Hierarchical Bayesian Quadrature

Numerical integration is a cornerstone of various scientific computing applications, such as engineering simulations and model evidence computations in probabilistic machine learning. Bayesian Quadrature uses Gaussian process surrogates that explicitly encode structural assumptions about the integrand to obtain integral estimates with quantified uncertainty. These surrogates are predominantly based on stationary covariance functions, which results in model misspecification for integrands exhibiting nonstationary behavior. We tackle this issue through an adaptively growing, tree-based partition of the integration domain into local stationary models. Our method recombines the local integral estimates through a hierarchy of GP conditioning that reintroduces cross-subdomain correlations, while model selection criteria control the tree growth to avoid unnecessary partitioning. The resulting algorithm is simple, requires no MCMC, and adapts its evaluation budget to local integrand complexity. On benchmark integration problems and a model evidence computation for an epidemiological model, Hierarchical Bayesian Quadrature achieves substantial gains over standard Bayesian Quadrature on nonstationary integrands while matching its performance on stationary ones.

Scalable Bayesian Inference for Nonlinear Conservation Laws

Nonlinear conservation laws are at the heart of many of the most important dynamical systems in science and engineering. In practical applications, such systems are often subject to various sources of uncertainty, e.g. due to sparse or noisy measurements. Inferring physical quantities and fields of interest then becomes an ill-posed problem which both classical numerical methods and modern deep learning-based methods struggle to treat appropriately. Recent work has framed classical numerical methods as Bayesian inference under Gaussian process priors, resulting in a physics-aware treatment of uncertainties. Following this line of work, we develop a novel numerically conservative method for uncertainty-aware simulations of nonlinear conservation laws. We use recent sparse approximation techniques to scale up to large-scale forward and inverse problems. For forward simulation, we inherit the accuracy of classical solvers while providing structured uncertainty quantification. On inverse problems, we recover posteriors over nonparametric source fields in seconds -- outperforming neural baselines that take minutes to produce a less accurate point estimate.

Flexible and Efficient Probabilistic PDE Solvers through Gaussian Markov Random Fields

Mechanistic knowledge about the physical world is virtually always expressed via partial differential equations (PDEs). Recently, there has been a surge of interest in probabilistic PDE solvers -- Bayesian statistical models mostly based on Gaussian process (GP) priors which seamlessly combine empirical measurements and mechanistic knowledge. As such, they quantify uncertainties arising from e.g. noisy or missing data, unknown PDE parameters or discretization error by design. Prior work has established connections to classical PDE solvers and provided solid theoretical guarantees. However, scaling such methods to large-scale problems remains a fundamental challenge primarily due to dense covariance matrices. Our approach addresses the scalability issues by leveraging the Markov property of many commonly used GP priors. It has been shown that such priors are solutions to stochastic PDEs (SPDEs) which when discretized allow for highly efficient GP regression through sparse linear algebra. In this work, we show how to leverage this prior class to make probabilistic PDE solvers practical, even for large-scale nonlinear PDEs, through greatly accelerated inference mechanisms. Additionally, our approach also allows for flexible and physically meaningful priors beyond what can be modeled with covariance functions. Experiments confirm substantial speedups and accelerated convergence of our physics-informed priors in nonlinear settings.

Scaling up Probabilistic PDE Simulators with Structured Volumetric Information

Modeling real-world problems with partial differential equations (PDEs) is a prominent topic in scientific machine learning. Classic solvers for this task continue to play a central role, e.g. to generate training data for deep learning analogues. Any such numerical solution is subject to multiple sources of uncertainty, both from limited computational resources and limited data (including unknown parameters). Gaussian process analogues to classic PDE simulation methods have recently emerged as a framework to construct fully probabilistic estimates of all these types of uncertainty. So far, much of this work focused on theoretical foundations, and as such is not particularly data efficient or scalable. Here we propose a framework combining a discretization scheme based on the popular Finite Volume Method with complementary numerical linear algebra techniques. Practical experiments, including a spatiotemporal tsunami simulation, demonstrate substantially improved scaling behavior of this approach over previous collocation-based techniques.