We study the problem of inferring edge flows and nodal injections in infrastructure networks. Leveraging the Thomson’s Principle from the electric circuits literature, we setup a framework to jointly learn network parameters and missing states. Despite being application agnostic, the proposed approach captures the fundamental physics of the infrastructure, and is able to handle partial observation, node and edge features as well as operational constraints. The physics inspired learning framework leads to a bilevel optimization problem, which is NP hard in general. By exploiting convexity properties, we reformulate the problem as a single level optimization, composed of a graph neural network and an additional implicit layer. The resulting architecture can be trained using standard gradient-based methods. We assess the validity of the proposed approach on two different infrastructure networks (power and traffic), and show it outperforms the current state of the art.