Bayesian Numerical Integration with Neural Networks
This work addresses scalability issues in Bayesian numerical integration for researchers and practitioners in fields like scientific computing and machine learning, offering a more efficient alternative to Gaussian process-based methods.
The paper tackles the high computational cost of Bayesian quadrature by proposing Bayesian Stein networks, a Bayesian neural network approach, achieving orders of magnitude speed-ups on benchmarks like Genz functions and real-world applications in dynamical systems and wind farm energy prediction.
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.