Relational inference aims to identify interactions between parts of a dynamical system from the observed dynamics. Current state-of-the-art methods fit a graph neural network (GNN) on a learnable graph to the dynamics. They use one-step message-passing GNNs -- intuitively the right choice since non-locality of multi-step or spectral GNNs may confuse direct and indirect interactions. But the \textit{effective} interaction graph depends on the sampling rate and it is rarely localized to direct neighbors, leading to local minima for the one-step model. In this work, we propose a \textit{dynamical graph prior} (DYGR) for relational inference. The reason we call it a prior is that, contrary to established practice, it constructively uses error amplification in high-degree non-local polynomial filters to generate good gradients for graph learning. To deal with non-uniqueness, DYGR simultaneously fits a ``shallow'' one-step model with shared graph topology. Experiments show that DYGR reconstructs graphs far more accurately than earlier methods, with remarkable robustness to under-sampling. Since appropriate sampling rates for unknown dynamical systems are not known a priori, this robustness makes DYGR suitable for real applications in scientific machine learning.

翻译：关系推断旨在从观测到的动力学行为中识别动力系统各组成部分之间的相互作用。现有最先进的方法通过可学习图上的图神经网络（GNN）拟合动力学过程。这些方法采用单步消息传递GNN——直观而言这是合理选择，因为多步或谱GNN的非局部性可能混淆直接与间接相互作用。但有效相互作用图依赖于采样率，且极少局限于直接邻域，导致单步模型陷入局部最优。本文提出一种用于关系推断的"动态图先验"（DYGR）。我们将其称为先验的理由在于：与现有惯例相反，该方法建设性地利用高次非局部多项式滤波器中的误差放大效应，为图学习生成优质梯度。为应对非唯一性问题，DYGR同时拟合了一个共享图拓扑的"浅层"单步模型。实验表明，DYGR能以极为准确的性能重构图结构，且对欠采样具有显著鲁棒性。由于未知动力系统的适当采样率无法先验获知，这种鲁棒性使DYGR适用于科学机器学习领域的实际应用。