Neural Eikonal Solver

Neural Eikonal Solver (NES) is framework for solving factored eikonal equation using physics-informed neural network, for details see our paper: early arXiv version and published final version. NES can simulate traveltimes of seismic waves in complex inhomogeneous velocity models.

Description

See quick introduction on Google Colab

NES has two solvers:

1. One-Point NES (NES-OP) is to solve conventional one-point eikonal (NES-OP tutorial)

$$\Vert \nabla \tau(\textbf{x}) \Vert = \frac{1}{v(\textbf{x})}$$

3. Two-Point NES (NES-TP) is to solve generalized two-point eikonal (NES-TP tutorial)

$$\Vert \nabla_r T(\textbf{x}_s, \textbf{x}_r) \Vert = \frac{1}{v(\textbf{x}_r)}$$

$$\Vert \nabla_s T(\textbf{x}_s, \textbf{x}_r) \Vert = \frac{1}{v(\textbf{x}_s)}$$

So far, NES outperforms all existing neural-network based solutions. Table shows average performance results on a smoothed part of Marmousi model (NES-OP vs. PINNeik and NES-TP vs. EikoNet). RMAE is relative mean-absolute error with respect to the reference solution (second-order factored Fast Marching Method). The tests were performed on GPU Tesla P100-PCIE.

|Solver |RMAE, % |Training time, sec |Network size | |--- |--- |--- |--- | |NES-OP (ours) |0.2 |240 |7856 | |PINNeik |12.4 |330 |4061 | |NES-TP (ours) |0.4 |300 |51308 | |EikoNet |5.4 |9600 |7913249 |

For detailed comparisons see our colab notebooks EikoNet and PINNeik.

Installation

pip install git+https://github.com/sgrubas/NES.git

Quick example

import NES

Vel = NES.velocity.MarmousiSmoothedPart()
Eik = NES.NES_TP(velocity=Vel)
Eik.build_model()
h = Eik.train(x_train=100000, epochs=1000, batch_size=25000)

grid = NES.utils.RegularGrid(Vel)
Xs = grid((5, 5)); Xr = grid((100, 100))
X = grid.sou_rec_pairs(Xs, Xr)
T = Eik.Traveltime(X)

2D examples of NES-OP

Isochrones of solutions. RMAE is shown above each figure. The NES solutions are white dashed isochrones, the reference solutions are black isochrones.

<img src="https://github.com/sgrubas/NES/blob/main/NES/data/NES_OP_Sinus_0.06.png" alt="0.06%" width="400"/> <img src="https://github.com/sgrubas/NES/blob/main/NES/data/NES_OP_GaussianPlus_0.12.png" alt="0.12%" width="400"/>

<img src="https://github.com/sgrubas/NES/blob/main/NES/data/NES_OP_Flower_0.42.png" alt="0.42%" width="400"/> <img src="https://github.com/sgrubas/NES/blob/main/NES/data/NES_OP_Boxes_0.28.png" alt="0.28%" width="400"/>

<img src="https://github.com/sgrubas/NES/blob/main/NES/data/NES_OP_Layered_0.33.png" alt="0.33%" width="400"/> <img src="https://github.com/sgrubas/NES/blob/main/NES/data/NES_OP_LayeredBoxGauss_0.34.png" alt="0.34%" width="400"/>

Citation

If you find NES useful for your research, please cite our paper and this repo:

@article{grubas2023NES,
title = {Neural Eikonal solver: Improving accuracy of physics-informed neural networks for solving eikonal equation in case of caustics},
journal = {Journal of Computational Physics},
volume = {474},
pages = {111789},
year = {2023},
issn = {0021-9991},
doi = {https://doi.org/10.1016/j.jcp.2022.111789},
url = {https://www.sciencedirect.com/science/article/pii/S002199912200852X},
author = {Serafim Grubas and Anton Duchkov and Georgy Loginov},
keywords = {Physics-informed neural network, Eikonal equation, Seismic, Traveltimes, Caustics}
}

@article{grubas2023NESpython,
title = {Neural Eikonal Solver},
journal = {GitHub},
url = {https://github.com/sgrubas/NES},
doi = {10.5281/zenodo.12588346},
year = {2023},
author = {Serafim Grubas and Anton Duchkov and Georgy Loginov}
}

Future plans

• Anisotropic eikonal
• Ray tracing
• Wave amplitudes
• Earthquake localization
• Traveltime tomography

Developers

Serafim Grubas (serafimgrubas@gmail.com) <br> Nikolay Shilov <br> Anton Duchkov <br> Georgy Loginov