One-Dimensional Elastic and Viscoelastic Full-Waveform Inversion in Heterogeneous Media Using Physics-Informed Neural Networks

In this study, we discuss a mathematical framework to handle the inverse problem for the applications of partial differential equations (PDEs). In particular, we focus on wave equations and attempt to identify the wave parameters such as wave velocity from scant measurements of the domain’s response to prescribed initial conditions. To this end, we need an algorithm to play the role of inverse PDE solver for full-waveform inversion. Over the past several years, multilayer neural networks have been developed and applied to a broad range of problems in applied mathematics and physics. Specifically, Physics Informed Neural Network (PINN) as a novel technique has been proposed recently for solving partial differential equations. In PINN’s algorithm, the mesh generation effort is not necessary as it is for any other numerical discretization method. The algorithm just needs a batch of points in which to apply the conditions set in the loss function. We employ PINN for solving the wave equations during the inversion process. The first objective of this research is to develop a robust and efficient algorithm based on PINN for the reconstruction of the wave velocity profile in heterogeneous media. Continuous and piecewise continuous functions are considered for the wave velocity target profiles. Next, we are interested in performing the inversion process of the wave equation in semi-infinite heterogeneous media which is one of the major advantages of the PINN in contrast to traditional numerical approaches. The last objective is to simultaneously recover the parameters of the viscously damped wave equation in heterogeneous domains. The effect of noisy measured response on the inversion process is also investigated.