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 approach to solving Eikonal equation, which characterizes first-arrival-time field heterogeneous 3-D velocity structures. Our grid-free allows rapid determination travel time between any two points within...

Abstract In this paper we prove the equivalence between some known notions of solutions to eikonal equation and more general analogs Hamilton-Jacobi equations in complete rectifiably connected metric spaces. The considered are that curve-based viscosity solutions, slope-based Monge solutions. By using induced intrinsic (path) metric, reduce space a length show these associated Dirichlet boundar...

We discuss in this Chapter a series of theoretical developments which motivate the introduction of a quantum evolution equation for which the eikonal approximation results in the geodesics of a four dimensional manifold. This geodesic motion can be put into correpondence with general relativity. The well-known problem of the self-interaction of a relativistic charged particle is studied from th...

We are concerned with the inverse problem for an eikonal equation of determining the speed function using observations of the arrival time on a fixed surface. This is formulated as an optimisation problem for a quadratic functional with the state equation being the eikonal equation coupled to the so-called Soner boundary condition. The state equation is discretised by a suitable finite differen...