In the paper, a variational Bayesian method is used to identify the reaction coefficient for space-time nonlocal diffusion equations using nonlocal averaged flux data. To show the posterior measure to be well-defined, we rigorously prove that the forward operator is continuous with respect to the unknown reaction field. Then, gradient-based prior information is proposed to explore oscillation features in the reaction coefficient. Moreover, the Bayesian inverse problem is shown to be well-posed in Hellinger distance. To accurately characterize the posterior density using uncorrelated samples, an efficient variational Bayesian method is used to estimate the reaction coefficient in the nonlocal models. A few numerical results are presented to illustrate the efficacy of the proposed approach and confirm some theoretic discoveries.

在本文中，使用非局部平均通量数据，使用变分贝叶斯方法来识别时空非局部扩散方程的反应系数。为了显示后验度量是明确定义的，我们严格证明了前向算子相对于未知反应场是连续的。然后，提出了基于梯度的先验信息，以探索反应系数中的振荡特征。此外，贝叶斯逆问题被证明在Hellinger距离中是恰当的。为了使用不相关的样本准确地表征后部密度，在非局部模型中使用了有效的变分贝叶斯方法来估计反应系数。一些数值结果被提出来说明所提出的方法的有效性并证实一些理论上的发现。