Topical Issue Scientific Machine Learning (1/2)

Scientific Machine Learning is a rapidly evolving field of research that combines and further develops techniques of scientific computing and machine learning. Special emphasis is given to the scientific (physical, chemical, biological, etc.) interpretability of models learned from data and their usefulness for robust predictions. On the other hand, this young field also investigates the utilization of Machine Learning methods for improving numerical algorithms in Scientific Computing. The name Scientific Machine Learning has been coined at a Basic Research Needs Workshop of the US Department of Energy (DOE) in January, 2018. It resulted in a report [2] published in February, 2019; see also [1] for a short brochure on this topic. The present special issue of the GAMM Mitteilungen, which is the first of a two-part series, contains contributions on the topic of Scientific Machine Learning in the context of complex applications across the sciences and engineering. Research in this new exciting field needs to address challenges such as complex physics, uncertain parameters, and possibly limited data through the development of new methods that combine algorithms from computational science and engineering and from numerical analysis with state of the art techniques from machine learning. At the GAMM Annual Meeting 2019, the activity group Computational and Mathematical Methods in Data Science (CoMinDS) has been established. Meanwhile, it has become a meeting place for researchers interested in all aspects of data science. All three editors of this special issue are founding members of this activity group. Because of the rapid development both in the theoretical foundations and the applicability of Scientific Machine Learning techniques, it is time to highlight developments within the field in the hope that it will become an essential domain within the GAMM and topical issues like this will have a frequent occurrence within this journal. We are happy that eight teams of authors have accepted our invitation to report on recent research highlights in Scientific Machine Learning, and to point out the relevant literature as well as software. The four papers in this first part of the special issue are: • Stoll, Benner: Machine Learning for Material Characterization with an Application for Predicting Mechanical Properties. This work explores the use of machine learning techniques for material property prediction. Given the abundance of data available in industrial applications, machine learning methods can help finding patterns in the data and the authors focus on the case of the small punch test and tensile data for illustration purposes. • Beck, Kurz: A Perspective on Machine Modelling Learning Methods in Turbulence. Turbulence modelling remains a humongous challenge in the simulation and analysis of complex flows. The authors review the use of data-driven techniques to open up new ways for studying turbulence and focus on the challenges and opportunities that machine learning brings to this field. • Heinlein, Klawonn, Lanser, Weber: Combining Machine Learning and Domain Decomposition Methods for the Solution of Partial Differential Equations – A Review. Domain decomposition (DD) has been a workhorse of solving complex simulation tasks. The authors review the combination of machine learning approaches with state-of-the-art DD-schemes. Their focus is on the use of ML techniques to improve the computational effort of adaptive domain decomposition schemes and the use of novel ML methods for the discretization and solution of subdomain problems. • Budd, van Gennip, Latz: Classification and image processing with a semi-discrete scheme for fidelity forced Allen–Cahn on graphs. Learning based on graphs provides exciting possibilities for discovering and using additional structure in data. In this work, the authors illustrate the use of a PDE-based learning technique relying on the graph Allen-Cahn equation for the segmentation of images. The authors illustrate that computational and mathematical advances can lead to efficiency and accuracy gains. Peter Benner1,2 Axel Klawonn3,4 Martin Stoll5