A modern retrospective on probabilistic numerics

CJ Oates, TJ Sullivan - Statistics and computing, 2019 - Springer
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 …

Probabilistic solutions to ordinary differential equations as nonlinear Bayesian filtering: a new perspective

F Tronarp, H Kersting, S Särkkä, P Hennig - Statistics and Computing, 2019 - Springer
We formulate probabilistic numerical approximations to solutions of ordinary differential
equations (ODEs) as problems in Gaussian process (GP) regression with nonlinear …

[图书][B] Probabilistic Numerics: Computation as Machine Learning

P Hennig, MA Osborne, HP Kersting - 2022 - books.google.com
Probabilistic numerical computation formalises the connection between machine learning
and applied mathematics. Numerical algorithms approximate intractable quantities from …

Bayesian ODE solvers: the maximum a posteriori estimate

F Tronarp, S Särkkä, P Hennig - Statistics and Computing, 2021 - Springer
There is a growing interest in probabilistic numerical solutions to ordinary differential
equations. In this paper, the maximum a posteriori estimate is studied under the class of ν ν …

Calibrated adaptive probabilistic ODE solvers

N Bosch, P Hennig, F Tronarp - International Conference on …, 2021 - proceedings.mlr.press
Probabilistic solvers for ordinary differential equations assign a posterior measure to the
solution of an initial value problem. The joint covariance of this distribution provides an …

Convergence rates of Gaussian ODE filters

H Kersting, TJ Sullivan, P Hennig - Statistics and computing, 2020 - Springer
A recently introduced class of probabilistic (uncertainty-aware) solvers for ordinary
differential equations (ODEs) applies Gaussian (Kalman) filtering to initial value problems …

Bayesian numerical methods for nonlinear partial differential equations

J Wang, J Cockayne, O Chkrebtii, TJ Sullivan… - Statistics and …, 2021 - Springer
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 …

Fenrir: Physics-enhanced regression for initial value problems

F Tronarp, N Bosch, P Hennig - International Conference on …, 2022 - proceedings.mlr.press
We show how probabilistic numerics can be used to convert an initial value problem into a
Gauss–Markov process parametrised by the dynamics of the initial value problem …

A probabilistic finite element method based on random meshes: A posteriori error estimators and Bayesian inverse problems

A Abdulle, G Garegnani - Computer Methods in Applied Mechanics and …, 2021 - Elsevier
We present a novel probabilistic finite element method (FEM) for the solution and uncertainty
quantification of elliptic partial differential equations based on random meshes, which we …

Differentiable likelihoods for fast inversion of'likelihood-free'dynamical systems

H Kersting, N Krämer, M Schiegg… - International …, 2020 - proceedings.mlr.press
Likelihood-free (aka simulation-based) inference problems are inverse problems with
expensive, or intractable, forward models. ODE inverse problems are commonly treated as …