TY - JOUR

T1 - Neural Eikonal solver: Improving accuracy of physics-informed neural networks for solving eikonal equation in case of caustics

AU - Grubas, Serafim

AU - Duchkov, Anton

AU - Loginov, Georgy

N1 - Funding Information:
NES package with tutorials and examples in 2D and 3D is available at https://github.com/sgrubas/NES . Duchkov A.A. was supported by grant No. FWZZ-2022-0017.
Publisher Copyright:
© 2022 Elsevier Inc.

PY - 2023/2/1

Y1 - 2023/2/1

N2 - The concept of physics-informed neural networks has become a useful tool for solving differential equations due to its flexibility. A few approaches use this concept to solve the eikonal equation that describes the first-arrival traveltimes of waves propagating in smooth heterogeneous velocity models. However, the challenge of the eikonal is exacerbated by the velocity models producing caustics, resulting in instabilities and deterioration of accuracy due to the non-smooth solution behavior. In this paper, we revisit the problem of solving the eikonal equation using neural networks to tackle caustic pathologies. We introduce the novel Neural Eikonal Solver (NES) for solving the isotropic eikonal equation in two formulations: the one-point problem is for a fixed source location; the two-point problem is for an arbitrary source-receiver pair. We present several techniques which provide stability in the case of caustics: improved factorization; non-symmetric loss function based on Hamiltonian; gaussian activation; symmetrization. In our tests, NES showed the relative mean-absolute error of 0.2-0.4% from the second-order factored Fast Marching Method with a similar inference time, and outperformed existing neural-network solvers giving 10-60 times lower errors and 2-30 times faster training. The one-point NES provides the most accurate solution, while the two-point NES gives slightly lower accuracy but an extremely compact representation with all spatial derivatives. It can be useful in many seismic problems: massive computations of traveltimes for millions of source-receiver pairs in Kirchhoff migration; modeling of ray amplitudes using spatial derivatives; traveltime tomography; earthquake localization; ray multipathing analysis.

AB - The concept of physics-informed neural networks has become a useful tool for solving differential equations due to its flexibility. A few approaches use this concept to solve the eikonal equation that describes the first-arrival traveltimes of waves propagating in smooth heterogeneous velocity models. However, the challenge of the eikonal is exacerbated by the velocity models producing caustics, resulting in instabilities and deterioration of accuracy due to the non-smooth solution behavior. In this paper, we revisit the problem of solving the eikonal equation using neural networks to tackle caustic pathologies. We introduce the novel Neural Eikonal Solver (NES) for solving the isotropic eikonal equation in two formulations: the one-point problem is for a fixed source location; the two-point problem is for an arbitrary source-receiver pair. We present several techniques which provide stability in the case of caustics: improved factorization; non-symmetric loss function based on Hamiltonian; gaussian activation; symmetrization. In our tests, NES showed the relative mean-absolute error of 0.2-0.4% from the second-order factored Fast Marching Method with a similar inference time, and outperformed existing neural-network solvers giving 10-60 times lower errors and 2-30 times faster training. The one-point NES provides the most accurate solution, while the two-point NES gives slightly lower accuracy but an extremely compact representation with all spatial derivatives. It can be useful in many seismic problems: massive computations of traveltimes for millions of source-receiver pairs in Kirchhoff migration; modeling of ray amplitudes using spatial derivatives; traveltime tomography; earthquake localization; ray multipathing analysis.

KW - Caustics

KW - Eikonal equation

KW - Physics-informed neural network

KW - Seismic

KW - Traveltimes

UR - http://www.scopus.com/inward/record.url?scp=85142910517&partnerID=8YFLogxK

UR - http://arxiv.org/abs/2205.07989

UR - https://www.mendeley.com/catalogue/ac88e0cc-4c5b-3161-9744-2f597217adb1/

U2 - 10.1016/j.jcp.2022.111789

DO - 10.1016/j.jcp.2022.111789

M3 - Article

AN - SCOPUS:85142910517

VL - 474

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

M1 - 111789

ER -