  The Eikonal equation plays a central role in seismic wave propagation and
hypocenter localization, a crucial aspect of efficient earthquake early warning
systems. Despite recent progress, real-time earthquake localization remains
challenging due to the need to learn a generalizable Eikonal operator. We
introduce a novel deep learning architecture, Enriched-DeepONet (En-DeepONet),
addressing the limitations of current operator learning models in dealing with
moving-solution operators. Leveraging addition and subtraction operations and a
novel `root' network, En-DeepONet is particularly suitable for learning such
operators and achieves up to four orders of magnitude improved accuracy without
increased training cost. We demonstrate the effectiveness of En-DeepONet in
earthquake localization under variable velocity and arrival time conditions.
Our results indicate that En-DeepONet paves the way for real-time hypocenter
localization for velocity models of practical interest. The proposed method
represents a significant advancement in operator learning that is applicable to
a gamut of scientific problems, including those in seismology, fracture
mechanics, and phase-field problems.
