L-BFGS [46, 47, 48, 49, 50]. The required gradients of the
loss function $F^{\textrm{L}}$ with respect to the ANN parameters
$\boldsymbol{\theta}$ can again be calculated by automatic differentiation. It
should be noted, that the implementation of $F^{\textrm{L}}$ in 25 is not
identical to the model formulation 24. However, $F^{\textrm{L}}=0$ implies
that $\mathbf{F}=\mathbf{0}$. Squaring the residuals in $F^{\textrm{L}}$
ensures that positive and negative deviations do not cancel out each other. In
addition, larger residuals are penalized more than smaller residuals.

##  4 Constitutive model calibration

In this contribution, we formulate the calibration from full-field
displacement data according to the reduced approach. Following the general
problem statement, we elaborate on the deterministic nonlinear least-squares
method. Afterwards, we address the calibration problem from a Bayesian
statistical point of view. In both the deterministic as well as the Bayesian
statistical setting, the parametric PINN is used as a surrogate for the
mechanical model.

###  4.1 Deterministic calibration

General problem statement: Recalling the notation from Sections 2–3, the
problem of constitutive model calibration is governed by the following set of
equations

$\begin{split}\mathbf{F}(\mathbf{u}^{\textrm{s}},\boldsymbol{\kappa})&=\mathbf{0}\quad\text{(state
equation)},\\\
\mathbf{O}(\mathbf{u}^{\textrm{s}})&=\mathbf{d}\quad\text{(observation
equation)},\end{split}$ (28)

with state vector
$\mathbf{u}^{\textrm{s}}\in\Omega_{\mathbf{u}}\subset\mathbb{R}^{2n_{\mathbf{u}}}$,
full-field displacement data
$\mathbf{d}{\,\in\mathbb{R}}^{2{n_{\textrm{{d}}}}}$, material parameter vector
$\boldsymbol{\kappa}\in\Omega_{\boldsymbol{\kappa}}\subset\mathbb{R}^{n_{\boldsymbol{\kappa}}}$
and observation operator $\mathbf{O}$. The latter relates the model state
$\mathbf{u}^{\textrm{s}}$ to the measurement data $\mathbf{d}$, such that
$\mathbf{O}(\mathbf{u}^{\textrm{s}}){\,\in\mathbb{R}}^{2{n_{\textrm{{d}}}}}$.
In principle, the observation operator can take many forms and may also
account for indirectly measured quantities, such as strains. If full-field
displacement measurements are available, it interpolates the model state
$\mathbf{u}^{\textrm{s}}$ to the ${n_{\textrm{{d}}}}$ sensor locations
$\left\\{\mathbf{X}^{(m)}\right\\}^{{n_{\textrm{{d}}}}}_{m=1}$. These are the
points where the displacement is measured. It is worth recalling that the PINN
is a global ansatz function that can be evaluated directly at the sensor
locations. Consequently, the observation operator becomes the identity
operator, i.e.,
$\mathbf{O}(\mathbf{u}^{\textrm{s}})=\boldsymbol{\mathsf{I}}\,\mathbf{u}^{\textrm{s}}=\mathbf{u}^{\textrm{s}}$,
where $\boldsymbol{\mathsf{I}}$ is the identity matrix of size
$2n_{\mathbf{u}}\times 2n_{\mathbf{u}}$. Hence, possible interpolation errors
are avoided.

Solution approach: As discussed earlier, (28) can be solved using the all-at-
once or the reduced approach. In the reduced formulation, the implicit
function theorem is applied, see, e.g., [51], and the state vector is
expressed directly as a function of the parameters via
$\mathbf{u}^{\textrm{s}}=\widehat{\mathbf{u}}^{\textrm{s}}(\boldsymbol{\kappa})$.
Accordingly, the displacement at a material point $\mathbf{X}$ is expressed
via
$\mathbf{u}(\mathbf{X})=\widehat{\mathbf{u}}(\mathbf{X},\boldsymbol{\kappa})$.

The parameters-to-state map, also known as solution map, is here provided by
the pre-trained PINN $\mathcal{U}$. The state vector is defined as

$\widehat{\mathbf{u}}^{\textrm{s}}(\boldsymbol{\kappa})=\begin{Bmatrix}\widehat{\mathrm{u}}_{x}(\mathbf{X}^{(1)},\boldsymbol{\kappa})\\\
\vdots\\\ \widehat{\mathrm{u}}_{x}(\mathbf{X}^{(m)},\boldsymbol{\kappa})\\\
\widehat{\mathrm{u}}_{y}(\mathbf{X}^{(1)},\boldsymbol{\kappa})\\\ \vdots\\\
\widehat{\mathrm{u}}_{y}(\mathbf{X}^{(m)},\boldsymbol{\kappa})\end{Bmatrix}=\begin{Bmatrix}\mathcal{U}_{x}(\mathbf{X}^{(1)},\boldsymbol{\kappa};\boldsymbol{\theta})\\\
\vdots\\\
\mathcal{U}_{x}(\mathbf{X}^{(m)},\boldsymbol{\kappa};\boldsymbol{\theta})\\\
\mathcal{U}_{y}(\mathbf{X}^{(1)},\boldsymbol{\kappa};\boldsymbol{\theta})\\\
\vdots\\\
\mathcal{U}_{y}(\mathbf{X}^{(m)},\boldsymbol{\kappa};\boldsymbol{\theta})\end{Bmatrix},$
(29)

where the subscript in $\widehat{\mathrm{u}}_{\bullet}$ and
$\mathcal{U}_{\bullet}$ denotes the dimension. With the parameters-to-state
map defined in 29, we obtain the following problem statement

$\displaystyle\widehat{\mathbf{u}}^{\textrm{s}}(\boldsymbol{\kappa})$
$\displaystyle=\mathbf{d},$ (30a) $\displaystyle\text{subject to
}\mathbf{F}(\widehat{\mathbf{u}}^{\textrm{s}}(\boldsymbol{\kappa}),\boldsymbol{\kappa})$
$\displaystyle=\mathbf{0}.$ (30b)

The parameters-to-state map
$\widehat{\mathbf{u}}^{\textrm{s}}(\boldsymbol{\kappa})$ implicitly satisfies
the state equation 30b by pre-training the PINN $\mathcal{U}$ to satisfy the
discrete model $\mathbf{F}$ 24 prior to the calibration for the parameter set
$\Omega_{\boldsymbol{\kappa}}$. After pre-training, the ANN parameters
$\boldsymbol{\theta}$ are frozen. Thus, in an online stage, the constitutive
model can be calibrated solely on 30a. The main advantage of the reduced
formulation is that the resulting optimization problem only needs to be solved
in the parameter domain $\Omega_{\boldsymbol{\kappa}}$. However, we note that
30b is not fulfilled exactly. Instead, the PINN training typically only
results in $\left\lVert\mathbf{F}\right\rVert$ being small, which is not
reflected in the notation, for simplicity. The important point is that the
PINN training introduces a parameters-to-state map that can be used in a
reduced approach to model calibration.

The deterministic, reduced calibration problem stated in 30b–30a can be
reformulated as a nonlinear least-squares (NLS) optimization problem.
Therefore, 30a is rearranged to define the residual $\mathbf{r}$ as

$\mathbf{r}(\boldsymbol{\kappa})=\widehat{\mathbf{u}}^{\textrm{s}}(\boldsymbol{\kappa})-\mathbf{d}.$
(31)

In order to account for different magnitudes of the displacements in each
dimension, we consider weighted residuals
$\tilde{\mathbf{r}}(\boldsymbol{\kappa})=\mathbf{W}\,\mathbf{r}(\boldsymbol{\kappa})$
with the diagonal weight matrix
$\mathbf{W}{\,\in\mathbb{R}}^{2{n_{\textrm{{d}}}}\times 2{n_{\textrm{{d}}}}}$,
see [52]. Especially in the context of parameter identification, a weight
matrix can also be introduced to take into account different physical
quantities or a meaningful scaling of observations that are not all equally
reliable [53]. The weight matrix is assembled as

$\mathbf{W}:=\begin{bmatrix}\mathbf{W}_{x}&\mathbf{0}\\\
\mathbf{0}&\mathbf{W}_{y}\end{bmatrix},\;\mathbf{W}{\,\in\mathbb{R}}^{2{n_{\textrm{{d}}}}\times
2{n_{\textrm{{d}}}}},$ (32)

where the sub-weight matrices
$\mathbf{W}_{x},\mathbf{W}_{y}{\,\in\mathbb{R}}^{{n_{\textrm{{d}}}}\times{n_{\textrm{{d}}}}}$
are defined as

$\mathbf{W}_{x}=\frac{1}{{u_{x}^{\textrm{mean}}}}\boldsymbol{\mathsf{I}}\;\;\text{and}\;\;\mathbf{W}_{y}=\frac{1}{{u_{y}^{\textrm{mean}}}}\boldsymbol{\mathsf{I}},$
(33)

with the identity matrix $\boldsymbol{\mathsf{I}}$ of size
${n_{\textrm{{d}}}}\times{n_{\textrm{{d}}}}$ and the mean absolute
displacements ${u_{x}^{\textrm{mean}}}$ and ${u_{y}^{\textrm{mean}}}$ in $x$\-
and $y$-direction determined as

${u_{x}^{\textrm{mean}}}=\frac{1}{{n_{\textrm{{d}}}}}\sum_{i=1}^{{n_{\textrm{{d}}}}}|u_{x}^{(i)}|\;\;\text{and}\;\;{u_{y}^{\textrm{mean}}}=\frac{1}{{n_{\textrm{{d}}}}}\sum_{i=1}^{{n_{\textrm{{d}}}}}|u_{y}^{(i)}|.$
(34)

The loss function $\phi(\boldsymbol{\kappa})$ is then given by the sum of the
squared, weighted residuals as

$\phi(\boldsymbol{\kappa})=\frac{1}{2}\left\lVert\tilde{\mathbf{r}}(\boldsymbol{\kappa})\right\rVert^{2}=\frac{1}{2}\left\lVert\mathbf{W}\,(\widehat{\mathbf{u}}^{\textrm{s}}(\boldsymbol{\kappa})-\mathbf{d})\right\rVert^{2}.$
(35)

A deterministic point estimate of the material parameters
$\boldsymbol{\kappa}^{*}$ can be determined by solving the minimization
problem

$\boldsymbol{\kappa}^{*}=\operatorname*{arg\,min}_{\boldsymbol{\kappa}}\phi(\boldsymbol{\kappa})\text{
subject to }\boldsymbol{\kappa}\in\Omega_{\boldsymbol{\kappa}},$ (36)

where $\boldsymbol{\kappa}^{*}$ must be a value from the set
$\Omega_{\boldsymbol{\kappa}}$ which contains only physically admissible
material parameters. The so-called normal equation is recovered from the
necessary condition of a vanishing gradient of the loss function
$\phi(\boldsymbol{\kappa})$ in the solution $\boldsymbol{\kappa}^{*}$,

$\left.\frac{\text{$\hskip 2.84544pt$d
$\phi(\boldsymbol{\kappa})$}}{\text{$\hskip 2.84544pt$d
$\boldsymbol{\kappa}$}}\right|_{\boldsymbol{\kappa}=\boldsymbol{\kappa}^{*}}=\left[\frac{\text{$\hskip
2.84544pt$d
$\widehat{\mathbf{u}}^{\textrm{s}}(\boldsymbol{\kappa}^{*})$}}{\text{$\hskip
2.84544pt$d
$\boldsymbol{\kappa}$}}\right]^{\top}\mathbf{W}^{\top}\mathbf{W}\,(\widehat{\mathbf{u}}^{\textrm{s}}(\boldsymbol{\kappa}^{*})-\mathbf{d})=\mathbf{0},$
(37)

which is in general a system of nonlinear equations. Here, $\text{$\hskip
2.84544pt$d
$\widehat{\mathbf{u}}^{\textrm{s}}(\boldsymbol{\kappa}^{*})$}/\text{$\hskip
2.84544pt$d $\boldsymbol{\kappa}$}{\,\in\mathbb{R}}^{2{n_{\textrm{{d}}}}\times
n_{\boldsymbol{\kappa}}}$ is the Jacobian of the parameters-to-state map
$\widehat{\mathbf{u}}^{\textrm{s}}$ with respect to the material parameters
$\boldsymbol{\kappa}$ and can be calculated with automatic differentiation
when using PINNs.

Problem 36 can be solved using well-established optimization procedures, such
as gradient-based or gradient-free techniques. In particular, we use the
L-BFGS algorithm. It should be noted that multiple global or local minima of
problem 36 may exist. In this case, $\boldsymbol{\kappa}^{*}$ is an arbitrary
element of the solution set of the minimization problem that depends, among
others, on the initial material parameter values. This leads to the concept of
local identifiability of material parameters and is addressed in [16] when
using full-field data.

###  4.2 Bayesian statistical inference

General problem statement: Constitutive model calibration can also be
addressed from a Bayesian statistical point of view. In this setting, the
unknown material parameters are treated as random variables with prior
probability distributions $p(\boldsymbol{\kappa})$. The prior distribution is
then updated according to Bayes’s law

$p(\boldsymbol{\kappa}|\mathbf{d})\propto
p(\mathbf{d}|\boldsymbol{\kappa})p(\boldsymbol{\kappa}),$ (38)

where $p(\boldsymbol{\kappa}|\mathbf{d})$ is the posterior probability density
and $p(\mathbf{d}|\boldsymbol{\kappa})$ represents the likelihood function
[54]. In analogy to the deterministic reduced formulation defined in 30b–30a,
the statistical counterpart reads

$\displaystyle\widehat{\mathbf{u}}^{\textrm{s}}(\boldsymbol{\kappa})$
$\displaystyle=\mathbf{d}+\mathbf{e},$ (39a) $\displaystyle\text{subject to
}\mathbf{F}(\widehat{\mathbf{u}}^{\textrm{s}}(\boldsymbol{\kappa}),\boldsymbol{\kappa})$
$\displaystyle=\mathbf{0},$ (39b)

with the observation noise vector $\mathbf{e}$.

Solution approach: We assume that the noise $\mathbf{e}$ in the measurement
data is normally distributed with zero mean and positive definite covariance
matrix $\boldsymbol{\Sigma}_{\mathbf{e}}$, i.e.,
$\mathbf{e}\sim\mathcal{N}(\mathbf{0},\boldsymbol{\Sigma}_{\mathbf{e}})$. In
addition, we assume the noise to be independent and identically distributed
(i.i.d), leading to a diagonal covariance matrix with entries
$\sigma_{e}^{2}$. Under these assumptions, the reduced observation equation in
39a implies the conditional probability density

$\displaystyle p(\mathbf{d}|\boldsymbol{\kappa})$
$\displaystyle=\mathcal{N}(\widehat{\mathbf{u}}^{\textrm{s}}(\boldsymbol{\kappa}),\boldsymbol{\Sigma}_{\mathbf{e}})$
(40) $\displaystyle=\frac{1}{(2\pi)^{2{n_{\textrm{{d}}}}/2}\text{$\hskip
2.84544pt$det$\left(\boldsymbol{\Sigma}_{\mathbf{e}}\right)$}^{1/2}}\mathrm{exp}\Bigl{(}-\frac{1}{2}(\widehat{\mathbf{u}}^{\textrm{s}}(\boldsymbol{\kappa})-\mathbf{d})^{\top}\boldsymbol{\Sigma}_{\mathbf{e}}^{-1}(\widehat{\mathbf{u}}^{\textrm{s}}(\boldsymbol{\kappa})-\mathbf{d})\Bigr{)},$

corresponding to the likelihood function of the data
$L_{\textrm{d}}(\boldsymbol{\kappa}):=p(\mathbf{d}|\boldsymbol{\kappa})$. The
likelihood function expresses the plausibility of observing the data
$\mathbf{d}$ for given material parameters $\boldsymbol{\kappa}$. The
posterior probability density $p(\boldsymbol{\kappa}|\mathbf{d})$ in 38 can be
determined by a sampling-based Markov chain Monte Carlo (MCMC) analysis. In
our numerical tests, we use a stable and well-tested implementation of the
affine-invariant ensemble sampler, also known as emcee [55]. This algorithm is
robust and in comparison to other MCMC algorithms, it does require hand-tuning
of only one hyperparameter, which is the stretch scale. For an in-depth
description of the algorithm behind emcee and an explanation of the
hyperparameter, please refer to [56].

Once the posterior distribution is determined, it provides both a point
estimate as well as a quantification of uncertainty. The maximum a posteriori
estimate is given by

$\displaystyle\boldsymbol{\kappa}^{*}$
$\displaystyle=\operatorname*{arg\,min}_{\boldsymbol{\kappa}}-\log
p(\boldsymbol{\kappa}|\mathbf{d})$ (41)
$\displaystyle=\operatorname*{arg\,min}_{\boldsymbol{\kappa}}-\bigl{(}\log
L_{\textrm{d}}(\boldsymbol{\kappa})+\log p(\boldsymbol{\kappa})\big{)}.$

Substituting the likelihood function $L_{\textrm{d}}(\boldsymbol{\kappa})$
from 40, we obtain

$\boldsymbol{\kappa}^{*}=\operatorname*{arg\,min}_{\boldsymbol{\kappa}}\Bigl{(}\frac{1}{2}\left\lVert\widehat{\mathbf{u}}^{\textrm{s}}(\boldsymbol{\kappa})-\mathbf{d}\right\rVert_{\boldsymbol{\Sigma}_{\mathbf{e}}^{-1}}^{2}-\log
p(\boldsymbol{\kappa})\Bigr{)},$ (42)

with the weighted norm
$\left\lVert\mathbf{b}\right\rVert_{\boldsymbol{\mathsf{A}}}^{2}=\mathbf{b}^{\top}\boldsymbol{\mathsf{A}}\mathbf{b}$
for any positive definite matrix $\mathbf{A}$. For a Gaussian prior, the
maximum a posteriori estimate naturally leads to a regularized NLS problem.

Uncertainty quantification from a frequentist perspective: The uncertainty of
a point estimate can be quantified through credible intervals which can be
derived on the basis of the posterior distribution, and are also referred to
as posterior intervals [54]. A credible interval is associated with an
interval in the parameter domain, containing an unknown parameter $\kappa_{i}$
with a certain probability. Provided that the posterior probability density of
the parameter $\kappa_{i}$ is normally distributed, such that
$p(\kappa_{i}|\mathbf{d})\approx\mathcal{N}(\mu_{p(\kappa_{i}|\mathbf{d})},\sigma_{p(\kappa_{i}|\mathbf{d})})$,
the unknown $\kappa_{i}$ has a value in the credible interval
$CI_{\textrm{95\%}}=\left[\mu_{p(\kappa_{i}|\mathbf{d})}\pm
1.96\cdot\sigma_{p(\kappa_{i}|\mathbf{d})}\right]$ with a probability of
approximately $95\text{\,}\mathrm{\char 37\relax}$.

In a Bayesian setting, a correct uncertainty quantification relies on an
accurate parameters-to-state map. However, if the parameters-to-state map is
misspecified, e.g., by simplifying modeling assumptions or simply by numerical
errors, it follows that
$\mathbf{F}(\widehat{\mathbf{u}}^{\textrm{s}}(\boldsymbol{\kappa}),\boldsymbol{\kappa})\neq\mathbf{0}$
in 39. This also leads to a misspecified statistical model represented by the
likelihood function $L_{\textrm{d}}(\boldsymbol{\kappa})$. As a consequence,
the quantification of uncertainty may not be valid [57]. The correctness and
reliability of the uncertainty quantification must therefore be verified. As
illustrated above, from a frequentist point of view, the uncertainty is valid
if for ${n_{\textrm{tests}}}\rightarrow\infty$ experiments the material
parameter has probability $\alpha$ to be within the credible interval
$CI_{\alpha}$, i.e., if the credible intervals are also confidence intervals.
The reliability of the uncertainty quantification from a frequentist
perspective can thus be determined by performing a coverage test. The coverage
test can be used to assess how well the credible interval covers the true
parameter and is described below in more detail. First, the posterior
distribution $p(\boldsymbol{\kappa}|\mathbf{d})$ is determined for a large
number of independent tests ${n_{\textrm{tests}}}$. Second, the probability
$\beta^{(i)}=n_{CI_{\alpha}}^{(i)}/{n_{\textrm{tests}}}$ of the true parameter
$\kappa_{i}$ to be within the credible interval $CI_{\alpha}^{(i)}$ is
calculated. Here, $n_{CI_{\alpha}}^{(i)}$ is the number of tests for which
$\kappa_{i}\in CI_{\alpha}^{(i)}$. Note that the coverage $\beta^{(i)}$ is
calculated separately for each parameter $\kappa_{i}$. Since the true
parameters $\boldsymbol{\kappa}$ must be known for the test, we use synthetic
data for which the parameters are then re-identified. Finally, the estimated
uncertainty for parameter $\kappa_{i}$ is valid if $\beta^{(i)}\approx\alpha$.

##  5 Results for synthetic full-field data

In the following, we demonstrate the calibration of constitutive models from
synthetic full-field displacement data using parametric PINNs. Both small
strain linear elasticity and finite strain hyperelasticity are considered.
First, we define the test cases and the hyperparameters of both the parametric
PINNs’ architecture and the training settings. We then start with the
deterministic calibration by solving the NLS problem. We further quantify the
uncertainty in the estimated material parameters by conducting Bayesian
statistical inference. All results are statistically analyzed.

###  5.1 Test cases and training of parametric PINNs

In this section, we describe the two test cases in more detail, specify the
hyperparameters of the parametric PINNs’ architecture and the training
settings, and report the accuracy of the parametric forward solutions. In both
test cases, we consider a plate with a hole. Since the geometry is two-fold
symmetric, we consider only the top left quadrant of the plate and define
symmetry boundary conditions on the bottom and right boundaries. We load the
plate on the left edge with
$\bar{\mathbf{t}}=[$-100\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$,$0\text{\,}\mathrm{}$]^{\top}$.
Furthermore, external specific body forces, such as gravity, are neglected.
The geometry and boundary conditions are shown in Fig. 1. The general workflow
including data generation, training and validation of the parametric PINN as
well as calibration is outlined in Fig. 2 and explained in more detail in the
following.

####  5.1.1 Test case 1: Linear elasticity

As our first synthetic test case, we assume isotropic, linear elastic material
and take construction steel as an example. Typical bulk and shear moduli for
construction steel are
$K=$175\,000\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$$ and
$G=$80\,769\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$$, respectively,
corresponding to a Young’s modulus
$E=$210\,000\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$$ and Poisson’s
ratio $\nu=$0.3\text{\,}\mathrm{}$$, respectively. The plate is assumed to be
under plane stress condition.

$\bar{\mathrm{u}}_{y}=$0\text{\,}\mathrm{}$,\bar{\mathrm{P}}_{xy}=$0\text{\,}\mathrm{}$$$\bar{\mathrm{u}}_{x}=$0\text{\,}\mathrm{}$$$\bar{\mathrm{P}}_{yx}=$0\text{\,}\mathrm{}$$$\bar{\mathbf{t}}=\begin{bmatrix}$-100\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$\\\
$0\text{\,}\mathrm{}$\end{bmatrix}$$\bar{\mathbf{t}}=\boldsymbol{0}$$\bar{\mathbf{t}}=\boldsymbol{0}$$L=$100\text{\,}\mathrm{mm}$$$L=$100\text{\,}\mathrm{mm}$$R
= $10\text{\,}\mathrm{mm}$yx Figure 1: Geometry and boundary conditions of the
top left quadrant of a plate with a hole under uniaxial tension. Body forces
are neglected. datagenerationusing FEMtrainingof parametricPINNvalidationof
parametricPINNdeterministiccalibrationstatisticalcalibrationNLSMCMCoffline
stageonline stage Figure 2: Flowchart of the entire process including the
offline as well as the online stage. In the offline stage, the data for both
training and validation is generated using FEM. The parametric PINN is then
trained and validated. In the online stage, the pre-trained parametric PINN
can be used to calibrate constitutive models in both a deterministic and
statistical setting. Note that in the synthetic test cases, the data for
calibration is also generated using FEM.

FE simulations: The synthetic displacement data for the training, validation
and calibration data sets are generated by FE simulations. For the FE
simulations, the geometry is meshed with triangular elements and we choose
linear ansatz functions with one point integration. The displacement field is
calculated and recorded at a total of $1\,148\,975\text{\,}\mathrm{}$ nodes.
Due to the high resolution of the computational grid, the discretization
errors are considered negligible.

PINN’s architecture and training: We use a fully-connected FFNN with six
hidden layers each with $128\text{\,}\mathrm{}$ neurons and a hyperbolic
tangent activation function. The PINN has further four input neurons for the
$x$\- and $y$-coordinate and the two material parameters which are the bulk
and shear modulus. Correspondingly, the PINN has two output neurons for the
displacement in $x$\- and $y$-direction. The weights and biases of the FFNN
are initialized according to Glorot normal initialization [58] and with zeros,
respectively.

For solving the resulting optimization problem that arises during training, we
choose the L-BFGS optimization algorithm [46, 47, 48, 49, 50]. The training
data set is composed as follows: We train the parametric PINN for bulk and
shear moduli within the range
$K_{\textrm{train}}=[$100\,000\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$,$200\,000\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$]$
and
$G_{\textrm{train}}=[$60\,000\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$,$100\,000\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$]$
corresponding to ranges for Young’s modulus and Poisson’s ratio of
$E_{\textrm{train}}=[$150\,000\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$,$257\,143\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$]$
and
$\nu_{\textrm{train}}=[$0.125\text{\,}\mathrm{}$,$0.3636\text{\,}\mathrm{}$]$,
respectively. Therefore, we collect collocation points within the domain and
on the boundaries for $1024\text{\,}\mathrm{}$ different combinations of bulk
and shear moduli. These material parameter samples are drawn by Sobol sampling
[59] from the material parameter domain. For each of the parameter samples, we
generate $64\text{\,}\mathrm{}$ collocation points to enforce the PDE
($\mathbf{T}_{\textrm{C}}$) within the domain and $64\text{\,}\mathrm{}$
collocation points on each of the five boundary segments
($\mathbf{T}_{\textrm{N}}$). While the collocation points on the boundaries
are distributed uniformly, the collocation points within the domain are again
drawn by Sobol sampling. The stress boundary conditions are enforced as
defined in Fig. 1. Since we consider the strong form of the PDE, it is
essential to explicitly account for the symmetry stress boundary conditions on
the bottom and right boundaries. Note that these symmetry boundary conditions
are also imposed in the Galerkin FEM. For the derivation of the correct
boundary conditions, please refer to Appendix A. We further enhance the
training by pre-simulated FE data ($\mathbf{T}_{\textrm{d}}$) for
$128\text{\,}\mathrm{}$ parameter samples drawn by Sobol sampling from the
parameter domain. For each of the parameter samples, we randomly pick
$128\text{\,}\mathrm{}$ nodes from the FE solution. In order to account for
the different scales of the loss terms, we weight the data loss term by a
constant factor $\lambda_{\textrm{d}}=10^{4}$.

Validation: For the validation of the parametric PINN and the subsequent
calibration, we generate a total of $100\text{\,}\mathrm{}$ different
synthetic displacement data sets for randomly selected combinations of bulk
and shear moduli using FE simulations. We do not expect the parametric PINNs
to approximate the displacements well beyond the training range of the
material parameters. To prevent the realizations of the material parameters
from being too close to the edges of the training range in calibration, we use
a slightly limited parameter range for the generation of the synthetic full-
field data. For the linear elastic constitutive model, we select bulk and
shear moduli within the ranges
$K_{\textrm{valid}}=[$101\,000\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$,$199\,000\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$]$
and
$G_{\textrm{valid}}=[$60\,500\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$,$99\,500\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$]$,
respectively. The validation is then performed on $1024\text{\,}\mathrm{}$
points randomly selected from each of the FE solutions. In comparison to the
high-fidelity FE solution, the mean absolute error (MAE) and the relative
$\text{rL}^{2}$ norm ($\text{rL}^{2}$) of the parametric PINN yield
$\text{MAE}=$1.32\text{\times}{10}^{-5}\text{\,}\mathrm{}$$ and
$\text{rL}^{2}=$9.98\text{\times}{10}^{-4}\text{\,}\mathrm{}$$, respectively.
Note that the calibration data is different from the data we use to enhance
the training. Please refer to Appendix B for a definition of the error
measures used in our numerical tests.

####  5.1.2 Test case 2: Hyperelasticity

In the second synthetic test case, we assume a weakly compressible Neo-Hookean
material. The geometry of the plate with a hole and the boundary conditions
are the same as in test case 1, see Fig. 1. We assume the plate to be under
plane strain condition.

FE simulations: For the generation of the FE data, we mesh the geometry with
triangular elements, but choose quadratic ansatz functions with four
quadrature points. The FE solution is computed and recorded at a total of
$1\,150\,118\text{\,}\mathrm{}$ nodes and we consider discretization errors to
be negligible.

PINN’s architecture and training: The hyperparameters of the parametric PINN
and the training settings as well as the number and composition of the
training and validation data sets are defined identically to test case 1
except for the training ranges of the material parameters. For the
hyperelastic material, we consider bulk and shear moduli within the range
$K_{\textrm{train}}=[$4000\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$,$8000\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$]$
and
$G_{\textrm{train}}=[$500\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$,$1500\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$]$.
It should be noted that a non-physical behavior due to the compressible part
of the strain-energy function 9 is not observable in the chosen parameter
range. For details, see [36, 37].

Validation: As in test case 1, we generate a total of $100\text{\,}\mathrm{}$
different synthetic displacement data sets using FE simulations. In parameter
space, we randomly sample bulk and shear moduli within the ranges
$K_{\textrm{valid}}=[$4020\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$,$7980\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$]$
and
$G_{\textrm{valid}}=[$505\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$,$1495\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$]$,
respectively. For validation, we use $1024\text{\,}\mathrm{}$ points randomly
selected from each of the FE solutions. In relation to the validation data,
the parametric PINN yields a MAE and a $\text{rL}^{2}$ of
$\text{MAE}=$4.92\text{\times}{10}^{-5}\text{\,}\mathrm{}$$ and
$\text{rL}^{2}=$1.04\text{\times}{10}^{-4}\text{\,}\mathrm{}$$, respectively.

###  5.2 Deterministic calibration

In the following, we present the results for the deterministic NLS calibration
for the two synthetic test cases. For the formulation of the NLS calibration
problem, please refer to Section 4.1. In order to make robust statements about
the accuracy of deterministic calibration, the accuracy of the identified
material parameters for a total of $100\text{\,}\mathrm{}$ synthetic full-
field displacement measurements is statistically analyzed. For the
deterministic calibration, we use the L-BFGS algorithm and initialize the
material parameters with the mean value of their training range, respectively.

We test the calibration for the same synthetic data sets that we used to
validate the performance of the parametric PINN, see Section 5.1. In contrast
to validation, however, we add artificial noise. First, we select
$128\text{\,}\mathrm{}$ data points at random from each of the
$100\text{\,}\mathrm{}$ synthetic full-field measurements. Second, in order to
emulate real DIC data, we add Gaussian noise $\mathcal{N}(0,\sigma^{2})$ with
zero mean to the clean synthetic displacement data. According to [60, 61], the
noise in DIC images has a standard deviation of
$\sigma=$4\text{\times}{10}^{-4}\text{\,}\mathrm{mm}$$. To take into account
that the optimal conditions required for this value are not always achieved in
practice, we assume a standard deviation of
$\sigma=$5\text{\times}{10}^{-4}\text{\,}\mathrm{mm}$$ instead.

In Table 1, the results for test cases 1 and 2 are listed. We report the mean
absolute relative errors of the identified parameters compared to the true
parameters used to calculate the synthetic data. In addition, to be able to
estimate the scatter of the results, we also provide the standard errors of
the means as well as the minimum and maximum AREs. For a definition of the
error measures used to evaluate the calibration results, please see Appendix
B.

The results show that for both the linear elastic and the hyperelastic
constitutive model, the material parameters can be identified with only small
AREs. In addition, the scatter of the AREs is small in both test cases, as
evidenced by the SEMs. However, for the hyperelastic constitutive model, the
errors are even significantly smaller than for the linear elastic constitutive
model. We suspect that one reason for this observation is different ratios
between the magnitude of the noise and the absolute displacements in the two
test cases. The order of magnitude of the maximum absolute displacements in
both $x$\- and $y$\- direction is $\mathcal{O}(10^{-2})$ in test case 1
(linear elasticity) and $\mathcal{O}(10^{0})$ in test case 2 (hyperelasticity)
and is thus two orders of magnitude higher. At the same time, the magnitude
and standard deviation of the noise remains constant, as these are only
associated with the device, not with the observations. Hence, in test case 1,
the noise has a significantly greater influence. Another reason for the larger
AREs and SEMs for calibrating the linear elastic constitutive model is that
the parametric PINN in test case 1 is trained for a significantly larger
parameter range. For both test cases, the NLS calibration takes less than five
seconds on average on a NVIDIA graphics processing unit (GPU) A100 with 80 GB
memory. The number of parametric PINN evaluations per calibration is
$\mathcal{O}(10^{1})$ in both test cases.

Table 1: Results of deterministic NLS calibration for the synthetic displacement data in test cases 1 and 2. We repeat the NLS calibration for $100\text{\,}\mathrm{}$ synthetic DIC measurements for different combinations of material parameters. From the obtained $100\text{\,}\mathrm{}$ identified material parameter sets, we calculate the mean absolute relative errors (AREs) with respect to the exact material parameters used for data generation. In addition, we provide the standard errors of the means (SEMs) as well as the minimum and maximum AREs to be able to estimate the scatter of the errors. |  | absolute relative error (ARE) [%]
---|---|---
|  |  mean |  SEM |  minimum |  maximum
test case 1: linear elasticity |  bulk modulus $K$ |  $7.20\text{\times}{10}^{-1}\text{\,}\mathrm{}$ |  $5.41\text{\times}{10}^{-2}\text{\,}\mathrm{}$ |  $1.09\text{\times}{10}^{-2}\text{\,}\mathrm{}$ |  $2.63\text{\,}\mathrm{}$
shear modulus $G$ |  $1.57\text{\times}{10}^{-1}\text{\,}\mathrm{}$ |  $1.18\text{\times}{10}^{-2}\text{\,}\mathrm{}$ |  $6.86\text{\times}{10}^{-4}\text{\,}\mathrm{}$ |  $4.79\text{\times}{10}^{-1}\text{\,}\mathrm{}$
test case 2: hyperelasticity |  bulk modulus $K$ |  $1.23\text{\times}{10}^{-2}\text{\,}\mathrm{}$ |  $1.03\text{\times}{10}^{-3}\text{\,}\mathrm{}$ |  $1.23\text{\times}{10}^{-5}\text{\,}\mathrm{}$ |  $5.83\text{\times}{10}^{-2}\text{\,}\mathrm{}$
shear modulus $G$ |  $1.64\text{\times}{10}^{-3}\text{\,}\mathrm{}$ |  $1.27\text{\times}{10}^{-4}\text{\,}\mathrm{}$ |  $7.47\text{\times}{10}^{-8}\text{\,}\mathrm{}$ |  $5.68\text{\times}{10}^{-3}\text{\,}\mathrm{}$

###  5.3 Bayesian statistical inference

In this subsection, we address the model calibration problem from a Bayesian
statistical point of view. We treat the material parameters as random
variables with a prior distribution that represents our estimate of the
material parameters before we have seen the data. We then perform Bayesian
statistical inference and sample the posterior distribution performing a MCMC
analysis. In order to validate the uncertainty of the estimated parameters
from a frequentist point of view, we further carry out a coverage test. For
the detailed formulation of the statistical calibration problem, we refer to
Section 4.2.

We carry out a coverage test for a total of $100\text{\,}\mathrm{}$ synthetic
full-field displacement measurements to validate the reliability of the
$95\text{\,}\mathrm{\char 37\relax}$-credible interval of the sampled
posterior distributions. We use the same synthetic data as in the
deterministic calibration. To emulate real DIC data, we add Gaussian noise
$\mathcal{N}(0,\sigma^{2})$ with zero mean and standard deviation
$\sigma=$5\text{\times}{10}^{-4}\text{\,}\mathrm{mm}$$ to the clean synthetic
displacement data. As we lack more detailed prior knowledge, we employ uniform
priors covering the parameter range in which the parametric PINNs were
trained. The MCMC analysis is performed using the emcee algorithm. For both
test cases, we employ an ensemble of $100\text{\,}\mathrm{}$ workers each with
a chain length of $200\text{\,}\mathrm{}$. The workers are initialized
randomly within the material parameter training ranges. Before the parameter
samples are recorded, we run a burn-in phase with a chain length of
$100\text{\,}\mathrm{}$ for each worker. In the burn-in phase, the Markov
chain explores the parameter space and the drawn samples are not
representative for the posterior distribution. We further choose a stretch
scale of $4\text{\,}\mathrm{}$ which results in sound acceptance ratios that
should be between $0.2\text{\,}\mathrm{}$ and $0.5\text{\,}\mathrm{}$ as a
rule of thumb [56].

The results of the Bayesian statistical inference are listed in Table 2. The
coverage test clearly shows that the estimated uncertainty is valid in the
sense of frequentist statistics. For both test cases 1 and 2, the coverage for
both material parameters is close to the expected $95\text{\,}\mathrm{\char
37\relax}$. We further report the average bias of the posterior mean values
with respect to the true material parameters and the standard deviations of
the posterior distributions. To calculate these quantities, we have made the
assumption that the sampled posterior probability density function (PDF) can
be approximated by a Gaussian distribution. As shown in Fig. 3 as an example,
this is a reasonable assumption. Furthermore, the runtime for the MCMC
analysis is less than $60\text{\,}\mathrm{}$ seconds on average on a NVIDIA
GPU A100 with 80 GB memory. According to the hyperparameters of the emcee
algorithm specified above, the parametric PINN is evaluated a total of
$3\text{\times}{10}^{5}\text{\,}\mathrm{}$ times in each MCMC analysis.

Table 2:  Results of Bayesian statistical inference for the synthetic displacement data in test cases 1 and 2. We carry out a coverage test comprising $100\text{\,}\mathrm{}$ synthetic DIC measurements each for different combinations of material parameters. The coverage indicates the percentage of test cases for which the true material parameter used to generate the synthetic data is within the $95\text{\,}\mathrm{\char 37\relax}$-credible interval. We further report the average bias of the posterior mean values with respect to the true material parameters and the standard deviations of the posterior distributions.  |  |  coverage |  average bias
of mean $[$\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$]$ |  standard deviation $[$\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$]$
---|---|---|---|---
test case 1: linear elasticity |  bulk modulus $K$ |  $94\text{\,}\mathrm{\char 37\relax}$ |  $-147.65\text{\,}\mathrm{}$ |  $1200.61\text{\,}\mathrm{}$
shear modulus $G$ |  $92\text{\,}\mathrm{\char 37\relax}$ |  $9.26\text{\,}\mathrm{}$ |  $138.84\text{\,}\mathrm{}$
test case 2: hyperelasticity |  bulk modulus $K$ |  $93\text{\,}\mathrm{\char 37\relax}$ |  $-2.26\text{\times}{10}^{-1}\text{\,}\mathrm{}$ |  $9.10\text{\times}{10}^{-1}\text{\,}\mathrm{}$
shear modulus $G$ |  $98\text{\,}\mathrm{\char 37\relax}$ |  $2.73\text{\times}{10}^{-3}\text{\,}\mathrm{}$ |  $2.44\text{\times}{10}^{-2}\text{\,}\mathrm{}$

(a)

(b)

Figure 3: Exemplary histograms of the posterior distribution of (a) bulk and
(b) shear modulus for the hyperelastic constitutive model determined by
Bayesian statistical inference. The illustration shows exemplary that the
assumption of normally distributed posteriors is reasonable.

##  6 Results for experimental full-field data

Finally, we showcase the calibration of the linear elastic material model from
real-world experimental full-field displacement data. As with the synthetic
data in Section 5, we perform both a deterministic and a statistical
calibration.

###  6.1 Setup and training of parametric PINN

We consider experimental full-field displacement data measured in a tensile
test using DIC. In the experiment, we used a specimen of S235 steel and assume
linear elastic material behaviour.

Experimental settings: The specimen was clamped on the left side and the
testing machine pulled on the right side in axial direction up to an averaged
axial strain of
${\varepsilon^{\textrm{mean}}}=$5.1\text{\times}{10}^{-2}\text{\,}\mathrm{\char
37\relax}$$. Thus, the strain is still in the linear elastic regime of the
material under consideration. After a maximum traction of
$\bar{\mathbf{t}}=[$106.26\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$,0]^{\top}$
has been applied, the displacements in the parallel area around the hole were
measured with a DIC system. For an illustration of the specimen geometry, the
boundary conditions and the measurement area, please refer to Fig. 4. The
full-field DIC measurement is published in [31].

FE simulations: To enhance the training process and to validate the parametric
PINN, we generate high fidelity displacement data using FEM. Therefore, the
simplified geometry is meshed with triangular elements and we choose linear
ansatz functions with one point integration. The displacement field is then
calculated and recorded for a total of $232\,984\text{\,}\mathrm{}$ nodes.
Discretization errors are neglected due to the high resolution of the
computational grid.

PINN’s architecture and training: The hyperparameters of the parametric PINN
and the training settings are identical to the two previous test cases. To
reduce the complexity, we train the parametric PINN not for the complete
specimen geometry but for a simplified one, see Fig. 4. For this purpose, we
transfer the stress boundary condition from the end of the clamped area where
the traction was actually applied to the end of the parallel area. As a
prerequisite, we make the assumption that the force is distributed
homogeneously over the height of the sample.

$\bar{\mathrm{u}}_{x}=$0\text{\,}\mathrm{}$$$\bar{\mathrm{u}}_{y}=$0\text{\,}\mathrm{}$$$\bar{\mathbf{t}}=\begin{bmatrix}$106.26\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$\\\
$0\text{\,}\mathrm{}$\end{bmatrix}$30
mm$20\text{\,}\mathrm{mm}$$50\text{\,}\mathrm{m}\mathrm{m}$$80\text{\,}\mathrm{m}\mathrm{m}$$90\text{\,}\mathrm{m}\mathrm{m}$$220\text{\,}\mathrm{m}\mathrm{m}$$25\text{\,}\mathrm{m}\mathrm{m}$$4\text{\,}\mathrm{m}\mathrm{m}$
Figure 4: Geometry and boundary conditions of the tensile test. The specimen
is clamped on the left side and subjected to traction on the right side (the
clamped areas are filled in gray). The displacements were measured by a DIC
system for the area filled in red. The parametric PINN is trained for the
boundary conditions shown in the figure and the simplified geometry defined by
the solid lines. Free Neumann boundary conditions were applied at the upper
and lower edge of the geometry and in the hole.

The training data is composed as follows: We train the parametric PINN for
bulk and shear moduli in the range
$K_{\textrm{train}}=[$100\,000\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$,$200\,000\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$]$
and
$G_{\textrm{train}}=[$60\,000\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$,$100\,000\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$]$
corresponding to ranges for Young’s modulus and Poisson’s ratio of
$E_{\textrm{train}}=[$150\,000\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$,$257\,143\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$]$
and
$\nu_{\textrm{train}}=[$0.125\text{\,}\mathrm{}$,$0.3636\text{\,}\mathrm{}$]$,
respectively. For training, we consider $1024\text{\,}\mathrm{}$ different
combinations of the material parameters drawn by Sobol sampling. For each of
the parameter samples, we generate $64\text{\,}\mathrm{}$ collocation points
within the domain ($\mathbf{T}_{\textrm{C}}$) and $64\text{\,}\mathrm{}$
collocation points on each of the six boundary segments
($\mathbf{T}_{\textrm{N}}$). In addition, we enhance the training data set by
pre-simulated FE data ($\mathbf{T}_{\textrm{d}}$). We randomly select
$128\text{\,}\mathrm{}$ data points from the FEM solution each for
$128\text{\,}\mathrm{}$ material parameter combinations drawn by Sobol
sampling. We further weight the data loss term by
$\lambda_{\textrm{d}}=10^{6}$ in order to account for the different loss term
scales.

Validation: As in the previous test cases, validation is performed on
$1024\text{\,}\mathrm{}$ data points directly and randomly taken from the FEM
solution each for $100\text{\,}\mathrm{}$ randomly sampled parameter
combinations within the training ranges. In relation to the validation data,
the parametric PINN yields a MAE and a $\text{rL}^{2}$ of
$\text{MAE}=$1.08\text{\times}{10}^{-6}\text{\,}\mathrm{}$$ and
$\text{rL}^{2}=$6.32\text{\times}{10}^{-5}\text{\,}\mathrm{}$$, respectively.

###  6.2 Deterministic calibration

The full-field displacement measurement comprises a total of
$5244\text{\,}\mathrm{}$ data points within the parallel area around the hole,
see Fig. 4 for the specimen geometry. For calibration, we again use the L-BFGS
algorithm and initialize the material parameters with the mean value of their
training range, respectively. As reference solution, we use the result of a
NLS-FEM calibration. In this approach, the parameters-to-state map is realized
by a FE simulation that is performed in each iteration instead of using the
parametric PINN. For solving the NLS-FEM problem, the lsqnonlin function in
Matlab is used. For more information on this approach when using full-field
displacement data, please refer to [16].

For the visualization of the DIC images in Fig. 5, the measured displacements
are interpolated onto a regular grid. The visualization shows that
particularly in the area around the hole and the clamping, displacements were
measured that deviate significantly from the expected displacement field.
Since the outliers also significantly distort the scale of the displacements
in $y$-direction, we therefore limit the scale of the displacements in
$y$-direction to
$\mathrm{u}_{y}^{\textrm{visual}}=[$-5\text{\times}{10}^{-3}\text{\,}\mathrm{mm}$,$5\text{\times}{10}^{-3}\text{\,}\mathrm{m}\mathrm{m}$]$
for visualization purposes only. In addition, it becomes clear that the
measured displacements in $y$-direction are superimposed by a lateral
displacement which may result from an eccentric clamping of the test specimen.
However, it should be noted that the expected magnitude of the displacements
in $y$-direction is small compared to the $x$-direction due to the material
properties and the experimental setup. The measurement in $y$-direction is
therefore more susceptible to external disturbances.

(a)

(b)

Figure 5: Visualization of the displacements in (a) $x$-direction and (b)
$y$-direction measured in the tensile test by DIC. For visualization purposes,
the measured displacements are interpolated onto a regular grid. Since the
outliers significantly distort the scale of the displacements in
$y$-direction, we limit the scale of the displacements in $y$-direction to
$\mathrm{u}_{y}^{\textrm{visual}}=[$-5\text{\times}{10}^{-3}\text{\,}\mathrm{mm}$,$5\text{\times}{10}^{-3}\text{\,}\mathrm{m}\mathrm{m}$]$
for visualization purposes only.

The results of the NLS calibration are listed in Table 3. The calibration
using the raw DIC data yields a bulk and shear modulus of
$K=$109\,343\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$$ and
$G=$71\,125\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$$, respectively. In
relation to the NLS-FEM results, the identified material parameters deviate by
relative deviations of $\text{RD}_{K}=$-14.63\text{\,}\mathrm{\char
37\relax}$$ and $\text{RD}_{G}=$-3.29\text{\,}\mathrm{\char 37\relax}$$. We
assume that the reason for the large deviation is that the displacement data
is pre-processed in NLS-FEM. The measured full-field displacement data is
linearly interpolated onto the FE mesh nodes. In this process, outliers in the
full-field measurement are smoothed out. For the linear interpolation, the
Matlab function scatteredInterpolant with default settings is used. The
parametric PINN, on the other hand, uses the raw measurement data without pre-
processing. For a fair comparison, we therefore also carry out the calibration
with the interpolated displacement measurements. After interpolation, the
full-field displacement measurement comprises a total of
$1124\text{\,}\mathrm{}$ data points. The calibration using the interpolated
data results in a bulk and shear modulus of
$K=$126\,679\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$$ and
$G=$73\,444\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$$, respectively,
which deviate by RDs of $\text{RD}_{K}=$-1.10\text{\,}\mathrm{\char
37\relax}$$ and $\text{RD}_{G}=$-0.13\text{\,}\mathrm{\char 37\relax}$$ from
the NLS-FEM results. Furthermore, with the parametric PINN, the runtime for
the NLS calibration is less than five seconds on a NVIDIA GPU A100 with 80 GB
memory. Both the parametric PINN and the FE model are evaluated
$\mathcal{O}(10^{1})$ times.

Table 3: Results of deterministic NLS calibration for the experimental displacement data. In addition to the material parameters identified by the parametric PINN, we also report the results of a NLS-FEM calibration as a reference solution. The parametric PINN is applied to both the raw full-field displacement data and the displacement data linearly interpolated to the FE mesh nodes. |  bulk modulus $K$ |  shear modulus $G$
---|---|---
FEM (interpolated data) |  $128\,085\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$ |  $73\,541\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$
PINN (raw data) |  $109\,343\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$ |  $71\,125\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$
$({\kappa_{i}^{\textrm{PINN}}}-{\kappa_{i}^{\textrm{FEM}}})/{\kappa_{i}^{\textrm{FEM}}}$ |  $-14.63\text{\,}\mathrm{\char 37\relax}$ |  $-3.29\text{\,}\mathrm{\char 37\relax}$
PINN (interpolated data) |  $126\,679\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$ |  $73\,444\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$
$({\kappa_{i}^{\textrm{PINN}}}-{\kappa_{i}^{\textrm{FEM}}})/{\kappa_{i}^{\textrm{FEM}}}$ |  $-1.10\text{\,}\mathrm{\char 37\relax}$ |  $-0.13\text{\,}\mathrm{\char 37\relax}$

###  6.3 Statistical calibration

Finally, we determine the posterior distribution of the material parameters in
the linear elastic constitutive model for the real-world experimental full-
field displacement data. A detailed description of the experimental setup is
given in Section 6.1. In order to validate our results for the parametric
PINN, we compare the posterior distributions to the results with FEM as
parameters-to-state map. As we found out in Section 6.2, for a fair
comparison, we need to use the interpolated displacement data. Furthermore,
for the MCMC analysis, we employ the emcee algorithm with an ensemble of
$100\text{\,}\mathrm{}$ workers each with a chain length of
$200\text{\,}\mathrm{}$ and a stretch scale of $4\text{\,}\mathrm{}$. Samples
are recorded after a burn-in phase with a chain length of
$100\text{\,}\mathrm{}$ for each worker. The workers are initialized randomly
within the material parameter training ranges.

In the first attempt, we assumed Gaussian noise $\mathcal{N}(0,\sigma^{2})$
with zero mean and standard deviation
$\sigma=$5\text{\times}{10}^{-4}\text{\,}\mathrm{mm}$$ just like with the
synthetic data. However, without further modifications, we have not obtained
reasonable results for this noise level. We suspect two possible reasons for
the failure of the MCMC analysis:

  1. (i)

First, the noise in the present data is superimposed by measurement artifacts,
such as lateral displacements due to a possibly eccentric clamping of the
specimen. Additionally, in Fig. 5, we can see some measurement outliers close
to the boundary caused by errors in the facet-matching in consequence of a
slightly incorrect placement of the tensile specimen with respect to the
camera alignment. The resulting measurement error which is made up of the
background noise and the measurement artifacts is therefore probably greater
than the assumed value of
$\sigma=$5\text{\times}{10}^{-4}\text{\,}\mathrm{mm}$$.

  2. (ii)

Second, we assume that the noise levels for the present data are actually
different in the $x$\- and $y$\- directions. One possible reason for this is
the different resolution of the DIC system in the different spatial
directions. In addition, in the deterministic setting, we have already
observed that weighting the residuals is essential for the calibration from
experimental data.

We therefore propose to use the diagonal covariance matrix obtained from
relating the NLS problem to the maximum a posteriori estimate, see 41–42. If
we use a uniform prior over the admissible set $\Omega_{\boldsymbol{\kappa}}$
of material parameters $\boldsymbol{\kappa}$, we restrict the statistical
calibration problem to the same parameter set as the deterministic NLS
problem, see 36. With a uniform prior, the logarithm of the prior $\log
p(\boldsymbol{\kappa})$ in 42 is constant and can be neglected in the
minimization problem. The maximum a posteriori estimate then simplifies to the
so-called maximum likelihood estimate

$\displaystyle\boldsymbol{\kappa}^{*}=\operatorname*{arg\,max}_{\boldsymbol{\kappa}}L_{\textrm{d}}(\boldsymbol{\kappa})$
$\displaystyle=\operatorname*{arg\,min}_{\boldsymbol{\kappa}}\big{(}-\log
L_{\textrm{d}}(\boldsymbol{\kappa})\big{)}$ (43)
$\displaystyle=\operatorname*{arg\,min}_{\boldsymbol{\kappa}}\big{(}\frac{1}{2}\left\lVert\widehat{\mathbf{u}}^{\textrm{s}}(\boldsymbol{\kappa})-\mathbf{d}\right\rVert_{\boldsymbol{\Sigma}_{\mathbf{e}}^{-1}}^{2}\big{)}.$

For uniform priors, the diagonal covariance matrix
$\boldsymbol{\Sigma}_{\mathbf{e}}$ can then be related to the weight matrix
$\mathbf{W}$ in the NLS problem 35 by

$\boldsymbol{\Sigma}_{\mathbf{e}}:=\begin{bmatrix}\boldsymbol{\Sigma}_{\mathbf{e}_{x}}&\mathbf{0}\\\
\mathbf{0}&\boldsymbol{\Sigma}_{\mathbf{e}_{y}}\end{bmatrix}=(\mathbf{W}^{\top}\mathbf{W})^{-1},\;\boldsymbol{\Sigma}_{\mathbf{e}}{\,\in\mathbb{R}}^{2{n_{\textrm{{d}}}}\times
2{n_{\textrm{{d}}}}},$ (44)

where the sub-covariance matrices
$\boldsymbol{\Sigma}_{\mathbf{e}_{x}},\boldsymbol{\Sigma}_{\mathbf{e}_{y}}{\,\in\mathbb{R}}^{{n_{\textrm{{d}}}}\times{n_{\textrm{{d}}}}}$
for i.i.d. noise are defined as

$\boldsymbol{\Sigma}_{\mathbf{e}_{x}}=\sigma_{x}^{2}\boldsymbol{\mathsf{I}}\;\;\text{and}\;\;\boldsymbol{\Sigma}_{\mathbf{e}_{y}}=\sigma_{y}^{2}\boldsymbol{\mathsf{I}},$
(45)

with the identity matrix of size ${n_{\textrm{{d}}}}\times{n_{\textrm{{d}}}}$
and standard deviations $\sigma_{x}$ and $\sigma_{y}$ of Gaussian noise
$\mathcal{N}(\mathbf{0},\sigma_{x})$ and $\mathcal{N}(\mathbf{0},\sigma_{y})$
in $x$\- and $y$-direction, respectively.

In the following, we use a uniform prior for the material parameters to be
inferred and derive the covariance matrix from 43, 44 and 45 as described
above. For the weight matrix used in the NLS problem, we finally obtain
standard deviations $\sigma_{x}=$0.0401\text{\,}\mathrm{mm}$$ and
$\sigma_{y}=$0.0017\text{\,}\mathrm{mm}$$ for i.i.d. Gaussian noise
$\mathcal{N}(\mathbf{0},\sigma_{x})$ and $\mathcal{N}(\mathbf{0},\sigma_{y})$
in $x$\- and $y$-direction, respectively.

The posterior probability densities for bulk and shear modulus obtained by a
MCMC analysis are illustrated in Fig. 7(a). The probability distributions show
a good concentration and small uncertainties for both material parameters.
Furthermore, the mean values of the posterior probability densities are close
to the values we obtain from the deterministic NLS-FEM calibration. This is
expected since we derive the covariance matrix from the relation between the
maximum a posteriori estimate and the NLS problem. For validation, we also
carry out the MCMC analysis with FEM as parameters-to-state map and the same
covariance matrix, see Fig. 7(b). The comparison shows that the posterior
probability densities obtained with the two different methods are in good
agreement. Moreover, with the parametric PINN, the runtime for the MCMC
analysis is less than $60\text{\,}\mathrm{}$ seconds on a NVIDIA GPU A100 with
80 GB memory. According to the hyperparameters of the emcee algorithm
specified above, the parametric PINN is evaluated a total of
$3\text{\times}{10}^{5}\text{\,}\mathrm{}$ times in the MCMC analysis.

(a) Parametric PINN

(b) FEM

Figure 7: Posterior probability densities of bulk and shear modulus determined
by a MCMC analysis for the experimental displacement measurements. The results
for the parametric PINN in (a) show a good concentration of the probability
density. For validation, in (b), we also provide the posterior probability
densities we obtain when using FEM as parameters-to-state map. The comparison
shows a good level of agreement.

Finally, we would like to make the following remarks: First, PINNs generally
do not well extrapolate beyond the training domain. We therefore recommend the
use of material parameter priors with at most weak support beyond the training
range of the parametric PINN. Otherwise, the Markov chain is more likely to
explore regions in the parameter domain for which the parametric PINN is not
trained and thus does not provide good prediction accuracy. As mentioned
before, in Bayesian inference, a correct uncertainty quantification relies on
a accurate parameters-to-state map. Second, it should be noted that the noise
levels derived from the weights used in the corresponding NLS problem are not
the actual noise levels of the measurements. The choice of the weights is
usually based on heuristics and not necessarily on a statistical analysis of
the measurement data. However, the chosen approach enables comparability
between the statistical and the deterministic calibration problems. Third, we
point out that Bayesian inference, in principle, also allows the noise level
to be estimated simultaneously with the material parameters. Therefore, the
noise can be modeled, e.g., by Gaussian distributions or by Gaussian processes
[62]. However, estimating the noise is beyond the scope of this work. For more
information on this approach, we refer, for instance, to [63].

##  7 Conclusion and outlook

Advances in the development of full-field measurement capabilities, such as,
e.g., digital image correlation (DIC), have recently led to an increasing
interest in appropriate methods for the calibration of constitutive models. In
experimental mechanics, the inverse problem of identifying the material
parameters is traditionally solved by numerical methods, such as nonlinear
least-squares finite element method (NLS-FEM) or virtual fields method (VFM).
However, the computational costs associated with these methods are oftentimes
to high, making them unsuitable for online applications. This results in an
urgent need for methods that enable rapid calibration in online applications,
even under severe time constraints.

In the present contribution, we demonstrate that the parametric PINN approach
enables an accurate and efficient model calibration and uncertainty
quantification of the inferred material parameters. In the offline stage, the
parametric PINN is trained to learn a parameterized solution of the underlying
parametric partial differential equation (PDE) by encoding the physics into a
loss function. In addition, training can be enhanced by high-fidelity
simulation data that can be easily integrated into the training process. In
the subsequent online stage, the parametric PINN then can be employed as a
surrogate for the parameters-to-state map in the calibration process. Due to
the low computational costs of artificial neural network (ANN) evaluations,
calibration can be performed in near real-time, even though ten thousands of
forward model evaluations are required.

We demonstrated the advantages of using parametric PINNs for constitutive
model calibration in deterministic nonlinear least-squares (NLS) calibration
as well as Markov chain Monte Carlo (MCMC)-based Bayesian inference in various
numerical tests. First, we considered the calibration of a small strain linear
elastic and a finite strain hyperelastic constitutive model using noisy
synthetic data. A statistical evaluation of the results showed both high
accuracy for the deterministic point estimate and valid uncertainty for the
Bayesian inference. In addition, we calibrated a small strain linear elastic
model using experimental full-field data from a tensile test. As reference, we
used the results obtained when using the finite element method (FEM) instead
of the parametric PINN as parameters-to-state map. The parametric PINN also
showed good results for the experimental data in both the deterministic and
statistical settings. At the same time, the runtime of the parametric PINN
needed for online calibration is considerably shorter, especially when it
comes to MCMC-based Bayesian inference.

To the best of the authors knowledge, this is the first contribution which
presents parametric PINNs for the calibration of constitutive models. While it
has often been stated that PINNs are especially suited for inverse problems,
the settings considered in the literature so far are often far away from
realistic applications. Herein, the authors have demonstrated the entire
process from parametric PINN training towards model calibration using real-
world experimental data. The achieved savings in the online calibration step
urge for further developments of parametric PINNs for more complex, history
dependent and anisotropic materials. The pre-training of parametric PINNs may
help to further establish full-field measurement techniques, such as DIC, in
materials development in both academia and industry as well as in online
applications, such as continuous structural health monitoring (SHM).

Although the parametric PINNs have already achieved good results in our
numerical tests, further work is necessary for real-world applications. In the
example with the experimental data, it became clear that the real measurement
data can also contain measurement artifacts in addition to the background
noise of the DIC system. In contrast to the background noise, the measurement
artifacts are difficult to characterize and make calibration more challenging.
This applies in particular to PINNs as they usually use the data directly,
without prior interpolation of the sensor data. For this reason, either a pre-
processing of the data is necessary before calibration, or the additional
uncertainties must be taken into account during calibration. Possible methods
for pre-processing are, among others, polynomial interpolation [64], ANN-based
interpolation [65] or kernel methods [66]. In a statistical setting, the
measurement error could also be considered as an additional error term in 39a
and modeled, e.g., by a Gaussian process [63].

The authors are aware that a reliable measurement of full-field displacement
data using, e.g., a DIC system, places very high demands on the measurement
system. These requirements are significantly higher for on-site online
applications in SHM compared to laboratory applications due to the
environmental impacts acting on the system. However, the use of DIC in the
context of SHM is an active field of research, see, e.g., [67, 68, 69].

From a modeling perspective, a further challenge arises as soon as the
displacement or load boundary conditions are not constant. This is
particularly likely for applications in the field of SHM. The load boundary
condition then needs to be inferred online using, e.g., load cells [70].
However, every boundary condition that is not exactly known before training
must be taken into account as a parameter and thus as an additional input to
the parametric PINN. This means that future work on methods for overcoming the
curse of dimensionality are also of great importance.

## Declarations

### Availability of data and materials

The research code for both training of parametric PINNs and calibration is
open-source and available both on GitHub and on Zenodo [30]. The experimental
dataset is available on Zenodo [31].

### Competing interests

The authors declare that they have no competing interests.

### Funding

DA, HW and UR acknowledge support by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) in the project DFG 255042459: ”GRK2075: Modeling
the constitutional evolution of building materials and structures with respect
to aging”. DA and HW also acknowledge support in the project DFG 501798687:
”Monitoring data driven life cycle management with AR based on adaptive, AI-
supported corrosion prediction for reinforced concrete structures under
combined impacts” which is a subproject of SPP 2388: ”Hundred plus - Extending
the Lifetime of Complex Engineering Structures through Intelligent
Digitalization” funded by the DFG. AH acknowledges support by an ETH Zurich
Postdoctoral Fellowship.

### Authors’ contributions

DA: conceptualization, data curation, formal analysis, investigation,
methodology, project administration, software, validation, visualization,
writing – original draft, writing – review and editing; JAT: data curation,
investigation, software, validation, writing – original draft, writing –
review and editing; HW: conceptualization, funding acquisition, methodology,
resources, supervision, writing – original draft, writing – review and
editing; UR: conceptualization, funding acquisition, methodology, supervision,
writing – review and editing; AH: conceptualization, supervision, writing –
review and editing; SH: resources, supervision, writing – review and editing.

### Acknowledgements

DA, HW and UR thank the members of the research training group GRK2075 for the
fruitful discussions.

##  Appendix A Boundary conditions in strong form PINNs

We consider the balance equation 2 with boundary conditions 3 for the top left
quadrant of a plate with a hole as described in Section 5.1. The same test
case has been considered earlier in [71], where it has been reported that the
accuracy of strong form PINNs was insufficient. Herein, we illustrate that the
reason for the unsatisfactory results is merely an incomplete imposition of
BCs in [71]. Note that in Galerkin Finite Element Methods, Neumann BCs are
treated via surface integrals, and zero traction BCs are automatically
fulfilled. This is not the case for methods relying on the strong form.

To this end, we exemplary consider the right boundary of the plate sketched in
Fig. 8, where the following BCs must be fulfilled:

$\displaystyle\mathrm{u}_{x}(x=0)$ $\displaystyle=0,$ (46a)
$\displaystyle\mathrm{P}_{yx}(x=0)$ $\displaystyle=0.$ (46b)

In [71], only the Dirichlet condition 46a has been considered, see also Fig.
8(a). However, since the balance of linear momentum 2 results in two coupled
PDEs for the considered 2D test case, at each boundary two BCs need to be
defined, one in each spatial dimension. With the surface normal of the right
boundary $\mathbf{n}_{\textrm{right}}=[1,0]^{\top}$, the Neumann BC 46b
follows directly from $t_{y}=$0\text{\,}\mathrm{}$$:

$\displaystyle\mathrm{t}_{y}=0$
$\displaystyle=\mathrm{P}_{yx}\mathrm{n}_{x}+\mathrm{P}_{yy}\mathrm{n}_{y},$
(47) $\displaystyle 0$ $\displaystyle=\mathrm{P}_{yx}.$

$\bar{\mathrm{u}}_{y}=$0\text{\,}\mathrm{}$$$\bar{\mathrm{u}}_{x}=$0\text{\,}\mathrm{}$$$\bar{\mathbf{t}}=\begin{bmatrix}$-100\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$\\\
$0\text{\,}\mathrm{}$\end{bmatrix}$$\bar{\mathbf{t}}=\boldsymbol{0}$$\bar{\mathbf{t}}=\boldsymbol{0}$$L=$100\text{\,}\mathrm{mm}$$$L=$100\text{\,}\mathrm{mm}$$R
= $10\text{\,}\mathrm{mm}$yx (a) BC as described in [71]. Only Dirichlet BCs
are applied.

$\bar{\mathrm{u}}_{y}=$0\text{\,}\mathrm{}$,\bar{\mathrm{P}}_{xy}=$0\text{\,}\mathrm{}$$$\bar{\mathrm{u}}_{x}=$0\text{\,}\mathrm{}$$$\bar{\mathrm{P}}_{yx}=$0\text{\,}\mathrm{}$$$\bar{\mathbf{t}}=\begin{bmatrix}$-100\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$\\\
$0\text{\,}\mathrm{}$\end{bmatrix}$$\bar{\mathbf{t}}=\boldsymbol{0}$$\bar{\mathbf{t}}=\boldsymbol{0}$$L=$100\text{\,}\mathrm{mm}$$$L=$100\text{\,}\mathrm{mm}$$R
= $10\text{\,}\mathrm{mm}$yx (b) BC as described in [72]. Beside the Dirichlet
BCs, the symmetry BCs also include Neumann BCs with respect to the shear
stresses.

Figure 8: BC in the test case plate with a hole as described in (a) [71] and
(b) our formulation presented earlier in [72].

To illustrate that the correct application of BCs is essential, we solve the
forward problem for the top left quadrant of a plate with a hole with and
without symmetry stress BCs and compare the results. The geometry and BCs are
shown in Fig. 8. We use the ansatz 23 with a fully connected feed-forward
neural network (FFNN) with six hidden layers each with $64\text{\,}\mathrm{}$
neurons and hyperbolic tangent activation functions. The weights and biases of
the FFNN are initialized according to Glorot normal initialization [58] and
with zeros, respectively. The training data set consists of
$8192\text{\,}\mathrm{}$ collocation points within the domain and
$256\text{\,}\mathrm{}$ points on each of the five boundary segments. No FE
data is used for training. We train the PINN for a predefined bulk modulus
$K=$175\,000\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$$ and shear modulus
$G=$80\,769\text{\,}\mathrm{N}\text{\,}{\mathrm{mm}}^{-2}$$, respectively. The
resulting optimization problem is solved using the L-BFGS optimization
algorithm [46, 47, 48, 49, 50].

The mean absolute error (MAE) and the relative $\text{rL}^{2}$ norm
($\text{rL}^{2}$) of the PINN solution with and without symmetry stress BCs
compared to a high-fidelity FE solution are summarized in Table 4. For
validation, we randomly select $2048\text{\,}\mathrm{}$ points from the FE
solution. In addition, we show the PINN solutions we obtained with and without
symmetry stress BCs as well as the FE reference solution in Fig. 9.

Table 4: MAE and relative $\text{rL}^{2}$ norm ($\text{rL}^{2}$) of the PINN for the test case with and without symmetry stress BCs compared to a high-fidelity FE solution. |  with symmetry stress BCs |  without symmetry stress BCs
---|---|---
MAE |  $5.3812\text{\times}{10}^{-6}\text{\,}\mathrm{}$ |  $5.7706\text{\times}{10}^{-3}\text{\,}\mathrm{}$
$\text{rL}^{2}$ |  $3.5649\text{\times}{10}^{-4}\text{\,}\mathrm{}$ |  $3.7064\text{\times}{10}^{-1}\text{\,}\mathrm{}$

(a) FEM solution: Displacement field in $x$.

(b) FEM solution: Displacement field in $y$.

(c) PINN solution: Displacement field in $x$ without symmetry stress BCs, see
Fig. 8(a).

(d) PINN solution: Displacement field in $y$ without symmetry stress BCs, see
Fig. 8(a).

(e) PINN solution: Displacement field in $x$ with symmetry stress BCs, see
Fig. 8(b).

(f) PINN solution: Displacement field in $y$ with symmetry stress BCs, see
Fig. 8(b).

Figure 9: Resulting displacement fields for the test case plate with a hole
with BCs as described in [71] (c, d) and our formulation presented earlier in
[72] (e, f). The reference solution (a, b) is provided by a high-fidelity FE
simulation.

##  Appendix B Error measures

In order to validate the performance of our parametric PINN formulation, we
compare the PINN predictions to the solutions of high-fidelity finite element
(FE) simulations. We consider the MAE as an absolute error measure and the
$\text{rL}^{2}$ as a relative error measure. In the following,
${\mathbf{u}^{\textrm{FEM}}}{\,\in\mathbb{R}}^{2{n_{\textrm{nodes}}}}$
represents the vector containing the displacements of all
${n_{\textrm{nodes}}}$ nodes with coordinates
$\\{\mathbf{X}^{(i)}\\}^{{n_{\textrm{nodes}}}}_{i=1}$ in the FE
discretization. The vector
${\mathbf{u}^{\textrm{PINN}}}{\,\in\mathbb{R}}^{2{n_{\textrm{nodes}}}}$
contains the displacements predicted by the parametric PINN where the PINN is
evaluated according to 29 at the coordinates
$\left\\{\mathbf{X}^{(i)}\right\\}^{{n_{\textrm{nodes}}}}_{i=1}$. The same
material parameters $\boldsymbol{\kappa}$ are used for both the FE simulation
and the evaluation of the parametric PINN.

The  mean absolute error (MAE) is then defined as

$\text{MAE}_{\mathbf{u}}=\frac{1}{2{n_{\textrm{nodes}}}}\sum_{i=0}^{2{n_{\textrm{nodes}}}}\left|{\mathrm{u}_{i}^{\textrm{PINN}}}-{\mathrm{u}_{i}^{\textrm{FEM}}}\right|,$
(48)

where $\left|\bullet\right|$ is the absolute value of the quantity $\bullet$.

The  relative $\text{rL}^{2}$ norm ($\text{rL}^{2}$) yields

$\text{rL}^{2}_{\mathbf{u}}=\frac{\left\lVert{\mathbf{u}^{\textrm{PINN}}}-{\mathbf{u}^{\textrm{FEM}}}\right\rVert}{\left\lVert{\mathbf{u}^{\textrm{FEM}}}\right\rVert},$
(49)

with $\left\lVert\bullet\right\rVert$ denoting the $\text{L}^{2}$-norm.

For the statistical evaluation of the calibration results, we consider the
absolute relative error (ARE). In addition to the mean, minimum and maximum
ARE, we also calculate the standard error of the mean (SEM) which gives us
information about the scatter of the ARE. Here, ${\kappa^{\textrm{true}}}$
represents the vector of true material parameters and
${\boldsymbol{\kappa}^{\textrm{identified}}}$ the vector of material
parameters identified by using the parametric PINN as parameters-to-state map
in the deterministic least-squares calibration.

The  absolute relative error (ARE) for material parameter $\kappa_{i}$ is
defined as

$\text{ARE}_{\kappa_{i}}=\frac{\left|{\kappa_{i}^{\textrm{identified}}}-{\kappa_{i}^{\textrm{true}}}\right|}{{\kappa_{i}^{\textrm{true}}}}.$
(50)

The  standard error of the mean (SEM) with respect to a certain error measure,
for instance, the ARE, is then calculated as

$SEM_{\kappa_{i}}=\frac{\sigma_{\kappa_{i}}}{\sqrt{{n_{\textrm{tests}}}}},$
(51)

where ${n_{\textrm{tests}}}$ is the number of test cases on which the
statistical evaluation is based and $\sigma_{\kappa_{i}}$ is the standard
deviation, e.g., of the ARE.

## References

  *   * Chang [1998] Chang, F.-K.: Structural Health Monitoring: A Summary Report on the First Stanford Workshop on Structural Health Monitoring, September 18-20, 1997. Technical report, Stanford University (1998). https://doi.org/10.21236/ADA350933
  * Entezami [2021] Entezami, A.: Structural Health Monitoring by Time Series Analysis and Statistical Distance Measures, 1st edn. SpringerBriefs in Applied Sciences and Technology. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-66259-2
  * Sutton et al. [2009] Sutton, M.A., Orteu, J.-J., Schreier, H.: Image Correlation for Shape, Motion and Deformation Measurements, 1st edn. Springer, New York (2009). https://doi.org/10.1007/978-0-387-78747-3
  * Yang and Ettemeyer [2003] Yang, L.X., Ettemeyer, A.: Strain measurement by three-dimensional electronic speckle pattern interferometry: potentials, limitations, and applications. Optical Engineering 42(5), 1257–1266 (2003) https://doi.org/10.1117/1.1566781
  * Mahnken and Stein [1996] Mahnken, R., Stein, E.: A unified approach for parameter identification of inelastic material models in the frame of the finite element method. Computer Methods in Applied Mechanics and Engineering 136(3), 225–258 (1996) https://doi.org/10.1016/0045-7825(96)00991-7
  * Rose and Menzel [2020] Rose, L., Menzel, A.: Optimisation based material parameter identification using full field displacement and temperature measurements. Mechanics of Materials 145, 103292 (2020) https://doi.org/%****␣main.bbl␣Line␣125␣****10.1016/j.mechmat.2019.103292
  * Avril et al. [2008] Avril, S., Bonnet, M., Bretelle, A.-S., Grédiac, M., Hild, F., Ienny, P., Latourte, F., Lemosse, D., Pagano, S., Pagnacco, E., Pierron, F.: Overview of Identification Methods of Mechanical Parameters Based on Full-field Measurements. Experimental Mechanics 48(4), 381–402 (2008) https://doi.org/10.1007/s11340-008-9148-y
  * Martins et al. [2018] Martins, J.M.P., Andrade-Campos, A., Thuillier, S.: Comparison of inverse identification strategies for constitutive mechanical models using full-field measurements. International Journal of Mechanical Sciences 145, 330–345 (2018) https://doi.org/10.1016/j.ijmecsci.2018.07.013
  * Raissi et al. [2019] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378, 686–707 (2019) https://doi.org/10.1016/j.jcp.2018.10.045
  * Karniadakis et al. [2021] Karniadakis, G.E., Kevrekidis, I.G., Lu, L., Perdikaris, P., Wang, S., Yang, L.: Physics-informed machine learning. Nature Reviews Physics 3(6), 422–440 (2021) https://doi.org/10.1038/s42254-021-00314-5
  * Psichogios and Ungar [1992] Psichogios, D.C., Ungar, L.H.: A hybrid neural network-first principles approach to process modeling. American Institute of Chemical Engineers Journal 38(10), 1499–1511 (1992) https://doi.org/10.1002/aic.690381003
  * Lagaris et al. [1998] Lagaris, I.E., Likas, A., Fotiadis, D.I.: Artificial neural networks for solving ordinary and partial differential equations. IEEE Transactions on Neural Networks 9(5), 987–1000 (1998) https://doi.org/%****␣main.bbl␣Line␣225␣****10.1109/72.712178
  * Baydin et al. [2018] Baydin, A.G., Pearlmutter, B.A., Radul, A.A., Siskind, J.M.: Automatic differentiation in machine learning: a survey. Journal of Machine Learning Research 18(1), 5595–5637 (2018)
  * Abadi et al. [2015] Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mané, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viégas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: TensorFlow: Large-Scale Machine Learning on Heterogeneous Systems. Software available from tensorflow.org (2015). https://www.tensorflow.org/
  * Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., Desmaison, A., Köpf, A., Yang, E., DeVito, Z., Raison, M., Tejani, A., Chilamkurthy, S., Steiner, B., Fang, L., Bai, J., Chintala, S.: PyTorch: An imperative style, high-performance deep learning library. arXiv Preprint (2019) https://doi.org/10.48550/arXiv.1912.01703 . Software available from pytorch.org
  * Römer et al. [2024] Römer, U., Hartmann, S., Tröger, J.-A., Anton, D., Wessels, H., Flaschel, M., De Lorenzis, L.: Reduced and all-at-once approaches for model calibration and discovery in computational solid mechanics. arXiv Preprint (2024) https://doi.org/10.48550/arXiv.2404.16980
  * Shukla et al. [2022] Shukla, K., Jagtap, A.D., Blackshire, J.L., Sparkman, D., Karniadakis, G.E.: A Physics-Informed Neural Network for Quantifying the Microstructural Properties of Polycrystalline Nickel Using Ultrasound Data: A Promising Approach for Solving Inverse Problems. IEEE Signal Processing Magazine 39(1), 68–77 (2022) https://doi.org/10.1109/MSP.2021.3118904
  * Rojas et al. [2023] Rojas, C.J.G., Boldrini, J.L., Bittencourt, M.L.: Parameter identification for a damage phase field model using a physics-informed neural network. Theoretical and Applied Mechanics Letters 13(3), 100450 (2023) https://doi.org/10.1016/j.taml.2023.100450
  * Zhang et al. [2022] Zhang, E., Dao, M., Karniadakis, G.E., Suresh, S.: Analyses of internal structures and defects in materials using physics-informed neural networks. Science Advances 8(7), 0644 (2022) https://doi.org/10.1126/sciadv.abk0644
  * Haghighat et al. [2021] Haghighat, E., Raissi, M., Moure, A., Gomez, H., Juanes, R.: A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics. Computer Methods in Applied Mechanics and Engineering 379, 113741 (2021) https://doi.org/10.1016/j.cma.2021.113741
  * Hamel et al. [2022] Hamel, C.M., Long, K.N., Kramer, S.L.B.: Calibrating constitutive models with full-field data via physics informed neural networks. Strain 59(2), 12431 (2022) https://doi.org/10.1111/str.12431
  * Zhang et al. [2020] Zhang, E., Yin, M., Karniadakis, G.E.: Physics-informed neural networks for nonhomogeneous material identification in elasticity imaging. arXiv Preprint (2020) https://doi.org/10.48550/arXiv.2009.04525
  * Anton and Wessels [2023] Anton, D., Wessels, H.: Physics-informed neural networks for material model calibration from full-field displacement data. arXiv Preprint (2023) https://doi.org/10.48550/arXiv.2212.07723
  * Hosseini et al. [2023] Hosseini, E., Scheel, P., Müller, O., Molinaro, R., Mishra, S.: Single-track thermal analysis of laser powder bed fusion process: Parametric solution through physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering 410, 116019 (2023) https://doi.org/10.1016/j.cma.2023.116019
  * Beltrán-Pulido et al. [2022] Beltrán-Pulido, A., Bilionis, I., Aliprantis, D.: Physics-Informed Neural Networks for Solving Parametric Magnetostatic Problems. IEEE Transactions on Energy Conversion 37(4), 2678–2689 (2022) https://doi.org/10.1109/TEC.2022.3180295
  * Sun et al. [2023] Sun, Y., Sengupta, U., Juniper, M.: Physics-informed deep learning for simultaneous surrogate modeling and PDE-constrained optimization of an airfoil geometry. Computer Methods in Applied Mechanics and Engineering 411, 116042 (2023) https://doi.org/10.1016/j.cma.2023.116042
  * Agarwal et al. [2024] Agarwal, G., Urrea-Quintero, J.-H., Wessels, H., Wick, T.: Model order reduction for transient coupled diffusion-deformation of hydrogels. arXiv Preprint (2024) https://doi.org/10.48550/arXiv.2403.08968
  * Stoter et al. [2022] Stoter, S.K.F., Jessen, E., Niedens, V., Schillinger, D.: A DEIM driven reduced basis method for the diffuse Stokes/Darcy model coupled at parametric phase-field interfaces. Computational Geosciences 26(6), 1465–1502 (2022) https://doi.org/10.1007/s10596-022-10164-4
  * Baratta et al. [2023] Baratta, I.A., Dean, J.P., Dokken, J.S., Habera, M., Hale, J.S., Richardson, C.N., Rognes, M.E., Scroggs, M.W., Sime, N., Wells, G.N.: DOLFINx: The next generation FEniCS problem solving environment. Zenodo. Software available from github.com/FEniCS/dolfinx (2023). https://doi.org/10.5281/zenodo.10447666
  * Anton [2024] Anton, D.: Code for the publication ”Deterministic and statistical calibration of constitutive models from full-field data with parametric physics-informed neural networks”. Zenodo. Code available from https://github.com/david-anton/CalibrationPINN (2024). https://doi.org/10.5281/zenodo.11368998
  * Tröger et al. [2024] Tröger, J.-A., Hartmann, S., Anton, D., Wessels, H.: Digital image correlation measurement of linear elastic steel specimen. Zenodo. Data set (2024). https://doi.org/10.5281/zenodo.11257192
  * Holzapfel [2000] Holzapfel, G.A.: Nonlinear Solid Mechanics: A Continuum Approach for Engineering, 1st edn. Wiley, Chichester (2000)
  * Wriggers [2008] Wriggers, P.: Nonlinear Finite Element Methods, 1st edn. Springer, Berlin (2008). https://doi.org/10.1007/978-3-540-71001-1
  * Marsden and Hughes [1983] Marsden, J., Hughes, T.J.R.: Mathematical Foundations of Elasticity, 1st edn. Dover Books on Mathematics. Dover Publications, New York (1983)
  * Lychev and Koifman [2019] Lychev, S., Koifman, K.: Geometry of Incompatible Deformations: Differential Geometry in Continuum Mechanics. De Gruyter Studies in Mathematical Physics. De Gruyter, Berlin (2019). https://doi.org/10.1515/9783110563214
  * Ehlers and Eipper [1998] Ehlers, W., Eipper, G.: The simple tension problem at large volumetric strains computed from finite hyperelastic material laws. Acta Mechanica 130(1), 17–27 (1998) https://doi.org/10.1007/BF01187040
  * Hartmann and Neff [2003] Hartmann, S., Neff, P.: Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility. International Journal of Solids and Structures 40(11), 2767–2791 (2003) https://doi.org/%****␣main.bbl␣Line␣650␣****10.1016/S0020-7683(03)00086-6
  * Goodfellow et al. [2016] Goodfellow, I., Bengio, Y., Courville, A.: Deep Learning. MIT Press, online (2016). https://www.deeplearningbook.org
  * Cybenko [1989] Cybenko, G.: Approximation by superpositions of a sigmoidal function. Mathematics of Control, Signals, and Systems 2(4), 303–314 (1989) https://doi.org/10.1007/BF02551274
  * Hornik et al. [1989] Hornik, K., Stinchcombe, M., White, H.: Multilayer feedforward networks are universal approximators. Neural Networks 2(5), 359–366 (1989) https://doi.org/10.1016/0893-6080(89)90020-8
  * Li [1996] Li, X.: Simultaneous approximations of multivariate functions and their derivatives by neural networks with one hidden layer. Neurocomputing 12(4), 327–343 (1996) https://doi.org/10.1016/0925-2312(95)00070-4
  * Berg and Nyström [2018] Berg, J., Nyström, K.: A unified deep artificial neural network approach to partial differential equations in complex geometries. Neurocomputing 317, 28–41 (2018) https://doi.org/10.1016/j.neucom.2018.06.056
  * LeCun et al. [2012] LeCun, Y.A., Bottou, L., Orr, G.B., Müller, K.-R.: Efficient BackProp. In: Montavon, G., Orr, G.B., Müller, K.-R. (eds.) Neural Networks: Tricks of the Trade, 2nd edn. Lecture Notes in Computer Science, pp. 9–48. Springer, Berlin (2012). https://doi.org/10.1007/978-3-642-35289-8_3