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EikoNet: Solving the Eikonal Equation With Deep Neural Networks

Smith, Jonathan D. and Azizzadenesheli, Kamyar and Ross, Zachary E. (2021) EikoNet: Solving the Eikonal Equation With Deep Neural Networks. IEEE Transactions on Geoscience and Remote Sensing, 59 (12). pp. 10685-10696. ISSN 0196-2892. doi:10.1109/TGRS.2020.3039165.

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The recent deep learning revolution has created enormous opportunities for accelerating compute capabilities in the context of physics-based simulations. In this article, we propose EikoNet, a deep learning approach to solving the Eikonal equation, which characterizes the first-arrival-time field in heterogeneous 3-D velocity structures. Our grid-free approach allows for rapid determination of the travel time between any two points within a continuous 3-D domain. These travel time solutions are allowed to violate the differential equation—which casts the problem as one of optimization—with the goal of finding network parameters that minimize the degree to which the equation is violated. In doing so, the method exploits the differentiability of neural networks to calculate the spatial gradients analytically, meaning that the network can be trained on its own without ever needing solutions from a finite-difference algorithm. EikoNet is rigorously tested on several velocity models and sampling methods to demonstrate robustness and versatility. Training and inference are highly parallelized, making the approach well-suited for GPUs. EikoNet has low memory overhead and further avoids the need for travel-time lookup tables. The developed approach has important applications to earthquake hypocenter inversion, ray multipathing, and tomographic modeling, as well as to other fields beyond seismology where ray tracing is essential.

Item Type:Article
Related URLs:
URLURL TypeDescription Paper ItemCode
Smith, Jonathan D.0000-0002-1684-1344
Azizzadenesheli, Kamyar0000-0001-8507-1868
Ross, Zachary E.0000-0002-6343-8400
Additional Information:© 2020 IEEE. Manuscript received March 25, 2020; revised August 11, 2020 and October 16, 2020; accepted November 6, 2020. Date of publication December 4, 2020; date of current version November 24, 2021. This work was supported in part by United States Geological Survey (USGS). The work of Kamyar Azizzadenesheli was supported in part by Raytheon and in part by Amazon Web Services. EikoNet is avaliable at github The authors would like to thank Jack Muir for interesting discussions about finite-difference methods and limitations.
Group:Center for Geomechanics and Mitigation of Geohazards (GMG), Division of Geological and Planetary Sciences, Seismological Laboratory
Funding AgencyGrant Number
Raytheon CompanyUNSPECIFIED
Amazon Web ServicesUNSPECIFIED
Subject Keywords:Geophysics, partial differential equations (PDEs), ray tracing, travel time
Issue or Number:12
Record Number:CaltechAUTHORS:20200526-084219717
Persistent URL:
Official Citation:J. D. Smith, K. Azizzadenesheli and Z. E. Ross, "EikoNet: Solving the Eikonal Equation With Deep Neural Networks," in IEEE Transactions on Geoscience and Remote Sensing, vol. 59, no. 12, pp. 10685-10696, Dec. 2021, doi: 10.1109/TGRS.2020.3039165.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:103439
Deposited By: Tony Diaz
Deposited On:26 May 2020 16:01
Last Modified:15 Nov 2022 19:12

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