Reply to: One neuron versus deep learning in aftershock prediction

Forecasting the spatial distribution of aftershocks in the aftermath of large seismic events is of great importance for improving both our understanding of earthquake triggering and post-disaster management. Recently, DeVries et al.1 attempted to solve this scientific problem by deep learning. Using the area under the curve (AUC) of receiver operating characteristic curves, the authors showed that a deep neural network (DNN) considerably outperformed classical Coulomb stress. Here we first clarify that similar performances had already been obtained (by the same authors, in 2017)2 for various scalar stress metrics, suggesting that deep learning does not actually improve prediction. Second, we reformulate the 2017 results2 using two-parameter logistic regression (that is, one neuron) and obtain the same performance as that of the 13,451-parameter DNN. We further show that a logistic regression based on the measured distance and mainshock average slip (instead of derived stresses) performs better than the DNN. This demonstrates that so far the proposed deep learning strategy does not provide any new insight (predictive or inferential) in this domain. Operational aftershock forecasting has been possible for decades thanks to well established empirical laws3–5. Spatial patterns of aftershocks are often described as a power-law decay5–8. Physical models based on the Coulomb stress paradigm only outperform statistical methods when considered in physical/statistical hybrids with high-quality mainshock rupture data9. Coulomb stress models can on their own be as performant as statistical methods when additionally including receiver fault heterogeneities10,11. Meade et al.2 performed a thorough analysis of various scalar stress metrics and showed that several of them outperform classic Coulomb failure stress. We are here concerned with their follow-up article1, which presented similar results, but via deep learning. In this study, we aim to demonstrate that while defining larger and deeper DNNs usually does not hurt model