Scientific Machine Learning
The Focus Group “Scientific Machine Learning” involves Hans Fischer Senior Fellow Prof. Wil Schilders (Eindhoven University of Technology) and his host Prof. Hans-Joachim Bungartz (Chair of Scientific Computing in Computer Science –SCCS, TUM Department of Informatics).
Computational scientific discovery is at an interesting juncture. While traditionally we have models of lots of different scientific phenomena, such as the famous Maxwell and Navier-Stokes equations, and abundant data being generated from experiments - our computational capabilities appear unable to keep up. Often, problems are too large for realistic simulation. Besides, problems are multiscale and very stiff. Solving such problems requires tedious work on suitable algorithms as well as getting code to run on GPUs and supercomputers. The next step forward is a combination of scientific computing and machine learning, combining mathematical models with data based reasoning, presented as a unified set of abstractions and a high performance implementation. This new area of research is referred to as scientific machine learning.
Scientific machine learning has been taking the academic world by storm as an interesting blend of traditional scientific modeling with machine learning methodologies like deep learning. While traditional deep learning methodologies have had difficulties with scientific issues like stiffness, interpretability, and enforcing physical constraints, this blend with numerical analysis and differential equations has evolved into a field of research with new methods, architectures, and algorithms which overcome these problems while adding the data-driven automatic learning features of modern deep learning. Many successes have already been found, with tools like physics-informed neural networks, universal differential equations, deep backward stochastic differential equation solvers for high dimensional partial differential equations, and neural surrogates showcasing how deep learning can greatly improve scientific modeling practice.
Mathematics will be essential in addressing the challenges that we encounter in the rapidly evolving field of scientific machine learning. The objective of the focus group is to develop new mathematical and computational methods combining physics-based and data-based models, especially also model order reduction methods suitable for application to such hybrid models. Important will be the incorporation of underlying properties, implying the development of mimetic numerical methods.