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With the recent success of representation learning methods, which includes deep learning as a special case, there has been considerable interest in developing techniques that incorporate known physical constraints into the learned representation. As one example, in many applications that involve a signal propagating through physical media (e.g., optics, acoustics, fluid dynamics, etc.), it is known that the dynamics of the signal must satisfy constraints imposed by the wave equation. Here we propose a matrix factorization technique that decomposes such signals into a sum of components, where each component is regularized to ensure that it nearly satisfies wave equation constraints. Although our proposed formulation is non-convex, we prove that our model can be efficiently solved to global optimality. Through this line of work we establish theoretical connections between wave-informed learning and filtering theory in signal processing. We further demonstrate the application of this work on modal analysis problems commonly arising in structural diagnostics and prognostics. 

Ultrasonic wavefields are widely employed in nondestructive testing and structural health monitoring to detect and evaluate structural damage. However, measuring wavefields continuously throughout space poses challenges and can be costly. To address this, we propose a novel approach that combines the wave equation with computer vision algorithms to visualize wavefields. Our algorithm incorporates the wave equation, which encapsulates our knowledge of wave propagation, to infer the wavefields in regions where direct measurement is not feasible. Specifically, we focus on reconstructing wavefields from partial measurements, where the wavefield data from large continuous regions are missing. The algorithm is tested on experimental data demonstrating its effectiveness in reconstructing the wavefields at unmeasured regions. This also benefits in reducing the need for expensive equipment and enhancing the accuracy of structural health monitoring at a lower cost. The results highlight the potential of our approach to advance ultrasonic wavefield imaging capabilities and open new avenues for Nondestructive testing and structural health monitoring. 

Recent advancements in physics-informed machine learning have contributed to solving partial differential equations through means of a neural network. Following this, several physics-informed neural network works have followed to solve inverse problems arising in structural health monitoring. Other works involving physics-informed neural networks solve the wave equation with partial data and modeling wavefield data generator for efficient sound data generation. While a lot of work has been done to show that partial differential equations can be solved and identified using a neural network, little work has been done the same with more basic machine learning (ML) models. The advantage with basic ML models is that the parameters learned in a simpler model are both more interpretable and extensible. For applications such as ultrasonic nondestructive evaluation, this interpretability is essential for trustworthiness of the methods and characterization of the material system under test. In this work, we show an interpretable, physics-informed representation learning framework that can analyze data across multiple dimensions (e.g., two dimensions of space and one dimension of time). The algorithm comes with convergence guarantees. In addition, our algorithm provides interpretability of the learned model as the parameters correspond to the individual solutions extracted from data. We demonstrate how this algorithm functions with wavefield videos. 

Modern machine learning has been on the rise in many scientific domains, such as acoustics. Many scientific problems face challenges with limited data, which prevent the use of the many powerful machine learning strategies. In response, the physics of wave-propagation can be exploited to reduce the amount of data necessary and improve performance of machine learning techniques. Based on this need, we present a physics-informed machine learning framework, known as wave-informed regression, to extract dispersion curves from a guided wave wavefield data from non-homogeneous media. Wave-informed regression blends matrix factorization with known wave-physics by borrowing results from optimization theory. We briefly derive the algorithm and discuss a signal processing-based interpretability aspect of it, which aids in extracting dispersion curves for non-homogenous media. We show our results on a non-homogeneous media, where the dispersion curves change as a function of space. We demonstrate our ability to use wave-informed regression to extract spatially local dispersion curves. 

Detecting and locating damage information from waves reflected off damage is a common practice in non-destructive structural health monitoring systems. Yet, the transmitted ultrasonic guided waves are affected by the physical and material properties of the structure and are often complicated to model mathematically. This calls for data-driven approaches to model the behaviour of waves, where patterns in wave data due to damage can be learned and distinguished from non-damage data. Recent works have used a popular dictionary learning algorithm, K-SVD, to learn an overcomplete dictionary for waves propagating in a metal plate. However, the domain knowledge is not utilized. This may lead to fruitless results in the case where there are strong patterns in the data that are not of interest to the domain. In this work, instead of treating the K-SVD algorithm as a black box, we create a novel modification by enforcing domain knowledge. In particular, we look at how regularizing the K-SVD algorithm with the one-dimensional wave equation affects the dictionary learned in the simple case of vibrating string. By adding additional non-wave patterns (noise) to the data, we demonstrate that the “wave-informed K-SVD” does not learn patterns which do not obey the wave equation hence learning patterns from data and not the noise. 

This work discusses an optimization framework to embed dictionary learning frameworks with the wave equation as a strategy for incorporating prior scientific knowledge into a machine learning algorithm. We modify dictionary learning to study ultrasonic guided wave-based defect detection for non-destructive structural health monitoring systems. Specifically, this work involves altering the popular-SVD algorithm for dictionary learning by enforcing prior knowledge about the ultrasonic guided wave problem through a physics-based regularization derived from the wave equation. We confer it the name “wave-informed K-SVD.” Training dictionary on data simulated from a fixed string added with noise using both K-SVD and wave-informed K-SVD, we show an improved physical consistency of columns of dictionary matrix with the known modal behavior of different one-dimensional wave simulations is observed.