## Abstract

Since the original algorithm by John Vidale in 1988 to numerically solve the isotropic eikonal equation, widely used in seismic wave propagation studies, there has been tremendous progress on the topic addressing an array of challenges, including improvement of the solution accuracy, incorporation of surface topography, adding more accurate physics by accounting for anisotropy/attenuation in the medium, and speeding up computations using multiple CPUs and GPUs. Despite these advances, there is no mechanism in these algorithms to carry information gained by solving one problem to the next. Moreover, these approaches may breakdown for certain complex forms of the eikonal equation, requiring simplification of the equations to estimate approximate solutions. Therefore, we seek an alternate approach to address the challenge in a holistic manner, i.e., a method that not only makes it simpler to incorporate topography, allows accounting for any level of complexity in physics, benefiting from computational speedup due to the availability of multiple CPUs or GPUs, but also able to transfer knowledge gained from solving one problem to the next. We develop an algorithm based on the emerging paradigm of physics-informed neural network to solve various forms of the eikonal equation. We show how transfer learning and surrogate modeling can be used to speed up computations by utilizing information gained from prior solutions. We also propose a two-stage optimization scheme to expedite the training process in the presence of sharper heterogeneity in the velocity model and recommend using a locally adaptive activation function for faster convergence. Furthermore, we demonstrate how the proposed approach makes it simpler to incorporate additional physics and other features in contrast to conventional methods that took years and often decades to make these advances. Such an approach not only makes the implementation of eikonal solvers much simpler but also puts us on a much faster path to progress. The method paves the pathway to solving complex forms of the eikonal equation that have remained unsolved using conventional algorithms or solved using some approximation techniques at best; thereby, creating new possibilities for advancement in the field of numerical eikonal solvers.

Original language | English (US) |
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Title of host publication | Advances in Subsurface Data Analytics |

Subtitle of host publication | Traditional and Physics-Based Machine Learning |

Publisher | Elsevier |

Pages | 251-278 |

Number of pages | 28 |

ISBN (Electronic) | 9780128223086 |

DOIs | |

State | Published - Jan 1 2022 |

## Keywords

- Anisotropy
- Eikonal equation
- Neural networks
- Scientific machine learning
- Traveltimes

## ASJC Scopus subject areas

- General Computer Science