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 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, 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.

1. INTRODUCTION

In many applications, ranging from road traffic to supply chains to power networks, the dynamics of flows on edges of a graph is governed by physical laws/models (Bressan et al., 2014; Garavello & Piccoli, 2006) . For instance, the LWR model describes equilibrium equations for road traffic Lighthill & Whitham (1955); Richards (1956) . However, it is often difficult to fully observe flows in these applications and, as a result, they rely on off-the-shelf machine learning models to make predictions about missing flows (Li et al., 2017; Yu et al., 2018) . A key limitation of these machine learning models is that they disregard the physics governing the flows. So, the question arises: can we combine physics and machine learning to make better flow predictions? This paper investigates the problem of predicting missing edge flows based on partial observations and the underlying domain-specific physics defined by flow conservation and edge features (Jia et al., 2019) . Edge flows depend on the graph topology due to a flow conservation law-i.e. the total inflow at every vertex is approximately its total out-flow. Moreover, the flow at an edge also depends on its features, which might regularize the space of possible flow distributions in the graph. Here, we propose a model that learns how to predict missing flows from data using bilevel optimization (Franceschi et al., 2017) and neural networks. More specifically, features are given as inputs to a neural network that produces edge flow regularizers. Weights of the network are then optimized via reverse-mode differentiation based on a flow estimation loss from multiple train-validation pairs. Our work falls under a broader effort towards incorporating physics knowledge to machine learning, which is relevant for natural sciences and engineering applications where data availability is limited (Rackauckas et al., 2020) . Conservation laws (of energy, mass, momentum, charge, etc.) are

