Seismicity induced by human activities poses a significant threat to public safety, emphasizing the need for accurate and timely earthquake hypocenter localization. In this study, we introduce X-DeepONet, a novel variant of deep operator networks (DeepONets), for learning moving-solution operators of parametric partial differential equations (PDEs), with application to real-time earthquake localization. Leveraging the power of neural operators, X-DeepONet learns to estimate traveltime fields associated with earthquake sources by incorporating information from seismic arrival times and velocity models. Similar to the DeepONet, X-DeepONet includes a trunk net and a branch net. Additionally, we introduce a root network that not only takes the standard DeepONet's multiplication operator as input, it also takes addition and subtraction operators. We show that for problems with moving fields, the standard multiplication operation of DeepONet is insufficient to capture field relocation, while addition and subtraction operators along with the eXtended root significantly improve its accuracy both under data-driven (supervised) and physics-informed (unsupervised) training. We demonstrate the effectiveness of X-DeepONet through various experiments, including scenarios with variable velocity models and arrival times. The results show remarkable accuracy in earthquake localization, even for heterogeneous and complex velocity models. The proposed framework also exhibits excellent generalization capabilities and robustness against noisy arrival times. The method provides a computationally efficient approach for quantifying uncertainty in hypocenter locations resulting from traveltime pick errors and velocity model variations. Our results underscore X-DeepONet's potential to improve seismic monitoring systems, aiding the development of early warning systems for seismic hazard mitigation.

翻译：人类活动引发的地震活动对公共安全构成重大威胁，因此准确、及时地确定地震震源位置至关重要。本研究提出X-DeepONet——深度算子网络(DeepONets)的一种新型变体，用于学习参数化偏微分方程(PDEs)的移动解算子，并应用于实时地震定位。通过利用神经算子的强大能力，X-DeepONet整合地震波到达时间和速度模型信息，学习估算与震源相关的走时场。与DeepONet类似，X-DeepONet包含主干网络和分支网络。此外，我们引入了一个根网络，它不仅接收标准DeepONet的乘法算子作为输入，还接收加法和减法算子。研究表明，对于存在移动场的问题，DeepONet的标准乘法运算不足以捕捉场的位置迁移，而加法和减法算子配合扩展根网络，在有监督(数据驱动)和无监督(物理信息约束)训练下均显著提升了定位精度。我们通过多种实验验证了X-DeepONet的有效性，包括不同速度模型和到达时间场景。结果表明，即使在非均匀复杂速度模型下，该方法仍能实现高精度地震定位。所提出的框架还展现出优异的泛化能力和对噪声到达时间的鲁棒性。该方法为量化走时拾取误差和速度模型变化导致的震源位置不确定性提供了高效计算途径。我们的研究结果凸显了X-DeepONet在改进地震监测系统、助力地震灾害缓解预警系统开发方面的潜力。