A Modern Retrospective on Probabilistic Numerics

Oates, C. J., Sullivan, T. J.

arXiv.org Machine Learning 

The field of probabilistic numerics (PN), loosely speaking, attempts to provide a statistical treatment of the errors and/or approximations that are made en route to the output of a deterministic numerical method, e.g. the approximation of an integral by quadrature, or the discretised solution of an ordinary or partial differential equation. This decade has seen a surge of activity in this field. In comparison with historical developments that can be traced back over more than a hundred years, the most recent developments are particularly interesting because they have been characterised by simultaneous input from multiple scientific disciplines: mathematics, statistics, machine learning, and computer science. The field has, therefore, advanced on a broad front, with contributions ranging from the building of overarching generaltheory to practical implementations in specific problems of interest. Over the same period of time, and because of increased interaction among researchers coming from different communities, the extent to which these developments were -- or were not -- presaged by twentieth-century researchers has also come to be better appreciated. Thus, the time appears to be ripe for an update of the 2014 Tübingen Manifesto on probabilistic numerics[Hennig, 2014, Osborne, 2014d,c,b,a] and the position paper[Hennig et al., 2015] to take account of the developments between 2014 and 2019, an improved awareness of the history of this field, and a clearer sense of its future directions. In this article, we aim to summarise some of the history of probabilistic perspectives on numerics (Section 2), to place more recent developments into context (Section 3), and to articulate a vision for future research in, and use of, probabilistic numerics (Section 4).