EikoNet: Solving the Eikonal Equation With Deep Neural Networks

Abstract

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 wellsuited 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. Index Terms-Geophysics, partial differential equations (PDEs), ray tracing, travel time. I. INTRODUCTIONT HE 3-D ray tracing is a fundamental component of modern seismology, having direct applications to earthquake hypocenter inversions [8], seismic tomography [29], and earthquake source properties [4]. These derived products further form the basis for many downstream seismological applications. The Eikonal equation is a well-known nonlinear partial differential equation (PDE) that characterizes the firstarrival-time field for a given source location in a 3-D medium [17]. The Eikonal formulation can be solved with several finite-difference algorithms [18], [20], [27], with varying computational demands and stabilities to the solutions.

Accelerating Bayesian microseismic event location with deep learning

“…Waheed et al, 2021). This is the route that has been recently followed by Smith et al (2020) in the development of EIKONET, a deep-learning solution of the eikonal equation based on PINNs (see also Song et al, 2021;Waheed et al, 2020). It would be interesting to apply a similar approach to our waveform emulation framework by solving the elastic wave partial differential equation by back projection over the network while simultaneously fitting the simulated waveforms.…”

HypoSVI: Hypocentre inversion with Stein variational inference and physics informed neural networks

“…where • 2 is the Euclidean norm, T s→r is the travel-time through the medium from a source location s to a receiver location r, V r is the velocity of the medium at the receiver location, S r is the slowness of the medium at the receiver location, and ∇r the gradient at the receiver location. Smith et al (2020) developed a PINN approach to solving the factored Eikonal equation (EikoNet), which trains a deep neural network to calculate the travel-time between any two points in a 3D medium for a given velocity model, satisfying the additional boundary condition that the travel-time at the source location equals zero, T s→s = 0. We leverage a factored eikonal formulations to mitigate the strong singularity affects at the source location, representing the travel-time as a deviation from a homogeneous medium with V = 1 (Treister et al, 2016).…”

“…Throughout this study we train EikoNet travel-time models using a set of constant training parameters and network architecture as described in Smith et al (2020) and supplied in Table 1. A model region is defined spanning our Longitude, Latitude, depth regional of interest, with xmin and xmax locations as [117 o 30 W , 32 o 30 N , −2km] and [115 o 30 W , 34 o 30 N , 50km] respectively.…”

HypoSVI: Hypocenter inversion with Stein Variational Inference and Physics Informed Neural Networks

“…Throughout this study we train EikoNet travel-time models using a set of constant training parameters and network architecture as described in Smith et al (2020) respectively. The grid is projected to a UTM coordinate system, with random source-reciever locations selected within the UTM model space.…”