We will start by presenting contemporary inverse problems in hydrology and briefly recall the basic principles of classical data assimilation methods, as well as their links in linear quadratic Gaussian cases. We will then take a look at some recent methods based on so-called "physically-informed" neural networks. Finally, we will show in a specific case how we can derive a (partially) physically consistent metric that improves the quality of optimization results and, consequently, the estimation of the desired parameters. The illustrations will focus on spatial hydrology problems. These studies have been conducted in collaboration, notably with T. Malou, H. Boulenc, P.-A. Garambois, and R. Bouclier.