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.

翻译：程函方程在地震波传播和震源定位（高效地震早期预警系统的关键环节）中起着核心作用。尽管近期取得进展，但由于需要学习可泛化的程函算子，实时地震定位仍然面临挑战。我们提出了一种新型深度学习架构——富集深度算子网络（En-DeepONet），以解决当前算子学习模型在处理移动解算子时的局限性。通过利用加法和减法运算以及一种新颖的"根"网络，En-DeepONet特别适用于学习此类算子，并在不增加训练成本的情况下实现了高达四个数量级的精度提升。我们展示了En-DeepONet在变波速和变到时条件下的地震定位有效性。结果表明，En-DeepONet为实际感兴趣速度模型的实时震源定位铺平了道路。所提出的方法代表了算子学习的重大进步，可应用于一系列科学问题，包括地震学、断裂力学和相场问题中的问题。