Multipole Graph Neural Operator for Parametric Partial Differential Equations

One of the main challenges in using deep learning-based methods for simulating physical systems and solving partial differential equations (PDEs) is formulating physics-based data in the desired structure for neural networks. Graph neural networks (GNNs) have gained popularity in this area since graphs offer a natural way of modeling particle interactions and provide a clear way of discretizing the continuum models. However, the graphs constructed for approximating such tasks usually ignore long-range interactions due to unfavorable scaling of the computational complexity with respect to the number of nodes. The errors due to these approximations scale with the discretization of the system, thereby not allowing for generalization under mesh-refinement. Inspired by the classical multipole methods, we purpose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Our multi-level formulation is equivalent to recursively adding inducing points to the kernel matrix, unifying GNNs with multi-resolution matrix factorization of the kernel. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time

Distributed learning of Gaussian graphical models via marginal likelihoods

We consider distributed estimation of the inverse covariance matrix, also called the concentration matrix, in Gaussian graphical models. Traditional centralized estimation often requires iterative and expensive global inference and is therefore difficult in large distributed networks. In this paper, we propose a general framework for distributed estimation based on a maximum marginal likelihood (MML) approach. Each node independently computes a local estimate by maximizing a marginal likelihood defined with respect to data collected from its local neighborhood. Due to the non-convexity of the MML problem, we derive and consider solving a convex relaxation. The local estimates are then combined into a global estimate without the need for iterative message-passing between neighborhoods. We prove that this relaxed MML estimator is asymptotically consistent. Through numerical experiments on several synthetic and real-world data sets, we demonstrate that the two-hop version of the proposed estimator is significantly better than the one-hop version, and nearly closes the gap to the centralized maximum likelihood estimator in many situations.