Using physics informed neural networks for computing the traveltime field with the eikonal equation

L'Institut Terre et Environnement de Strasbourg (ITES) (UMR7063) a démarré au 1er janvier 2021 sous les tutelles du CNRS, de l'Université de Strasbourg et de l'ENGEES. Cette nouvelle UMR s'appuie sur les 4 piliers disciplinaires d'étude de la Terre et de son environnement de surface : Hydrologie, Géochimie, Géologie et Géophysique. L'ITES compte 7 équipes résultant de la fusion du LHYGES et de l'IPGS impliquées dans 5 axes thématiques :

Déformations et aléas

Dynamique des surfaces continentales

Ressources en eau et transferts associés

Structure et dynamique de la Terre

Systèmes géologiques et (géo)réservoirs

The eikonal equation is widely applied in various fields. In seismic imaging, under the high-frequency approximation, eikonal solvers are often used as forward modeling engines. Resolving the latter fundamental non-linear partial differential equation has been treated in many ways with the most common strategy employing finite-differences for the local solution followed by Gauss-Siedel iterations for the global solutions. Moreover, a plethora of tweaks are introduced to handle the source singularity and other challenges in specific cases.
Physis informed neural networks (PINNs) have emerged as a new technique for integrating solving partial differential equations with several advantages (Lehmann et al. 2023). PINN can be used to solve forward problems, but it can be also used for meta-modeling and for inverse problems. Evaluating the potential of PINN in dealing with different types of physics is currently an important research topic of machine learning. The main goal of this work is to develop a physics-informed neural networks (PINNs) framework that is capable of approximate any variant of eikonal solvers in 3D anisotropic media. The neural network is trained to predict travel times consistent with the governing partial differential equation. Advanced learning techniques will be implemented to improve the efficiency of PINNs in this specific application, where the mathematical problem is highly non-linear. The fact that the solution could be sharp will also require implementing a specific activation function. The results of PINN will be validated against a finite element solution. The PINN will be evaluated on usual seismic imaging benchmarks and plugged in an existing reflection travel time tomography framework.

Methodology:

1- Learning the physical processes and the governing equations
2- Learning neural networks, Python and Physics informed neural networks
3- Implementing PINN-based eikonal solver
4- Comparing the results with finite differences solutions

Machine learning- Neural network- sesmic waves