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\title{A Stable Neural Network-Based Eikonal Tomography using Hard-Constrained
Measurements}



\author[1]{Mohammad Taufik}%
\author[1]{Tariq Alkhalifah}%
\author[2]{Umair Waheed}%
\affil[1]{King Abdullah University of Science and Technology (KAUST), Physical Sciences and Engineering Division, Thuwal, Saudi Arabia}%
\affil[2]{King Fahd University of Petroleum and Minerals}%


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  \date{May 08, 2026}


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\begin{abstract}
Eikonal tomography, or travel time inversion, has been one of the
primary seismological tools for decades and has been used to understand
Earth's properties and dynamic processes. At the heart of the inversion
process is the need for an accurate, and preferably flexible, eikonal
solver to compute the travel time field. Most of the conventional
eikonal solvers, however, suffer from first-order convergence errors and
difficulties in dealing with irregular computational grids.
Physics-informed neural networks (PINNs) have been introduced to tackle
these problems and have shown great success in addressing those
challenges. Nevertheless, these approaches still suffer from slow
convergence and unstable training dynamics due to the multi-term nature
of the loss function. To improve on this, we propose a new formulation
for the isotropic eikonal equation, which imposes boundary conditions as
hard constraints. We employ the theory of functional connections to the
eikonal tomography problem, which allows for the utilization of a single
loss term for training the PINN model. Through rigorous numerical tests,
its efficiency, stability, and flexibility in tackling a variety of
cases, including topography-dependent and 3D models, are attested, thus
providing an efficient and stable PINN-based eikonal tomography.%
\end{abstract}%



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