Relational inference aims to identify interactions between parts of a dynamical system from the observed dynamics. Current state-of-the-art methods fit the dynamics with a graph neural network (GNN) on a learnable graph. 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 poor local optima for the one-step model. In this work, we propose a \textit{graph dynamics prior} (GDP) for relational inference. GDP constructively uses error amplification in non-local polynomial filters to steer the solution to the ground-truth graph. To deal with non-uniqueness, GDP simultaneously fits a ``shallow'' one-step model and a polynomial multi-step model with shared graph topology. Experiments show that GDP 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 GDP suitable for real applications in scientific machine learning. Reproducible code is available at https://github.com/DaDaCheng/GDP.

翻译：关系推断旨在根据观测到的动力学过程识别动态系统中各部件之间的相互作用。当前最先进的模型利用可学习的图结构通过图神经网络（GNN）来拟合动力学过程，采用单步消息传递的GNN——直观而言这是合理的选择，因为多步或谱域GNN的非局部特性可能混淆直接交互与间接交互。然而，有效交互图取决于采样率且极少局限于直接邻域，这导致单步模型容易陷入较差的局部最优解。本文提出一种用于关系推断的图动态先验（GDP），该先验建设性地利用非局部多项式滤波器中的误差放大机制，将解引导至真实图结构。为解决解的非唯一性问题，GDP同步拟合共享图拓扑的“浅层”单步模型与多项式多步模型。实验表明，GDP重构图的精度远超先前方法，且对欠采样具有显著鲁棒性。由于未知动态系统的合适采样率无法先验获知，这种鲁棒性使得GDP适用于科学机器学习中的真实应用场景。可复现代码见https://github.com/DaDaCheng/GDP。