# Physics-informed neural networks for solving forward and inverse problems in
complex beam systems

Taniya Kapoor , Hongrui Wang , , Alfredo Núñez , , Rolf Dollevoet Manuscript
submitted April 1, 2022, Revised February 24, 2023, May 19, 2023, July 22,
2023, Accepted August 23, 2023 (Corresponding author: Hongrui Wang)T. Kapoor,
H. Wang, A. Núñez, R. Dollevoet are with the Section of Railway Engineering,
Department of Engineering Structures, Delft University of Technology, The
Netherlands. (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>r.p.b.j.dollevoet@tudelft.nl).
0000-0002-6361-446X 0000-0001-7194-4728 0000-0001-5610-6689
0000-0002-8279-8284

###### Abstract

This paper proposes a new framework using physics-informed neural networks
(PINNs) to simulate complex structural systems that consist of single and
double beams based on Euler-Bernoulli and Timoshenko theory, where the double
beams are connected with a Winkler foundation. blackIn particular, forward and
inverse problems for the Euler-Bernoulli and Timoshenko partial differential
equations (PDEs) are solved using nondimensional equations with the physics-
informed loss function. Higher-order complex beam PDEs are efficiently solved
for forward problems to compute the transverse displacements and cross-
sectional rotations with less than $1e-3$ percent error. Furthermore, inverse
problems are robustly solved to determine the unknown dimensionless model
parameters and applied force in the entire space-time domain, even in the case
of noisy data. The results suggest that PINNs are a promising strategy for
solving problems in engineering structures and machines involving beam
systems.

###### Index Terms:

PINNs, complex system, Euler-Bernoulli beam, Timoshenko beam, double-beam
system.

black

##  I Introduction

Complex engineering issues in real-life scenarios are often characterized by
the connection between various subsystems and uncertainty in behavior caused
by internal and external variables and their interactions. Furthermore, the
design and maintenance of complex systems, such as engineering structures and
machines, is made challenging by the unpredictable collective behaviors and
properties of these concurrently operating and interacting components. These
issues are typically difficult to analyze through conventional methods [1].
Most of these complex engineering systems are continuous, and partial
differential equation (PDE) models are used to characterize and understand
their behavior. These PDE models are used to simulate a wide range of
engineering phenomena, ranging from multiple beam systems in suspension bridge
cables (Timoshenko beam equations)[2] to catenary-pantograph interactions in
railways (damped beam equations) [3] to simulating air turbulence that
disrupts flight (Navier-Stokes equations) [4, 5], among many others [6, 7, 8,
9, 10, 11, 12, 13]. Solutions to governing PDEs enable real challenges such as
structural health monitoring [14, 15, 16] and optimal structural design [17,
18] to be addressed.

The development of algorithms for diagnostics and prognosis is an issue in
maintaining complex engineering systems[1]. Insights could be obtained by
solving the forward and inverse problems for the governing PDEs of interest to
forecast the system’s behavior and minimize unexpected downtimes of complex
systems. These equations range in complexity from being extremely nonlinear
(Navier-Stokes equation [19]) to incorporating intricate higher-order boundary
conditions (fourth-order beam equations [20]). In practice, these equations
are too complicated to be solved analytically and must be solved numerically.
Numerical methods such as the finite-difference and finite-element methods
have been used to approximate the solutions of these PDEs. Despite their
success in practice, these methods encounter some difficulties, such as mesh
creation, which is more difficult for complex geometries in higher
dimensions[21, 22].

In recent years, scientific machine learning, which combines scientific
computing with machine learning methodologies to estimate PDEs solutions, has
made remarkable developments and has emerged as a viable alternative to the
aforementioned numerical methods. The review papers [21, 23, 24] extensively
discuss state-of-the-art breakthroughs in scientific machine learning,
including works on real-world engineering problems. However, data-driven
methods require a large amount of data, which is possibly computationally
expensive and susceptible to noise in some engineering systems [25]. One
possible way to mitigate the effects of these problems is to utilize the known
physical knowledge of the underlying system in the learning procedure [26, 27,
28]. Prior physical knowledge could be incorporated into the learning
procedure by collocating the PDE residual at training points, similar to
leveraging the physical equation in the training process. The underlying
neural networks proposed in [25] are called physics-informed neural networks
(PINNs).

PINNs utilize neural networks’ universal function approximation property [29]
and embed the well-posed physical equations modeled by PDEs in the loss
function. Prior knowledge of physical principles works as a regularization
agent in neural network training, restricting the space of admissible
solutions and improving function approximation accuracy. As a result, given
some knowledge of the physical features of the problem and some training data,
PINN can be utilized to identify a high-fidelity solution. PINNs have already
proven to be a very effective paradigm for approximating solutions of PDEs for
real-world problems [30, 31], as discussed in the review papers [23, 21].

However, several challenges for PINNs have also been found [32]. One such
challenge for PINNs is to learn relevant physical phenomena for more complex
problems with large coefficients in the physical equation [33]. A sequence-to-
sequence learning task was proposed in [33] as a remedy to this problem.
However, this can be computationally expensive when the scale is large. In
[34], the importance of using nondimensional equations in the PINN framework
was highlighted for cardiovascular blood flow. We build on these works and
address the challenge of multiscale complex beam systems. Accordingly, this
paper uses nondimensional PDEs instead of dimensional PDEs in the loss
function. This provides a way to simulate realistic physical equations with
computational tractability.

Accurate prediction of the dynamics of structures [35] and structural
elements, such as plates [36], and beams [37, 38], is crucial in the field of
structural engineering. However, measuring quantities of interest in beam
systems through lab experiments can prove to be difficult, as it necessitates
specialized prototypes, training, and safety during the testing process,
increasing the overall cost of the experiment. PINNs offer a simulation-based
solution as a mesh-free method that does not require discretizing the domain
into a finite number of elements, making it computationally inexpensive
compared to numerical methods. PINNs can effectively integrate incomplete or
noisy information with prior physical knowledge. The proposed framework
converts dimensionalized PDEs to a nondimensionalized form, increasing the
suitability for neural networks and enabling the prediction of deflections and
rotations for any material, resulting in a more generalizable method.

This paper provides a framework to simulate complex structural systems
consisting of two or more basic structural systems connected by an elastic
layer. In particular, the forced vibration of two elastically connected beams
is studied, which is commonly encountered in the mechanical, construction, and
aeronautical industries [6]. These double-beam systems in engineering
structures have received significant attention in the scientific community and
are considered complex systems. Studies have been conducted to predict the
dynamics of these systems under various loading and force conditions, such as
those found in papers [39, 40, 41, 42, 43, 44, 45, 46, 47], among others.
These studies include the use of analytical and closed-form solutions [43, 48,
49, 50, 51]; however, analytical methods have limitations in applicability, as
they may be useful only for specific types of problems and can become complex
for systems with many variables or nonlinear equations. Other approaches, such
as the state-space method presented in [52, 45], may also be computationally
expensive for systems with a large number of states. Additionally, modal
analysis methods as presented in [53, 6] have been used to study the natural
frequencies and modes of vibration, but they do not provide information on the
full response of the system and cannot be used to predict the time-domain
response at any instant.

The considered governing equations are modeled using Euler-Bernoulli and
Timoshenko theory. In addition to solving the forward problem and computing
the physical quantities of interest, we also solve the inverse problem. For
the inverse problem, one may not necessarily have complete information about
the inputs to the PDEs, such as initial or boundary data, coefficients [54,
55, 56] or applied forces. This lack of knowledge makes the forward problem
ill-posed, and subsequently, the forward problem cannot be solved uniquely. In
this paper, access to data for quantities of interest is leveraged to
determine the PDEs’ unknown inputs, for instance, the model parameters and
applied forces.

The main contributions of the current paper are as follows,

  * •

To the best of the authors’ knowledge, this is the first work to use physics-
informed machine learning to solve the forward and inverse problems of Euler-
Bernoulli and Timoshenko complex beam models.

  * •

We address a challenge for PINNs in solving multiscale complex beam PDEs and
propose a framework for using nondimensional equations in the loss function.

  * •

The proposed nondimensional PINN framework is employed to address ill-posed
inverse problems for complex systems and to identify the unknown model
parameters and the applied force on the beam components. This is achieved by
utilizing data from indirect measurements such as the displacement and cross-
sectional rotations of the beams.

  * •

The presented methodology is robust to noise and can accomodate potential
uncertainty in the measurement data, making it well suited for real-world
applications where data are incomplete or uncertain.

The rest of the article is organized as follows. In Section II, the PINN
method is presented to simulate the dimensional Euler-Bernoulli beam equation.
Due to the limitations of PINNs in simulating the dimensional Euler-Bernoulli
beam equation, an alternative approach of using nondimensional equations in
the PINN’s loss function is proposed and successfully used to solve the
dimensionless Euler-Bernoulli equation in Section III. Section IV first
applies the proposed framework to simulate the Timoshenko beam model for
solving forward and inverse problems. The forward problem of the Euler-
Bernoulli double-beam equation is then solved. Additionally, Section IV covers
forward and inverse Timoshenko double-beam system problems. Section V
concludes this paper.

##  II PINNs for Dimensional PDEs

In this section, the method of PINNs to simulate PDEs is presented in brief
using an abstract dimensional PDE. The method is then used to simulate the
dimensional Euler-Bernoulli equation. The following abstract dimensional PDE
is considered with implicit initial and boundary conditions:

$\mathcal{\bar{K}}(\bar{x},\bar{t}):=\mathcal{D}[\bar{u}](\bar{x},\bar{t};\bar{\lambda})-\bar{f}(\bar{x},\bar{t})\quad\forall(\bar{x},\bar{t})\in\bar{\Omega}\times\bar{T}\subset\mathbb{R}^{\mathrm{d}}\times\mathbb{R}$
(1)

where $\mathcal{D}[^{.}]$ denotes the differential operator, $\bar{u}$ is the
quantity of interest, $\bar{x}\in$ $\bar{\Omega}$ $\subset$
$\mathbb{R}^{\mathrm{d}}$, $\bar{t}\in\bar{T}$ $\subset$ $\mathbb{R}$ for
$d\geq 1$, $\bar{\Omega}$ denotes the spatial boundary contained in the
d-dimensional Cartesian spatial space and $\bar{T}$ denotes the temporal
domain, $\bar{\lambda}\in\mathbb{R}$ is the model parameter,
$\bar{f}(\bar{x},\bar{t})$ is the external force, and $\bar{K}$ is the
notation for the abstract physical equation.

Deep neural networks are the core for PINNs in which inputs
$(\bar{x},\bar{t})$ map to output ($\bar{u}$) through an iterative composition
of hidden layers. The composition consists of weights ($w$), biases ($b$), and
linear or nonlinear activation function(s) ($\sigma$). The inputs undergo a
linear composition within a neuron, where they are multiplied by respective
weights and summed along with a bias term. Subsequently, this combined input
is passed through a nonlinear activation function $(\sigma)$ as presented in
Fig. 4. This allows the neural network to introduce nonlinearity, enabling the
network to capture intricate relationships between inputs and outputs.

To train the neural network, one needs training set ($\Delta$), consisting of
spatial boundary points ($\Delta_{\mathrm{b}}$), temporal boundary points
($\Delta_{\mathrm{i}}$) and interior points ($\Delta_{\mathrm{int}}$). As a
result, the training set can be written as
$\Delta=\Delta_{\mathrm{i}}\cup\Delta_{\mathrm{b}}\cup\Delta_{\mathrm{int}}$.
In this work, $\Delta_{\mathrm{i}}$, $\Delta_{\mathrm{b}}$, and
$\Delta_{\mathrm{int}}$ are considered to have $N_{\mathrm{i}}$,
$N_{\mathrm{b}}$ and $N_{\mathrm{int}}$ training points respectively. The
total number of training points is denoted by $N_{\mathrm{train}}$. To
approximate the quantity of interest $\bar{u}$, one needs to minimize the loss
function containing the physical model in the form of a PDE with initial and
boundary conditions of (1). No additional data are required in the loss
function for forward problems. The loss function $\bar{\mathcal{L}}$ is
defined as follows:

$\bar{\mathcal{L}}(\theta)=\mathrm{\underset{\mathrm{\theta}}{Min}}(\frac{1}{N_{\mathrm{train}}}\sum_{n=1}^{N_{\mathrm{train}}}||\mathcal{\bar{K}}(\bar{x}_{\mathrm{n}},\bar{t}_{\mathrm{n}})||^{2})$
(2)

where $(\bar{x}_{\mathrm{n}},\bar{t}_{\mathrm{n}})$ represents the training
tuple for each n. Minimizing this loss function using a suitable optimization
algorithm provides optimal parameters $\theta=\\{w,b\\}$.

Now, we employ the PINN algorithm for the dimensional Euler-Bernoulli beam
equation and evaluate the corresponding performance. The dynamic Euler-
Bernoulli beam equation is given by

$\rho
A\bar{u}_{\mathrm{\bar{t}\bar{t}}}+EI\bar{u}_{\mathrm{\bar{x}\bar{x}\bar{x}\bar{x}}}=\bar{f}(\bar{x},\bar{t})\quad\bar{x}\in[0,\bar{l}],\bar{t}\in[0,t_{\mathrm{end}}]$
(3)

Figure 1: Simply supported beam with varying transverse force.

Here, $\bar{l}$ and $t_{\mathrm{end}}$ refer to the length of the beam and
final time, respectively. This equation models the transverse displacement of
beam $\bar{u}$ in the space-time domain subject to the external transverse
force $\bar{f}$ as shown in Fig. 1. This work considers a uniform cross-
sectioned beam with constant material properties throughout the beam. The
parameters $\rho$ and $A$ denote the density and cross-sectional area of the
beam, respectively. The parameters $E$ and $I$ are Young’s modulus and the
moment of inertia of the beam, respectively. The external force $\bar{f}$ acts
nonuniformly on the body, and $\bar{u}$ is the transverse displacement of the
beam, which is the only unknown in the governing PDE. In addition,
$u_{\mathrm{tt}}$ represents the second order partial derivative of u with
respect to t, and $u_{\mathrm{xxxx}}$ represents the fourth order partial
derivative of u with respect to x. The goal of the forward problem is to
compute the transverse displacement of the beam supplemented with the initial
and boundary conditions. For this study, simply supported beams are
considered, which rest on two supports and are free to move horizontally.
Real-world applications of simply supported beams include railway tracks, and
bridges, to name a few. Mathematically, the simply supported boundary
condition for (3) is given by

$\bar{u}(0,\bar{t})=\bar{u}(\bar{l},\bar{t})=\bar{u}_{\mathrm{\bar{x}\bar{x}}}(0,\bar{t})=\bar{u}_{\mathrm{\bar{x}\bar{x}}}(\bar{l},\bar{t})=0$

For the numerical experiment, the parameter values of aluminium-like material
are considered in the physical equation, which are widely used for making
beams. The parameter values taken for the problem are $\rho=2\times
10^{3}$kg/$\mathrm{m}^{3}$, $A=5\times 10^{-2}$ $\mathrm{m}^{2}$,
$E=10^{10}$N/$\mathrm{m^{2}}$, and $I=4\times 10^{-4}$$\mathrm{m^{4}}$.
Additionally, the beam is taken to be $\pi^{2}$ meters long, and the external
force $\bar{f}$ is taken to be
$EI(1-16\pi^{2})\sin{(\bar{x}/\pi)}\cos(4c\bar{t}/\pi)/\bar{l}^{3}$N, where
$c=\sqrt{\frac{EI}{\rho A}}$. Taking the final time to be $\pi^{2}/200$, the
PDE to be solved takes the form

$10^{2}\bar{u}_{\mathrm{\bar{t}\bar{t}}}+4\times
10^{6}\bar{u}_{\mathrm{\bar{x}\bar{x}\bar{x}\bar{x}}}=\\\ 4\times
10^{6}(1-16\pi^{2})\sin{(\bar{x}/\pi)}\cos(800\bar{t}/\pi)/\pi^{3}$ (4)

in the domain $\bar{x}\in[0,\pi^{2}]$ and $\bar{t}\in[0,\pi^{2}/200]$. For (4)
to be well-posed the initial condition of the beam is taken to be
$\sin(\bar{x}/l)$ with zero initial velocity, where $l=\sqrt{\bar{l}}$.

For training the neural network, $16000$ random training points are generated
with the distribution $N_{\mathrm{i}}=2000$, $N_{\mathrm{b}}=4000$, and
$N_{\mathrm{int}}=10000$. The neural network consists of $4$ hidden layers
with $20$ neurons in each hidden layer. The $\tanh$ activation function, which
is one of the most commonly used activation functions in the PINN literature,
as described in the review paper [23], is chosen. The loss function (2)
consists of the initial condition, boundary condition and PDE. The PDE is
regularized in the loss function with the residual parameter $0.1$[57]. The
L-BFGS optimizer, which is again one of the most commonly used optimizers in
the PINN literature[23] is used to minimize the loss function. As shown in
Fig. 2 $15000$ epochs are performed. However, the figure clearly illustrates
that the optimizer does not converge to the solution, and a vast training loss
of $10^{14}$ is obtained. Additionally, the graph shows that the optimizer is
stuck in the local minima and hence will not converge even if the number of
epochs is increased for the same neural network configuration.

In [16, 58], the problem of free vibrations in the Euler-Bernoulli single-beam
equation was successfully solved by PINNs, where the coefficients of the PDE
were taken to be unity. This shows that PINNs can simulate the beam equations,
and the challenge lies in the multiscale coefficient values that arise when
dealing with a real-life physical equation. The nonconvergence in our case is
due to the high value of coefficients, which is due to the dimensional
equation. Consequently, a pressing need arises to transform the dimensional
form of the equation into a nondimensional form. It may be possible that for
some configurations containing hundreds of hidden layers and neurons, this
problem may be solved without the need to non-dimensionalizing the PDE.
However, nondimensionalization aims to provide computational tractability.

Figure 2: L-BFGS training loss vs. the number of epochs for the dimensional
Euler-Bernoulli beam equation.

##  III PINNs for NonDimensional PDEs

This section presents the proposed framework of using nondimensional equations
in the PINN loss function. The method for nondimensionalizing the governing
PDE is described first. Then, the algorithms for forward and inverse problems
using dimensionless equations in PINNs are presented. To nondimensionalize the
abstract PDE given by (1), the following transformations are performed

$\bar{x}=\xi_{1}(x);\quad\bar{t}=\xi_{2}(t);\quad\bar{u}=\xi_{3}(u);\quad\bar{f}=\xi_{4}(f)$
(5)

where, $\xi_{1}$, $\xi_{2}$, $\xi_{3}$, and $\xi_{4}$ are suitable functions
that map the dimensional quantities $\bar{x}$, $\bar{t}$, $\bar{u}$, and
$\bar{f}$ to the corresponding nondimensional quantities. After substituting
the above transformations in (1) and introducing the dimensionless parameter
$\lambda$, one obtains

$\mathcal{K}(x,t):=\mathcal{D}[u](x,t;\lambda)-f(x,t)\quad\forall(x,t)\in\Omega\times
T\subset\mathbb{R}^{d}\times\mathbb{R}$ (6)

The proposed framework uses dimensionless equations to simplify and stabilize
the problem computationally. By nondimensionalizing the variables and
parameters, they are kept within a specific range, resulting in improved
performance and generalization of the neural network. Furthermore,
dimensionless equations generate more interpretable solutions by eliminating
the units of measure, making it easier to understand the underlying physical
phenomena and to compare results across different physical systems in the form
of ratios and parameters. Hence, using dimensionless equations in PINNs can
enhance the neural network’s computational stability, generalization, and
interpretability.

###  III-A PINN Framework for Forward Problems

$\mathcal{K}$, the nondimensional PDE corresponding to the dimensional PDE
$\mathcal{\bar{K}}$, is now used in the loss function $\mathcal{L}$ defined as
follows:

$\mathcal{L}(\theta)=\mathrm{\underset{\mathrm{\theta}}{Min}}(\frac{1}{N_{\mathrm{train}}}\sum_{n=1}^{N_{\mathrm{train}}}||\mathcal{K}(x_{\mathrm{n}},t_{\mathrm{n}})||^{2})$
(7)

Figure 3: Nondimensional Euler-Bernoulli beam equation Color bar represents
Left: Predicted solution ($u^{*}$); Right: Absolute error in prediction
($|u-u^{*}|$)

A schematic representation of the proposed PINN-based framework is illustrated
in Fig 4.

Figure 4: PINN framework for beam systems: For forward problems, the loss
function comprises the nondimensional PDEs and the boundary and initial
conditions. For inverse problems, the nondimensional PDEs are supplemented
with extra data and potential initial/boundary conditions.

###  III-B Nondimensional Euler-Bernoulli Beam Equation

We now test the nondimensional equation in the PINN framework and evaluate the
corresponding performance. To nondimensionalize (3), following transformations
are used:

$u=\bar{u}/l;\quad x=\bar{x}/l;\quad t=c\bar{t}/l^{2};\quad
f=\bar{f}l^{3}/(EI)$ (8)

Upon substituting these values in (3), one obtains

$u_{\mathrm{tt}}+u_{\mathrm{xxxx}}=f(x,t)\quad x\in[0,\pi],t\in[0,1]$ (9)

where $f(x,t)=(1-16\pi^{2})\sin{(x)}\cos(4\pi t)$, with initial and boundary
conditions

$u(x,0)=\sin(x),\quad u_{t}(x,0)=0$
$u(0,t)=u(\pi,t)=u_{\mathrm{xx}}(0,t)=u_{\mathrm{xx}}(\pi,t)=0$

For the error estimation, the relative percentage error ($\mathcal{R}$) used
in [57] is chosen. Here, $u^{*}$ is the prediction and $u$ is the analytical
solution.

$\displaystyle\mathcal{R}=\frac{||u^{*}-u||_{2}}{||u||_{2}}\times 100$

The same neural network architecture as the previous case is chosen to solve
this resulting nondimensional PDE. A low training loss is obtained, indicating
that the PINN is trained successfully. The analytical solution for this case
is $u(x,t)=\sin(x)\cos(4\pi t)$, which is used to quantify the error in the
approximated solution. The nondimensional displacement of the Euler-Bernoulli
beam is computed within $\mathcal{R}=5.3e-4$ percent. The nondimensional
displacement prediction using PINN is shown in Fig. 3.(a). Fig. 3.(b) shows
the absolute error between the exact and predicted solutions.

The contour plot for the approximate solution shows the dynamics of a simply
supported beam under a force, where the x-axis represents the time, the y-axis
represents the position along the length of the beam, and the colors represent
the displacement of the beam. In Fig. 3.(b) the red regions indicate high
displacement, while the blue regions indicate low displacement. There is a
strong displacement at the position of the beam when a substantial force is
applied, which is consistent with the known physics of this system. The
network accurately captures the displacement behavior of the beam, which is
evident by the smooth and continuous transition of colors across the plot.

The contour plot for the error in Fig. 3.(b) shows the difference between the
approximate solution obtained from the network and the true solution. The
x-axis represents the time, the y-axis represents the position along the
length of the beam, and the colors represent the error. The red regions
indicate high error, while the blue regions indicate low error. The areas
where the training point concentration is low account for more error, and
areas where the concentration of training points is more have relatively low
error. One approach to reduce the error is to have more training points in the
regions of high error. However, the overall error is low, which indicates that
the network accurately captures the displacement behavior of the beam.

From Fig. 3.(b), the PINNs are found to solve the dimensionless Euler-
Bernoulli beam equation accurately and hence, for all further experiments,
nondimensional PDEs are simulated using PINNs. Additionally, the
nondimensional displacement is henceforth referred to as displacement for
conciseness. The presented methodology predicts the dimensionless quantities
and hence all the plots of results and their associated error plots are
dimensionless. Consequently no units are mentioned in the plots of the
presented results. Next, we describe the inverse problem-solving strategy
using nondimensional equations.

###  III-C PINN Framework for Inverse Problems

The abstract dimensionless PDE described by (6) is well-posed, and the forward
problem can be solved uniquely. However, in the case of an inverse problem,
the problem is ill-posed and either the initial/boundary conditions or the
parameters/forces are unknown. Hence the generic abstract PDE can be re-
written as

$\mathcal{K^{{}^{\prime}}}(x,t):=\mathcal{D}[u](x,t;\lambda)-f(x,t)\quad\forall(x,t)\in\Omega\times
T\subset\mathbb{R}^{d}\times\mathbb{R}$ (10)

The algorithm for the PINN framework is presented to solve inverse problems.

Algorithm 1 Inverse PINN algorithm

  1. Goal:

To predict the unknown parameter $\bar{\lambda}$ or function
$\bar{f}(\bar{x},\bar{t})$.

  2. Step 1:

Nondimensionalize the governing PDE to approximate the dimensionless parameter
$\lambda$ or function $f(x,t)$.

  3. Step 2:

Choose the training set from the space-time domain $\Omega\times{T}$, and
augment with ($x_{\mathrm{data}},t_{\mathrm{data}}$) at which additional data
($u_{\mathrm{data}}$) are provided.

  4. Step 3:

Construct a feedforward deep neural network with inputs $(x,t)$ and outputs
$u$, $\lambda$ or $f(x,t)$.

  5. Step 4:

Minimize the loss function (11) with a suitable optimization algorithm, and
find the optimal parameters.

  6. Step 5:

Use the optimal parameters to approximate the parameter $\lambda^{*}$ or the
function $f^{*}(x,t)$.

The aim of the inverse problem is to predict the unknown parameter $\lambda$
or the force function $f(x,t)$, when data are provided for the observable $u$
in some part of the training domain. In this paper, $u_{\mathrm{data}}$
denotes the available data for the inverse problem at $N_{\mathrm{data}}$
points. The prediction of the unknown parameter requires additional
information in the loss function as shown in Fig 4. It is essential for the
Jacobian matrix utilized in the inverse operation study employing neural
networks to exhibit a nonzero determinant, to be invertible, and to possess a
reasonable ratio between its largest and smallest eigenvalues to guarantee a
unique solution and ensure computational stability. The algorithm for the
inverse problem is the same as for the forward problem with a minor
modification in the loss function. In addition to the output $u$, the PINNs
now predict the unknown parameter, force, initial or boundary conditions of
the physical problems by leveraging the known data. The loss function for the
inverse problem can be defined as

Figure 5: Timoshenko single beam; Color bar represents Left: Cross-sectional
rotation ($\theta^{\mathrm{*}}$); Right: Transverse displacement
($w^{\mathrm{*}}$).

$\mathcal{L^{{}^{\prime}}}(\theta)=\mathrm{\underset{\theta}{Min}}(\frac{1}{N_{\mathrm{train}}}\sum_{n=1}^{N_{\mathrm{train}}}||\mathcal{K}(x_{\mathrm{n}},t_{\mathrm{n}})||^{2}+\\\
\frac{1}{N_{\mathrm{data}}}\sum_{n=1}^{N_{\mathrm{data}}}||u_{\mathrm{data}}(x_{\mathrm{n}},t_{\mathrm{n}})-u_{\mathrm{pred}}(x_{\mathrm{n}},t_{\mathrm{n}})||^{2})$
(11)

Here, $u_{\mathrm{{pred}}}$ denotes the prediction of u by the neural network
section implementing the PINN algorithm for forward and inverse problems of
dimensionless beam equations.

##  IV Numerical EXPERIMENTS AND DISCUSSION

In the following subsections, five numerical experiments are presented. The
experiments are conducted in a progressive manner, beginning with simple
models such as a single beam system and then progressing to more complex ones
such as a double beam connected to a Winkler foundation. To verify the
proposed method, we first investigate forward and inverse problems for a
single beam, which serves as the proof of the concept. Then, we apply the
method to more intricate cases of double-beam systems to simulate forward and
inverse problems.

###  IV-A Timoshenko Beam Forward Problem

The Euler-Bernoulli theory of beams is widely used in the literature and has
been successfully applied in structures such as the Eiffel Tower and Ferris
wheels. However, it does not consider the effects of transverse shear
deformations, which are often significant in the vertical displacements of
short and thick beams[59]. Timoshenko beam theory provides a mathematical
framework for analyzing thick-beam bending[59]. According to Timoshenko
theory, upon the action of an external force, the beam undergoes some cross-
sectional rotation in addition to transverse displacement. Mathematically, the
dynamics are modeled by a coupled system of PDEs with two variables:
transverse displacement and cross-sectional rotation. The model is given by

$\displaystyle\rho
I\bar{\theta}_{\mathrm{\bar{t}\bar{t}}}-EI\bar{\theta}_{\mathrm{\bar{x}\bar{x}}}-kAG(\bar{w}_{\bar{x}}-\mathrm{\bar{\theta}})=0$
(12) $\displaystyle\rho
A\bar{w}_{\mathrm{\bar{t}\bar{t}}}-kAG(\bar{w}_{\mathrm{\bar{x}\bar{x}}}-\bar{\theta}_{\mathrm{\bar{x}}})=\bar{g}(\bar{x},\bar{t})$

where $\rho$, $A$, $E$ and $I$ have the usual meaning as in the case of the
Euler-Bernoulli beam; $k$ is called the Timoshenko shear coefficient; $G$ is
the shear modulus; and $\bar{g}(\bar{x},\bar{t})$ is the external force acting
on the beam. The transverse displacement is $\bar{w}(\bar{x},\bar{t})$ and
$\bar{\theta}(\bar{x},\bar{t})$ is the cross-sectional rotation of the beam at
position $\bar{x}$ and time $\bar{t}$. After nondimensionalizing (12) and
taking the resulting parameters [60] to be unity, the nondimensional equation
can be written as follows:

$\displaystyle\theta_{\mathrm{tt}}-\theta_{\mathrm{xx}}+(\theta-
w_{\mathrm{x}})=0$ (13) $\displaystyle w_{\mathrm{tt}}+(\theta-
w_{\mathrm{x}})_{\mathrm{x}}=g(x,t)$

We consider the external force [61] to be $g(x,t)$ =
$\cos(t)-\frac{\pi}{2}\sin(x)\cos(t)$ and the computational domain to be
$x\in[0,\pi]$ and $t\in[0,1]$. To make (13) well-posed, the initial and
boundary conditions are supplemented as:

Figure 6: Timoshenko single beam absolute error in predictions Left:
$|\theta-\theta^{\mathrm{*}}|$; Right: Absolute error $|w-w^{\mathrm{*}}|$.

$\theta(x,0)=\frac{\pi}{2}\cos(x)+\left(x-\frac{\pi}{2}\right),\quad\theta_{t}(x,0)=0$
$w(x,0)=\frac{\pi}{2}\sin(x),\quad w_{t}(x,0)=0$
$\theta(0,t)=\theta(\pi,t)=w(0,t)=w(\pi,t)=0$

To estimate the error in the approximated solutions, the analytical solution
for the considered problem is used, which is

$\theta(x,t)=\left(\frac{\pi}{2}\cos(x)+\left(x-\frac{\pi}{2}\right)\right)\cos(t)$
$w(x,t)=\frac{\pi}{2}\sin(x)\cos(t)$

When analytical solutions are not available, there are various ways to
validate the PINN solution. One approach is to compare the solutions with
those obtained using numerical methods such as finite difference, finite
element, finite volume or spectral methods. This can be done by comparing the
predicted solutions from the PINNs with the solutions from the numerical
simulation for the same physical equation. Another approach is to compare the
solutions obtained through PINNs with experimental data. One can compare the
predicted solutions from the PINNs with values experimentally measured over
space and time. Finally, one can validate the solutions obtained through PINNs
by checking if they satisfy the known physical constraints of the system. In
summary, one can use available experimental data, numerical methods or
physical constraints to evaluate the accuracy of the solution obtained using
PINNs.

The difficulty of solving a system of PDEs is greater than that solving a
single PDE, but the neural network structure used for the Euler-Bernoulli
equation is successful in approximating solutions for Timoshenko beams. In
particular, the transverse displacement of the beam is computed within
$\mathcal{R}=3.3e-4$ percent, and the cross-sectional rotation is approximated
within $\mathcal{R}=2.8e-3$ percent. Approximated solutions and absolute
errors in predicting the transverse displacement and cross-sectional rotation
are presented in Figs. 5 and 6. Fig. 5 demonstrates that when a sinusoidal
force is applied to a Timoshenko beam, the beam bends more than it rotates. As
indicated by the scale in the figures, the maximum deflection is $1.44$ and
the maximum rotation is $0.32$. Additionally, the low error in predictions
demonstrates that even with the increase in the PDE complexity, the PINN
successfully solves the Timoshenko PDE with comparable results to the Euler-
Bernoulli equation.

We compare the results obtained from our method with three other methods. The
first method we consider is the widely used numerical technique called the
finite difference method (FDM). The other two methods are neural network-based
approaches, namely physics-guided neural networks (PGNN) [28, 62, 63, 64, 65]
and gradient-enhanced physics-informed neural networks (gPINN) [66]. First,
for FDM we employ a central difference scheme to approximate space derivatives
and a leapfrog scheme to approximate time derivatives. This approach allows us
to solve problems with second-order accuracy in space and time. The results
for the Timoshenko beam show that PINNs can achieve a higher level of accuracy
than the FDM even with a smaller number of training points. Specifically,
$30,000$ points are used in the FDM scheme while only $16,000$ points were
used for training with PINNs and Table I indicates that PINNs perform better
than FDM.

Second, the performance of PINN is compared to a neural network-based approach
PGNN, which leverages physical knowledge embedded in the available data, for
instance, the relationship between beam acceleration and displacement for the
Timoshenko beam problem. Accelerometers can be employed at discrete locations
along the beam to obtain acceleration data. Acceleration data at five
equidistant points along the beam are used, with $2000$ data points at each
location. This dataset is augmented with the boundary and initial conditions
of displacement to match the training data size of PINN. PGNN is a deep neural
network-based architecture with inputs: position ($x$), time ($t$), and
acceleration. Displacement ($w$) is taken as the output of this neural
network. Training PGNN with identical hyperparameters to those used in PINN,
PGNN predicts the displacement ($w$) with an error of approximately
$0.002739$$\%$, as shown in Table I.

Furthermore, utilizing the displacement values ($w$), the neural network’s
auto differentiation and (13), we derived $\theta_{\mathrm{x}}$. Subsequently,
a second neural network was constructed to predict $\theta$, where
$\theta_{\mathrm{x}}$ is used as the input. Boundary and initial conditions
for cross sectional rotation ($\theta$) are also used to guide the PGNN
towards the optimal solution. After training the PGNN, cross-sectional
rotation is predicted with approximately $3.486727$$\%$ error. It can be
inferred from Table I that both displacement and rotation predictions
exhibited higher errors than PINN. This discrepancy can be attributed to the
restricted availability of acceleration data at only discrete spatial
locations within the interior domain rather than a random distribution across
the entire domain. Furthermore, the second neural network, employed for
rotation prediction, demonstrated inferior performance potentially due to
error propagation.

Third, we perform another comparison with a neural network-based method to
simulate PDEs, gradient-enhanced PINN (gPINN) [66], which differs from PINN in
terms of the loss function. The acronym ”gPINN” proposed in [66] is used in
this work instead of ”GPINN” as it is used for another method [67]. In
addition to the loss function of PINN, gPINN leverages gradient information of
the PDE residual and embed the gradient into the loss function. For the
Timoshenko beam problem, derivatives of the system of PDE (13) with respect to
space ($x$) and time ($t$) are supplemented in the loss function. Table I
shows that gPINN exhibits higher relative error percentages in learning
displacement and cross-sectional rotation than PINN. The high-order
derivatives of the physical equations in the loss function of gPINN make it
challenging for autodifferentiation [68] and backpropagation of the loss
function, resulting in poor predictions of deflection and rotation for the
Timoshenko beam. Table I demonstrates that PINN outperforms FDM, PGNN, and
gPINN in accurately predicting displacement and cross-sectional rotation for
the Timoshenko beam, emphasizing its superior performance compared to the
three alternative methods.

TABLE I: Timoshenko beam: $\mathcal{R}$ at $t=1$ $w$, $\theta$ | PINN | FDM | PGNN | gPINN
---|---|---|---|---
$w$ (%) | 3.3e-4 | 0.005615 | 0.002739 | 0.249849
$\theta$ (%) | 2.8e-3 | 0.004733 | 3.486727 | 5.498449

###  IV-B Timoshenko Beam Inverse Problem

This section addresses the inverse problem for the Timoshenko beam, with the
aim to determine the material properties of a beam leveraging the PDE and
beam’s displacement and rotation data. In structural engineering, the inverse
problem of a Timoshenko beam PDE is significant for determining the beam
system’s structural behavior and for health monitoring. This helps engineers
infer the internal material properties and unknown forces from observed
responses such as displacement and rotation measurements. The PINN solves this
problem by combining the knowledge of physics and deep learning. The PINN uses
a neural network to learn the mapping between the unknown parameters of the
PDE and observed data while incorporating the constraints of physics in the
form of PDEs. This parameter identification aids in providing crucial
information for structural diagnosis and repair and helps engineers ensure the
safety and stability of structures. The Timoshenko model for parameter
estimation is presented as follows.

Figure 7: Data to learn the parameters for the Timoshenko single-beam: Blue
dots Collocation points. Red dots Additional data points of rotations
($\theta$) and displacement (u). Black dots Initial and boundary points.

$\displaystyle\alpha\theta_{\mathrm{tt}}-\theta_{\mathrm{xx}}+(\theta-
w_{\mathrm{x}})=0$ (14) $\displaystyle w_{\mathrm{tt}}+(\theta-
w_{\mathrm{x}})_{\mathrm{x}}=g(x,t)$

In the context of the inverse problem of the Timoshenko beam, the PINN is
trained on the observed deflections and rotations of the beam, and the
material properties are treated as the unknowns to be estimated. In this case,
the force g(x, t) applied to the beam is considered to be known, and the only
unknown in the model is $\alpha$. This makes the problem ill-posed, requiring
additional data at a priori to predict the unknown parameter. For $\alpha=1$,
the transverse displacement and cross-sectional rotation data obtained from
the forward problem is supplied to approximate the parameter value. This data
is not error-free and comes with $10^{-3}$ percent error for transverse
displacement and with $10^{-4}$ percent error for cross-sectional rotation. As
shown in Fig. 7 the additional data is supplied on $5000$ points (red dots) at
five positions on the beam $(x=0.2,0.8,1.8,2.6,3)$. In practice, this data can
be collected using sensors installed at the corresponding locations on the
beam as shown in Fig. 7.

To solve the inverse problem, the neural network consist of $1600$ random
training points with the distribution $N_{\mathrm{i}}=200$,
$N_{\mathrm{b}}=400$, and $N_{\mathrm{int}}=1000$. To regularize the PDE term
in the loss function, a regularization parameter of $1$ was chosen [25]. Using
the L-BFGS optimizer $5000$ iterations are performed and the other parameters
are kept the same as in the forward Timoshenko problem. At $t=0.5$, the
unknown parameter $\alpha=1.0136$ is learned.

Figure 8: Euler-Bernoulli double-beam: color bar represents absolute error in
predictions Left: $|w_{1}-w_{1}^{\mathrm{*}}|$; Right:
$|w_{2}-w_{2}^{\mathrm{*}}|$.

Figure 9: Derived quantities for the Euler-Bernoulli double beam. Scattered
points represent the exact solution and the continuous line refers to the
derived solution. Top: First beam Left Bending moment; Mid Velocity; Right
Acceleration. Bottom: Second beam Left Bending moment; Mid Velocity; Right
Acceleration.

Figure 10: Double beam system connected by a Winkler foundation.

We perform a comparison between the PINN and DNNs, as using a numerical
iterative method for inverse problems is computationally expensive. From
PINNs, at $t=0.5$, the unknown parameter $\alpha=1.0136$ is learned. We
utilize DNNs to identify the parameters of a Timoshenko single beam. We use
the same architecture for DNN as used by the PINN. The predicted value of
alpha is $0.6124$ using DNN. PINN is more accurate than DNNs for the inverse
problem of beam systems.

However, there are several issues that one may need to take care of while
solving inverse problems through the presented framework. First, to avoid
overfitting, the minimum training data points required to solve the problem
should be determined empirically by gradually increasing the number of
training points until the model’s performance is satisfactory. Second, for
some physical problems, noisy data may lead to nonconvergence of the
optimization algorithm. Hence, suitable filtering or preprocessing of data may
be required before using the PINN framework. Finally, for every run of the
neural network, one may learn a different parameter or function value; due to
the convergence of the optimizers at different local minima, it may be useful
to find the statistics of the inverse problem solution through multiple runs.

Experimental results for single beam equations illustrate that PINNs can
efficiently solve forward and inverse problems for single beams. In this
study, we investigate the ability of PINNs to handle more complex systems,
specifically double-beam systems connected by a Winkler foundation, as
depicted in Fig. 10.

###  IV-C Euler-Bernoulli Double-Beam Forward Problem

In this section, and for all further experiments, forced transverse vibrations
of two parallel beams are studied. Structurally, two parallel beams of equal
lengths joined by a Winkler massless foundation are considered. Both beams are
considered slender and have homogeneous material properties. The transverse
displacement of both beams is governed by the following system of PDEs [41]:

$\displaystyle
m_{1}\bar{w}_{1_{\mathrm{\bar{t}\bar{t}}}}+K_{1}\bar{w}_{1_{\mathrm{\bar{x}\bar{x}\bar{x}\bar{x}}}}+k(\bar{w}_{1}-\bar{w}_{2})=\bar{f}_{1}(\bar{x},\bar{t})$
(15) $\displaystyle
m_{2}\bar{w}_{2_{\mathrm{\bar{t}\bar{t}}}}+K_{2}\bar{w}_{2_{\mathrm{\bar{x}\bar{x}\bar{x}\bar{x}}}}+k(\bar{w}_{2}-\bar{w}_{1})=\bar{f}_{2}(\bar{x},\bar{t})$

Here, $\bar{w}_{1}$ and $\bar{w}_{2}$ are the beam displacements for the first
and the second beams respectively. The distributed continuous forces acting
transversely on the beams are $\bar{f}_{1}$ and $\bar{f}_{2}$ as shown in Fig.
10. The product of the density and the cross-sectional area of the beams is
given by $m_{1}=\rho_{1}A_{1}$ for the first beam and $m_{2}=\rho_{2}A_{2}$
for the second beam. The parameters $K_{1}$ and $K_{2}$ denote the flexural
rigidity of the beams and are given by $K_{1}=E_{1}I_{1}$ and
$K_{2}=E_{2}I_{2}$. The stiffness modulus of the Winkler elastic layer
connecting both beams is given by $k$. For simplicity, we consider
$m_{1}=m_{2}$, and $K_{1}=K_{2}$, and nondimensionalize (15). After taking all
the resulting parameters to be unity, the nondimensional equation has the same
form as (15) with unit coefficients. The initial conditions are

Figure 11: Timoshenko double beam. Scattered points represent the exact
solution, and the continuous line refers to the predicted solution. Top: First
beam Left Displacement $(w_{1})$; Right Rotation $(\theta_{1})$. Bottom:
Second beam Left Displacement $(w_{2})$; Right Rotation $(\theta_{2})$

.

$\displaystyle w_{1}(x,0)=\sin(x),\quad w_{1_{\mathrm{t}}}(x,0)=0$
$\displaystyle w_{2}(x,0)=\frac{\pi}{2}\sin(x),\quad
w_{2_{\mathrm{t}}}(x,0)=0$

All four ends of the beams are assumed to be simply supported, expressed as

$\displaystyle
w_{1}(0,t)=w_{1}(\pi,t)=w_{1_{\mathrm{xx}}}(0,t)=w_{1_{\mathrm{xx}}}(\pi,t)=0$
$\displaystyle
w_{2}(0,t)=w_{2}(\pi,t)=w_{2_{\mathrm{xx}}}(0,t)=w_{2_{\mathrm{xx}}}(\pi,t)=0$

The external acting force is

$\displaystyle f_{1}(x,t)=\left(1-\frac{\pi}{2}\right)\sin(x)\cos(t)$
$\displaystyle f_{2}(x,t)=\left(\frac{\pi}{2}-1\right)\sin(x)\cos(t)$

For the considered problem, the analytical solution is given by

$\displaystyle w_{1}(x,t)=\sin(x)\cos(t),\quad
w_{2}(x,t)=\frac{\pi}{2}\sin(x)\cos(t)$

In addition to computing the beam displacements, derived quantities such as
velocity, acceleration, and bending moment are also computed for this problem.
These derived quantities also help in the prognosis and diagnostics of the
system. For instance, the bending moment estimates the bending effect when an
external force is applied to a structural element. Estimating the bending
moment can be used to quantify the bending upon the action of applied forces.
The beam is the most common structural member vulnerable to bending moments
because it can bend at any point along its length when subjected to an
external force.

TABLE II: Euler-Bernoulli double-beam: $\mathcal{R}$ at $t=1$ | First beam | Second beam
---|---|---
Displacement (%) | $1.9348\times 10^{-5}$ | $4.3253\times 10^{-5}$
Bending Moment (%) | $9.6112\times 10^{-4}$ | $6.5506\times 10^{-4}$
Velocity (%) | $1.9043\times 10^{-3}$ | $2.0161\times 10^{-3}$
Acceleration (%) | $1.9011\times 10^{-2}$ | $1.4442\times 10^{-2}$

For simulating Euler-Bernoulli double beams, the same neural network
architecture as for the single Euler-Bernoulli beam is considered. The only
change is in the residual parameter, which is $1$ for this case. The results
are illustrated in Figs. ( 8\- 9) and Table II. The absolute difference
between the PINN predicted solution and the exact solution for the first beam
is approximately $10^{-4}$, and for the second beam, it is approximately
$10^{-3}$, as shown in Fig. 8. The bending moment, velocity and acceleration
are computed using the neural network’s autodifferentiation and
backpropagation features. Table II describes the efficiency in the computation
of these quantities at $t=1$ for both beams. The relative percent error in
computing the transverse displacement of the beams on the order of $10^{-5}$,
and for acceleration, this error is on the order of $10^{-2}$, which is very
low and shows the potential of physics-informed learning. Fig. 9 illustrates
the computed velocity, bending moment, and acceleration of both beams.

###  IV-D Timoshenko Double-Beam Forward Problem

The double-beam system modeled by Euler-Bernoulli theory can also be modelled
using Timoshenko theory under the same assumptions as described for the single
Timoshenko equations [40]. In addition to providing the transverse
displacement of the beams, Timoshenko theory also provides the cross-sectional
rotation of both beams through the system of PDEs [40] given by

$\displaystyle
kA_{1}G(\bar{\theta}_{1_{\mathrm{\bar{x}}}}-\bar{w}_{1_{\mathrm{\bar{x}\bar{x}}}})+\rho
A_{1}\bar{w}_{1_{\mathrm{\bar{t}\bar{t}}}}+K(\bar{w}_{1}-\bar{w}_{2})=\bar{f}_{1}(\bar{x},\bar{t})$
$\displaystyle
EI_{2}\bar{\theta}_{2_{\mathrm{\bar{x}\bar{x}}}}+GA_{2}k(\bar{w}_{2_{\mathrm{\bar{x}}}}-\bar{\theta}_{2})-\rho
I_{2}\bar{\theta}_{2_{\mathrm{\bar{t}\bar{t}}}}=0$ $\displaystyle
kA_{2}G(\bar{\theta}_{2_{\mathrm{\bar{x}}}}-\bar{w}_{2_{\mathrm{\bar{x}\bar{x}}}})+\rho
A_{2}\bar{w}_{2_{\mathrm{\bar{t}\bar{t}}}}+K(\bar{w}_{2}-\bar{w}_{1})=\bar{f}_{2}(\bar{x},\bar{t})$
$\displaystyle
EI_{1}\bar{\theta}_{1_{\mathrm{\bar{x}\bar{x}}}}+GA_{1}k(\bar{w}_{1_{\mathrm{\bar{x}}}}-\bar{\theta}_{1})-\rho
I_{1}\bar{\theta}_{1_{\mathrm{\bar{t}\bar{t}}}}=0$ (16)

where $\bar{w}_{\mathrm{i}}(\bar{x},\bar{t})$ and
$\bar{\theta}_{\mathrm{i}}(\bar{x},\bar{t})$, $i=1,2$ denote the transverse
displacement and cross-sectional rotation of the beams respectively. $K$ is
the stiffness modulus of the Winkler elastic layer. G is the shear modulus and
$k$ is the Timoshenko shear coefficient. The rest of the parameters have the
usual meanings as described earlier. For simplicity, we consider
$A_{1}=A_{2}$, and $I_{1}=I_{2}$ and nondimensionalize (16). With some
additional assumptions, the non-dimensional equation has the same form as (16)
with unit coefficients. For the numerical experiment the initial state of the
double beam system is taken to be

Figure 12: Timoshenko double beam absolute errors in prediction; $\theta_{1}$
and $w_{1}$ are the rotation and displacement of the first beam, $\theta_{2}$
and $w_{2}$ are the rotation and displacement of the second beam Top $16000$
training points (a) $|\theta_{1}-\theta_{1}^{\mathrm{*}}|$; (b)
$|w_{1}-w_{1}^{*}|$; (c) $|\theta_{2}-\theta_{2}^{\mathrm{*}}|$; (d)
$|w_{2}-w_{2}^{*}|$. Bottom $1600$ training points (a)
$|\theta_{1}-\theta_{1}^{\mathrm{*}}|$; (b) $|w_{1}-w_{1}^{*}|$; (c)
$|\theta_{2}-\theta_{2}^{\mathrm{*}}|$; (d) $|w_{2}-w_{2}^{*}|$.

$\displaystyle\theta_{1}(x,0)$
$\displaystyle=\left(\frac{\pi}{2}\cos(x)+\left(x-\frac{\pi}{2}\right)\right),\quad\theta_{1_{t}}(x,0)=0$
$\displaystyle w_{1}(x,0)$ $\displaystyle=\frac{\pi}{2}\sin(x),\quad
w_{1_{t}}(x,0)=0$ $\displaystyle\theta_{2}(x,0)$
$\displaystyle=\frac{2}{\pi}\left(\frac{\pi}{2}\cos(x)+\left(x-\frac{\pi}{2}\right)\right),\quad\theta_{2_{t}}(x,0)=0$
$\displaystyle w_{2}(x,0)$ $\displaystyle=\sin(x),\quad
w_{2_{\mathrm{t}}}(x,0)=0$

Simply supported boundary conditions are provided to make the problem
wellposed

$\displaystyle\theta_{1}(0,t)$
$\displaystyle=\theta_{1}(\pi,t)=w_{1}(0,t)=w_{1}(\pi,t)=0$
$\displaystyle\theta_{2}(0,t)$
$\displaystyle=\theta_{2}(\pi,t)=w_{2}(0,t)=w_{2}(\pi,t)=0$

Here, $f_{1}(x,t)$, $f_{2}(x,t)$ and the analytic solutions are as follows

$\displaystyle f_{1}(x,t)$ $\displaystyle=\cos(t)(1-\sin(x))$ $\displaystyle
f_{2}(x,t)$ $\displaystyle=\frac{2}{\pi}\cos(t)-\frac{\pi}{2}\sin(x)\cos(t)$

$\displaystyle\theta_{1}(x,t)$
$\displaystyle=\left(\frac{\pi}{2}\cos(x)+\left(x-\frac{\pi}{2}\right)\right)\cos(t)$
$\displaystyle\theta_{2}(x,t)$
$\displaystyle=\frac{2}{\pi}\left(\frac{\pi}{2}\cos(x)+\left(x-\frac{\pi}{2}\right)\right)\cos(t)$
$\displaystyle w_{1}(x,t)$ $\displaystyle=\frac{\pi}{2}\sin(x)\cos(t),\quad
w_{2}(x,t)=\sin(x)\cos(t)$

TABLE III: Timoshenko double-beam: hyperparameters No. of points | $N_{\mathrm{i}}$ | $N_{\mathrm{b}}$ | $N_{\mathrm{int}}$ | Layers | Neurons | Epochs
---|---|---|---|---|---|---
16000 | 2000 | 2000 | 10000 | 4 | 20 | 15K
1600 | 200 | 200 | 1000 | 4 | 20 | 15K
TABLE IV: Timoshenko double-beam: $\mathcal{R}$ at $t=1$ |  $16000$ points |  $1600$ points
---|---|---
$\theta_{1}$ (%) | $1.6038\times 10^{-3}$ | $2.6211\times 10^{-3}$
$w_{1}(\%)$ | $3.9302\times 10^{-5}$ | $2.503\times 10^{-4}$
$\theta_{2}$ (%) | $1.0826\times 10^{-3}$ | $4.9405\times 10^{-3}$
$w_{2}(\%)$ | $7.8614\times 10^{-5}$ | $3.4904\times 10^{-4}$

Two experiments are performed, varying the number of training points, as shown
in Table III. Table IV shows the relative percent error in approximating the
transverse displacement and cross-sectional rotations for both beams. For
cross-sectional rotations $\theta_{1}$ and $\theta_{2}$, the magnitude of the
percent error remains the same even for fewer training points.

Using a large number of training points can increase the training time and may
not be feasible for problems with many parameters. In these cases, using fewer
training points can lead to less accurate solutions, but they can be obtained
relatively faster. This approach allows engineers to make informed decisions
about the parameters, and once optimal parameters have been identified,
forward solutions can be recalculated with higher accuracy by using more
training points. This is referred to as training with fewer points for the
forward problem. The absolute difference between the predicted and exact
solutions of $\theta_{1}$, $w_{1}$, $\theta_{2}$ and $w_{2}$, even for $1600$
training points is very small as shown in Fig. 11 Fig. 12. Fig. 11 presents
the PINNs prediction for a double Timoshenko beam. The scattered points refer
to the exact solution, and the continuous line represents the predicted
solution. The force is applied uniformly in both beams; however, the
deflection and rotation of the first beam are greater than those of the second
beam. The results in Fig. 12 indicate that, for the second beam, a larger
number of training points (16000) results in a more accurate prediction of
deflection and rotation than a smaller number of training points (1600).
Conversely, for the first beam, a smaller number of training points (1600)
results in a more accurate prediction of the quantity of interest than a
larger number of training points (16000). In any case, the difference in
absolute error is relatively small, demonstrating that even with fewer
training points, PINNs can still produce accurate predictions.

###  IV-E Timoshenko Double-Beam Inverse Problem

The applied force on structural systems is critical for structural design and
condition assessment. In design, control, and diagnosis, accurate estimation
of dynamic forces acting on a structure is essential. These details can be
used to evaluate the structural condition. For example, understanding the
impact of heavy vehicles on bridge structures can aid in detecting early
damage to them. Indirect force determination is of special interest when the
applied forces cannot be measured directly, while the responses can be
measured easily.

For the inverse problem, three distinct experiments are performed on (16).
First, the unknown parameter is learned from the Timoshenko double-beam
system. We consider the unknown parameter to be $\rho A_{1}$ from (16). For
the value of $\rho A_{1}=1$, the data for transverse displacement and cross-
sectional rotation are provided at some points in the computational domain.
Second, the unknown applied function on the first beam is learned by providing
noise-free simulated displacement and cross-sectional rotation data. For this
case, all other parameters, initial and boundary conditions are considered to
be known, and only the function $f_{1}(x,t)$ is unknown. Third, the same force
function is predicted by providing noisy displacement and cross-sectional
rotation data. The data generated for learning the function in the second case
are corrupted with noise to be used in the third case. The exact solution for
the function to be learned in the second and third cases is
$\cos({t})(1-\sin({x}))$.

Figure 13: Timoshenko double-beam inverse problem: absolute error in the
prediction of force when the additional data of rotation and deflections
provided at five locations has left: no noise right: 20 percent Gaussian
noise.

Figure 14: Data to learn material properties for the Timoshenko double beam:
Blue dots Collocation points. Red dots Additional data points of displacement
and rotation for the double beam at one location. Black dots Initial and
boundary points.

Figure 15: Data to learn force for the Timoshenko double beam: Blue dots
Collocation points. Red dots Additional data points of displacement and
rotation for the double beam at six different locations. Black dots Initial
and boundary points.

The inverse problem in engineering refers to the process of estimating unknown
parameters or functions from a set of measured data. In PINNs, the inverse
problem is usually solved by training a neural network to fit the measured
data and the known physical laws. However, the measured data can be affected
by various sources of noise, which can make estimation of the quantity of
interest more challenging. The noise can make the measured data unreliable,
and the neural network may not be able to accurately estimate the unknown
parameters or functions. In such a scenario, the optimizer of the neural
network does not necessarily converge to local minima.

The same neural network architecture is used as in the forward double-beam
Timoshenko problem, with residual parameter $1$ to regularize the physical
equation in the loss function. Here, $2500$ epochs are performed using the
L-BFGS optimizer to train the neural network. For learning the parameter,
$5000$ data points are provided at $x=1.8$, as shown in Fig. 14. The exact
value of the unknown parameter is $\rho A_{1}=1$ in (16), and the predicted
value of the parameter using the PINN framework is 1.0208, which is close to
the desired value. Even for a system of four PDEs, by only providing data at
one particular beam location, the unknown parameter is learned successfully
using PINNs. This shows that PINNs can handle large complex systems of PDEs
efficiently.

TABLE V: Timoshenko double-beam inverse problem: noise vs. $\mathcal{R}$ Noise percent (%) | Relative error percent (%)
---|---
$0$ | $4.3271\times 10^{-2}$
$10$ | $4.8688\times 10^{-2}$
$20$ | $1.1123\times 10^{-1}$

The function $f_{1}(x,t)$, the applied force on the first beam is predicted in
the second experiment. As illustrated in Fig. 15, the data for transverse
displacement and cross-sectional rotation are provided at $6$ different
locations with $5000$ data points at each location.

For the third experiment, the data provided for learning the unknown function
$f_{1}(x,t)$ are provided with $10\%$ and $20\%$ Gaussian noise and the
corresponding performance in learning the function is shown in Table V. Even
with $10\%$ and $20\%$ noise, the relative error percent between analytic and
predicted force is lower, as seen in TABLE V. Fig. 13 shows the force
prediction along the beam when rotation and deflection observations are
available at five points. The results demonstrate that the PINN is more
precise in its predictions when the data are free from noise compared to when
they are noisy. Despite the presence of noise in the data, the absolute error
remains within the magnitude of $10^{-2}$, which is comparable to the error
observed when data are not noisy. To be more precise, Fig. 13 shows the
absolute difference error of the PINN predicted and exact force at $t=0.5$
with $0$ percent and $20$ percent noise. Even with $20$ percent noise, the
unknown force is learned with less than $1\%$ error over the entire space-time
domain, demonstrating that PINN is a very accurate and robust approach.

The minimum number of data points required to estimate the model parameters
depends on several factors, such as the complexity of the physics, the number
of physical parameters in the model, and the quality of the data. More data
points and more complex physics require more neural network capacity,
resulting in a larger neural network with more hyperparameters. In practice,
more data points lead to overfitting. The minimum training data points
required for a PINN framework are determined empirically by gradually
increasing the number of training points until the model’s performance is
satisfactory.

Finally, a sensitivity analysis is carried out to examine the influence of
input variables, specifically the displacement and rotation, on the output
variable, which is the force. The analysis involves adding $20\%$ Gaussian
noise to the displacement data while no noise is added to the rotation data.
The resulting mean accuracy of the force is 0.14313413. In contrast, when
$20\%$ noise is introduced to the rotation data with the displacement data
remaining unaltered, the mean accuracy of the force is $0.204627$. The results
of this analysis show that the force is more sensitive to rotations than the
displacement data.

##  V Conclusions

The design and maintenance of complex structural systems are challenging due
to the multiscale interaction of their components. It is desirable to predict
the behavior of these complex systems by solving the governing model of
interest. Recently, PINNs have emerged as a viable method for simulating PDEs.
In this work, we propose using the PINN algorithm with the
nondimensionalization step aiding in the learning procedure for complex beam
systems. The PINN framework successfully solves the forward and inverse
problems for nondimensional single and double-beam systems. Based on the
numerical experiments, the following conclusions are drawn.

First, the relative percent error in computing the beam displacement does not
increase with increasing model complexity when solving the forward problem. In
fact, for both Euler-Bernoulli and Timoshenko theory, the error decreases by
an order of magnitude for double-beam systems compared to single-beam systems.
In addition, the error in computing the bending rotation is comparable for
single and double Timoshenko beam systems. This nonincrease in error as the
model complexity increases suggests that the PINN framework is appropriate for
simulating large-scale systems with multiple connected components.

Second, it is demonstrated that PINNs precisely discover the unknown force
function and model parameters through their inverse problem-solving
capability. The proposed algorithm successfully learns the model parameter
with less than $3\%$ error for the single Timoshenko beam. In addition, for
the double beam Timoshenko system, the unknown function is approximated on the
whole space-time domain with less than $0.05\%$ error, demonstrating the
algorithm’s effectiveness for solving inverse problems.

Third, physical quantities such as velocity, acceleration, and bending moment
characterize the system’s behavior. Even though the derived quantities are not
directly trained in the neural network, they are approximated with less than
$2e-2\%$ error for the Euler-Bernoulli double-beam system.

Fourth, the algorithm’s ability to use fewer training points in forward
problems and to accomodate noisy data in inverse problems is exploited. The
obtained results show that even with $1600$ training points, the double
Timoshenko beam displacement is predicted on the entire space-time domain with
less than $5e-3\%$ error. In the case of the inverse problem, the force
function is discovered with less than $0.2\%$ error even when the data used in
the learning procedure contains $20\%$ Gaussian noise. These findings imply
that the algorithm is accurate and robust under the tested noise levels.

To summarize, PINNs enable the simulation of complex structural systems with
multiple interacting components efficiently, accurately, and robustly. In the
future, this approach could be extended to estimate displacements for various
input forces and mechanical vibration modes and incorporate robust methods to
account for stochasticities. Additionally, future works on PINNs could be
focused on reducing the computational cost and developing methodologies to
augment their generalizability, thereby expanding the applicability of PINNs
beyond the training domain.

## Acknowledgment

The authors would like to express their appreciation to the anonymous
reviewers and editors for their valuable comments and feedback, which have
significantly improved the quality of this work. The authors extend their
appreciation to Prof. Siddhartha Mishra (ETH Zürich) for his insightful
suggestions in comparisons with numerical methods.

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|  Taniya Kapoor is currently pursuing Ph.D. degree in the department of
Engineering Structures at Delft University of Technology, The Netherlands.
Before this, she was an intern in the seminar of applied mathematics at ETH
Zürich, Switzerland. She received her M.Sc. degree in applied mathematics from
South Asian University, India and in scientific computing from Université de
Lille, France. She currently works on scientific machine learning with
applications in engineering structures.
---|---
|  Hongrui Wang  (Member, IEEE) received the Ph.D. degree from the Section of
Railway Engineering, Delft University of Technology, Delft, The Netherlands,
in 2019. He was a Post-Doctoral Researcher with the Delft University of
Technology, until November 2020, where he is currently an Assistant Professor
with the Department of Engineering Structures. His research interests include
signal processing, artificial intelligence, and their applications in the
structural health monitoring and digital modeling and the design of railway
infrastructures. Dr. Wang is an Associate Editor of the IEEE TRANSACTIONS ON
INSTRUMENTATION AND MEASUREMENT.
---|---
|  Alfredo Núñez (Senior Member, IEEE) received the Ph.D. degree in electrical
engineering from the University of Chile, Santiago, Chile, in 2010. He is
currently an Associate Professor in the field of data-based maintenance for
railway infrastructure with the Section of Railway Engineering, Department of
Engineering Structures, Delft University of Technology, Delft, The
Netherlands. He was a Postdoctoral Researcher with the Delft Center for
Systems and Control, Delft University of Technology. He has authored or
coauthored more than a hundred international journal and international
conference papers. His current research interests include railway
infrastructures, intelligent conditioning monitoring and maintenance of
engineering structures, computational intelligence, big data, risk analysis,
and optimization. Dr. Núñez is on the Editorial Board of IEEE TRANSACTIONS ON
INTELLIGENT TRANSPORTATION SYSTEMS and Applied Soft Computing.
---|---
|  Rolf Dollevoet received the M.Sc. degree in mechanical engineering from the
Eindhoven University of Technology, Eindhoven, The Netherlands, in 2003, and
the Ph.D. degree in rail research on rolling contact fatigue from the
University of Twente, Enschede, The Netherlands, in 2010. Since 2003, he has
been with the Railway Sector, ProRail, Utrecht, The Netherlands. Since 2012,
he has been appointed as a part-time Professor with the Section of Railway
Engineering, Delft University of Technology, Delft, The Netherlands. He was
also a Railway System Expert with ProRail, where he was responsible for all
the scientific research and innovation with the Civil Engineering Division,
ProRail Asset Management. Dr. Dollevoet was a recipient of the Jan van Stappen
Spoorprijs 2010 Award (a yearly prize for contributions to the travel quality
and service for passengers in The Netherlands) from the railway sector for his
Ph.D. research and its huge potential to reduce track maintenance costs.
---|---