Title: Combining Physics and Machine Learning for Network Flow Estimation. Abstract: The flow estimation problem consists of predicting missing edge flows in a network (e.g., traffic, power, and water) based on partial observations. These missing flows depend both on the underlying \textit{physics} (edge features and a flow conservation law) as well as the observed edge flows. This paper introduces an optimization framework for computing missing edge flows and solves the problem using bilevel optimization and deep learning. More specifically, we learn regularizers that depend on edge features (e.g., number of lanes in a road, the resistance of a power line) using neural networks. Empirical results show that our method accurately predicts missing flows, outperforming the best baseline, and is able to capture relevant physical properties in traffic and power networks.