Graph Neural Flows for Unveiling Systemic Interactions Among Irregularly Sampled Time Series

TL;DR

A graph-based model is developed that unveils the systemic interactions of time series observed at irregular time points by using a directed acyclic graph to model the conditional dependencies of the system components and learning this graph in tandem with a continuous-time model that parameterizes the solution curves of ordinary differential equations (ODEs).

Abstract

Interacting systems are prevalent in nature. It is challenging to accurately predict the dynamics of the system if its constituent components are analyzed independently. We develop a graph-based model that unveils the systemic interactions of time series observed at irregular time points, by using a directed acyclic graph to model the conditional dependencies (a form of causal notation) of the system components and learning this graph in tandem with a continuous-time model that parameterizes the solution curves of ordinary differential equations (ODEs). Our technique, a graph neural flow, leads to substantial enhancements over non-graph-based methods, as well as graph-based methods without the modeling of conditional dependencies. We validate our approach on several tasks, including time series classification and forecasting, to demonstrate its efficacy.

Figures (3)

• Figure 1: Left two: Trajectories of a non-interacting system and an interacting system (using interaction matrix $\mathbf{A}$), under the same initial conditions. Right two: Replica of the left two systems but the initial conditions are changed. Trajectories change on the rightmost plot.
• Figure 2: Comparison with neural flow for forecasting on synthetic systems. (ResNet flow).
• Figure 3: Graph learning quality and forecast quality. Top two rows: Sink (20 nodes); bottom row: all four datasets (20 nodes).

Theorems & Definitions (4)

• Theorem 1
• Theorem 2
• proof : Proof of Theorem \ref{['thm:GCN']}
• proof : Proof of Theorem \ref{['thm:gru.flow']}