Probabilistic Numerical Methods - From Theory to Implementation

Summary

Probabilistic Numerical algorithms frame a numerical task as a statistical inference problem, expressed in the language of probabilistic inference. The key advantage of this approach is that it allows quantification of uncertainty arising from finite computational resources, and to combine thus with other forms of uncertainty, in particular those arising from model misspecification, finite observational data, and measurement errors. In recent years, algorithms arising from this formalism have repeatedly shown that they can enrich and improve upon classic methods in tasks where

• hyperparameter adaptation is not straightforward;
• computational stochasticity and low precision play a prominent role;
• limited data make uncertainty quantification a key functionality;
• related problems have to be solved repeatedly;
• and where extreme scale or tight budgets call for rough approximations at low cost.

Probabilistic Numerics lies at the intersection of machine learning within computer science and numerical analysis within applied mathematics. This interdisciplinary nature raises an exciting and challenging set of viewpoints with regards to goals and challenges of the field. The first goal of this seminar was to rekindle our community following two years of pandemic lockdown, to provide an opportunity to update others on one's own research, and to discuss new directions and ideas together. We were lucky to assemble - both in-person and remote - a diverse group of people from computer science, machine learning, from statistics, optimization, and from numerical analysis.

The second key goal of this seminar was to take the next step in the development of probabilistic numerical methods by focusing on their implementation. From Lapack to SciPy to PyTorch, open-source software libraries have driven scientific advancement in their respective domains. Such libraries accelerate research, enable benchmarking and promote the development of new methods via rapid prototyping. Most importantly, they are a necessary step towards their use in applications. While considerable advances in the theoretical understanding of probabilistic numerical methods have been made, the lack of high-quality implementations is holding back their adoption. In response, we recently started a community effort to develop an open-source framework named ProbNum (http://probnum.org).

A central theme of Dagstuhl seminars is the open, collaborative atmosphere with a focus on new ideas and tangible outcomes as opposed to existing work. The seminar stimulated multiple focussed discussions around software and additions to ProbNum. Examples included how to best include automatic differentiation functionality, or how to expand the package's Bayesian quadrature functionalities. It also set the starting point for potential new research collaborations on probabilistic linear solvers and probabilistic numerical methods for PDEs. Even at this point, shortly after the seminar's conclusion, two tangible products are already available: the seminar's participants jointly created a Probabilistic Numerics Wikipedia page https://en.wikipedia.org/wiki/Probabilistic_numerics, and the implementation sessions culminated with a preprint for the community library ProbNum [1].

As the organizers we want to thank all participants, both physical and virtual, for their interesting talks, the stimulating discussions and the collaborative overall atmosphere. We also want to thank Schloss Dagstuhl for their technical support that made the challenging hybrid format possible.

References

1. Jonathan Wenger, Nicholas Krämer, Marvin Pförtner, Jonathan Schmidt, Nathanael Bosch, Nina Effenberger, Johannes Zenn, Toni Karvonen Alexandra Gessner, François-Xavier Briol, Maren Mahsereci, and Philipp Hennig. ProbNum: Probabilistic numerics in Python. arxiv preprint, 2021. URL http://arxiv.org/abs/2112.02100