Subhayan De
email: subhayan.de@colorado.edu

Bhuiyan Shameem Mahmood

Ebna Hai
email: ebnahaib@hsu-hh.de

Alireza Doostan
email: doostan@colorado.edu

Markus Bause
email: bause@hsu-hh.de

Bhuiyan Shameem

Mahmood Ebna

Prediction of ultrasonic guided wave propagation in solid-fluid and their interface under uncertainty using machine learning

Keywords: Fluid-structure interaction, Ultrasonic guided waves, Gaussian process regression, Super-resolution. A PREPRINT -APRIL 24, 2021

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Introduction

In the last few decades, maintenance and quality control of infrastructures have come to the forefront of engineering. As such, implementation of a damage identification system not only improves safety, but also minimizes the operational cost. Here, structural damages are defined as changes in the physical or geometric properties, boundary conditions, including those in the system connectivity, which adversely affect the performance of an infrastructure. In general, damage identification systems inform the user about any defect such as corrosion, delamination, or fatigue cracks in the structure in real time. This information can then be used for reactive measures. Additionally, these systems can also inform the user about the state of the structure, including an estimation of the remaining lifetime of the structure.

Structural health monitoring (SHM) is the key component to implementing a damage identification system for a wide range of engineering infrastructures due to its ability to quickly respond to structural damages, improving life-cycle management and reliability by means of timely maintenance. The SHM system examines the state of structural health, providing the data for processing and interpretation, which can be used to estimate the remaining life-time of a structure. There are two types of SHM systems distinguished based on their operation: (a) active SHM, and (b) passive SHM [START_REF] Giurgiutiu | Structural Health Monitoring with Piezoelectric Wafer Active Sensors[END_REF]. The active SHM system uses an approach similar to the one in non-destructive testing or evaluation (NDT or NDE) [START_REF]Non-Destructive Evaluation (NDE) of Polymer Matrix Composites[END_REF][START_REF] He Ren | Reliability Based Aircraft Maintenance Optimization and Applications[END_REF], except that sensors in the active SHM system are permanently embedded into the structure [START_REF] Giurgiutiu | Structural Health Monitoring with Piezoelectric Wafer Active Sensors[END_REF][START_REF] Srinivasan Gopalakrishnan | Computational Techniques for Structural Health Monitoring[END_REF][START_REF] Yuan | Structural Health Monitoring (SHM) in Aerospace Structures[END_REF]. Based on the data provided by these sensors, the active SHM system may provide a recommendation for actions to be undertaken, e.g., to replace a specific part or repair the damage. While passive SHM does not require real-time data, the active SMH system is based on the continuous direct monitoring of the physical behaviour in real time [START_REF] Srinivasan Gopalakrishnan | Computational Techniques for Structural Health Monitoring[END_REF][START_REF] Gharaibeh | Investigation of the behaviour of selected ultrasonic guided wave modes to inspect rails for long-range testing and monitoring[END_REF]. For a deeper discussion on SHM systems, we refer to [START_REF] Ostachowicz | New Trends in Structural Health Monitoring[END_REF][START_REF]New Developments in Sensing Technology for Structural Health Monitoring[END_REF][START_REF]New Trends in Vibration Based Structural Health Monitoring[END_REF], and the references therein.

Currently, guided waves (GW) propagation approach is considered to be one of the most efficient NDT techniques. These techniques are sensitive to damage/inclusion, and can scan large areas. Accordingly, some studies considered ultrasonic guided waves (UGWs) [START_REF] Cara | Guided waves in anisotropic and quasi-isotropic aerospace composites: Three-dimensional simulation and experiment[END_REF][START_REF] Gresil | Guided wave propagation and damage detection in composite pipes using piezoelectric sensors[END_REF][START_REF] Yan | Ultrasonic guided wave imaging techniques in structural health monitoring[END_REF][START_REF] Rizzo | Structural health monitoring of immersed structures by means of guided ultrasonic waves[END_REF], while others Lamb wave propagation [START_REF] Yuan | Structural Health Monitoring (SHM) in Aerospace Structures[END_REF][START_REF] De Luca | Numerical simulation of the lamb wave propagation in impacted cfrp laminate[END_REF][START_REF] Lamb | On waves in an elastic plate[END_REF]. There is an extensive literature that addresses issues related to the GW-based SHM systems from an experimental perspective [START_REF] Rose | Ultrasonic Guided Waves in Solid Media[END_REF][START_REF] Lee | Fully noncontact wave propagation imaging in an immersed metallic plate with a crack[END_REF][START_REF] Raghavan | Review of guided-wave structural health monitoring[END_REF][START_REF] Huang | Quantitative modeling of coupled piezo-elastodynamic behavior of piezoelectric actuators bonded to an elastic medium for structural health monitoring: A review[END_REF][START_REF] Mitra | Guided wave based structural health monitoring: A review[END_REF]. Historically, a GW-based SHM system has been represented by simulating a theoretical wave propagation model (e.g., Helmholtz, Lamé-Navier equations). Such an SHM system can identify and localize damage/inclusion by comparing the signal received at receiver sensors and the output of the physical model [START_REF] Raghavan | Review of guided-wave structural health monitoring[END_REF][START_REF] Huang | Quantitative modeling of coupled piezo-elastodynamic behavior of piezoelectric actuators bonded to an elastic medium for structural health monitoring: A review[END_REF][START_REF] Mitra | Guided wave based structural health monitoring: A review[END_REF][START_REF]Lamb-Wave Based Structural Health Monitoring in Polymer Composites[END_REF]. However, there is a disproportionately small amount of modeling studies for this problem. This is despite the inherent advantage of mathematical modeling in structural damage identification. More specifically, it would allow evaluating a vast amount of alternative scenarios to prevent and manage any potential structural damage in a timely manner. The scarcity of studies, in part, can be explained by the associated modeling complexity. Recently, a subset of the authors [START_REF] Shameem | Finite element approximation of ultrasonic wave propagation under fluid-structure interaction for structural health monitoring systems[END_REF][START_REF] Shameem | Numerical study and comparison of time discretization schemes for an ultrasonic guided wave propagation problem coupled with fluid-structure interaction[END_REF][START_REF] Shameem | Modeling and simulation of ultrasonic guided waves propagation in the fluid-structure domain by a monolithic approach[END_REF][START_REF] Shameem | Modeling concept and numerical simulation of ultrasonic wave propagation in a moving fluid-structure domain based on a monolithic approach[END_REF] proposed a monolithically coupled system of acoustic and elastic wave propagation problem with or without fluid-structural interaction (FSI) effect to understand quantitatively and phenomenologically UGWs propagation and influence of the geometrical and mechanical properties of the inclusion and media.

In the current paper, we focus on an ultrasonic wave propagation problem for a rigid solid plate in the absence of vibrations. That is, we consider the wave propagation in fluid-solid and their interface (henceforth, the WpFSI problem) [START_REF] Shameem | Finite element approximation of ultrasonic wave propagation under fluid-structure interaction for structural health monitoring systems[END_REF][START_REF] Shameem | Modeling and simulation of ultrasonic guided waves propagation in the fluid-structure domain by a monolithic approach[END_REF]. To address the WpFSI problem in a rigid/ideal fluid-solid domain, we need to consider the fluid-solid interface effect throughout the modeling process, where the acoustic and elastic wave propagation models are fully coupled. Furthermore, the WpFSI problem considered in this paper allows gaining deeper understanding of the behavior of the UGWs propagation in solid-fluid, solid-solid, and their interface with or without any damage or inclusion through the solution of the associated nonlinear multiphysics problems. In particular, UGWs propagation pattern changes at the solid-fluid or solid-solid interface based on the number, location, size and material properties of the inclusion. However, due to the variations in external factors, such as material properties of the medium and inclusion, damage location, humidity, temperature [START_REF] Lanza | Temperature effects in ultrasonic lamb wave structural health monitoring systems[END_REF][START_REF] Salamone | Guided-wave health monitoring of aircraft composite panels under changing temperature[END_REF], vibration [START_REF] Shameem | Modeling concept and numerical simulation of ultrasonic wave propagation in a moving fluid-structure domain based on a monolithic approach[END_REF], random noise, etc., uncertainties in UGWs propagation persist. Because of the complex nature of UGWs propagation, these uncertainties are challenging and computationally expensive to incorporate in a physical wave propagation model.

In the age of data, predictions using data-driven methods have found immense popularity. Among these methods is Gaussian process regression built from data and used for interpolation [START_REF] Edward | Gaussian processes for machine learning[END_REF]. This approach for building a data-driven model is also known as kriging and has been applied to a variety of scientific problems, such as, geostatistics [START_REF] Daniel G Krige | A statistical approach to some basic mine valuation problems on the Witwatersrand[END_REF], machine learning [START_REF] Edward | Gaussian processes for machine learning[END_REF][START_REF] Koch | Efficient multicriteria optimization on noisy machine learning problems[END_REF], design optimization [START_REF] Michael Tm Emmerich | Single-and multiobjective evolutionary optimization assisted by Gaussian random field metamodels[END_REF][START_REF] Gary | Review of metamodeling techniques in support of engineering design optimization[END_REF][START_REF] Forrester | Engineering design via surrogate modelling: a practical guide[END_REF], and so on. On the other hand, neural networks used for approximating functions [START_REF] Goodfellow | Deep learning[END_REF] have found recent popularity within the scientific community [START_REF] Baker | Workshop report on basic research needs for scientific machine learning: Core technologies for artificial intelligence[END_REF]. In [START_REF] Raissi | Physics informed deep learning (part I): Datadriven solutions of nonlinear partial differential equations[END_REF][START_REF] Raissi | Hidden physics models: Machine learning of nonlinear partial differential equations[END_REF] neural networks are trained by minimizing a loss function that adds a contribution from the error in satisfying the governing differential equations. This approach was used for the propagation of seismic waves in [START_REF] Sun | Physics-guided deep learning for seismic inversion with hybrid training and uncertainty analysis[END_REF]. Convolutional neural networks and recurrent neural networks were used to simulate the wave dynamics in [START_REF] Lino | Simulating surface wave dynamics with convolutional networks[END_REF] and [START_REF] Wilhelm E Sorteberg | Approximating the solution to wave propagation using deep neural networks[END_REF], respectively. For seismic wave simulation, [START_REF] Moseley | Fast approximate simulation of seismic waves with deep learning[END_REF] used convolutional neural networks. The procedure to generate a high-quality image with more details from a low-quality image is known as the super-resolution [START_REF] Yang | Deep learning for single image super-resolution: A brief review[END_REF]. Convolutional neural networks were shown to perform this procedure efficiently in [START_REF] Dong | Learning a deep convolutional network for image super-resolution[END_REF][START_REF] Dong | Image super-resolution using deep convolutional networks[END_REF]. [START_REF] Stengel | Physics-informed super resolution of climatological wind and solar resource data[END_REF] used generative adversarial networks (GANs) [START_REF] Goodfellow | Deep learning[END_REF] for super-resolution to generate high-resolution meteorological data from low-resolution images. GANs were trained in [START_REF] Zhu | Wave-dynamics simulation using deep neural networks[END_REF] to generate images of the seismic wave propagation. [START_REF] Melville | Structural damage detection using deep learning of ultrasonic guided waves[END_REF][START_REF] Ishan | Accounting for physics uncertainty in ultrasonic wave propagation using deep learning[END_REF][START_REF] Rautela | Ultrasonic guided wave based structural damage detection and localization using model assisted convolutional and recurrent neural networks[END_REF] used neural networks for UGW based damage identification and localization. For a detailed list of publications on machine learning assisted modeling for physical problems, we refer the readers to [START_REF] Willard | Integrating physics-based modeling with machine learning: A survey[END_REF].

In uncertainty quantification of physical systems, [START_REF] Zhang | Quantifying total uncertainty in physicsinformed neural networks for solving forward and inverse stochastic problems[END_REF] used the dropout strategy [START_REF] Hinton | Improving neural networks by preventing co-adaptation of feature detectors[END_REF], which ignores some of the connections in the networks with a probability for model and parametric uncertainty. Recently, [START_REF] De | On transfer learning of neural networks using bi-fidelity data for uncertainty propagation[END_REF] and [START_REF] De | Uncertainty quantification of locally nonlinear dynamical systems using neural networks[END_REF] used transfer learning techniques and convolutional neural networks for uncertainty quantification of physical systems. However, these studies do not address the prediction of UGW patterns when uncertainty is present in a WpFSI problem. There are few studies on applied fluid-structure interaction (FSI) problems that use machine learning techniques to predict the flow. Among them, [START_REF] Miyanawala | A hybrid data-driven deep learning technique for fluid-structure interaction[END_REF] developed a hybrid data-driven technique for unsteady FSI problems, relying on the deep learning framework for long-term prediction of unsteady flow fields in a freely vibrating bluff body subjected to a wake-body synchronization. [START_REF] Whisenant | Galerkin-free technique for the reduced-order modeling of fluidstructure interaction via machine learning[END_REF] employed neural networks in the Galerkin-free reduced-order modeling of the FSI applications.

As the solution of the associated multiphysics model for UGW propagation in solid-fluid and their interface imposes large computational costs, it limits the use of structural health monitoring methods for practical systems. To alleviate this issue, herein, we propose a method that improves on existing modeling approach by using Gaussian process regression and convolutional neural network for predicting UGW propagation in a solid, fluid, and their interface, in the presence of uncertainty in material and geometric properties. First, a set of training images for a small number of realizations of the uncertain properties of the fluid inclusion inside a solid plate is generated using a monolithicallycoupled system of acoustic and elastic wave equations [START_REF] Shameem | Modeling and simulation of ultrasonic guided waves propagation in the fluid-structure domain by a monolithic approach[END_REF]. For a new realization of the uncertainty, we use an interpolation scheme using Gaussian process regression to generate an approximate image of the propagated wave through the plate. The quality of the propagated wave pattern is further improved through super-resolution using convolutional neural networks. When validated with a separate dataset of propagating UGW, the results indicate that the proposed approach provides an accurate prediction for the WpFSI problem.

Background

In this section, we first briefly describe the multiphysics model of the coupled wave propagation problem in the FSI domain, i.e., the WpFSI problem. Next, we introduce on Gaussian process regression and convolutional neural networks that are used in this paper to predict high-resolution images of propagating wave patterns.

Physics Model

𝛤 ! 𝛤 ! 𝛤 ! 𝛤 ! 𝛺 " 𝑓 " 𝛤 # 𝛺 $
Figure 1: Schematic representation of the WpFSI computational solid domain with a liquid inclusion at time t = 0 (i.e., the reference domain). Here, Ω s represents a solid plate with a fluid inclusion Ω f with a common solid-fluid interface Γ i . Homogeneous Dirichlet boundaries are defined by Γ D and f s represent a non-vanishing disc-shaped piezoelectric actuator.

Let Ω := Ω(t) ⊂ R d , d = {2, 3} be a time-dependent bounded domain with Lipschitz boundary, Γ ⊂ R. The computational domain Ω(t) is comprised of two time-dependent sub-domains Ω s (t) and Ω f (t), i.e. Ω(t) = Ω s (t) ∪ Ω f (t), where, suffixes "f " and "s" are used to indicate the fluid and solid related terms, respectively. These two sub-domains do not overlap, i.e., Ω s (t) ∩ Ω f (t) = ∅, and the corresponding solid-fluid common interface is denoted by Γ i = Γ s ∩ Γ f ⊂ R. The outer boundaries Γ D are fixed (homogeneous Dirichlet boundary condition). Wave signal displacements and velocities on the outer boundaries are set to zero. For brevity, to highlight time-dependent variables, we use a circumflex symbol (i.e., " "), and omit an explicit indication of the time-dependence in the rest of this paper.

The initial configuration (namely the reference domain at time t = 0) is illustrated in Figure 1. In this paper, we denote the reference domain, boundaries, and common solid-fluid interface without any circumflex symbol. Furthermore, a non-vanishing disc-shaped piezoelectric actuator f s is embedded into the geometric center (0, 0) of a solid plate Ω s . The corresponding outward normal vectors at the solid-fluid interface are henceforth denoted by n s and n f . For the coupled elastic-acoustic wave propagation (WpFSI) problem in the solid or fluid domain [START_REF] Shameem | Finite element approximation of ultrasonic wave propagation under fluid-structure interaction for structural health monitoring systems[END_REF][START_REF] Shameem | Modeling and simulation of ultrasonic guided waves propagation in the fluid-structure domain by a monolithic approach[END_REF], the following spaces1 are employed as

L X := L 2 (X), L 0 X := L 2 (X)/R, V X := H 1 (X), V 0 X := H 1 0 (X)
, where X can be used as X := Ω s or X := Ω f . Since we use the variational-monolithic coupling method [START_REF] Shameem | Finite element approximation of ultrasonic wave propagation under fluid-structure interaction for structural health monitoring systems[END_REF][START_REF] Wick | Coupling fluid-structure interaction with phase-field fracture[END_REF][START_REF] Richter | Fluid-structure Interactions: Models, Analysis and Finite Elements[END_REF] to generate the training dataset, the wave signal displacements and velocity spaces are extended to the entire domain Ω = Ω s ∪ Ω f , which makes it more convenient to work with global H 1 function spaces. For the WpFSI problem, we work with the global

H 1 function space V 0 Ω := {u ∈ H 1 (Ω) d : u = 0 on Γ D }.
Therefore, coupling conditions for the WpFSI problem in the variational formulation are automatically satisfied. Furthermore, for the time-dependent domains, we denote the corresponding spaces with a circumflex symbol (" ") notation.

Next, we introduce the vector-valued elastic wave equation for the UGWs propagation in the structure, and the acoustic wave propagation in terms of wave displacement in the fluid (see Appendix II). We formulate a symmetric system of equations, which is characterized by the displacement in both domains, for direct coupling of acoustic and elastic wave propagation problems (see Figure 2). In addition, to ensure that the problem is well-posed and is physically sensible

Elastic wave propagation in Ωs ρs∂tvs -div JσsF -T = Jfs ρs (∂tus -vs) = 0 Coupling Condition on ∂Ωi us = uf JσsF -T ns + c 2 Jρf ∇uf F -1 F -T nf = 0 Acoustic wave propagation in Ωf Jρf ∂tuf -F -1 w • ∇ uf -vf = 0 Jρf ∂tvf -F -1 w • ∇ vf -c 2 Jρf div ∇uf F -1 = 0 Mesh Motion PDEs Model ζ = -αu∆u, -αu∆ζ = 0
The ALE Mapping (w = ∂tA)

A (x, t) : (Ω × I) → Ω A (x, t) = x + u (x, t)
Figure 2: For the physics model of the WpFSI problem a monolithic coupling in the arbitrary Lagrangian-Eulerian (ALE) framework is used (see Appendix I). Here, ρ is density, u is the wave displacement, v is the wave velocity, c is the wave speed, ζ is auxiliary variables from mesh motion PDEs model (see Appendix II), F is the deformation gradient, J is the deformation determinant gradient, and suffixes "s" and "f " are used to indicate the solid and fluid related terms, respectively. we introduce coupling conditions. For the coupled formulation of the UGWs propagation problem in solid-fluid and their common interface, it is the balance between the normal components of displacements and forces on the common interface Γ i between the solid and fluid domains that needs to be ensured. This is achieved via the conditions

u s = u f on Γ i , Jσ s F -T n s + c 2 Jρ f (∇u f F -1 )F -T n f = 0 on Γ i , (1)
which relate to a Neumann-like boundary condition for the structure. To summarize, by combining the acoustics and elastic wave equations with the interface and boundary conditions, the coupled wave propagation problem in fluid, solid and their interface (cf. the WpFSI problem) is formulated. Here, the WpFSI problem describes the UGWs propagation in the computational domain using the principle of nondestructive evaluation. The associated variational form of the WpFSI problem is given below [START_REF] Shameem | Numerical study and comparison of time discretization schemes for an ultrasonic guided wave propagation problem coupled with fluid-structure interaction[END_REF][START_REF] Shameem | Modeling and simulation of ultrasonic guided waves propagation in the fluid-structure domain by a monolithic approach[END_REF][START_REF] Shameem | Modeling concept and numerical simulation of ultrasonic wave propagation in a moving fluid-structure domain based on a monolithic approach[END_REF].

A PREPRINT -APRIL 24, 2021

The Coupled Wave Propagation (WpFSI) Problem

Find the global ultrasonic wave signal displacement u ∈ V 0 Ω , velocity v ∈ L Ω , and auxiliary variables ζ ∈ V Ω , such that the initial conditions u(0) = u 0 and ∂ t u(0) = v 0 are satisfied, and for almost all time t ∈ I t it holds that

(ρ∂ t v, φ v ) Ωs + (Jσ s F -T , ∇φ v ) Ωs -(Jf s , φ v ) Ωs + (Jρ∂ t v, φ v ) Ω f -(Jρ(F -1 w • ∇)v, φ v ) Ω f + (c 2 Jρ(∇uF -1 )F -T , ∇φ v ) Ω f = 0 ∀φ v ∈ V 0 Ω , (ρ∂ t u, φ u ) Ωs -(ρv, φ u ) Ωs + (Jρ∂ t u, φ u ) Ω f -(Jρv, φ u ) Ω f -(Jρ(F -1 w • ∇)u, φ u ) Ω f + (α u ∇ζ, ∇φ u ) Ω f -(Jf a , φ u ) Ω f = 0 ∀φ u ∈ L Ω , (α u ζ, φ ζ ) Ω f -(α u ∇u, ∇φ ζ ) Ω f + (α u ζ, φ ζ ) Ωs -(α u ∇u, ∇φ ζ ) Ωs = 0 ∀φ ζ ∈ V Ω , (2)
where φ u , φ v , and φ ζ are test-functions, and w = ∂ t u f is a mesh velocity driven by the external loads f a . This allows us to link the WpFSI problem to the additional multiphysics problem, e.g. the FSI problem that can describe the solid deformation due to the ambient or internal fluid flow [START_REF] Shameem | Numerical study and comparison of time discretization schemes for an ultrasonic guided wave propagation problem coupled with fluid-structure interaction[END_REF][START_REF] Shameem | Modeling and simulation of ultrasonic guided waves propagation in the fluid-structure domain by a monolithic approach[END_REF][START_REF] Shameem | Modeling concept and numerical simulation of ultrasonic wave propagation in a moving fluid-structure domain based on a monolithic approach[END_REF]. All the quantities can be found in Appendices I-III. The test-functions for the fluid and the solid sub-domains belong to the global test space φ v ∈ V 0 Ω , which implies that they coincide on the interface Γ i . Accordingly, the Neumann coupling conditions at the interface are satisfied (in a variational way) and the following condition is implicitly accounted for in the WpFSI problem (see ( 2)) as

g s , φ v Γi + Jg f F -T , φ v Γi = 0 on Γ i . (3)

Gaussian Process Regression or Kriging

In Gaussian process regression, we assume that the measured data y = [y 1 , . . . , y N ] T is generated according to a Gaussian process and can be given by

y i = f (ξ i ) + ε i , i = 1, . . . , N, (4)
where ε is independent zero-mean Gaussian noise with variance σ 2 n . A zero-mean Gaussian process f (ξ) ∼ GP 0, κ(ξ, ξ ) with covariance function κ(ξ, ξ ) is written using a Gaussian distribution [START_REF] Edward | Gaussian processes for machine learning[END_REF]

f (ξ) f (ξ ) ∼ N 0, κ(ξ, ξ) κ(ξ, ξ ) κ(ξ , ξ) κ(ξ , ξ ) . (5)
Hence, we can write the joint probability density for prediction at ξ as

y f (ξ ) ∼ N 0, K + σ 2 n I k f k T f κ f f , ( 6
)
where I is the identity matrix, K is the N × N covariance matrix for the input {ξ i } N i=1 with K ij = κ(ξ i , ξ j ), k f is the N × 1 covariance vector between ξ and ξ with k f ,i = κ(ξ i , ξ ), and

κ f f = κ(ξ , ξ ). The posterior density of f (ξ ) is given by f (ξ ) y ∼ N f , σ 2 f , where f = k T f K + σ 2 n I -1 y and σ 2 f = κ f f -k T f K + σ 2 n I -1 k f .
Table 1: Examples of covariance kernels used in Gaussian process regression.

Kernel κ(ξ, ξ ) Hyperparameters

Radial basis

γ 2 exp -ξ-ξ 2 2τ 2 [γ, τ ] Rational quadratic γ 2 exp 1 + ξ-ξ 2 2τ l 2 -τ [γ, l, τ ] Matern γ 2 2 1-ν Γ(ν) √ 2ν ξ-ξ τ ν B ν √ 2ν ξ-ξ τ * [γ, τ, ν] White noise γ 2 δ ξ,ξ
There are many choices for the covariance function. Some of them that are tried in this paper are listed in Table 1. The hyperparameters in the kernel κ(•, •) and the noise variance σ 2 n are estimated from the measurement data by maximizing the likelihood function or its logarithm given by log p(y|θ gp ) = -

1 2 y T K + σ 2 n I -1 y + 1 2 log|K + σ 2 n I| + N 2 log(2π), (7)
where θ gp is a vector consisting of the hyperparamters of the covariance kernel and σ 2 n .

the output z. In one layer of CNN, the convolution operation is followed by the use of an activation function and then the maxpool operation as shown in Figure 4. In one variation of the network, known as the residual network or ResNet, the output from a previous layer is directly added to the output from the current layer. This architecture is shown to be e↵ective for image processing [START_REF] Goll | Dopelib: Differential equations and optimization environment; a goal oriented software library for solving pdes and optimization problems with pdes[END_REF]. In this paper, we use this residual architecture of CNN to further enhance the images obtained from the Gaussian process regression. Fig. 3: An example of cross-correlation as performed in [START_REF] Cara | Guided waves in anisotropic and quasi-isotropic aerospace composites: Three-dimensional simulation and experiment[END_REF].

Training of a neural network The mean-squared error between the prediction from the network and the training dataset D tr = {x i , y i } Ntr i=1 given by

J = 1 N tr Ntr X i=1 y i M NN (x i ; ✓) 2 (11)
is used as the cost function to train a neural network to optimize the parameters ✓. In the case of a CNN, the parameter is the kernel . Stochastic gradient descent and its variants [START_REF] Bangerth | deal.II -a general purpose object-oriented finite element library[END_REF][START_REF] De | Bi-fidelity stochastic gradient descent for structural optimization under uncertainty[END_REF] are common choice for the optimization algorithm, where at kth iteration, we update the parameters as follows

✓ (k+1) ✓ (k) ⌘ @J @✓ (k) , (12)
where ⌘ is the step size also known as the learning rate. In this paper, we train the neural network used for super-resolution using the Adam algorithm [START_REF] Ciarlet | The finite element method for elliptic problems[END_REF][START_REF] Ciarlet | A mixed finite element method for the biharmonic equation[END_REF], a variant of the standard stochastic gradient descent.

Neural Networks

Neural networks are popularly used for approximating functions in the scientific machine learning community. In its standard form (i.e., a feed-forward neural network), a neural network uses an affine transformation followed by a nonlinear one in every neuron. Multiple neurons are then arranged in layers which are then placed between the input and output layers. Hence, a neural network with N H hidden layers models the relation between the input x and output y as follows

y = F 0 (σ(F N H (. . . σ(F 1 (x; θ 1 )) . . . ); θ N H ); θ 0 ), (8)
where {θ i } N H i=1 are parameters of the N H hidden layers; θ 0 is the parameter of the output layer; F i for a standard neural network is an affine transform but for a convolutional neural network it involves a convolution operation and for a residual neural network involves inputs from previous layers; and σ is a nonlinear function known as activation. Most commonly used activation function is rectified linear unit (ReLU) given by max(0, z) for an input z. In this section, the convolution neural network used in this paper is briefly discussed next.

Convolutional Neural Network (CNN)

In convolutional neural networks, a kernel Ψ corresponding to a layer is learned for a two-dimenional input x such that the output of the layer is given by z ij = q r

x i-q,j-r Ψ qr .

This representation reduces the number of parameters in the network for a kernel of size smaller than the size of the input resulting in sparse connectivity and reducing chances of overfitting. Note that in software tools, such as PyTorch [START_REF] Adam | Automatic differentiation in PyTorch[END_REF], cross-correlation is used instead of the convolution, which is given by

z ij = q r
x i+q,j+r Ψ qr .

A schematic of this procedure is shown in Figure 3. Note that the kernel in [START_REF] Cara | Guided waves in anisotropic and quasi-isotropic aerospace composites: Three-dimensional simulation and experiment[END_REF] is two-times reflected version of the kernel in [START_REF]New Trends in Vibration Based Structural Health Monitoring[END_REF]. We denote the above operation in short using the * symbol, i.e., z = x * Ψ. Often the convolution in [START_REF]New Trends in Vibration Based Structural Health Monitoring[END_REF] or cross-correlation in ( 10) is followed by a maxpooling, where the maximum over a window is selected to downsample the output z. In one layer of CNN, the convolution operation is followed by the use of an activation function and then the maxpool operation as shown in Figure 4. In one variation of the network, known as the residual network or ResNet, the output from a previous layer is directly added to the output from the current layer. This architecture is shown to be effective for image processing [START_REF] Tai | Image super-resolution via deep recursive residual network[END_REF]. In this paper, we use this residual architecture of CNN to further enhance the images obtained from the Gaussian process regression.

Training of a Neural Network

The mean-squared error between the prediction from the network and the training dataset D tr = {x i , y i } Ntr i=1 given by

J = 1 N tr Ntr i=1 y i -M NN (x i ; θ) 2 , (11)
where M NN (•; •) is the prediction from the neural network, is used as the cost function to train a neural network to optimize the parameters θ. In the case of a CNN, the parameter consists of the kernel Ψ and any bias if present.

Figure 4: A typical single layer of convolutional neural network, where the operation in ( 11) is followed by the application of the activation function and a maxpooling operation.

Stochastic gradient descent and its variants [START_REF] Bottou | Optimization methods for large-scale machine learning[END_REF] are common choices for the optimization algorithm, where at kth iteration, we update the parameters as follows

θ (k+1) ← θ (k) -η ∂J ∂θ (k) . (12)
Here, η is the step size also known as the learning rate. In this paper, we train the neural network used for superresolution using the Adam algorithm [START_REF] Diederik | Adam: A method for stochastic optimization[END_REF][START_REF] De | Topology optimization under uncertainty using a stochastic gradient-based approach[END_REF], a variant of the standard stochastic gradient descent.

Image Quality Assessment

In this paper, we compare the quality of the predicted images with the true results using the structural similarity (SSIM) index, a commonly used metric in image quality assessment. In [START_REF] Wang | Image quality assessment: from error visibility to structural similarity[END_REF] the SSIM index between images I 1 and I 2 is defined as

S(I 1 , I 2 ) = (2µ 1 µ 2 + c l )(2σ 12 + c c ) (µ 2 1 + µ 2 2 + c l )(σ 2 1 + σ 2 2 + c c ) , (13)
where µ 1 and µ 2 are the mean intensity of the images I 1 and I 2 , respectively; σ 1 and σ 2 are the standard deviation of intensity of the images I 1 and I 2 , respectively; and c l and c c are small constants to avoid division by zero. This index has a range of [-1, 1] with +1 for a perfect match and -1 for an imperfect match. For images with multiple channels, an average over all the channels is used. We use this index to compare the images obtained from the proposed method with the true images in the validation dataset.

Machine Learning based Framework for Prediction of Ultrasonic Guided Wave Propagation

In this paper, we propose a method for generating images of ultrasonic guided wave propagation at discrete time instances in the presence of an inclusion inside the medium. We further assume that there are uncertainties in the medium properties and the inclusion's geometry. The steps of the proposed machine learning based approach that includes the use of Gaussian process regression and deep neural networks are outlined next. Note that once trained, the proposed framework can produce high-resolution images of the propagated wave patterns for different realizations of the uncertain parameters at a negligible computational cost compared to solving the multiphysics model in the previous section.

Step I: Prediction of Field Variables using Gaussian Process Regression

A training dataset D tr = {ξ i , I i } Ntr i=1 of either images or field variables from simulations of the multiphysics model of the ultrasonic guided wave propagation is generated using the method described in the "Training and Validation Datasets" section, where each of these instances I has values at H × W coordinates and in total for C field variables or channels. A family of Gaussian processes {GP i } Ngp i=1 for N gp = CHW is trained for predicting an I pred of the propagated wave for a realization of the uncertain variables ξ pred . Hence, the computational cost of this step is proportional to CHW . To avoid a large CHW and significant prediction cost we only generate field variables in this step with moderate resolution. Then, in the next step, we use neural networks to generate more enhanced high-fidelity results in a procedure known as super-resolution as described below. Apart from reducing the computational cost of training the family of Gaussian process, super-resolution also helps in identifying the wave pattern better, as can be seen in the numerical examples. Further, it removes the requirement of generating high-quality images from a finer mesh for training the Gaussian processes. Note that a single GP for the whole domain does not produce accurate predictions for the WpFSI problem studied here and hence we choose to use a family of Gaussian process {GP i } Ngp i=1 .

3.2

Step II: Use of a Cascading Residual Network To further enhance the images and produce a high-fidelity prediction of the propagated wave pattern, the images obtained from Step I are further enhanced in this step using a procedure, known as the super-resolution (SR), with the help of a cascading residual network (CARN) following [START_REF] Namhyuk Ahn | Fast, accurate, and lightweight super-resolution with cascading residual network[END_REF]. Herein, a global cascading is used to generate the output image as

y k = y 0 , . . . , y k-1 , F local N l (y k-1 ) * Ψ k + b k ; k = 1, . . . , N g , (14)
where * denotes the convolution or cross-correlation operation as defined in the background section; y 0 = x * Ψ 0 + b 0 is the output from the first convolution operation; and F local N l (•) is the output from the local cascade. In the local cascade the output is

F local j (y k ) = I, F local 0 , . . . , F local j-1 , F res F local j-1 * Ψ local j + b local j ; j = 1, . . . , N l , (15)
where F local 0 = y k ; F res is the output from the residual layer; and Ψ local j and b local j are kernel and bias for the jth layer of the local cascade, respectively. Herein, the residual output is obtained after two convolution operations as follows

F res F local j = σ . . . σ F local j * Ψ res 1 + b res 1 . . . * Ψ res i + b res i + F local j ; i = 1, . . . , N r , (16)
where Ψ res j and b res j are the kernel and bias of the jth convolution, respectively. Figure 5 shows the local cascade with N l = 3 implemented herein that uses residual blocks with N r = 3 consisting of two 3 × 3 group convolution followed by a 1 × 1 convolution as suggested in [START_REF] Namhyuk Ahn | Fast, accurate, and lightweight super-resolution with cascading residual network[END_REF]. The global cascade similarly use three 1 × 1 convolution (i.e., N g = 3) after every local cascade block. In the absence of a large quantity of high-quality images of the WpFSI problem, we use 1000 diverse 2K resolution (DIV2K) images from the dataset described in [START_REF] Agustsson | NTIRE 2017 challenge on single image super-resolution: Dataset and study[END_REF], a commonly used dataset to train super-resolution neural networks, to train the CARN. Note that the CARN needs to be trained once. The same network is used for all numerical examples in this paper.

Training and Validation Datasets

To generate the training and validation datasets, we solve the WpFSI problem (i.e, (2) in the "Physics Model" section). In this work, we focus on an idealized (simplified) case, where the structure remains in its initial position and the acting force, e.g., the FSI effect, is negligible. Thereby, there is no solid mesh deformation due to the additional force. As a result, the deformation gradient and its determinant become F := I, and J := det(F ) = 1. The variational (or weak) formulation is subsequently prescribed in an arbitrary reference domain. In line with [START_REF] Shameem | Finite element approximation of ultrasonic wave propagation under fluid-structure interaction for structural health monitoring systems[END_REF], [START_REF] Richter | Fluid-structure Interactions: Models, Analysis and Finite Elements[END_REF], and [START_REF] Shameem | Modeling concept and numerical simulation of ultrasonic wave propagation in a moving fluid-structure domain based on a monolithic approach[END_REF], we apply the Rothe method for the problem, where finite difference method is used for the temporal discretization and spatial discretization is performed based on a standard Galerkin finite element approach. More specifically, the time discretization is done by using the well-known shifted Crank-Nicolson scheme [START_REF] Richter | Fluid-structure Interactions: Models, Analysis and Finite Elements[END_REF][START_REF] Wick | Fluid-structure interactions using different mesh motion techniques[END_REF] with the time step size, ∆t = 10 µs and the total time T = 3 ms. Space-discretization is done via global refinement iteration into 67, 450 quadratic-mesh cells. Here, each cell has 73 local degrees of freedom (DOFs) [START_REF] Shameem | Numerical study and comparison of time discretization schemes for an ultrasonic guided wave propagation problem coupled with fluid-structure interaction[END_REF]. This nonlinear problem is solved Ultrasonic guided wave propagation under uncertainty using machine learning 9

! !"# ! ! ! !$#

Reference fluid domain

Reference Solid domain Find by using Newton-like method Here, in fully discretized form U n,j h = {u n,j h , v n,j h , ⇣ n,j h }, and the nonlinear residual R(•) is defined in [START_REF] Salamone | Guided-wave health monitoring of aircraft composite panels under changing temperature[END_REF][START_REF] Edward | Gaussian processes for machine learning[END_REF][START_REF] Daniel G Krige | A statistical approach to some basic mine valuation problems on the Witwatersrand[END_REF].

! ! ",% "#$ Tolerance "#$ Increment & " ⟶ & "&' Set j = 0 ( ) ! ",$%& -( ) ! ",$ ≥ ,-. ( ) ! ",$%& -( ) ! ",$ ≤ ,-. Increment, j < j+1 (b)
the residual-based Newton's method [START_REF] Wick | Fluid-structure interactions using different mesh motion techniques[END_REF]79,77] is set to 10 16 . The implementation is accomplished by means of the software library package DOpElib [80] and deal.ii [81].

Numerical Tests: The ultimate goal of tests conducted in this section is the tracking of the UGWs propagation in solid-fluid and their interface. The geometry of a square-shaped solid plate that is used for the numerical simulation is defined as follows. The plate has a length L and a breadth B aligned to the x-axis and the y-axis, respectively.

Here, we assume L = B = 0.35 m. This plate is aligned in a Cartesian coordinate system; the solid domain is defined by ⌦ s := ( 0.175, 0.175) ⇥ (0.175, 0.175). To foster comparison, the numerical simulations share the same configuration as above. Besides, all feature the non-vanishing disc-shaped piezo burst signal force f s [START_REF] Daniel G Krige | A statistical approach to some basic mine valuation problems on the Witwatersrand[END_REF][START_REF] Salamone | Guided-wave health monitoring of aircraft composite panels under changing temperature[END_REF]79] (see Appendix C), which has a radius of 7.5 mm and is located at the geometric center (0, 0) of the solid plate.

For the purpose of visualisation of the wave displacement field component, a normalized color gradient with three channels: green, blue, and red is used. The displacements are normalized between [ 1, +1]. For a schematic of di↵erent component of a propagated UGW patterns see Figure 13a in Appendix C. Figure 7(a) illustrates the wave signal displacement field component u x of a five-cycle symmetric S 0 Lamb wave burst signal on the top surface (e.g., xy plane) of a solid plate with a circular shape fluid inclusion to mimic mechanical damage under uncertainties in the base material properties. To understand the consequences of alternative base material properties, we compare results at a given time t = 3.0 ms. In Figure 7(a), we depict the propagation of a symmetric wave package through a circular-shaped inclusion under four di↵erent base material specifications (e.g., E Ys , ⇢ s , and ⌫ s ). Among these cases, we observe distinct wave propagation speeds, frequencies, wavelength, and amplitudes of the same Lamb wave burst signal as it passes from its origin to the boundary. In addition, as evident from the Figure 7, when wave signal approaches the fluid inclusion, it is partly reflected, part of it propagates further through the inclusion, turning into acoustic wave signals. It is worth noting that the wave signal reflections from the solid to a fluid exceed those from the fluid to the solid. Here, the magnitude of the reflected wave signal frequency is dependent upon the di↵erence between the densities and the wave speeds of the materials; the wave signal reflections are relatively stronger for the sti↵er structures. Due to these additional (a) A monolithic coupling in the ALE framework at time tn Ultrasonic guided wave propagation under uncertainty using machine learning 9

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Reference fluid domain

Reference Solid domain

! !

Acoustic wave Propagation Problem

Elastic wave Propagation Problem

Common solid-fluid interface

(a)

Choose an initial Newton guess Here, in fully discretized form U n,j h = {u n,j h , v n,j h , ⇣ n,j h }, and the nonlinear residual R(•) is defined in [START_REF] Salamone | Guided-wave health monitoring of aircraft composite panels under changing temperature[END_REF][START_REF] Edward | Gaussian processes for machine learning[END_REF][START_REF] Daniel G Krige | A statistical approach to some basic mine valuation problems on the Witwatersrand[END_REF].

! ! ",$ Update ! ! ",% Find by using Newton-like method ! ! ",% "#$ Tolerance "#$ Increment & " ⟶ & "&' Set j = 0 ( ) ! ",$%& -( ) ! ",$ ≥ ,-. ( ) ! ",$%& -( ) ! ",$ ≤ ,-. Increment, j < j+1 (b)
the residual-based Newton's method [START_REF] Wick | Fluid-structure interactions using different mesh motion techniques[END_REF]79,77] is set to 10 16 . The implementation is accomplished by means of the software library package DOpElib [80] and deal.ii [81].

Numerical Tests: The ultimate goal of tests conducted in this section is the tracking of the UGWs propagation in solid-fluid and their interface. The geometry of a square-shaped solid plate that is used for the numerical simulation is defined as follows. The plate has a length L and a breadth B aligned to the x-axis and the y-axis, respectively.

Here, we assume L = B = 0.35 m. This plate is aligned in a Cartesian coordinate system; the solid domain is defined by ⌦ s := ( 0.175, 0.175) ⇥ (0.175, 0.175). To foster comparison, the numerical simulations share the same configuration as above. Besides, all feature the non-vanishing disc-shaped piezo burst signal force f s [START_REF] Daniel G Krige | A statistical approach to some basic mine valuation problems on the Witwatersrand[END_REF][START_REF] Salamone | Guided-wave health monitoring of aircraft composite panels under changing temperature[END_REF]79] (see Appendix C), which has a radius of 7.5 mm and is located at the geometric center (0, 0) of the solid plate.

For the purpose of visualisation of the wave displacement field component, a normalized color gradient with three channels: green, blue, and red is used. The displacements are normalized between [ 1, +1]. For a schematic of di↵erent component of a propagated UGW patterns see Figure 13a in Appendix C. Figure 7(a) illustrates the wave signal displacement field component u x of a five-cycle symmetric S 0 Lamb wave burst signal on the top surface (e.g., xy plane) of a solid plate with a circular shape fluid inclusion to mimic mechanical damage under uncertainties in the base material properties. To understand the consequences of alternative base material properties, we compare results at a given time t = 3.0 ms. In Figure 7(a), we depict the propagation of a symmetric wave package through a circular-shaped inclusion under four di↵erent base material specifications (e.g., E Ys , ⇢ s , and ⌫ s ). Among these cases, we observe distinct wave propagation speeds, frequencies, wavelength, and amplitudes of the same Lamb wave burst signal as it passes from its origin to the boundary. In addition, as evident from the Figure 7, when wave signal approaches the fluid inclusion, it is partly reflected, part of it propagates further through the inclusion, turning into acoustic wave signals. It is worth noting that the wave signal reflections from the solid to a fluid exceed those from the fluid to the solid. Here, the magnitude of the reflected wave signal frequency is dependent upon the di↵erence between the densities and the wave speeds of the materials; the wave signal reflections are relatively stronger for the sti↵er structures. Due to these additional h , v n,j h , ζ n,j h }, and the nonlinear residual R(•) are defined in [START_REF] Shameem | Finite element approximation of ultrasonic wave propagation under fluid-structure interaction for structural health monitoring systems[END_REF][START_REF] Shameem | Modeling and simulation of ultrasonic guided waves propagation in the fluid-structure domain by a monolithic approach[END_REF][START_REF] Shameem | Modeling concept and numerical simulation of ultrasonic wave propagation in a moving fluid-structure domain based on a monolithic approach[END_REF].

by using a Newton-like method (see Figure 6). We apply the direct solver UMFPACK [START_REF] Timothy | An unsymmetric-pattern multifrontal method for sparse lu factorization[END_REF] for the solution of the linear system of equation at each Newton step (see Figure 6). The tolerance (TOL) for the residual-based Newton's method [START_REF] Wick | Fluid-structure interactions using different mesh motion techniques[END_REF][START_REF] Shameem | Finite element approximation of fluid-structure interaction with coupled wave propagation[END_REF] is set to 10 -16 . The implementation is accomplished by means of the software library package DOPELIB [START_REF] Goll | Dopelib: Differential equations and optimization environment; a goal oriented software library for solving pdes and optimization problems with pdes[END_REF] and deal.ii [START_REF] Bangerth | deal.II -a general purpose object-oriented finite element library[END_REF].

The ultimate goal of tests conducted in this section is to present the tracking of the UGWs propagation in solid-fluid and their interface under uncertainty. The geometry of a square-shaped solid plate that is used for the numerical simulation is defined as follows. The plate has a length L and a breadth B aligned to the x-axis and the y-axis, respectively. Here, we assume L = B = 0.35 m. This plate is aligned in a Cartesian coordinate system; the solid domain is defined by Ω s := (-0.175, -0.175) × (0.175, 0.175). To foster comparison, the numerical simulations share the same configuration as above. Besides, all feature the non-vanishing disc-shaped piezo burst signal force f s [START_REF] Shameem | Modeling and simulation of ultrasonic guided waves propagation in the fluid-structure domain by a monolithic approach[END_REF][START_REF] Shameem | Finite element approximation of fluid-structure interaction with coupled wave propagation[END_REF] (see Appendix III), which has a radius of 7.5 mm and is located at the geometric center (0, 0) of the solid plate. For the purpose of visualisation of the wave displacement field component, a normalized color gradient with three channels: green, blue, and red is used. The displacements are normalized between [-1, +1]. For a schematic of different component of a propagated UGW patterns see Figure 13a in Appendix III.

Figure 7a illustrates the wave signal displacement field component u x of a five-cycle symmetric S 0 Lamb wave burst signal on the top surface (e.g., xy plane) of a solid plate with a circular shape fluid inclusion to mimic mechanical damage under uncertainty in the base material properties. To understand the consequences of alternative base material properties, we compare results at a given time t = 3.0 ms. In Figure 7a, we depict the propagation of a symmetric wave package through a circular-shaped inclusion under four different base material specifications (e.g., E Ys , ρ s , and ν s ). Among these cases, we observe distinct wave propagation speeds, frequencies, wavelength, and amplitudes of the same Lamb wave burst signal as it passes from its origin to the boundary. In addition, as evident from Figure 7, when wave signal approaches the fluid inclusion, it is partly reflected, part of it propagates further through the De, Ebna Hai, and Bause

(i) (ii) (iii) (iv) (a)
Wave propagation in a solid plate with a circular shape inclusion under uncertainties in the material properties (see Table

(i) (ii) (iii) (iv) (b)
Wave propagation in a solid plate with a circular inclusion under uncertainties in the material properties (see Table 2) and circular inclusion's radius, rc (see Table (a) Wave propagation in a solid plate with a circular shape inclusion under uncertainties in the material properties (see Table

(i) (ii) (iii) (iv) (b)
Wave propagation in a solid plate with a circular inclusion under uncertainties in the material properties (see Table 2) and circular inclusion's radius, rc (see Table (b) Wave propagation in a solid plate with a inclusion under uncertainties in the material properties (see Table 2) and circular inclusion's radius, rc (see Table

(i) (ii) (iii) (iv) (b)
Wave propagation in a solid plate with a circular inclusion under uncertainties in the material properties (see Table 2) and circular inclusion's radius, rc (see Table inclusion, turning into acoustic wave signals. It is worth noting that the wave signal reflections from the solid to a fluid exceed those from the fluid to the solid. Here, the magnitude of the reflected wave signal frequency is dependent upon the difference between the densities and the wave speeds of the materials; the wave signal reflections are relatively stronger for the stiffer structures. Due to these additional reflected wave signals implementation of machine learning technologies becomes more challenging. For more details about wave propagation behaviour in a solid-fluid domain, we refer the readers to [START_REF] Shameem | Finite element approximation of ultrasonic wave propagation under fluid-structure interaction for structural health monitoring systems[END_REF], and [START_REF] Shameem | Modeling and simulation of ultrasonic guided waves propagation in the fluid-structure domain by a monolithic approach[END_REF].

Next, Figure 7b illustrates the wave propagation behaviour under uncertainty in the base material properties and the dimension of the circular-shaped fluid inclusion. In this case, the Lamb wave burst signal propagates with different wave propagation speed, frequency, amplitude, etc. In addition, we observe the reflection of an additional wave package with different patterns due to distinct dimensions of the inclusion. In Figure 7c, we illustrated the displacement component u x of a five cycle Lamb wave burst signal in a solid plate with different types of fluid inclusions. Here, the material properties of the fluid inclusion is characterized by density, ρ f = 1×10 3 kgm -3 , and viscosity, ν f = 1×10 -3 m 2 s -1 . Due to the different types of inclusions and different shapes of the solid-fluid common interface, the wave A PREPRINT -APRIL 24, 2021 signal passes through the inclusion with distinct amplitudes and wavelengths. In addition, as the wave propagates from the base domain to the inclusion through the solid-fluid common interface, strong reflections occur.

Numerical Illustrations

Two numerical examples are used to illustrate the proposed approach in this paper. In the first example, we assume uncertainty is present in the material properties of the solid plate. In the second example, we further add uncertainty in the geometry of the inclusion. The dataset that we use here consists of images of the displacement component u x of the propagating wave.

Example I: Uncertainty in material properties

In our first example, the uncertainty is assumed in the material properties of the solid plate with known probability distribution functions as given in Table 2. We use three different inclusion geometry, namely, circle, square, and rectangle to illustrate the proposed approach. The training dataset D tr and the validation dataset D val , contain 50 and 30 sets of images for a time instance and a particular geometry of inclusion, which corresponds to (in total) 80 realizations of the material properties according to their probability distributions given in Table 2. The training dataset D tr is used to train the Gaussian processes {GP} Ngp i=1 for N gp = CHW . For these images, we have C = 3, H = W = 270 to keep the computational cost of Step I reasonable. On the other hand, the validation dataset D val contains images that are used to compare the output of the proposed approach. We use a Matern kernel with γ = 1.0, τ = 1.0, and ν = 1.5 (see Table 1) for the Gaussian process. In Step II, we use the neural network described in the previous section that is trained with 1000 open-source high-resolution images from the DIV2K dataset [START_REF] Agustsson | NTIRE 2017 challenge on single image super-resolution: Dataset and study[END_REF]. that the neural network is trained only once and is used for the next example as well. This trained network can further be used for future exercises. This step of performing a super-resolution on the Gaussian process output reduces the cost of Step I as well as remove the need for high-quality training images that are generated in this paper using the multiphysics model described in the background section to train the Gaussian processes. Figure 8 compares the true images in the first and last columns for a circular inclusion with images from the Gaussian process regression (i.e., Step I) in the second column for two random realizations of the uncertain parameters. In the third column, we show an enlarged image of the Gaussian process output near inclusion to show the reflected wave. These images are of poor resolution due to the use of small H and W . The fourth column shows the output from Step II, i.e., output from the CARN, which improves the image quality from Step I and shows the reflected wave in high resolution with two-fold increase in H and W each. Due to the uncertainty in the properties of the medium at t = 3.0 ms the position of the wave including the reflected wave is different in these images. However, the proposed method shows high accuracy in the predicted images for the position of the wave as measured in Figure 8a. images, as before. Table 3 lists the average SSIM index for the validation dataset, which is around 0.6 for all three inclusions. Note that the SSIM index can vary from -1 for a mismatch to +1 for a perfect match. Hence, on average, the predicted wave patterns show reasonable agreement with their respective true images. To predict the propagation of wave for multiple time steps we repeat the procedure. Figure 11 shows the predicted images for t = 2.7, and 3.0 ms using the proposed method for one realization of the uncertain parameters and compares with the true wave patterns, which again shows reasonable agreement. The predictions for all these cases, however, show some errors near the zero displacement but the magnitude of these errors is small. Hence, this example shows that the proposed method can be used to efficiently predict the wave patterns for different shapes of inclusions and under uncertainty in medium's material properties and for multiple time-steps.

Example II: Uncertainty in material and geometric properties

In the second example, we increase the stochastic dimension of the problem by considering uncertainty in the inclusion geometry as well. In particular, we assume the diameter of the circular inclusion is uniformly distributed between 0.005 m and 0.015 m. The uncertainty in material properties of the solid plate are assumed same as before and listed in Table 4. Two datasets, namely, the training dataset D tr and the validation dataset D val , contain 50 and 30 sets of images for t = 3.0 ms, for realizations of the geometry and material properties according to their probability distributions given in Table 4. We used the training dataset D tr to train the Gaussian processes {GP} Ngp i=1 for N gp = CHW . For these images, we have C = 3, H = W = 270 to keep the computational cost of Step I reasonable. A Matern kernel with γ = 1.0, τ = 1.0, and ν = 1.5 is used for the Gaussian processes. We use these trained Gaussian processes to predict the propagation of waves and the same trained CARN from previous example for generating high-quality images. These predicted images are then compared with images in the validation dataset D val .

True

Step I: Output of GP

Step II: Output of CARN (b) EY s = 509.57 kN/m 2 , s = 0.3525, s = 997.8 kg/m 3 Fig. 8: Prediction of the propagated UGW through a solid plate with uncertainties in the material properties at t = 3.0 ms in the presence of a circular inclusion using the proposed approach for two realizations of the uncertainty.

True

Step I: Output of GP

Step II: Output of CARN Fig. 9: Prediction of the propagated UGW through a solid plate with uncertainties in the material properties at t = 3.0 ms in the presence of a square shaped inclusion using the proposed approach for two realizations of the uncertainty.

True

Step I: Output of GP

Step II: Output of CARN True for all these cases, however, show some errors near the zero displacement but the magnitude of these 421 errors is small. Hence, this example shows that the proposed method can be used to e ciently Fig. 9: Prediction of the propagated UGW through a solid plate with uncertainties in the material properties at t = 3.0 ms in the presence of a square shaped inclusion using the proposed approach for two realizations of the uncertainty.

True

Step I: Output of GP

Step II: Output of CARN True for all these cases, however, show some errors near the zero displacement but the magnitude of these 421 errors is small. Hence, this example shows that the proposed method can be used to e ciently Fig. 8: Prediction of the propagated UGW through a solid plate with uncertainties in the material properties at t = 3.0 ms in the presence of a circular inclusion using the proposed approach for two realizations of the uncertainty.

True

Step I: Output of GP

Step II: Output of CARN Fig. 9: Prediction of the propagated UGW through a solid plate with uncertainties in the material properties at t = 3.0 ms in the presence of a square shaped inclusion using the proposed approach for two realizations of the uncertainty.

True

Step I: Output of GP

Step II: Output of CARN True for all these cases, however, show some errors near the zero displacement but the magnitude of these 421 errors is small. Hence, this example shows that the proposed method can be used to e ciently

Results:

Figure 12 compares the true images in the first and last columns with images from the Gaussian process regression (i.e., Step I) in the second column for two random realizations of the uncertain parameters. Since the computational cost of Step I is proportional to CHW the resolution of the images obtained from Step I is low as illustrated in the third column with an enlarged image of the GP output near inclusion to show the reflected wave. The fourth column shows the output from Step II, i.e., output from the CARN, which improves the image resolution from Step I and shows the reflected wave better. In particular, the last two columns of Figure 12b shows that the results from the proposed approach can even produce the wave patterns with more clarity than the solution of the multiphysics problem with a reasonable mesh resolution. Note that due to the uncertainty in the properties of the medium and inclusion at t = 3.0 ms the position of the wave including the reflected wave is different for different realizations of the uncertain parameters in these images. However, the proposed method is able to predict the position of the reflected wave as shown in Figure 12a. Again, there are some errors in the predicted images near zero displacements but the magnitude of this error is small, e.g., in Figure 12b. For the validation dataset D val the predicted images get an average SSIM value of 0.57, which is expected as we increase the stochastic dimension of the problem. Note that the SSIM index Fig. 10: Prediction of the propagated UGW through a solid plate with uncertainties in the material properties at t = 3.0 ms in the presence of a rectangular inclusion using the proposed approach for two realizations of the uncertainty.

Figure 11 shows the predicted images for t = 2.7, 2.8, 2.9, and 3.0 ms using the proposed method for one realization of the uncertain parameters and compares with the true wave patterns, which shows good agreement. The predictions for all these cases, however, show some errors near the zero displacement but the magnitude of these errors are small. Hence, this example shows that the proposed method can be used to e ciently predict the wave patterns for di erent shapes of inclusions and under uncertainty in medium's material properties and for multiple time-steps.

Example II: Uncertainty in material and geometric properties

In the second example, we increase the stochastic dimension of the problem by considering uncertainty in the inclusion geometry as well. In particular, we assume the diameter of the circular inclusion is uniformly distributed between 0.005 m and 0.015 m. The uncertainty in material properties of the solid plate are assumed same as before and listed in Table 4. Two datasets, namely, the training dataset D tr and the validation dataset D val , contain 50 and 30 sets of images for t = 3.0 ms, for realizations of the geometry and material properties according to their probability distributions given in Table 4. We used the training dataset D tr to train the Gaussian processes {GP} CHW i=1 . For these images, we have C = 3, H = W = 270 to keep the computational cost of step I reasonable. A Matern kernel with = 1.0, = 1.0, and = 1.5 is used for the Gaussian processes. We then use these trained Gaussian processes to predict the propagation of waves and the same trained CARN from previous example for generating high-quality images. These predicted images are then compared with images in the validation dataset D val .

True

Step I: Output of GP

Step II: Output of CARN True Figure 11 shows the predicted images for t = 2.7, 2.8, 2.9, and 3.0 ms using the proposed method for one realization of the uncertain parameters and compares with the true wave patterns, which shows good agreement. The predictions for all these cases, however, show some errors near the zero displacement but the magnitude of these errors are small. Hence, this example shows that the proposed method can be used to e ciently predict the wave patterns for di erent shapes of inclusions and under uncertainty in medium's material properties and for multiple time-steps.

Example II: Uncertainty in material and geometric properties

In the second example, we increase the stochastic dimension of the problem by considering uncertainty in the inclusion geometry as well. In particular, we assume the diameter of the circular inclusion is uniformly distributed between 0.005 m and 0.015 m. The uncertainty in material properties of the solid plate are assumed same as before and listed in Table 4. Two datasets, namely, the training dataset D tr and the validation dataset D val , contain 50 and 30 sets of images for t = 3.0 ms, for realizations of the geometry and material properties according to their probability distributions given in Table 4. We used the training dataset D tr to train the Gaussian processes {GP} CHW i=1 . For these images, we have C = 3, H = W = 270 to keep the computational cost of step I reasonable. A Matern kernel with = 1.0, = 1.0, and = 1.5 is used for the Gaussian processes. We then use these trained Gaussian processes to predict the propagation of waves and the same trained CARN from previous example for generating high-quality images. These predicted images are then compared with images in the validation dataset D val .

True

Step I: Output of GP

Step Fig. 10: Prediction of the propagated UGW through a solid plate with uncertainties in the material properties at t = 3.0 ms in the presence of a rectangular inclusion using the proposed approach for two realizations of the uncertainty.

Figure 11 shows the predicted images for t = 2.7, 2.8, 2.9, and 3.0 ms using the proposed method for one realization of the uncertain parameters and compares with the true wave patterns, which shows good agreement. The predictions for all these cases, however, show some errors near the zero displacement but the magnitude of these errors are small. Hence, this example shows that the proposed method can be used to e ciently predict the wave patterns for di erent shapes of inclusions and under uncertainty in medium's material properties and for multiple time-steps.

Example II: Uncertainty in material and geometric properties

In the second example, we increase the stochastic dimension of the problem by considering uncertainty in the inclusion geometry as well. In particular, we assume the diameter of the circular inclusion is uniformly distributed between 0.005 m and 0.015 m. The uncertainty in material properties of the solid plate are assumed same as before and listed in Table 4. Two datasets, namely, the training dataset D tr and the validation dataset D val , contain 50 and 30 sets of images for t = 3.0 ms, for realizations of the geometry and material properties according to their probability distributions given in Table 4. We used the training dataset D tr to train the Gaussian processes {GP} CHW i=1 . For these images, we have C = 3, H = W = 270 to keep the computational cost of step I reasonable. A Matern kernel with = 1.0, = 1.0, and = 1.5 is used for the Gaussian processes. We then use these trained Gaussian processes to predict the propagation of waves and the same trained CARN from previous example for generating high-quality images. These predicted images are then compared with images in the validation dataset D val .

True

Step I: Output of GP

Step II: Output of CARN True Fig. 10: Prediction of the propagated UGW through a solid plate with uncertainties in the material properties at t = 3.0 ms in the presence of a rectangular inclusion using the proposed approach for two realizations of the uncertainty.

Figure 11 shows the predicted images for t = 2.7, 2.8, 2.9, and 3.0 ms using the proposed method for one realization of the uncertain parameters and compares with the true wave patterns, which shows good agreement. The predictions for all these cases, however, show some errors near the zero displacement but the magnitude of these errors are small. Hence, this example shows that the proposed method can be used to e ciently predict the wave patterns for di erent shapes of inclusions and under uncertainty in medium's material properties and for multiple time-steps.

Example II: Uncertainty in material and geometric properties

In the second example, we increase the stochastic dimension of the problem by considering uncertainty in the inclusion geometry as well. In particular, we assume the diameter of the circular inclusion is uniformly distributed between 0.005 m and 0.015 m. The uncertainty in material properties of the solid plate are assumed same as before and listed in Table 4. Two datasets, namely, the training dataset D tr and the validation dataset D val , contain 50 and 30 sets of images for t = 3.0 ms, for realizations of the geometry and material properties according to their probability distributions given in Table 4. We used the training dataset D tr to train the Gaussian processes {GP} CHW i=1 . For these images, we have C = 3, H = W = 270 to keep the computational cost of step I reasonable. A Matern kernel with = 1.0, = 1.0, and = 1.5 is used for the Gaussian processes. We then use these trained Gaussian processes to predict the propagation of waves and the same trained CARN from previous example for generating high-quality images. These predicted images are then compared with images in the validation dataset D val .

True

Step I: Output of GP

Step II: Output of CARN True Fig. 10: Prediction of the propagated UGW through a solid plate with uncertainties in the material properties at t = 3.0 ms in the presence of a rectangular inclusion using the proposed approach for two realizations of the uncertainty.

Figure 11 shows the predicted images for t = 2.7, 2.8, 2.9, and 3.0 ms using the proposed method for one realization of the uncertain parameters and compares with the true wave patterns, which shows good agreement. The predictions for all these cases, however, show some errors near the zero displacement but the magnitude of these errors are small. Hence, this example shows that the proposed method can be used to e ciently predict the wave patterns for di erent shapes of inclusions and under uncertainty in medium's material properties and for multiple time-steps.

Example II: Uncertainty in material and geometric properties

In the second example, we increase the stochastic dimension of the problem by considering uncertainty in the inclusion geometry as well. In particular, we assume the diameter of the circular inclusion is uniformly distributed between 0.005 m and 0.015 m. The uncertainty in material properties of the solid plate are assumed same as before and listed in Table 4. Two datasets, namely, the training dataset Dtr and the validation dataset Dval, contain 50 and 30 sets of images for t = 3.0 ms, for realizations of the geometry and material properties according to their probability distributions given in Table 4. We used the training dataset Dtr to train the Gaussian processes {GP} CHW i=1 . For these images, we have C = 3, H = W = 270 to keep the computational cost of step I reasonable. A Matern kernel with = 1.0, = 1.0, and = 1.5 is used for the Gaussian processes. We then use these trained Gaussian processes to predict the propagation of waves and the same trained CARN from previous example for generating high-quality images. These predicted images are then compared with images in the validation dataset Dval.

True

Step I: Output of GP

Step II: Output of CARN True Fig. 10: Prediction of the propagated UGW through a solid plate with uncertainties in the material properties at t = 3.0 ms in the presence of a rectangular inclusion using the proposed approach for two realizations of the uncertainty.

Figure 11 shows the predicted images for t = 2.7, 2.8, 2.9, and 3.0 ms using the proposed method for one realization of the uncertain parameters and compares with the true wave patterns, which shows good agreement. The predictions for all these cases, however, show some errors near the zero displacement but the magnitude of these errors are small. Hence, this example shows that the proposed method can be used to e ciently predict the wave patterns for di erent shapes of inclusions and under uncertainty in medium's material properties and for multiple time-steps.

Example II: Uncertainty in material and geometric properties

In the second example, we increase the stochastic dimension of the problem by considering uncertainty in the inclusion geometry as well. In particular, we assume the diameter of the circular inclusion is uniformly distributed between 0.005 m and 0.015 m. The uncertainty in material properties of the solid plate are assumed same as before and listed in Table 4. Two datasets, namely, the training dataset Dtr and the validation dataset Dval, contain 50 and 30 sets of images for t = 3.0 ms, for realizations of the geometry and material properties according to their probability distributions given in Table 4. We used the training dataset Dtr to train the Gaussian processes {GP} CHW i=1 . For these images, we have C = 3, H = W = 270 to keep the computational cost of step I reasonable. A Matern kernel with = 1.0, = 1.0, and = 1.5 is used for the Gaussian processes. We then use these trained Gaussian processes to predict the propagation of waves and the same trained CARN from previous example for generating high-quality images. These predicted images are then compared with images in the validation dataset Dval.

True

Step I: Output of GP

Step II: Output of CARN True Fig. 10: Prediction of the propagated UGW through a solid plate with uncertainties in the material properties at t = 3.0 ms in the presence of a rectangular inclusion using the proposed approach for two realizations of the uncertainty.

Figure 11 shows the predicted images for t = 2.7, 2.8, 2.9, and 3.0 ms using the proposed method for one realization of the uncertain parameters and compares with the true wave patterns, which shows good agreement. The predictions for all these cases, however, show some errors near the zero displacement but the magnitude of these errors are small. Hence, this example shows that the proposed method can be used to e ciently predict the wave patterns for di erent shapes of inclusions and under uncertainty in medium's material properties and for multiple time-steps.

Example II: Uncertainty in material and geometric properties

In the second example, we increase the stochastic dimension of the problem by considering uncertainty in the inclusion geometry as well. In particular, we assume the diameter of the circular inclusion is uniformly distributed between 0.005 m and 0.015 m. The uncertainty in material properties of the solid plate are assumed same as before and listed in Table 4. Two datasets, namely, the training dataset Dtr and the validation dataset Dval, contain 50 and 30 sets of images for t = 3.0 ms, for realizations of the geometry and material properties according to their probability distributions given in Table 4. We used the training dataset Dtr to train the Gaussian processes {GP} CHW i=1 . For these images, we have C = 3, H = W = 270 to keep the computational cost of step I reasonable. A Matern kernel with = 1.0, = 1.0, and = 1.5 is used for the Gaussian processes. We then use these trained Gaussian processes to predict the propagation of waves and the same trained CARN from previous example for generating high-quality images. These predicted images are then compared with images in the validation dataset Dval.

True

Step I: Output of GP

Step II: Output of CARN True Figure 11 shows the predicted images for t = 2.7, 2.8, 2.9, and 3.0 ms using the proposed method for one realization of the uncertain parameters and compares with the true wave patterns, which shows good agreement. The predictions for all these cases, however, show some errors near the zero displacement but the magnitude of these errors are small. Hence, this example shows that the proposed method can be used to e ciently predict the wave patterns for di erent shapes of inclusions and under uncertainty in medium's material properties and for multiple time-steps.

Example II: Uncertainty in material and geometric properties

In the second example, we increase the stochastic dimension of the problem by considering uncertainty in the inclusion geometry as well. In particular, we assume the diameter of the circular inclusion is uniformly distributed between 0.005 m and 0.015 m. The uncertainty in material properties of the solid plate are assumed same as before and listed in Table 4. Two datasets, namely, the training dataset Dtr and the validation dataset Dval, contain 50 and 30 sets of images for t = 3.0 ms, for realizations of the geometry and material properties according to their probability distributions given in Table 4. We used the training dataset Dtr to train the Gaussian processes {GP} CHW i=1 . For these images, we have C = 3, H = W = 270 to keep the computational cost of step I reasonable. A Matern kernel with = 1.0, = 1.0, and = 1.5 is used for the Gaussian processes. We then use these trained Gaussian processes to predict the propagation of waves and the same trained CARN from previous example for generating high-quality images. These predicted images are then compared with images in the validation dataset Dval.

True

Step I: Output of GP

Step II: Output of CARN True can vary from -1 for a mismatch to +1 for a perfect match. However, the proposed method shows accuracy in the predicted images for the position of the wave.

Conclusions

In non-destructive tests, as part of structural health monitoring, the ultrasonic guided waves (UGWs) are tracked and contrasted against benchmark wave pattern, which is then used to inform about the presence of structural damages. However, if the properties of the medium are uncertain, the wave pattern can change for different realizations of the uncertainty. A multiphysics approach can be used to solve the associated mathematical model as discussed in the "Background" section but for many realizations of the uncertain parameters this requires significant computational budget. In this paper, we propose a two-step procedure that uses a Gaussian process regression first to predict the propagated wave pattern at a time instance. However, to keep the computational cost of this step reasonable the resolution of the predicted images are kept low. In the next step, we address this shortcoming by using a pre-trained neural Fig. 12: Prediction of the propagated UGW through a solid plate with uncertainty in the material properties of the medium and radius of the circular inclusion at t = 3.0 ms using the proposed approach for two realizations of the uncertainty.

Conclusions

In non-destructive tests as part of structural health monitoring, the ultrasonic guided waves (UGWs) are tracked and contrasted against benchmark wave pattern, which is then used to inform about the presence of the structural damage. However, if the properties of the medium are uncertain, the wave pattern can change for di erent realizations of the uncertainty. A multiphysics approach can be used to solve the associated mathematical model as discussed in Section 2.1 but for many realizations of the uncertain parameters this requires significant computational budget. In this paper, we propose a two-step procedure that uses a Gaussian process regression first to predict the propagated wave pattern at a time instance. However, to keep the computational cost of this step reasonable the resolution of the predicted images are kept low. In the next step, we address this shortcoming by using a pre-trained neural network for increasing the resolution, a procedure known as super-resolution. Once trained, using this procedure the wave patterns can be estimated with a significantly small computational e ort for many realizations of the uncertainty. We tested this procedure with two examples, where we assume uncertainty in the material and geometric properties of the medium and inclusions. Further, we consider three di erent shapes for the inclusions, namely, circle, square, and rectangle. The proposed method demonstrates high accuracy in predicting the wave patterns as well as the position of the reflected wave. In the future, the proposed method will be extended further with the use of a bi-fidelity dataset [82], to reduce the computational cost of training.

True

Step I: Output of GP

Step II: Output of CARN True Fig. 12: Prediction of the propagated UGW through a solid plate with uncertainty in the material properties of the medium and radius of the circular inclusion at t = 3.0 ms using the proposed approach for two realizations of the uncertainty.

Conclusions

In non-destructive tests as part of structural health monitoring, the ultrasonic guided waves (UGWs) are tracked and contrasted against benchmark wave pattern, which is then used to inform about the presence of the structural damage. However, if the properties of the medium are uncertain, the wave pattern can change for di erent realizations of the uncertainty. A multiphysics approach can be used to solve the associated mathematical model as discussed in Section 2.1 but for many realizations of the uncertain parameters this requires significant computational budget. In this paper, we propose a two-step procedure that uses a Gaussian process regression first to predict the propagated wave pattern at a time instance. However, to keep the computational cost of this step reasonable the resolution of the predicted images are kept low. In the next step, we address this shortcoming by using a pre-trained neural network for increasing the resolution, a procedure known as super-resolution. Once trained, using this procedure the wave patterns can be estimated with a significantly small computational e ort for many realizations of the uncertainty. We tested this procedure with two examples, where we assume uncertainty in the material and geometric properties of the medium and inclusions. Further, we consider three di erent shapes for the inclusions, namely, circle, square, and rectangle. The proposed method demonstrates high accuracy in predicting the wave patterns as well as the position of the reflected wave. In the future, the proposed method will be extended further with the use of a bi-fidelity dataset [82], to reduce the computational cost of training.

True

Step Figure 12a. Again, there are some errors in the predicted images near zero displacements but the 453 magnitude of this error is small. For the validation dataset D val the predicted images get an average 454 SSIM value of 0.57, which is expected as we increase the stochastic dimension of the problem.

455

Note that the SSIM index can vary from 1 for a mismatch to +1 for a perfect match. However, the 456 proposed method shows high accuracy in the predicted images for the position of the wave. Fig. 12: Prediction of the propagated UGW through a solid plate with uncertainty in the material properties of the medium and radius of the circular inclusion at t = 3.0 ms using the proposed approach for two realizations of the uncertainty.

Conclusions

In non-destructive tests as part of structural health monitoring, the ultrasonic guided waves (UGWs) are tracked and contrasted against benchmark wave pattern, which is then used to inform about the presence of the structural damage. However, if the properties of the medium are uncertain, the wave pattern can change for di erent realizations of the uncertainty. A multiphysics approach can be used to solve the associated mathematical model as discussed in Section 2.1 but for many realizations of the uncertain parameters this requires significant computational budget. In this paper, we propose a two-step procedure that uses a Gaussian process regression first to predict the propagated wave pattern at a time instance. However, to keep the computational cost of this step reasonable the resolution of the predicted images are kept low. In the next step, we address this shortcoming by using a pre-trained neural network for increasing the resolution, a procedure known as super-resolution. Once trained, using this procedure the wave patterns can be estimated with a significantly small computational e ort for many realizations of the uncertainty. We tested this procedure with two examples, where we assume uncertainty in the material and geometric properties of the medium and inclusions. Further, we consider three di erent shapes for the inclusions, namely, circle, square, and rectangle. The proposed method demonstrates high accuracy in predicting the wave patterns as well as the position of the reflected wave. In the future, the proposed method will be extended further with the use of a bi-fidelity dataset [82], to reduce the computational cost of training.

True

Step I: Output of GP

Step II: Output of CARN True Fig. 12: Prediction of the propagated UGW through a solid plate with uncertainty in the material properties of the medium and radius of the circular inclusion at t = 3.0 ms using the proposed approach for two realizations of the uncertainty.

Conclusions

In non-destructive tests as part of structural health monitoring, the ultrasonic guided waves (UGWs) are tracked and contrasted against benchmark wave pattern, which is then used to inform about the presence of the structural damage. However, if the properties of the medium are uncertain, the wave pattern can change for di erent realizations of the uncertainty. A multiphysics approach can be used to solve the associated mathematical model as discussed in Section 2.1 but for many realizations of the uncertain parameters this requires significant computational budget. In this paper, we propose a two-step procedure that uses a Gaussian process regression first to predict the propagated wave pattern at a time instance. However, to keep the computational cost of this step reasonable the resolution of the predicted images are kept low. In the next step, we address this shortcoming by using a pre-trained neural network for increasing the resolution, a procedure known as super-resolution. Once trained, using this procedure the wave patterns can be estimated with a significantly small computational e ort for many realizations of the uncertainty. We tested this procedure with two examples, where we assume uncertainty in the material and geometric properties of the medium and inclusions. Further, we consider three di erent shapes for the inclusions, namely, circle, square, and rectangle. The proposed method demonstrates high accuracy in predicting the wave patterns as well as the position of the reflected wave. In the future, the proposed method will be extended further with the use of a bi-fidelity dataset [82], to reduce the computational cost of training.

True

Step Figure 12a. Again, there are some errors in the predicted images near zero displacements but the 453 magnitude of this error is small. For the validation dataset D val the predicted images get an average 454 SSIM value of 0.57, which is expected as we increase the stochastic dimension of the problem.

455

Note that the SSIM index can vary from 1 for a mismatch to +1 for a perfect match. However, the 456 proposed method shows high accuracy in the predicted images for the position of the wave. network for increasing the resolution, a procedure known as super-resolution. Once trained, using this procedure the wave patterns can be estimated with a significantly small computational effort for many realizations of the uncertainty. We verified the accuracy of this procedure on two examples, where we assume uncertainty in the material and geometric properties of the medium and inclusions. Further, we considered three different shapes for the inclusions, namely, circle, square, and rectangle. The proposed method demonstrates high accuracy in predicting the wave patterns as well as the position of the reflected wave. In the future, the proposed method will be extended further with the use of a bi-fidelity dataset [START_REF] De | Bi-fidelity stochastic gradient descent for structural optimization under uncertainty[END_REF], to reduce the computational cost of training.

function of F . Moreover, function values in Eulerian and Lagrangian coordinates are given by x = x( x, t) + u(x, t) ⇔ u( x, t) = u(x, t) = xx( x, t), [START_REF] Lee | Fully noncontact wave propagation imaging in an immersed metallic plate with a crack[END_REF] where x = A(x, t). We define the ALE time derivative as follows [START_REF] Richter | Fluid-structure Interactions: Models, Analysis and Finite Elements[END_REF]

∂ ∂t A f ( x, t) = ∂ ∂t f ( x, t) + w • ∇ f ( x, t), (18)
where w(x, t) = ∂ t A is the mesh/domain velocity for all x ∈ Ω with w(•, t) = w(•, t) • A -1 . The mesh displacements are computed with the help of an additional partial differential equations (PDEs), which are referred to as mesh motion PDEs (or MMPDEs). We use the MMPDEs model based on the biharmonic equation. The alternative mesh motion techniques are elucidated in [START_REF] Richter | Fluid-structure Interactions: Models, Analysis and Finite Elements[END_REF][START_REF] Wick | Fluid-structure interactions using different mesh motion techniques[END_REF][START_REF] Ciarlet | The finite element method for elliptic problems[END_REF] and the references cited therein. The mixed formulation of the biharmonic mesh motion model with a control parameter α u [START_REF] Ciarlet | The finite element method for elliptic problems[END_REF] for the WpFSI problem is given by [START_REF] Richter | Fluid-structure Interactions: Models, Analysis and Finite Elements[END_REF] Biharmonic Mesh Motion Models

Find an auxiliary variable ζ f ∈ V Ω f and fluid displacement u f ∈ V 0 Ω f ,u f , such that for almost all times t ∈ I t it holds that (α u ∇ζ f ,

∇φ u f ) = 0 ∀φ u f ∈ V 0 Ω f ,φ u f , (α u ζ f , φ ζ f ) -(α u ∇u f , ∇φ ζ f ) = 0 ∀φ ζ f ∈ V Ω f , (19)
where φ u f and φ ζ f are test-function. This mixed formulation allows avoiding the use of H 2 -conforming finite elements for the spatial discretization (see Ciarlet-Raviart mixed formulation [START_REF] Ciarlet | A mixed finite element method for the biharmonic equation[END_REF]). We emphasize that the biharmonic model does not require a careful choice of a mesh-dependent parameter α u . Using this model, we simply choose a small number α u > 0 [START_REF] Richter | Fluid-structure Interactions: Models, Analysis and Finite Elements[END_REF][START_REF] Wick | Fluid-structure interactions using different mesh motion techniques[END_REF]. Let A be a C 1 -diffeomorphism; f ∈ H 1 ( Ω f (t)) be a differentiable function; and v ∈ H 1 ( Ω f (t)) d a differentiable vector-field. Then the transformation between Eulerian and Lagrangian coordinate systems becomes [START_REF] Richter | Fluid-structure Interactions: Models, Analysis and Finite Elements[END_REF]:

∂ t f = ∂ t f -F -1 ∂ t A • ∇ f, d t f = ∂ t f + F -1 (v -∂ t A) • ∇ f, ∇ f = F -T ∇f, ∇ v = ∇vF -1 , ( v • ∇) f = F -1 v • ∇ f. (20)

Appendix II. Governing Equations

In this Section, we introduce the elastic wave equation for the UGWs propagation in the structure, and the acoustic wave propagation in terms of displacement in the fluid so that we can present a symmetric system of equations in the FSI domain; see [START_REF]Non-Destructive Evaluation (NDE) of Polymer Matrix Composites[END_REF].

Elastic Wave Equations in the Solid

Find the linear elastic wave signal displacement u s ∈ V 0 Ωs,Γi and velocity v s ∈ L Ωs , such that the initial conditions u s (0) = u 0 s and ∂ t u s (0) = v 0 s are satisfied, and for almost all time t ∈ I t it holds that ρ s (∂ t u sv s ) = 0 in Ω s ,

ρ s ∂ t v s -∇ • (Jσ s F -T ) = Jf s in Ω s , u s = u D s = 0 on Γ Ds , (Jσ s F -T )n s = g s on Γ Ns , (21)
where V 0 Ωs,Γi := {u s ∈ H (Ω s ) d : u s = u f on Γ i , u s = 0 on Γ D }, g s is a vector-valued function, ρ s is the density of the solid, and I t is the time interval. Accordingly, the variational (or weak) form of the elastic wave propagation problem in the Lagrangian coordinates is given by

Elastic Wave Propagation in Lagrangian Coordinates

Find the elastic wave signal displacement u s ∈ V 0 Ωs,Γi and velocity v s ∈ L Ωs , such that the initial conditions u s (0) = u 0 s and ∂ t u s (0) = v 0 s are satisfied, and for almost all time t ∈ I t it holds that (ρ s (∂ t u sv s ), φ u s ) Ωs = 0 ∀φ u s ∈ L Ωs , (ρ s ∂ t v s , φ v s ) Ωs + (Jσ s F -T , ∇φ v s ) Ωsg s , φ v s Γi -(Jf s , φ v s ) Ωs = 0 ∀φ v s ∈ V 0 Ωs ,

with the linearized stress tensor given by σ s = 1 J F (2µ s E s + λ s tr(E s )I) F T , E s = 1 2 ∇u s F -1 + F -T ∇u T s , where φ u s and φ v s are test-function; F is the deformation gradient, and J is the determinant of the deformation gradient. Furthermore, µ s and λ s are the Lamé coefficients for the solid. The relationship between the two material parameters µ s , λ s , Poisson ratio ν s and the Young modulus E Ys reads: µ s = E Ys 2(1+νs) , λ s = νsE Ys (1+νs) (1-2νs) .

Acoustic Wave Propagation

Find the acoustic wave signal displacement u f ∈ V 0 Ω f and velocity v f ∈ L Ω f , such that the initial conditions u f (0) = u 0 f and ∂ t u f (0) = v 0 f are satisfied, and for almost all time t ∈ I t it holds that

ρ f (∂ t u f -v f ) = 0 in Ω f , ρ f ∂ t v f -c 2 ρ f ∇ • ( ∇ u f ) = 0 in Ω f , c 2 ρ f ( ∇ u f ) n f = g f on Γ N f , (23)
where g f is a vector-valued function by the normal stress from the elastic wave propagation problem and c is the wave speed. Note, the elastic wave propagation problems [START_REF] Shameem | Finite element approximation of ultrasonic wave propagation under fluid-structure interaction for structural health monitoring systems[END_REF] are formulated in the ALE framework (see Appendix I). Thereby, the acoustic wave equations [START_REF] Shameem | Numerical study and comparison of time discretization schemes for an ultrasonic guided wave propagation problem coupled with fluid-structure interaction[END_REF] have to be transferred to an arbitrary reference domain Ω f . Accordingly, the variational formulation reads

Acoustic Wave Propagation in the ALE Coordinates

Find the acoustic wave signal displacement u f ∈ V 0 Ω f and velocity v f ∈ L Ω f , such that the initial conditions u f (0) = u 0 f and ∂ t u f (0) = v 0 f are satisfied, and for almost all time t ∈ I t it holds that

(Jρ f (∂ t u f -(F -1 w • ∇)u f -v f ), φ u f ) Ω f = 0 ∀φ u f ∈ L Ω f , (Jρ f ∂ t v f , φ v f ) Ω f -(Jρ f (F -1 w • ∇)v f , φ v f ) Ω f +(c 2 Jρ f (∇u f F -1 )F -T , ∇φ v f ) Ω f -Jg f F -T , φ v f Γi = 0 ∀φ v f ∈ V 0 Ω f . (24)
Appendix III. The non-vanishing Lamb wave burst signal

The non-vanishing burst signal force is given by f s (x, t) = r(t) cos(ϕ(x)) r(t) sin(ϕ(x)) , x ∈ Ω fs ,

where r(t) is radius of the non-vanishing signal at time t. The time evolution of r(t) can be approximated by

r(t) := 1 4 cos (2πf c t) • sign(t) -sign t - n c f c • 1 -cos 2π f c n c t , (26)
where, carrier frequency f c = 5 kHz and modulation frequency f m = 1 kHz, which corresponds to n c = 5 cycles of the carrier signal in [0, 1] ms (see Figure 13). Futhermore, Ω fs = x 1 , x 2 ∈ Ω s x 2 1 + x 2 2 ≤ r 2 fs , and ϕ(x) is given by

ϕ(x) =                              arctan x 2 x 1 , x 1 > 0 ∧ x 2 ≥ 0, π 2 , x 1 = 0 ∧ x 2 > 0, π + arctan x 2 x 1 , x 1 < 0, 3π 2 , x 1 = 0 ∧ x 2 < 0, 2π + arctan x 2 x 1 , x 1 > 0 ∧ x 2 < 0.

Figure 3 :

 3 Figure 3: An example of cross-correlation as performed in (11).

Figure 5 :

 5 Figure 5: A schematic for the local cascading residual neural network used in Step II of the proposed method. Note that this cascade is used inside a global cascade (see (14)).

Fig. 6 :

 6 Fig. 6: (a) Schematic representation of a monolithic coupling of the WpFSI problem in the ALE framework at time t n . (b) Solution technique for the WpFSI problem with the Newton-like method. Here, in fully discretized form U n,j

Fig. 6 :

 6 Fig. 6: (a) Schematic representation of a monolithic coupling of the WpFSI problem in the ALE framework at time t n . (b) Solution technique for the WpFSI problem with the Newton-like method. Here, in fully discretized form U n,j

Figure 6 :

 6 Figure 6: Schematic representation of a monolithic coupling and solution technique for the WpFSI problem. Here, in fully discretized form U n,j h = {u n,jh , v n,j h , ζ n,j h }, and the nonlinear residual R(•) are defined in[START_REF] Shameem | Finite element approximation of ultrasonic wave propagation under fluid-structure interaction for structural health monitoring systems[END_REF][START_REF] Shameem | Modeling and simulation of ultrasonic guided waves propagation in the fluid-structure domain by a monolithic approach[END_REF][START_REF] Shameem | Modeling concept and numerical simulation of ultrasonic wave propagation in a moving fluid-structure domain based on a monolithic approach[END_REF].

2

 2

  ) at time, t = 3 ms. (i) EY s = 500 kN/m 2 , ⌫s = 0.4, ⇢s = 1000 kg/m 3 (ii) EY s = 508.43 kN/m 2 , ⌫s = 0.3086, ⇢s = 999.78 kg/m 3 (iii) EY s = 489.28 kN/m 2 , ⌫s = 0.3338, ⇢s = 1002.48 kg/m 3 (iv) EY s = 510.33 kN/m 2 , ⌫s = 0.4489, ⇢s = 1008.27 kg/m 3

4

 4

  ) at time, t = 3 ms. (i) rc = 0.0060, EY s = 489.28 kN/m 2 , ⌫s = 0.3338, ⇢s = 1002.48 kg/m 3 (ii) rc = 0.0053, EY s = 506.89 kN/m 2 , ⌫s = 0.4174, ⇢s = 1001.53 kg/m 3 (iii) rc = 0.0146, EY s = 501.19 kN/m 2 , ⌫s = 0.3878, ⇢s = 1006.36 kg/m 3 (iv) rc = 0.0085, EY s = 494.34 kN/m 2 , ⌫s = 0.4461, ⇢s = 997.99 kg/m 3 . Wave propagation in a solid plate with uncertainties in the inclusion shape at time, t = 3 ms. The material properties of the solid plate are EY s = 500 kN/m 2 , ⌫s = 0.4, ⇢s = 1000 kg/m 3 . (i) No inclusion, (ii) circular inclusion, (iii) square shape inclusion, and (iv) rectangular inclusion.

Fig. 7 :

 7 Fig. 7: Uncertainties behaviour of the UGW propagation through a solid plate.

  [START_REF]Non-Destructive Evaluation (NDE) of Polymer Matrix Composites[END_REF]

  ) at time, t = 3 ms. (i) EY s = 500 kN/m 2 , νs = 0.4, ρs = 1000 kg/m 3 (ii) EY s = 508.43 kN/m 2 , νs = 0.3086, ρs = 999.78 kg/m 3 (iii) EY s = 489.28 kN/m 2 , νs = 0.3338, ρs = 1002.48 kg/m 3 (iv) EY s = 510.33 kN/m 2 , νs = 0.4489, ρs = 1008.27 kg/m 3 10 De, Ebna Hai, and Bause (i) (ii) (iii) (iv) (a) Wave propagation in a solid plate with a circular shape inclusion under uncertainties in the material properties (see Table

2

 2

  ) at time, t = 3 ms. (i) EY s = 500 kN/m 2 , ⌫s = 0.4, ⇢s = 1000 kg/m 3 (ii) EY s = 508.43 kN/m 2 , ⌫s = 0.3086, ⇢s = 999.78 kg/m 3 (iii) EY s = 489.28 kN/m 2 , ⌫s = 0.3338, ⇢s = 1002.48 kg/m 3 (iv) EY s = 510.33 kN/m 2 , ⌫s = 0.4489, ⇢s = 1008.27 kg/m 3

  [START_REF] Srinivasan Gopalakrishnan | Computational Techniques for Structural Health Monitoring[END_REF]

  ) at time, t = 3 ms. (i) rc = 0.0060, EY s = 489.28 kN/m 2 , ⌫s = 0.3338, ⇢s = 1002.48 kg/m 3 (ii) rc = 0.0053, EY s = 506.89 kN/m 2 , ⌫s = 0.4174, ⇢s = 1001.53 kg/m 3 (iii) rc = 0.0146, EY s = 501.19 kN/m 2 , ⌫s = 0.3878, ⇢s = 1006.36 kg/m 3 (iv) rc = 0.0085, EY s = 494.34 kN/m 2 , ⌫s = 0.4461, ⇢s = 997.99 kg/m 3 . (i) (ii) (iii) (iv) (c) Wave propagation in a solid plate with uncertainties in the inclusion shape at time, t = 3 ms. The material properties of the solid plate are EY s = 500 kN/m 2 , ⌫s = 0.4, ⇢s = 1000 kg/m 3 . (i) No inclusion, (ii) circular inclusion, (iii) square shape inclusion, and (iv) rectangular inclusion.

Fig. 7 :

 7 Fig. 7: Uncertainties behaviour of the UGW propagation through a solid plate.

  [START_REF] Srinivasan Gopalakrishnan | Computational Techniques for Structural Health Monitoring[END_REF]

  ) at time, t = 3 ms. (i) rc = 0.0060, EY s = 489.28 kN/m 2 , νs = 0.3338, ρs = 1002.48 kg/m 3 (ii) rc = 0.0053, EY s = 506.89 kN/m 2 , νs = 0.4174, ρs = 1001.53 kg/m 3 (iii) rc = 0.0146, EY s = 501.19 kN/m 2 , νs = 0.3878, ρs = 1006.36 kg/m 3 (iv) rc = 0.0085, EY s = 494.34 kN/m 2 , νs = 0.4461, ρs = 997.99 kg/m 3 10 De, Ebna Hai, and Bause(i) (ii) (iii) (iv) (a)Wave propagation in a solid plate with a circular shape inclusion under uncertainties in the material properties (see Table2

  ) at time, t = 3 ms. (i) EY s = 500 kN/m 2 , ⌫s = 0.4, ⇢s = 1000 kg/m 3 (ii) EY s = 508.43 kN/m 2 , ⌫s = 0.3086, ⇢s = 999.78 kg/m 3 (iii) EY s = 489.28 kN/m 2 , ⌫s = 0.3338, ⇢s = 1002.48 kg/m 3 (iv) EY s = 510.33 kN/m 2 , ⌫s = 0.4489, ⇢s = 1008.27 kg/m 3

  [START_REF] Srinivasan Gopalakrishnan | Computational Techniques for Structural Health Monitoring[END_REF]

  ) at time, t = 3 ms. (i) rc = 0.0060, EY s = 489.28 kN/m 2 , ⌫s = 0.3338, ⇢s = 1002.48 kg/m 3 (ii) rc = 0.0053, EY s = 506.89 kN/m 2 , ⌫s = 0.4174, ⇢s = 1001.53 kg/m 3 (iii) rc = 0.0146, EY s = 501.19 kN/m 2 , ⌫s = 0.3878, ⇢s = 1006.36 kg/m 3 (iv) rc = 0.0085, EY s = 494.34 kN/m 2 , ⌫s = 0.4461, ⇢s = 997.99 kg/m 3 . (i) (ii) (iii) (iv) (c) Wave propagation in a solid plate with uncertainties in the inclusion shape at time, t = 3 ms. The material properties of the solid plate are EY s = 500 kN/m 2 , ⌫s = 0.4, ⇢s = 1000 kg/m 3 . (i) No inclusion, (ii) circular inclusion, (iii) square shape inclusion, and (iv) rectangular inclusion.

Fig. 7 :

 7 Fig. 7: Uncertainties behaviour of the UGW propagation through a solid plate.

Figure 7 :

 7 Figure 7: The wave displacement field component u x in a solid plate with a fluid inclusion.

Figures 9 3 Fig. 8 : 3 Fig. 9 : 3 Fig. 8 : 3 Fig. 9 : 3 Fig. 8 . 3 Fig. 8 : 3 Fig. 9 : 3 Fig. 8 : 3 Fig. 9 : 3 Fig. 8 .

 938393839383839383938 Figures 9 and 10 show similar images for a square and a rectangular inclusion, respectively, for two realizations of the uncertainty. The propagated waves as predicted by the proposed method show a high level of similarity with the true

Figure 8 :

 8 Figure8: Prediction of the propagated UGW through a solid plate with uncertainties in the properties at t = 3.0 ms in the presence of circular inclusion using the proposed approach for two realizations of the uncertainty.

( a )

 a EY s = 506.97 kN/m 2 , s = 0.3977, s = 1000.5 kg/m 3 True Step I: Output of GP Step II: Output of CARN (b) EY s = 509.57 kN/m 2 , s = 0.3525, s = 997.8 kg/m 3

( 3 Fig. 8 :

 38 Fig.8: Prediction of the propagated UGW through a solid plate with uncertainties in the material properties at t = 3.0 ms in the presence of a circular inclusion using the proposed approach for two realizations of the uncertainty.

Fig. 9 : 3 Fig. 9 .

 939 Fig.9: Prediction of the propagated UGW through a solid plate with uncertainties in the material properties at t = 3.0 ms in the presence of a square shaped inclusion using the proposed approach for two realizations of the uncertainty.TrueStep I: Output of GPStep II: Output of CARN True (b) ⇢ . B = 509.57 kN/m 2 , a B = 0.3525, d B = 997.8 kg/m 3 Fig. 9. Prediction of the propagated UGW through a solid plate with uncertainties in the material properties at C = 3.0 ms in the presence of a square shaped inclusion using the proposed approach for two realizations of the uncertainty.

422

  predict the wave patterns for di erent shapes of inclusions and under uncertainty in medium's 423 material properties and for multiple time-steps. 424 Example II: Uncertainty in material and geometric properties 425 In the second example, we increase the stochastic dimension of the problem by considering 426 uncertainty in the inclusion geometry as well. In particular, we assume the diameter of the circular 427 inclusion is uniformly distributed between 0.005 m and 0.015 m. The uncertainty in material 428 properties of the solid plate are assumed same as before and listed in Table 4. Two datasets, 429 namely, the training dataset D tr and the validation dataset D val , contain 50 and 30 sets of images for 430 C = 3.0 ms, for realizations of the geometry and material properties according to their probability 431 23 De et al., 2021

( a ) 3 Fig. 8 :

 a38 Fig.8: Prediction of the propagated UGW through a solid plate with uncertainties in the material properties at t = 3.0 ms in the presence of a circular inclusion using the proposed approach for two realizations of the uncertainty.

( 3 Fig. 8 : 3 Fig. 9 : 3 Fig. 9 .

 383939 Fig.8: Prediction of the propagated UGW through a solid plate with uncertainties in the material properties at t = 3.0 ms in the presence of a circular inclusion using the proposed approach for two realizations of the uncertainty.

422

  predict the wave patterns for di erent shapes of inclusions and under uncertainty in medium's 423 material properties and for multiple time-steps. 424 Example II: Uncertainty in material and geometric properties 425 In the second example, we increase the stochastic dimension of the problem by considering 426 uncertainty in the inclusion geometry as well. In particular, we assume the diameter of the circular 427 inclusion is uniformly distributed between 0.005 m and 0.015 m. The uncertainty in material 428 properties of the solid plate are assumed same as before and listed in Table 4. Two datasets, 429 namely, the training dataset D tr and the validation dataset D val , contain 50 and 30 sets of images for 430 C = 3.0 ms, for realizations of the geometry and material properties according to their probability 431 23 De et al., 2021 (a) ⇢ . B = 506.97 kN/m 2 , a B = 0.3977, d B = 1000.5 kg/m 3 True Step I: Output of GP Step II: Output of CARN (b) EY s = 509.57 kN/m 2 , s = 0.3525, s = 997.8 kg/m 3

( a )

 a EY s = 506.97 kN/m 2 , s = 0.3977, s = 1000.5 kg/m 3 True Step I: Output of GP Step II: Output of CARN (b) EY s = 509.57 kN/m 2 , s = 0.3525, s = 997.8 kg/m 3

( 3 Fig. 8 : 3 Fig. 9 : 3 Fig. 9 .

 383939 Fig.8: Prediction of the propagated UGW through a solid plate with uncertainties in the material properties at t = 3.0 ms in the presence of a circular inclusion using the proposed approach for two realizations of the uncertainty.

422 3 Fig. 9 .TABLE 3 .Figure 8 3 Fig. 9 .

 393839 Fig.9. Prediction of the propagated UGW through a solid plate with uncertainties in the material properties at C = 3.0 ms in the presence of a square shaped inclusion using the proposed approach for two realizations of the uncertainty.

Figure 8

 8 Figure 8 compares the true images in the first and last columns for a circular inclusion with

Figure 9 :

 9 Figure9: Prediction of the propagated UGW through a solid plate with uncertainties in the material properties at t = ms in the presence of a square shaped inclusion using the proposed approach for two realizations of the uncertainty.

A

  PREPRINT -APRIL 24, 2021 Ultrasonic guided wave propagation under uncertainty using machine learning 13 True Step I: Output of GP Step II: Output of CARN (a) EY s = 506.97 kN/m 2 , s = 0.3977, s = 1000.5 kg/m 3 True Step I: Output of GP Step II: Output of CARN (b) EY s = 509.57 kN/m 2 , s = 0.3525, s = 997.8 kg/m 3

( 3 Fig. 10 :

 310 Fig.10: Prediction of the propagated UGW through a solid plate with uncertainties in the material properties at t = 3.0 ms in the presence of a rectangular inclusion using the proposed approach for two realizations of the uncertainty.

3 Fig. 10 .Fig. 11 .

 31011 Fig. 10. Prediction of the propagated UGW through a solid plate with uncertainties in the material properties at C = 3.0 ms in the presence of a rectangular inclusion using the proposed approach for two realizations of the uncertainty.

( a )

 a EY s = 506.97 kN/m 2 , νs = 0.3977, ρs = 1000.5 kg/m 3 Ultrasonic guided wave propagation under uncertainty using machine learning 13 True Step I: Output of GP Step II: Output of CARN (a) EY s = 506.97 kN/m 2 , s = 0.3977, s = 1000.5 kg/m 3 True Step I: Output of GP Step II: Output of CARN (b) EY s = 509.57 kN/m 2 , s = 0.3525, s = 997.8 kg/m 3

(

  a) ⇢ . B = 506.97 kN/m 2 , a B = 0.3977, d B = 1000.5 kg/m 3 Ultrasonic guided wave propagation under uncertainty using machine learning 13 True Step I: Output of GP Step II: Output of CARN (a) EY s = 506.97 kN/m 2 , s = 0.3977, s = 1000.5 kg/m 3 True Step I: Output of GP Step II: Output of CARN (b) EY s = 509.57 kN/m 2 , s = 0.3525, s = 997.8 kg/m 3

( 3 Fig. 10 .Fig. 11 .Figure 10 :

 3101110 Fig.10. Prediction of the propagated UGW through a solid plate with uncertainties in the material properties at C = 3.0 ms in the presence of a rectangular inclusion using the proposed approach for two realizations of the uncertainty.

(

  a) ⇢ .B = 506.97 kN/m 2 , a B = 0.3977, d B = 1000.5 kg/m 3 Ultrasonic guided wave propagation under uncertainty using machine learning 13 True Step I: Output of GP Step II: Output of CARN (a) EY s = 506.97 kN/m 2 , s = 0.3977, s = 1000.5 kg/m 3 True Step I: Output of GP Step II: Output of CARN (b) EY s = 509.57 kN/m 2 , s = 0.3525, s = 997.8 kg/m 3

3 Fig. 10 .Fig. 11 .

 31011 Fig.10. Prediction of the propagated UGW through a solid plate with uncertainties in the material properties at C = 3.0 ms in the presence of a rectangular inclusion using the proposed approach for two realizations of the uncertainty.

( 3 Fig. 10 :

 310 Fig.10: Prediction of the propagated UGW through a solid plate with uncertainties in the material properties at t = 3.0 ms in the presence of a rectangular inclusion using the proposed approach for two realizations of the uncertainty.

3 Fig. 10 .Fig. 11 .Figure 11 :

 3101111 Fig.10. Prediction of the propagated UGW through a solid plate with uncertainties in the material properties at C = 3.0 ms in the presence of a rectangular inclusion using the proposed approach for two realizations of the uncertainty.

  rc = 0.0084 m, EY s = 489.13 kN/m 2 , s = 0.3222, s = 1006.4 kg/m 3 True Step I: Output of GP Step II: Output of CARN (b) rc = 0.0075 m, EY s = 509.57 kN/m 2 , s = 0.3525, s = 997.8 kg/m 3

(a) A 2

 2 = 0.0084 m, ⇢ . B = 489.13 kN/m 2 , a B = 0.3222, d B = 1006.4 kg/m 3 Ultrasonic guided wave propagation under uncertainty using machine learning 15 True Step I: Output of GP Step II: Output of CARN 0.2167 m 0.2184 m (a) rc = 0.0084 m, EY s = 489.13 kN/m 2 , s = 0.3222, s = 1006.4 kg/m 3 True Step I: Output of GP Step II: Output of CARN (b) rc = 0.0075 m, EY s = 509.57 kN/m 2 , s = 0.3525, s = 997.8 kg/m 3

3 Fig. 12 .

 312 Fig. 12. Prediction of the propagated UGW through a solid plate with uncertainty in the material properties of the medium and radius of the circular inclusion at C = 3.0 ms using the proposed approach for two realizations of the uncertainty.

457 CONCLUSIONS 458

 458 In non-destructive tests, as part of structural health monitoring, the ultrasonic guided waves 459 (UGWs) are tracked and contrasted against benchmark wave pattern, which is then used to inform 460 about the presence of structural damages. However, if the properties of the medium are uncertain, 461 the wave pattern can change for di erent realizations of the uncertainty. A multi-physics approach 462 can be used to solve the associated mathematical model as discussed in the "Background" section 463 26 De et al., 2021

  (a) rc = 0.0084 m, EY s = kN/m 2 , νs = 0.3222, ρs = 1006.4 kg/m 3 Ultrasonic guided wave propagation under uncertainty using machine learning 15 True Step I: Output of GP Step II: Output of CARN 0.2167 m 0.2184 m (a) rc = 0.0084 m, EY s = 489.13 kN/m 2 , s = 0.3222, s = 1006.4 kg/m 3 True Step I: Output of GP Step II: Output of CARN (b) rc = 0.0075 m, EY s = 509.57 kN/m 2 , s = 0.3525, s = 997.8 kg/m 3

(a) A 2

 2 = 0.0084 m, ⇢ . B = 489.13 kN/m 2 , a B = 0.3222, d B = 1006.4 kg/m 3 Ultrasonic guided wave propagation under uncertainty using machine learning 15 True Step I: Output of GP Step II: Output of CARN 0.2167 m 0.2184 m (a) rc = 0.0084 m, EY s = 489.13 kN/m 2 , s = 0.3222, s = 1006.4 kg/m 3 True Step I: Output of GP Step II: Output of CARN (b) rc = 0.0075 m, EY s = 509.57 kN/m 2 , s = 0.3525, s = 997.8 kg/m 3

3 Fig. 12 .

 312 Fig. 12. Prediction of the propagated UGW through a solid plate with uncertainty in the material properties of the medium and radius of the circular inclusion at C = 3.0 ms using the proposed approach for two realizations of the uncertainty.

457 CONCLUSIONS 458

 458 In non-destructive tests, as part of structural health monitoring, the ultrasonic guided waves 459 (UGWs) are tracked and contrasted against benchmark wave pattern, which is then used to inform 460 about the presence of structural damages. However, if the properties of the medium are uncertain, 461 the wave pattern can change for di erent realizations of the uncertainty. A multi-physics approach 462 can be used to solve the associated mathematical model as discussed in the "Background" section 463 26 De et al., 2021 (b) rc = 0.0075 m, EY s = 509.57 kN/m 2 , νs = 0.3525, ρs = 997.8 kg/m 3

Figure 12 :

 12 Figure12: Prediction of the propagated UGW through a solid plate with uncertainty in the material properties of the medium and radius of the circular inclusion at t = 3.0 ms using the proposed approach for two realizations of the uncertainty.

Fig. 13 :

 13 Fig.13: The non-vanishing Lamb wave burst signal propagation in a solid plate. The displacement are normalized between[ 1, +1]

Fig. 13 :Fig. 13 :

 1313 Fig. 13: The non-vanishing Lamb wave burst signal propagation in a solid plate. The displacement are normalized between [ 1, +1] (b) Five-cycle Hanning windowed Lamb wave burst signal with a center frequency of 5 kHz

Figure 13 :

 13 Figure 13: The non-vanishing Lamb wave burst signal propagation in a solid The displacement are normalized between [-1, +1].

Table 2 :

 2 Specification of the uncertain parameters in Example I.

	Parameter	Distribution	Mean	Std. Dev.
	Elastic modulus, E Ys Truncated Gaussian * 500 kN/m 2 10 kN/m 2
	Poisson's ratio, ν s Density, ρ s	Uniform Uniform	0.4 1000 kg/m 3 5.77 kg/m 3 0.0577

*

Truncated below at zero.

Table 3 :

 3 Image quality assessment using average SSIM (see the "Background" section) for the validation dataset D val .

	Inclusion shape	Average SSIM
	Circle	0.61
	Square	0.61
	Rectangle	0.60
	4.1.1 Results:

TABLE 3 .

 3 Image quality assessment using average SSIM (see the "Background" section) for the validation dataset D val .

		Inclusion shape	Average SSIM
		Circle Square Rectangle	0.61 0.61 0.60
	405	Results:

Table 4 :

 4 Specification of the uncertain parameters in Example II.

	Part	Parameter	Distribution	Mean	Std. Dev.
	Inclusion	Radius, r c	Uniform	0.01 m	0.0029 m
		Elastic modulus, E Ys Truncated Gaussian * 500 kN/m 2 10 kN/m 2
	Solid plate	Possion's ratio, ν s Density, ρ s	Uniform Uniform	0.4 1000 kg/m 3 5.77 kg/m 3 0.0577

*

Truncated below at zero.

The standard Lebesgue space L p (X) where 1 p ∞ consists of measurable functions u, which are Lebesgue-integrable to the p-th power. The Sobolev space W m,p (X), m ∈ N , 1 p ∞ is the space of functions in L p (X) with distributional derivatives of an order up to m, and which belongs to L p (X). For p =

2, H m (X) := W m,2 (X) is a Hilbert space equipped with the norm • H m (X) . The Hilbert space with zero trace on ΓD is defined as H 1 0 (X) = {u ∈ H 1 (X) : u|Γ D = 0}.

Data Availability StatementTrained Gaussian processes and neural networks that support the findings of this study are available from the corresponding author upon reasonable request.

Acknowledgments

This paper describes the technical results and analysis of the research conducted during the visit of the corresponding author to Smead Aerospace Engineering Sciences Department, University of Colorado, Boulder, CO, USA. The author would like to acknowledge his host Prof. Kurt Maute and support from the Helmut Schmidt University -University of the Federal Armed Forces Hamburg, Germany. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the Federal Armed Forces Germany.

Acknowledgments

This paper describes the technical results and analysis of the research conducted during the visit of the corresponding author to Smead Aerospace Engineering Sciences Department, University of Colorado, Boulder, CO, USA. The author would like to acknowledge his host Prof. Kurt Maute and support from the Helmut Schmidt University -University of the Federal Armed Forces Hamburg, Germany. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the Federal Armed Forces Germany.

Acknowledgments

This paper describes the technical results and analysis of the research conducted during the visit of the corresponding author to Smead Aerospace Engineering Sciences Department, University of Colorado, Boulder, CO, USA. The author would like to acknowledge his host Prof. Kurt Maute and support from the Helmut Schmidt University -University of the Federal Armed Forces Hamburg, Germany. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the Federal Armed Forces Germany.

Acknowledgments

This paper describes the technical results and analysis of the research conducted during the visit of the corresponding author to Smead Aerospace Engineering Sciences Department, University of Colorado, Boulder, CO, USA. The author would like to acknowledge his host Prof. Kurt Maute and support from the Helmut Schmidt University -University of the Federal Armed Forces Hamburg, Germany. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the Federal Armed Forces Germany.

Acknowledgments

This paper describes the technical results and analysis of the research conducted during the visit of the corresponding author to Smead Aerospace Engineering Sciences Department, University of Colorado, Boulder, CO, USA. The author would like to acknowledge his host Prof. Kurt Maute and support from the Helmut Schmidt University -University of the Federal Armed Forces, Hamburg, Germany. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the Federal Armed Forces of Germany.

Appendix I. The Arbitrary Lagrangian-Eulerian (ALE) transformation

The Arbitrary Lagrangian-Eulerian (ALE) mapping [START_REF] Richter | Fluid-structure Interactions: Models, Analysis and Finite Elements[END_REF] is defined in terms of any mesh displacement u : Ω → R d such that A(x, t) : (Ω×I) → Ω, where A(x, t) = x+u(x, t). The ALE mapping is specified through the deformation gradient as F := ∇A = I + ∇u, J := det(F ), where I is the identity matrix, and deformation determinant J is a