Deep learning for full-field ultrasonic characterization

Consider ultrasonic tests where the specimen $\Uppi\subset\mathbb{R}^{d}$, $d=2,3$, is subject to (boundary or internal) excitation over the incident surface $S^{\text{inc}}\!\subset\overline{\Uppi}$ and the induced (particle displacement or velocity) field $\text{\bf{u}}\colon\overline{\Uppi}\times[0\,\,T]\to\mathbb{R}^{N_{\Lambda}}\!$ ($N_{\Lambda}\leqslant d$) is captured over the observation surface $S^{\text{obs}}\!\subset\Uppi$ in a timeframe of length $T$. Here, $\Uppi$ is an open set whose closure is denoted by $\overline{\Uppi}$, and the sensing configuration is such that $\overline{S^{\text{inc}}}\cap\overline{S^{\text{obs}}\!}=\emptyset$. In this setting, the spectrum of observed waveforms $\hat{\text{\bf{u}}}\colon S^{\text{obs}}\times\Omega\to\mathbb{C}^{N_{\Lambda}}\!$ is governed by

where $\Lambda$ of size $N_{\Lambda}\times 1$ designates a differential operator in frequency-space; $\mathscr{F}$ represents the temporal Fourier transform; ${\boldsymbol{\vartheta}}$ of dimension $N_{\vartheta}\times 1$ is the vector of relevant geometric and elastic parameters e.g., Lamé constants and mass density; $\boldsymbol{\xi}\in\mathbb{R}^{d}$ is the position vector; and $\omega>0$ is the frequency of wave motion within the specified bandwidth $\Omega$.

All quantities in (1) are rendered dimensionless by identifying $\rho_{\circ}$, $\sigma_{\circ}$, and $\ell_{\circ}$ as the respective reference scales [33] for mass density, elastic modulus, and length whose explicit values will be later specified.

Given the full waveform data $\hat{\text{\bf{u}}}$ on $S^{\text{obs}}\times\Omega$, the goal is to identify the distribution of material properties over $S^{\text{obs}}$. For this purpose, two reconstruction paradigms based on neural networks are adopted in this study, namely: (i) direct inversion, and (ii) physics-based neural networks. Inspired by the elastography method [18, 19], quantities of interest in (i) are identified by neural maps over $S^{\text{obs}}\times\Omega$ that minimize a regularized measure of $\Lambda$ in (1). The neural networks in (ii), however, are by design predictive maps of the waveform data (i.e., $\hat{\text{\bf{u}}}$) obtained by minimizing the data mismatch subject to (1) as a soft or hard constraint. In this setting, the unknown properties of $\Lambda$ may be recovered as distributed parameters of the (data) network during training via multitask optimization. In what follows, a detailed description of the deployed cost functions in (i) and (ii) is provided after a brief review of the affiliated networks.

Plane-strain wave motion in two linear, elastic, piecewise homogeneous, and isotropic samples is modeled according to Fig. 3 (a). On denoting the frequency of excitation by $\omega$, let $\ell_{r}=\frac{2\pi}{\omega}\sqrt{\mu_{r}/\rho_{r}}$, $\rho_{r}=1$, and $\mu_{r}=1$ be the reference scales for length, mass density, and stress, respectively. In this framework, both specimens are of size $16$ $\!\times\!$ $16$ and uniform density $\rho=1$. The first sample $\Uppi_{1}\subset\mathbb{R}^{2}$ is characterized by the constant Lamé parameters $\mu_{\circ}=1$ and $\lambda_{\circ}=0.47$, while the second sample $\Uppi_{2}\subset\mathbb{R}^{2}$ is comprised of four perfectly bonded homogenous components $\Uppi_{2_{j}}\!$ of $\mu_{j}=j$ and $\lambda_{j}=2j/3$, $j=\{1,2,3,4\}$ such that $\overline{\Uppi_{2}}=\bigcup_{j=1}^{4}\overline{\Uppi_{2_{j}}}$. Accordingly, the shear and compressional wave speeds read $\textrm{c}_{\textrm{s}}^{\circ}=1$, $\textrm{c}_{\textrm{p}}^{\circ}=1.57$ in $\Uppi_{1}$, and $\textrm{c}_{\textrm{s}}^{j}=\sqrt{j}$, $\textrm{c}_{\textrm{p}}^{j}=1.63\sqrt{j}$ in $\Uppi_{2_{j}}$. Every numerical experiment entails an in-plane harmonic excitation at $\omega=3.91$ via a point source on $S^{\text{inc}}$ (the perimeter of a $14$ $\!\times\!$ $14$ square centered at the origin). The resulting displacement field $\boldsymbol{u}^{\upalpha}=(u^{\upalpha}_{1},u^{\upalpha}_{2})$, $\upalpha=1,2$, is then computed in $\Uppi_{\upalpha}$ over $S^{\text{obs}}$ (a concentric square of dimension $8$ $\!\times\!$ $8$) such that

where $\boldsymbol{x}$ and $\boldsymbol{d}$ respectively indicate the source location and polarization vector; $\boldsymbol{n}$ is the unit outward normal to the specimen’s exterior, and

When $\upalpha=2$, the first of (6) should be understood as a shorthand for the set of four governing equations over $\Uppi_{2_{j}}$, $j=\{1,2,3,4\}$, supplemented by the continuity conditions for displacement and traction across $\partial\Uppi_{2_{j}}\!\!\setminus\!\partial\Uppi_{2}$ as applicable.

In this setting, the generic form (1) may be identified as the following

wherein $\boldsymbol{I}_{2}\!$ is the second-order identity tensor; ${\boldsymbol{\tau}}=(\boldsymbol{x},\boldsymbol{d})\in S^{\text{inc}}\times\mathscr{B}_{1}=\mathscr{T}$ with $\mathscr{B}_{1}$ denoting the unit circle of polarization directions. Note that $\rho$ is treated here as a known parameter.

In the numerical experiments, $S^{\text{inc}}$ (resp. $S^{\text{obs}}$) is discretized by a uniform grid of 32 (resp. 50 $\!\times\!$ 50) points, while $\Omega$ and $\mathscr{B}_{1}$ are respectively sampled at $\omega=3.91$ and $\boldsymbol{d}=(1,0)$.

All inversions in this study are implemented within the PyTorch framework [50].

The three-tier logic of Section 2.3.2 is employed to reconstruct the distribution of $\mu_{\upalpha}$ and $\lambda_{\upalpha}$, $\upalpha=1,2$, over $S^{\text{obs}}$, entailing: (a) spectral differentiation of the displacement field $\boldsymbol{u}^{\upalpha}$ in order to compute $\Delta\boldsymbol{u}^{\upalpha}$ and $\nabla\nabla\cdot\boldsymbol{u}^{\upalpha}$ as per (6), (b) construction of three positive-definite MLP networks $\mu^{\star}$, $\lambda^{\star}$, and $\alpha^{\star}$; each of which is comprised of one hidden layer of 64 neurons, and (c) training the MLPs by minimizing $\mathscr{L}_{\varepsilon}$ as in (3) and (7) by way of the ADAM algorithm [51]. To avoid near-boundary errors affiliated with the one-sided FFT differentiation in $\Delta\boldsymbol{u}^{\upalpha}$ and $\nabla\nabla\cdot\boldsymbol{u}^{\upalpha}$, a concentric $40\times 40$ subset of collocation points sampling $S^{\text{obs}}$ is deployed for training purposes. It should also be mentioned that in the heterogeneous case, i.e., $\upalpha=2$, the discontinuity of derivatives across $\partial\Uppi_{2_{j\in\{1,2,3,4\}}}\!$ calls for piecewise spectral differentiation. According to Section 2.3.1, the input to $\mathscr{P}^{\star}=\mathscr{N}_{\mathscr{P}}(\boldsymbol{\xi},\omega)$, $\mathscr{P}=\mu,\lambda$, and $\alpha^{\star}=\mathscr{N}_{\alpha}(\boldsymbol{\xi},\omega)$ is of size $N_{\xi}N_{\tau}\times N_{\omega}=1600N_{s}\times 1$ where $N_{s}\leqslant 32$ is the number of simulations i.e., source locations used to generate distinct waveforms for training. In this setting, since the physical quantities of interest are independent of ${\boldsymbol{\tau}}$, the real-valued output of MLPs is of dimension $1600\times 1$ furnishing a local estimate of the Láme and regularization parameters at the specified sampling points on $S^{\text{obs}}$. Each epoch makes use of the full dataset and the learning rate is $0.005$.

In this work, the reconstruction error is measured in terms of the normal misfit

where $\text{q}^{\star}$ is an MLP estimate for a quantity with the “true” value q.

Let $S^{\text{inc}}$ be sampled at one point i.e., $N_{s}=1$ so that a single forward simulation in $\Uppi_{\upalpha}$, $\upalpha=1,2$, generates the training dataset. The resulting reconstructions are shown in Figs. 4 and 5. It is evident from both figures that the single-source reconstruction fails at the loci of near-zero displacement which may explain the relatively high values of the recovered regularization parameter $\alpha^{\star}$. Table 1 details the true values as well as mean and standard deviation of the reconstructed Láme distributions $\boldsymbol{\vartheta}^{\star}=(\mu^{\star},\lambda^{\star})$ in $\Uppi_{1}$ (resp. $\Uppi_{2_{j}}$ for $j=1,2,3,4$) according to Fig. 4 (resp. Fig. 5).

This problem may be addressed by enriching the training dataset e.g., via increasing $N_{s}$. Figs. 6 and 7 illustrate the reconstruction results when $S^{\text{inc}}$ is sampled at $N_{s}=5$ source points. The mean and standard deviation of the reconstructed distributions are provided in Table 2. It is worth noting that in this case the identified regularization parameter $\alpha^{\star}$ assumes much smaller values – compared to that of Figs. 4 and 5. This is closer to the scale of computational errors in the forward simulations.

To examine the impact of noise on the reconstruction, the multisource dataset used to generate Figs. 6 and 7 are perturbed with $5\%$ white noise. The subsequent direct inversions from noisy data are displayed in Figs. 8 and 9, and the associated statistics are presented in Table 3. Note that spectral differentiation as the first step in direct inversion plays a critical role in denoising the waveforms, and subsequently regularizing the reconstruction process. This may substantiate the low magnitude of MLP-recovered $\alpha^{\star}$ in the case of noisy data in Figs. 8 and 9. The presence of noise, nonetheless, affects the magnitude and thus composition of terms in the Fourier representation of the processed displacement fields in space which is used for differentiation. This may in turn lead to the emergence of fluctuations in the reconstructed fields.

The learning process of Section 2.3.3 is performed as follows: (a) the MLP network ${\boldsymbol{u}^{\upalpha}}^{\star}=\mathscr{N}_{\boldsymbol{u}^{\upalpha}}(\boldsymbol{\xi},\omega,\boldsymbol{x}\hskip 1.13809pt|\hskip 1.13809pt\gamma,\boldsymbol{\vartheta}^{\star})$ endowed with the positive-definite parameters $\gamma$ and $\boldsymbol{\vartheta}^{\star}=(\mu^{\star},\lambda^{\star})$ is constructed such that the input $\boldsymbol{x}$ labels the source location and the auxiliary weight ${\gamma}$ is a nonadaptive scaler, (b) $\mu^{\star}$ and $\lambda^{\star}$ may be specified as scaler or distributed parameters of the network according to Fig. 2 (i), and (c) ${\boldsymbol{u}^{\upalpha}}^{\star}$ is trained by minimizing $\mathscr{L}_{\varpi}$ in (4) via the ADAM optimizer using the synthetic waveforms of Section 3.1. Reconstructions are performed on the same set of collocation points sampling $S^{\text{obs}}\!\times\Omega\times\!\mathscr{T}$ as in Section 3.2. Accordingly, the input to ${\boldsymbol{u}^{\upalpha}}^{\star}$ is of size $N_{\xi}\!\times\!N_{\omega}\!\times\!N_{\tau}=1600\!\times\!1\!\times\!N_{s}$, while its output is of dimension $(1600\!\times\!1\!\times\!N_{s})^{2}$ modeling the displacement field along $\xi_{1}$ and $\xi_{2}$ in the sampling region. Similar to Section 3.2, each epoch makes use of the full dataset for training and the learning rate is $0.005$. The PyTorch implementation of PINNs in this section is accomplished by building upon the available codes on the Github repository [52].

The MLP network ${\boldsymbol{u}^{1}}^{\star}={\boldsymbol{u}^{1}}^{\star}(\boldsymbol{\xi},\omega,\boldsymbol{x}\hskip 1.13809pt|\hskip 1.13809pt\gamma,\boldsymbol{\vartheta}^{\star})$ with three hidden layers of respectively $20$, $40$, and $20$ neurons is employed to map the displacement field ${\boldsymbol{u}^{1}}$ (in $\Uppi_{1}$) associated with a single point source of frequency $\omega=3.91$ at $\boldsymbol{x}=\boldsymbol{x}_{1}\in S^{\text{inc}}$. The Láme constants are defined as the unknown scaler parameters of the network i.e., $\boldsymbol{\vartheta}^{\star}=\{\mu^{\star},\lambda^{\star}\}$, and the Lagrange multiplier $\gamma$ is specified per the following argument. Within the dimensional framework of this section and with reference to (7), observe that on setting $\gamma=\frac{1}{\rho\omega^{2}}$ (i.e., $\gamma=0.065$), both (the PDE residue and data misfit) components of the loss function $\mathscr{L}_{\varpi}$ in 4 emerge as some form of balance in terms of the displacement field. This may naturally facilitate maintaining of the same scale for the loss terms during training, and thus, simplifying the learning process by dispensing with the need to tune an additional parameter $\gamma$. Keep in mind that the input to ${\boldsymbol{u}^{1}}^{\star}$ is of size $1600\!\times\!1\!\times\!1$, while its output is of dimension $(1600\!\times\!1\!\times\!1)^{2}$. In this setting, the training objective is two-fold: (a) construction of a surrogate map for ${\boldsymbol{u}^{1}}$, and (b) identification of $\mu^{\star}$ and $\lambda^{\star}$.

Fig. 10 showcases (i) the accuracy of PINN estimates based on noiseless data in terms of the vertical component of displacement field ${u}^{1}_{2}$ in $\Uppi_{1}$, and (ii) the performance of automatic differentiation [42] in capturing the field derivatives in terms of components that appear in the governing PDE 7 i.e., ${u}^{1}_{2,ij}={\partial^{2}{u}^{1}_{2}}/(\partial\xi_{i}\partial\xi_{j})$, $i,j=1,2$. The comparative analysis in (ii) is against the spectral derivates of FEM fields according to Section 2.3.2. It is worth noting that similar to Fourier-based differentiation, the most pronounced errors in automatic differentiation occur in the near-boundary region i.e., the support of one-sided derivatives. It is observed that the magnitude of such discrepancies may be reduced remarkably by increasing the number of epochs. Nonetheless, the loci of notable errors remain at the vicinity of specimen’s external boundary or internal discontinuities such as cracks or material interfaces. Fig. 10 is complemented with the reconstruction

Next, the PINN ${\boldsymbol{u}^{2}}^{\star}={\boldsymbol{u}^{2}}^{\star}(\boldsymbol{\xi},\omega,\boldsymbol{x}\hskip 1.13809pt|\hskip 1.13809pt\boldsymbol{\vartheta}^{\star})$ with three hidden layers of respectively $120$, $120$, and $80$ neurons is created to reconstruct (i) displacement field ${\boldsymbol{u}^{2}}$ in the heterogeneous specimen $\Uppi_{2}$, and (ii) distribution of the Láme parameters over the observation surface. In this vein, synthetic waveform data associated with five point sources $\{\boldsymbol{x}_{i}\}\in S^{\text{inc}}$, $i=1,2,\ldots,5$ at $\omega=3.91$ is used for training. Here, $\boldsymbol{\vartheta}^{\star}$ is the network’s unknown distributed parameter, of dimension $(40\!\times\!40)^{2}$, and the nonadaptive scaler weight $\gamma=0.065$ in light of the sample’s uniform density $\rho=1$. In this setting, the input to ${\boldsymbol{u}^{2}}^{\star}$ is of size $1600\!\times\!1\!\times\!5$, while its output is of dimension $(1600\!\times\!1\!\times\!5)^{2}$. Fig. 13 provides a comparative analysis between the FEM and PINN maps of horizontal displacement ${u}^{1}_{2}$ in $\Uppi_{2}$ and its spatial derivatives computed by spectral and automatic differentiation respectively.

The PINN-reconstructed distribution of PDE parameters is illustrated in Fig. 14 whose statistics is detailed in Table 4. It is worth mentioning that the learning process is repeated for a suit of weights $\gamma=\{0.01,0.025,0.1,0.25,0.5,1.5,2,5,10,15\}$. In all cases, the results are either quite similar or worse than that of Figs. 13 and 14.

Experiments are performed on two (homogeneous and heterogeneous) specimens: $\Uppi_{1}^{{}^{\text{exp}}}\!$ which is a $27$ cm $\!\times 27$ cm $\!\times 1.5$ mm sheet of T6 6061 aluminum, and $\Uppi_{2}^{{}^{\text{exp}}}\!$ composed of (a) $5$ cm $\!\times\hskip 0.56905pt27$ cm $\!\times 1.5$ mm sheet of Grade 2 titanium, (b) $2.5$ cm $\!\times\hskip 0.56905pt27$ cm $\!\times 1.5$ mm sheet of 4130 steel, and (c) $5$ cm $\!\times\hskip 0.56905pt27$ cm $\!\times 1.5$ mm sheet of 260-H02 brass, connected via metal epoxy. For future reference, the density $\uprho_{\mu}$, Young’s modulus $\textsf{E}_{\mu}$, and Poisson’s ratio $\nu_{\mu}$ for $\mu=\{\text{Al, Ti, St, Br}\}$ are listed in Table 5 as per the manufacturer.

Ultrasonic experiments on both samples are performed in a similar setting in terms of the sensing configuration and illuminating wavelet. In both cases, the specimen is excited by an antiplane shear wave from a designated source location $S^{\text{inc}}$, shown in Fig. 15, by a $0.5$ MHz p-wave piezoceramic transducer (V101RB by Olympus Inc.). The incident signal is a five-cycle burst of the form

where $H$ denotes the Heaviside step function, and the center frequency ${\sf f_{c}}\!$ is set at $165$ kHz (resp. $\{80,300\}$ kHz) in $\Uppi_{1}^{{}^{\text{exp}}}\!$ (resp. $\Uppi_{2}^{{}^{\text{exp}}}$). The induced wave motion is measured in terms of the particle velocity ${\sf{v}^{\upbeta}}$, $\upbeta=1,2$, on the scan grids $\mathscr{G}_{\upbeta}$ sampling $S^{\text{obs}}$ where $S^{\text{obs}}\cap S^{\text{inc}}=S^{\text{obs}}\cap\partial\Uppi_{\upbeta}^{{}^{\text{exp}}}\!=\emptyset$. A laser Doppler vibrometer (LDV) which is mounted on a 2D robotic translation frame (for scanning) is deployed for measurements. The VibroFlex Xtra VFX-I-120 LDV system by Polytec Inc. is capable of capturing particle velocity within the frequency range $\sim\text{DC}-24$ MHz along the laser beam which in this study is normal to the specimen’s surface.

The scanning grid $\mathscr{G}_{1}\subset\Uppi_{1}^{{}^{\text{exp}}}\!$ is identified by a $2$ cm $\!\times\hskip 0.56905pt2$ cm square sampled by $100\!\times\!100$ uniformly spaced measurement points. This amounts to a spatial resolution of $0.2$ mm in both spatial directions. In parallel, $\mathscr{G}_{2}\subset\Uppi_{2}^{{}^{\text{exp}}}\!$ is a $2.5$ cm $\!\times\hskip 0.56905pt7.5$ cm rectangle positioned according to Fig. 15 (b) and sampled by a uniform grid of $180\!\times\!60$ scan points associated with the spatial resolution of $0.42$ mm. At every scan point, the data acquisition is conducted for a time period of $400$ $\mu$s at the sampling rate of $250$ MHz. To minimize the impact of optical and mechanical noise in the system, the measurements are averaged over an ensemble of 80 realizations at each scan point. Bear in mind that both the direct inversion and PINNs deploy the spectra of normalized velocity fields $v^{\text{obs}}$ for data inversion. Such distributions of out-of-plane particle velocity at $165$ kHz (resp. $80$ kHz) in $\Uppi_{1}^{{}^{\text{exp}}}\!$ (resp. $\Uppi_{2}^{{}^{\text{exp}}}$) is displayed in Fig. 15.

It should be mentioned that in the above experiments, the magnitude of measured signals in terms of displacement is of $O(\text{nm})$ so that it may be appropriate to assume a linear regime of propagation. The nature of antiplane wave motion is dispersive nonetheless. Therefore, to determine the relevant length scales in each component, the associated dispersion curves are obtained as in Fig. 19 via a set of complementary experiments described in Section 4.4.1. Accordingly, for excitations of center frequency $\{{\sf f_{c_{1}}},{\sf f_{c_{2}}},{\sf f_{c_{3}}}\}=\{165,80,300\}$ kHz, the affiliated phase velocity $\textsf{c}_{\mu}$ and wavelength $\uplambda_{\mu}$ for $\mu=\{\text{Al, Ti, St, Br}\}$ is identified in Table 6.

On recalling Section 2.2, let $\ell_{r}\colon\!\!\!=\uplambda_{\text{Al}}=0.01$ m, $\mu_{r}\colon\!\!\!=\textsf{E}_{\text{Al}}=68.9$ GPA, and $\rho_{r}\colon\!\!\!=\uprho_{\text{Al}}=2700$ kg/m³ be the reference scales for length, stress, and mass density, respectively. In this setting, the following maps take the physical quantities to their dimensionless values

where ${\sf h}=1.5$ mm and ${\sf f}$ respectively indicate the specimen’s thickness and cyclic frequency of wave motion. Table 5 (resp. Table 6) details the normal values for the first (resp. second) of (10). The normal thickness and center frequencies are as follows,

In light of (11) and Table 6, observe that in all tests the wavelength-to-thickness ratio $\frac{\lambda_{\mu}}{h}\in[3.33\,\,\,9.33]$, $\mu=\{\text{Al, Ti, St, Br}\}$. Therefore, one may invoke the equation governing flexural waves in thin plates [53] to approximate the physics of measured data. In this framework, (1) may be recast as

where $\rho_{\upbeta},E_{\upbeta},\nu_{\upbeta}$ respectively denote the normal density, Young’s modulus, and Poisson’s ratio in $\Uppi_{\upbeta}^{{}^{\text{exp}}}\!$, $\upbeta\,=\,1,2$, and ${\boldsymbol{\tau}}$ indicates the source location. Note that $\nu_{\upbeta}\sim 0.32$ according to Table 5 and $\Lambda$, related to $1-\nu_{\upbeta}^{2}$, shows little sensitivity to small variations in the Poisson’s ratio. Thus, in what follows, $\nu_{\upbeta}$ is treated as a known parameter. Provided $v^{\upbeta}(\boldsymbol{\xi},f;{\boldsymbol{\tau}})$, the objective is to reconstruct $\chi_{\upbeta}(\boldsymbol{\xi},f)$.

Following the reconstruction procedure of Section 3.2, the distribution of $\chi_{\upbeta}$ in $\mathscr{G}_{\upbeta}$, $\upbeta=1,2$, is obtained at specific frequencies. In this vein, the positive-definite MLP networks $\chi^{\star}_{\upbeta}=\mathscr{N}_{\chi_{\upbeta}}(\boldsymbol{\xi},\omega)$ and $\alpha^{\star}=\mathscr{N}_{\alpha}(\boldsymbol{\xi},\omega)$ comprised of three hidden layers of respectively $20$, $40$, and $20$ neurons are constructed according to Fig. 1. In all MLP trainings of this section, each epoch makes use of the full dataset and the learning rate is $0.005$.

When $\upbeta=1$, the inversion is conducted at $f_{1}=0.336$. $S^{\text{inc}}$ is sampled at one point i.e., the piezoelectric transducer remains fixed during the test on Al plate, and thus, $N_{\tau}=1$, while a concentric $60\!\times\!60$ subset of collocation points sampling $S^{\text{obs}}$ is deployed for training. In this setting, the input to $\chi^{\star}_{1}$ and $\alpha^{\star}$ is of size $N_{\xi}N_{\tau}\times N_{\omega}=3600\times 1$, and their real-valued outputs are of the same size. The results are shown in Fig. 16. When $\upbeta=2$, the direct inversion is conducted at $f_{2}=0.17$ and $f_{3}=0.61$. For the low-frequency reconstruction, $S^{\text{inc}}$ is sampled at one point, while a $40\!\times\!120$ subset of scan points in $\mathscr{G}_{2}$ is used for training so that the input/output size for $\chi^{\star}_{2}$ and $\alpha^{\star}$ is $4600\!\times\!1$. The recovered fields and associated normal error are provided in Fig. 17. Table 7 enlists the true values as well as mean and standard deviation of the reconstructed distributions $\chi^{\star}_{\upbeta}$ in $\Uppi_{\upbeta}^{{}^{\text{exp}}}\!$, $\upbeta=1,2$, according to Figs. 16 and 17. For the high-frequency reconstruction, when $\upbeta=2$, $S^{\text{inc}}$ is sampled at three points i.e., experiments are performed for three distinct positions of the piezoelectric transducer, while the same subset of scan points is used for training. In this case, the input to $\chi^{\star}_{2}$ and $\alpha^{\star}$ is $13800\!\times\!1$, while their output is of dimension $4600\!\times\!1$. The high-frequency reconstruction results are illustrated in Fig. 18, and the affiliated means and standard deviations are provided in Table 8. It should be mentioned that the computed normal errors in Figs. 16, 17, and 18 are with respect to the verified values of Section 4.4.1. Note that the recovered $\alpha^{\star}$s from laboratory test data are much smoother than the ones reconstructed from synthetic data in Section 3.2. This could be attributed to the scaler nature of (12) with a single unknown parameter – as opposed to the vector equations governing the in-plane wave motion with two unknown parameters. More specifically, here, $\alpha^{\star}$ controls the weights and biases of a single network $\chi^{\star}_{\upbeta}$, while in Section 3.2, $\alpha^{\star}$ simultaneously controls the parameters of two separate networks $\mu^{\star}$ and $\lambda^{\star}$. A comparative analysis of Figs. 17 and 18 reveals that (a) enriching the waveform data by increasing the number of sources remarkably decrease the reconstruction error, (b) the regularization parameter $\alpha$ in (3) is truly distributed in nature as the magnitude of the recovered $\alpha^{\star}$ in brass is ten times greater than that of titanium and steel which is due to the difference in the level of noise in measurements related to distinct material surfaces, and (c) the recovered field $\chi^{\star}_{2}$ – which according to (12) is a material property ${E_{2}}/{\rho_{2}}$, demonstrates a significant dependence to the reconstruction frequency. The latter calls for proper verification of the results which is the subject of Section 4.4.1.