In this article, we focus on spatio-temporal point patterns observed on linear networks and recorded continuously over time. We present a nonparametric methodology for spatio-temporal intensity estimation in inhomogeneous Poisson point processes. The approach combines maximum likelihood estimation with roughness penalties based on differential operators in time and in space, the latter defined over the linear network domains. This balances data adaptation and smoothness of the estimate. We establish theoretical properties related to the proposed estimator. 

For the implementation, we rely on advanced techniques coming from optimization and numerical analysis. The discretization of the estimation problem combines finite elements in space, designed for linear networks, and B–splines in time, ensuring flexibility at feasible computational costs. We present an application to real data concerning road accidents occurred in Bergamo, Italy, in 2015-2022. This offers the opportunity to validate the proposed method, and to assess its performance in comparison with state-of-the-art techniques.