发表机构
IMDEA Materials Institute; University of Oulu（IMDEA材料研究所; 奥卢大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出B-GRASP贝叶斯框架，从含噪节点观测推断图边权重，结合扩散反应动力学与随机强迫，用MAP和NUTS量化不确定性，在逆热传导、反应扩散及COVID-19数据上验证。
AI 中文摘要
我们提出B-GRASP（带有SPDE先验的贝叶斯图推断），这是一个贝叶斯框架，用于从节点状态的含噪观测中推断图上随机动力系统中不确定的边权重。未知的边权重参数化图微分算子，因此直接控制图过程的演化。受PDE和SPDE中的微分算子与其图对应物之间联系的启发，我们构建了包含扩散、反应动力学和随机强迫的随机图模型。由此产生的分层公式联合表示图结构和随机强迫中的不确定性。潜在图变量确定正边权重和相应的图拉普拉斯算子，而潜在布朗变量表示随机强迫。以这些变量为条件，图动力学定义了一个确定性前向映射，由此构建似然函数和后验分布。我们使用最大后验估计和No-U-Turn采样器来表征后验，从而实现点估计和不确定性量化。我们在一个一维逆热传导问题、具有非线性动力学的平稳和非平稳图反应-扩散系统以及美国州级COVID-19数据上展示了该框架。数值结果表明，后验不确定性提供了点估计未捕获的信息，特别是对于弱可识别性或高导电性的边，并使得图连通性和随机强迫中的不确定性能够在贝叶斯框架内被量化。
英文摘要
We present B-GRASP (Bayesian GRAph inference with SPDE priors), a Bayesian framework for inferring uncertain edge weights in stochastic dynamical systems on graphs from noisy observations of nodal states. The unknown edge weights parameterize the graph differential operator and therefore directly govern the evolution of the graph process. Motivated by connections between differential operators in PDEs and SPDEs and their graph counterparts, we construct stochastic graph models incorporating diffusion, reaction dynamics, and stochastic forcing. The resulting hierarchical formulation jointly represents uncertainty in the graph structure and stochastic forcing. Latent graph variables determine positive edge weights and the corresponding graph Laplacian, while latent Brownian variables represent the stochastic forcing. Conditional on these variables, the graph dynamics define a deterministic forward map from which the likelihood and posterior distribution are constructed. We characterize the posterior using maximum a posteriori estimation and the No-U-Turn Sampler, enabling both point estimation and uncertainty quantification. We demonstrate the framework on a one-dimensional inverse heat-conduction problem, stationary and nonstationary graph reaction-diffusion systems with nonlinear dynamics, and state-level COVID-19 data in the United States. The numerical results show that posterior uncertainty provides information not captured by point estimates, particularly for weakly identifiable or highly conductive edges, and enables uncertainty in both graph connectivity and stochastic forcing to be quantified within a Bayesian framework.
Comments33 pages, 16 figures, main article and supplementary material
