With the availability of data and computational technologies in the modern world, machine
learning (ML) has emerged as a preferred methodology for data analysis and prediction …

We present a deep learning framework for quantifying and propagating uncertainty in
systems governed by non-linear differential equations using physics-informed neural …

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

For more than two centuries, solutions of differential equations have been obtained either
analytically or numerically based on typically well-behaved forcing and boundary conditions …

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 …

Gaussian processes are ubiquitous in machine learning, statistics, and applied
mathematics. They provide a exible modelling framework for approximating functions, whilst …

We build on the view of the Exact Renormalization Group (ERG) as an instantiation of
Optimal Transport described by a functional convection–diffusion equation. We provide a …

We propose a fast, simple and robust algorithm for computing shortest paths and distances
on Riemannian manifolds learned from data. This amounts to solving a system of ordinary …

The numerical solution of differential equations can be formulated as an inference problem
to which formal statistical approaches can be applied. However, nonlinear partial differential …