GKNet: Graph Kalman Filtering and Model Inference via Model-Based Deep Learning

Inference tasks involving time series over graphs are of importance in applications such as urban water networks, economics, and networked neuroscience. Addressing these tasks typically relies on identifying a computationally affordable model that jointly captures the graph-temporal patterns of the data. In this work, we propose a graph-aware state space model for graph time series, where both the latent state and the observation equation are parametric graph-induced models. More specifically, we consider the state equation to follow a stochastic partial differential equation driven by noise over the edges accounting not only for potential uncertainties thereof, but also for increasing the degrees of freedom in a tractable manner. The graph structure conditioning of the noise dispersion allows the state variable to deviate from the stochastic process in certain neighborhoods. The observation model is a sampled and graph-filtered version of the state capturing multi-hop neighboring influence. The goal is to learn the parameters in both state and observation models from the partially observed data for downstream tasks such as prediction and imputation. The model is inferred first through a maximum likelihood approach that provides theoretical tractability but is limited in expressivity and scalability. To improve on the latter, we use the state space formulation to build a principled deep learning architecture that jointly learns the parameters and tracks the state in an end-to-end manner in the spirit of Kalman neural networks. The proposed models are evaluated on controlled and four real-world scenarios, showing the effectiveness of the proposed approach when the model lacks training data. Finally, we conduct a case study on water networks, where the proposed model is used to predict the water levels in the network when the data is extremely scarce.