2020

EikoNet: Solving the Eikonal equation with Deep Neural Networks

Smith, Jonathan D., Azizzadenesheli, Kamyar, Ross, Zachary E.

Understand

The recent deep learning revolution has created an enormous opportunity for accelerating compute capabilities in the context of physics-based simulations.

Here, we propose EikoNet, a deep learning approach to solving the Eikonal equation, which characterizes the first-arrival-time field in heterogeneous 3D velocity structures.
Our grid-free approach allows for rapid determination of the travel time between any two points within a continuous 3D 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.

Built on

L. Bar, and N. Sochen, 2019, Unsupervised deep learning algorithm for PDE-based forward and inverse problems , arXiv preprint arXiv:1904.05417
Original
1904
Earlier work this paper cites.
S. Das, 2012, Three-dimensional spontaneous rupture propagation and implications for the earthquake source mechanism , Geophysical Journal International, Volume 67, Issue 2, November 1981, Pages 375-393
1981
Earlier work this paper cites.
P. Podvin, and I. Lecomte, 1991. Finite difference computation of traveltimes in very contrasted velocity models: a massively parallel approach and its associated tools , Geophys. J. Int., 105, 271-284
1991
Earlier work this paper cites.
R. Versteeg, 1994, The Marmousi experience: Velocity model determination on a synthetic complex data set , The Leading Edge
1994
Earlier work this paper cites.
J.A. Sethian, 1996, A fast marching level set method for monotonically advancing fronts , Proceedings of the National Academy of Sciences, Feb 1996, 93 (4) 1591-1595
1996
Earlier work this paper cites.
M. Droske, B. Meyer, M. Rumpf, and C.Schaller, 2001, An Adaptive Level Set Method for Medical Image Segmentation , Information Processing in Medical Imaging, Springer Berlin Heidelberg, 416–422
2001
Earlier work this paper cites.
Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, and A. Anandkumar, 2020, Neural Operator: Graph Kernel Network for Partial Differential equations , arXiv preprint arXiv:2003.03485
Original
2003
Earlier work this paper cites.
N. Rawlinson, and M. Sambridge, 2004, Wave front evolution in strongly heterogeneous layered media using the fast marching method , Geophysical Journal International, Volume 156, Issue 3, Pages 631-647
2004
Earlier work this paper cites.
H. Zhao, 2004, A Fast Sweeping Method for Eikonal equations, Mathematics and Computation , Volume 74, Number 250, Pages 603-627,
2004
Earlier work this paper cites.
G.S. Martin, R. Wiley, and K.J. Marfurt, 2006, Marmousi2: An elastic upgrade for Marmousi , Leading Edge, Volume 25, pages 156–166, 10.1190/1.2172306
2006
Earlier work this paper cites.
I. Ihrke, G. Ziegler, A. Tevs, C. Theobalt, M. Magnor, and H.P. Seidel, 2007, Eikonal rendering: Efficient light transport in refractive objects , ACM Transactions on Graphics (TOG), 26(3), 59-es
2007
Earlier work this paper cites.

Similar

H. Zhang, C. Thurber, and P.A. Bedrosian, 2009, Joint inversion for Vp, Vs, and Vp/Vs at SAFOD, Parkfield, California. , Geochem. Geophys. Geosyst.,10(11):Q11002, doi:10.1029/2009GC002709
2009
Cited alongside, same era.
V. Nair, and G.E. Hinton, 2010, Rectified linear units improve restricted boltzmann machines , Proceedings of the 27th International Conference on International Conference on Machine Learning (ICML10), 807-814
2010
Cited alongside, same era.
N. Rawlinson, M. Sambridge and J. Hauser, 2010, Multipathing, reciprocal traveltime fields and raylets , Geophysical Journal International, 181, 2, 1077–1092
2010
Cited alongside, same era.
J.A. Hoffnagle, and D.L. Shealy, 2011, Refracting the k-function: Stavroudis solution to the eikonal equation for multielement optical systems , JOSA, 28, 1312-1321
2011
Cited alongside, same era.
E. Hauksson, W. Yang, and P.M. Shearer, 2012, Waveform Relocated Earthquake Catalog for Southern California (1981 to 2011) , Bull. Seismol. Soc. Am., Vol. 2012, doi: 10.1785/0120120010
2012
Cited alongside, same era.
A. Capozzoli, C. Curcio, A. Liseno and S. Savarese, 2013, A comparison of Fast Marching, Fast Sweeping and Fast Iterative Methods for the solution of the eikonal equation , 2013 21st Telecommunications Forum Telfor (TELFOR), 685–688
2013
Cited alongside, same era.
DA. Clevert, T. Unterthiner, and S. Hochreiter, 2015, Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs) ,arXiv,arXiv:1511.07289
Original
2015
Cited alongside, same era.
D. Kingma, and J. Ba, 2015, Adam: A method for stochastic optimization , International Conference on Learning Representations (ICLR)
2015
Cited alongside, same era.
X. Guo, W. Li, and F. Iorio, 2016, Convolutional Neural Networks for Steady Flow Approximation ,P roceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 481-490, 10.1145/2939672.2939738
2016
Cited alongside, same era.
E. Treister and E. Halder, 2016, A fast marching algorithm for the factored eikonal equation , Journal of Computational physics, 324, 210-225
2016
Cited alongside, same era.

Then

P.M.R. DeVries, T. Thompson, T. Ben, and B.J. Meade, 2017, Enabling large-scale viscoelastic calculations via neural network acceleration , Geophysical Research Letters, 44, 2662 – 2669, 10.1002/2017GL072716
2017
Later among the works it cites.
F. Huang, J.T. Ash, J. Langford, and R.E. Schapire, 2017, Learning deep resnet blocks sequentially using boosting theory . CoRR abs/1706.04964
Original
2017
Later among the works it cites.
M.M. Noack and S. Clark, 2017, Acoustic wave and eikonal equations in a transformed metric space for various types of anisotropy Heliyon, 3
2017
Later among the works it cites.
S.H. Rudy, S. L. Burton, J. L. Proctor and J. N. Kutz, 2017, Data-driven discovery of partial differential equations , Science Advances, 3
2017
Later among the works it cites.
B. Moseley, A. Markman,and T. Nissen-Meyer, 2018, Fast approximate simulation of seismic waves with deep learning , ArXiv, 1807.06873
2018
Later among the works it cites.
E. Weinan, and Y. Bing, 2018, The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems , Communications in Mathematics and Statistics, 6, 1–12
2018
Later among the works it cites.
Y. Zhu, and N. Zabaras, 2018, Bayesian deep convolutional encoderdecoder networks for surrogate modeling and uncertainty quantification , Journal of Computational Physics, 366, 415 – 447
2018
Later among the works it cites.
M. Raissi, P. Perdikaris and G. E. Karniadakis, 2019, physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations , Journal of Computational physics, 378, 686-707
2019
Later among the works it cites.
E. F. Williams, M. R. Fernández-Ruiz, R. Magalhaes, R. Vanthillo, Z. Zhan, M. González-Herráez, and H. F. Martins, Nature Communications, 2019, 10 (1) 1-11
2019
Later among the works it cites.
J.E. Vidale, 1988, Finite-difference traveltime calculation , Seis. Sot. Am., 78, 2062-2076
2076
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