An integrable neural network for marked hawkes process with an application in earthquake modelling

Self-exciting point process models are widely used to describe events whose initial occurrence triggers subsequent events, leading to event clustering. These models are characterized by their conditional intensity function, which explicitly incorporates event history through a self-excitation kernel. While pure temporal models can capture basic clustering behaviour, they often fail to represent variations introduced by marks associated with events. However, the law governing the influence of marks on subsequent events may be complex and difficult to model directly through parametric formulations, whereas nonparametric models, although flexible, are computationally expensive for large datasets. To address this challenge, we propose a neural-network–based method combined with stochastic-gradient estimation, enabling fast and efficient estimation of the unknown intensity function through minibatch training. This approach employs a separable kernel for the temporal and mark components. Experiments on both simulated data and an earthquake dataset demonstrate strong performance. The proposed neural network model achieves substantially improved likelihood-based performance compared to the nonparametric kernel density estimation approach, while providing a flexible representation of the triggering structure.