Probabilistic Meshless Methods for Partial Differential Equations and Bayesian Inverse Problems

By Jon Cockayne, Chris J. Oates, T. Sullivan, Mark A. Girolami. 2016

47 citations recorded by the source corpus.

This paper develops a probabilistic numerical method for solution of partial differential equations (PDEs) and studies application of that method to PDE-constrained inverse problems. This approach enables the solution of challenging inverse problems whilst accounting, in a statistically principled way, for the impact of discretisation error due to numerical solution of the PDE. In particular, the approach confers robustness to failure of the numerical PDE solver, with statistical inferences driven to be more conservative in the presence of substantial discretisation error. Going further, the problem of choosing a PDE solver is cast as a problem in the Bayesian design of experiments, where the aim is to minimise the impact of solver error on statistical inferences; here the challenge of non-linear PDEs is also considered. The method is applied to parameter inference problems in which discretisation error in non-negligible and must be accounted for in order to reach conclusions that are statistically valid.

Subfields: Artificial Intelligence, Computational Theory and Mathematics, Statistics and Probability

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Paris Perdikaris
2. Yibo Yang
3. G. Em Karniadakis
4. Maizar Raissi
5. Chris J. Oates
6. Liu Yang
7. Jon Cockayne
8. Stefania Fresca

In Computational Theory and Mathematics

1. Paris Perdikaris
2. G. Em Karniadakis
3. Yibo Yang
4. Maizar Raissi
5. Chris J. Oates
6. Mark A. Girolami
7. T. Sullivan
8. Jon Cockayne

In Statistics and Probability

1. Paris Perdikaris
2. Yibo Yang
3. G. Em Karniadakis
4. Chris J. Oates
5. Maizar Raissi
6. Mark A. Girolami
7. Jon Cockayne
8. T. Sullivan

Citations in the corpus

• Adversarial Uncertainty Quantification in Physics-Informed Neural Networks (2018, methodology): “In contrast to previous approaches to inferring solutions of partial differential equations from data [44, 45, 46, 47], the proposed methodology does not rely on Gaussian assumptions, and it can directly tackle nonlinear problems without any need for linearization.”
• Machine learning of linear differential equations using Gaussian processes (2017, methodology): “Along exactly the same lines, the methodology outlined in this work is related to and yet fundamentally differentiated from the so-called “meshless” meth-ods (see [12, 13, 14, 15]) for differential equations.”
• Bayesian Probabilistic Numerical Methods (2017, methodology): “For Examples 2.3 and 2.4 of Bayesian PNMs, Briol et al. (2016) and Cockayne et al. (2016) established rates of contraction for particular prior distributions µ; we refer the reader to those papers for details.”
• Inferring solutions of differential equations using noisy multi-fidelity data (2016, background): “In the same spirit, the work of [13, 14, 15, 16] has highlighted the possibility of solving linear partial differential equations and quantifying parameter and discretization uncertainty using Gaussian process priors.”