Fatemeh Pourahmadian

Yang Xu, Fatemeh Pourahmadian, Jian Song et al.

This study takes advantage of recent advances in machine learning to establish a physics-based data analytic platform for distributed reconstruction of mechanical properties in layered components from full waveform data. In this vein, two logics, namely the direct inversion and physics-informed neural networks (PINNs), are explored. The direct inversion entails three steps: (i) spectral denoising and differentiation of the full-field data, (ii) building appropriate neural maps to approximate the profile of unknown physical and regularization parameters on their respective domains, and (iii) simultaneous training of the neural networks by minimizing the Tikhonov-regularized PDE loss using data from (i). PINNs furnish efficient surrogate models of complex systems with predictive capabilities via multitask learning where the field variables are modeled by neural maps endowed with (scaler or distributed) auxiliary parameters such as physical unknowns and loss function weights. PINNs are then trained by minimizing a measure of data misfit subject to the underlying physical laws as constraints. In this study, to facilitate learning from ultrasonic data, the PINNs loss adopts (a) wavenumber-dependent Sobolev norms to compute the data misfit, and (b) non-adaptive weights in a specific scaling framework to naturally balance the loss objectives by leveraging the form of PDEs germane to elastic-wave propagation. Both paradigms are examined via synthetic and laboratory test data. In the latter case, the reconstructions are performed at multiple frequencies and the results are verified by a set of complementary experiments highlighting the importance of verification and validation in data-driven modeling.

Fatemeh Pourahmadian, Bojan B. Guzina, Houssem Haddar

A non-iterative waveform sensing approach is proposed toward (i) geometric reconstruction of penetrable fractures, and (ii) quantitative identification of their heterogeneous contact condition by seismic i.e. elastic waves. To this end, the fracture support $Γ$ (which may be non-planar and unconnected) is first recovered without prior knowledge of the interfacial condition by way of the recently established approaches to non-iterative waveform tomography of heterogeneous fractures, e.g. the methods of generalized linear sampling and topological sensitivity. Given suitable approximation $\breveΓ$ of the fracture geometry, the jump in the displacement field across $\breveΓ$ i.e. the fracture opening displacement (FOD) profile is computed from remote sensory data via a regularized inversion of the boundary integral representation mapping the FOD to remote observations of the scattered field. Thus obtained FOD is then used as input for solving the traction boundary integral equation on $\breveΓ$ for the unknown (linearized) contact parameters. In this study, linear and possibly dissipative interactions between the two faces of a fracture are parameterized in terms of a symmetric, complex-valued matrix $\boldsymbol{K}(\boldsymbolξ)$ collecting the normal, shear, and mixed-mode coefficients of specific stiffness. To facilitate the high-fidelity inversion for $\boldsymbol{K}(\boldsymbolξ)$, a 3-step regularization algorithm is devised to minimize the errors stemming from the inexact geometric reconstruction and FOD recovery. The performance of the inverse solution is illustrated by a set of numerical experiments where a cylindrical fracture, endowed with two example patterns of specific stiffness coefficients, is illuminated by plane waves and reconstructed in terms of its geometry and heterogeneous (dissipative) contact condition.

Abigail C. Schmid, Alireza Doostan, Fatemeh Pourahmadian

This work leverages laser vibrometry and the weak form of the sparse identification of nonlinear dynamics (WSINDy) for partial differential equations to learn macroscale governing equations from full-field experimental data. In the experiments, two beam-like specimens, one aluminum and one IDOX/Estane composite, are subjected to shear wave excitation in the low frequency regime and the response is measured in the form of particle velocity on the specimen surface. The WSINDy for PDEs algorithm is applied to the resulting spatio-temporal data to discover the effective dynamics of the specimens from a family of potential PDEs. The discovered PDE is of the recognizable Euler-Bernoulli beam model form, from which the Young's modulus for the two materials are estimated. An ensemble version of the WSINDy algorithm is also used which results in information about the uncertainty in the PDE coefficients and Young's moduli. The discovered PDEs are also simulated with a finite element code to compare against the experimental data with reasonable accuracy. Using full-field experimental data and WSINDy together is a powerful non-destructive approach for learning unknown governing equations and gaining insights about mechanical systems in the dynamic regime.

Fatemeh Pourahmadian, Irene de Teresa

A theoretical platform is developed for active elastic-wave sensing of (stationary and advancing) fractures along bi-material interfaces in layered composites. Damaged contact surfaces are characterized by a heterogeneous distribution of (elastic) stiffness $\boldsymbol{K}$ which is a-priori unknown. The proposed imaging functional takes advantage of sequential wavefield measurements for non-iterative and concurrent reconstruction of multiple fractures in heterogeneous domains, with an exceptional spatial resolution and with minimal sensitivity to measurement errors and uncertain elasticity of damage zones. This is accomplished through adaptation of the $F_\sharp$-factorization method (FM) applied to elastodynamic wavefields in a sensing sequence i.e. a measurement campaign performed before and after the formation (or evolution) of interfacial fractures. The direct scattering problem is formulated in the frequency domain where the damage support is illuminated by a set of incident P- and S-waves, and thus-induced scattered waves are monitored at the far field. In this setting, the germane mixed reciprocity principle is introduced, and $F_\sharp$-factorization of the differential scattering operator is rigorously established for composite backgrounds. These results lead to the development of an FM-based fracture indicator whose affiliated cost functional is inherently convex, thus, its minimizer can be computed without iterations. This attribute coupled with the reduced sampling space, proposed in this work, remarkably expedite the data inversion process. The performance of proposed imaging functional is demonstrated by a set of numerical experiments.

Yang Xu, Fatemeh Pourahmadian

A deep learning framework is developed for multiscale characterization of poroelastic media from full waveform data which is known as poroelastography. Special attention is paid to heterogeneous environments whose multiphase properties may drastically change across several scales. Described in space-frequency, the data takes the form of focal solid displacement and pore pressure fields in various neighborhoods furnished either by reconstruction from remote data or direct measurements depending on the application. The objective is to simultaneously recover the six hydromechanical properties germane to Biot equations and their spatial distribution in a robust and efficient manner. Two major challenges impede direct application of existing state-of-the-art techniques for this purpose: (i) the sought-for properties belong to vastly different and potentially uncertain scales, and~(ii) the loss function is multi-objective and multi-scale (both in terms of its individual components and the total loss). To help bridge the gap, we propose the idea of \emph{network scaling} where the neural property maps are constructed by unit shape functions composed into a scaling layer. In this model, the unknown network parameters (weights and biases) remain of O(1) during training. This forms the basis for explicit scaling of the loss components and their derivatives with respect to the network parameters. Thereby, we propose the physics-based \emph{dynamic scaling} approach for adaptive loss balancing. The idea is first presented in a generic form for multi-physics and multi-scale PDE systems, and then applied through a set of numerical experiments to poroelastography. The results are presented along with reconstructions by way of gradient normalization (GradNorm) and Softmax adaptive weights (SoftAdapt) for loss balancing. A comparative analysis of the methods and corresponding results is provided.

Fatemeh Pourahmadian, Yang Xu

A supervised learning approach is proposed for regularization of large inverse problems where the main operator is built from noisy data. This is germane to superresolution imaging via the sampling indicators of the inverse scattering theory. We aim to accelerate the spatiotemporal regularization process for this class of inverse problems to enable real-time imaging. In this approach, a neural operator maps each pattern on the right-hand side of the scattering equation to its affiliated regularization parameter. The network is trained in two steps which entails: (1) training on low-resolution regularization maps furnished by the Morozov discrepancy principle with nonoptimal thresholds, and (2) optimizing network predictions through minimization of the Tikhonov loss function regulated by the validation loss. Step 2 allows for tailoring of the approximate maps of Step 1 toward construction of higher quality images. This approach enables direct learning from test data and dispenses with the need for a-priori knowledge of the optimal regularization maps. The network, trained on low-resolution data, quickly generates dense regularization maps for high-resolution imaging. We highlight the importance of the training loss function on the network's generalizability. In particular, we demonstrate that networks informed by the logic of discrepancy principle lead to images of higher contrast. In this case, the training process involves many-objective optimization. We propose a new method to adaptively select the appropriate loss weights during training without requiring an additional optimization process. The proposed approach is synthetically examined for imaging damage evolution in an elastic plate. The results indicate that the discrepancy-informed regularization networks not only accelerate the imaging process, but also remarkably enhance the image quality in complex environments.