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：贝叶斯概率数值方法在数值积分中相比非贝叶斯方法具有显著优势：它们可以对被积函数的先验信息进行编码，并能量化积分估计的不确定性。然而，这类方法中最流行的算法——贝叶斯求积——基于高斯过程模型，因此计算成本较高。为提高可扩展性，我们提出一种基于贝叶斯神经网络的替代方法，称为贝叶斯施泰因网络。其关键要素是基于施泰因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近似方法。我们证明，该方法在流行的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型风电场能量生产预测中，实现了数量级的加速。