# Bayesian Numerical Integration with Neural Networks

arXiv: https://arxiv.org/abs/2305.13248 · HTML: https://mlsanity.com/paper/2305.13248/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 72 references that are in the map.
- First posted on arXiv 22 May 2023.

## Builds on (most cited first, 30 of 33)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [Bayesian Deep Learning and a Probabilistic Perspective of Generalization](https://mlsanity.com/paper/2002.08791/) (2020)
- [Laplace Redux -- Effortless Bayesian Deep Learning](https://mlsanity.com/paper/2106.14806/) (2021)
- [What Are Bayesian Neural Network Posteriors Really Like?](https://mlsanity.com/paper/2104.14421/) (2021)
- [Functional Variational Bayesian Neural Networks](https://mlsanity.com/paper/1903.05779/) (2019)
- [Bayesian Neural Network Priors Revisited](https://mlsanity.com/paper/2102.06571/) (2021)
- [Continuously Differentiable Exponential Linear Units](https://mlsanity.com/paper/1704.07483/) (2017)
- [Probabilistic Numerics and Uncertainty in Computations](https://mlsanity.com/paper/1506.01326/) (2015)
- [Control functionals for Monte Carlo integration](https://mlsanity.com/paper/1410.2392/) (2014)
- [Minimum Stein Discrepancy Estimators](https://mlsanity.com/paper/1906.08283/) (2019)
- [Bayesian Probabilistic Numerical Methods](https://mlsanity.com/paper/1702.03673/) (2017)
- [Sampling for Inference in Probabilistic Models with Fast Bayesian Quadrature](https://mlsanity.com/paper/1411.0439/) (2014)
- [Convergence Guarantees for Gaussian Process Means With Misspecified Likelihoods and Smoothness](https://mlsanity.com/paper/2001.10818/) (2020)
- [Stein Point Markov Chain Monte Carlo](https://mlsanity.com/paper/1905.03673/) (2019)
- [Optimal Thinning of MCMC Output](https://mlsanity.com/paper/2005.03952/) (2020)
- [Measuring Sample Quality with Diffusions](https://mlsanity.com/paper/1611.06972/) (2016)
- [Emulation of physical processes with Emukit](https://mlsanity.com/paper/2110.13293/) (2021)
- [Gradient Estimation with Discrete Stein Operators](https://mlsanity.com/paper/2202.09497/) (2022)
- [Expressive Priors in Bayesian Neural Networks: Kernel Combinations and Periodic Functions](https://mlsanity.com/paper/1905.06076/) (2019)
- [Convergence Guarantees for Adaptive Bayesian Quadrature Methods](https://mlsanity.com/paper/1905.10271/) (2019)
- [Frank-Wolfe Bayesian Quadrature: Probabilistic Integration with Theoretical Guarantees](https://mlsanity.com/paper/1506.02681/) (2015)
- [A Modern Retrospective on Probabilistic Numerics](https://mlsanity.com/paper/1901.04457/) (2019)
- [Active Multi-Information Source Bayesian Quadrature](https://mlsanity.com/paper/1903.11331/) (2019)
- [Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings](https://mlsanity.com/paper/1709.00147/) (2017)
- [Bayesian Quadrature for Multiple Related Integrals](https://mlsanity.com/paper/1801.04153/) (2018)
- [Scalable Control Variates for Monte Carlo Methods via Stochastic Optimization](https://mlsanity.com/paper/2006.07487/) (2020)
- [Meta-learning Control Variates: Variance Reduction with Limited Data](https://mlsanity.com/paper/2303.04756/) (2023)
- [Probabilistic Models for Integration Error in the Assessment of Functional Cardiac Models](https://mlsanity.com/paper/1606.06841/) (2016)

## Built on by (most cited first, 3 of 3)

- [Conditional Bayesian Quadrature](https://mlsanity.com/paper/2406.16530/) (2024)
- [Training neural control variates using correlated configurations](https://mlsanity.com/paper/2505.07719/) (2025)
- [BayesSum: Bayesian Quadrature in Discrete Spaces](https://mlsanity.com/paper/2512.16105/) (2025)

## Abstract (arXiv)

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.