  Non-homogeneous Poisson processes are used in a wide range of scientific
disciplines, ranging from the environmental sciences to the health sciences.
Often, the central object of interest in a point process is the underlying
intensity function. Here, we present a general model for the intensity function
of a non-homogeneous Poisson process using measure transport. The model is
built from a flexible bijective mapping that maps from the underlying intensity
function of interest to a simpler reference intensity function. We enforce
bijectivity by modeling the map as a composition of multiple simple bijective
maps, and show that the model exhibits an important approximation property.
Estimation of the flexible mapping is accomplished within an optimization
framework, wherein computations are efficiently done using recent technological
advances in deep learning and a graphics processing unit. Although we find that
intensity function estimates obtained with our method are not necessarily
superior to those obtained using conventional methods, the modeling
representation brings with it other advantages such as facilitated point
process simulation and uncertainty quantification. Modeling point processes in
higher dimensions is also facilitated using our approach. We illustrate the use
of our model on both simulated data, and a real data set containing the
locations of seismic events near Fiji since 1964.
