Learning Symbolic Models for Graph-structured Physical Mechanism

Graph-structured physical mechanisms are ubiquitous in real-world scenarios, thus revealing underneath formulas is of great importance for scientific discovery. However, classical symbolic regression methods fail on this task since they can only handle input-output pairs that are not graph-structured. In this paper, we propose a new approach that generalizes symbolic regression to graph-structured physical mechanisms. The essence of our method is to model the formula skeleton with a message-passing flow, which helps transform the discovery of the skeleton into the search for the message-passing flow. Such a transformation guarantees that we are able to search a message-passing flow, which is efficient and Pareto-optimal in terms of both accuracy and simplicity. Subsequently, the underneath formulas can be identified by interpreting component functions of the searched message-passing flow, reusing classical symbolic regression methods. We conduct extensive experiments on datasets from different physical domains, including mechanics, electricity, and thermology, and on real-world datasets of pedestrian dynamics without ground-truth formulas. The experimental results not only verify the rationale of our design but also demonstrate that the proposed method can automatically learn precise and interpretable formulas for graph-structured physical mechanisms.