We describe and compare two formulations of inverse problems for a physics-based process
model in the context of uncertainty and random variability: the Bayesian inverse problem …

This work leverages recent advances in probabilistic machine learning to discover
governing equations expressed by parametric linear operators. Such equations involve, but …

We introduce a simple, rigorous, and unified framework for solving nonlinear partial
differential equations (PDEs), and for solving inverse problems (IPs) involving the …

Similar to many fields of sciences, recent deep learning advances have been applied
extensively in geosciences for both small-and large-scale problems. However, the necessity …

Over forty years ago average-case error was proposed in the applied mathematics literature
as an alternative criterion with which to assess numerical methods. In contrast to worst-case …

This article attempts to place the emergence of probabilistic numerics as a mathematical–
statistical research field within its historical context and to explore how its gradual …

We introduce a priori Sobolev-space error estimates for the solution of arbitrary nonlinear,
and possibly parametric, PDEs that are defined in the strong sense, using Gaussian process …

Partial differential equations (PDEs) are important tools to model physical systems and
including them into machine learning models is an important way of incorporating physical …

Inverse problems describe the task of blending a mathematical model with observational
data---a fundamental task in many scientific and engineering disciplines. The solvability of …

Probabilistic numerical computation formalises the connection between machine learning
and applied mathematics. Numerical algorithms approximate intractable quantities from …