Bayesian probabilistic numerical methods for numerical integration offer significant advantages over their non-Bayesian counterparts: they can encode prior information about the integrand, and can quantify uncertainty over estimates of an integral. However, the most popular algorithm in this class, Bayesian quadrature, is based on Gaussian process models and is therefore associated with a high computational cost. To improve scalability, we propose an alternative approach based on Bayesian neural networks which we call Bayesian Stein networks. The key ingredients are a neural network architecture based on Stein operators, and an approximation of the Bayesian posterior based on the Laplace approximation. We show that this leads to orders of magnitude speed-ups on the popular Genz functions benchmark, and on challenging problems arising in the Bayesian analysis of dynamical systems, and the prediction of energy production for a large-scale wind farm.

翻译：贝叶斯概率数值积分方法相较于非贝叶斯方法具有显著优势：既能够编码被积函数的先验信息，又可量化积分估计的不确定性。然而，该类方法中最流行的贝叶斯求积算法基于高斯过程模型，导致计算成本高昂。为提升可扩展性，我们提出一种基于贝叶斯神经网络的替代方法——贝叶斯斯坦网络。其核心组件包括基于斯坦算子的神经网络架构，以及基于拉普拉斯近似的贝叶斯后验逼近方法。研究表明，该方法在经典Genz函数基准测试、动态系统贝叶斯分析中的挑战性问题以及大规模风电场能量预测任务中，实现了数量级的计算加速。