I started studying computer science at the University of Stuttgart in 2013. I became interested in algorithms early on, in particular in machine learning. After I realized that there is a lot of mathematics behind these algorithms, I decided to start studying for a parallel degree in mathematics in 2015. Fortunately, this is possible at the University of Stuttgart and it is also possible to have some modules recognised for both degrees. I then completed my Bachelor in Computer Science in 2016 and my Bachelor in Mathematics and Master in Computer Science in 2019. Since 2020, I am pursuing a PhD at the Department of Mathematics with Prof. Steinwart. I am also part of the Cluster of Excellence for Data-Integrated Simulation Science (SimTech) and the International Max Planck Research School for Intelligent Systems (IMPRS-IS), which is part of the Cyber-Valley Initiative of the State of Baden-Württemberg.
What are you researching on?
I study (artificial) neural networks, a broad class of methods from machine learning, which in turn is a subfield of artificial intelligence (AI). The beauty of neural networks is that they are extremely versatile. You can combine inputs of different kinds in different ways, as long as you can calculate derivatives. An umbrella term for this is "Differentiable Programming". In most cases these derivatives can be calculated automatically by software like PyTorch or Tensorflow. With the help of the derivatives, optimization methods can often make a neural network solve a given task very well. A defining characteristic of neural networks is that they convert the input into a result not in one but in several successive steps. In the last decade there has been a trend to increase the number of these steps, which led to the name "deep learning". The flexibility and nonlinear structure also mean that neural networks are not easy to study mathematically. Moreover, it is not always clear when and why a certain "architecture" of a neural network works better than another. With my research I would like, among other things, to contribute to a better understanding of these problems.
I also want to investigate how neural networks can be applied in simulation science. Simulations are widely used in science and industry - anything can be simulated: Weather, underground currents, muscles, air resistance, traffic, chemical reactions, the formation of galaxies and much more. Methods of machine learning can be helpful here to accelerate simulations, which sometimes require a lot of computing power, or to identify unknown environmental parameters.
How does mathematics help?
First of all, mathematics provides a language to talk about the different methods of machine learning and get a rough idea of how they work. A solid basic knowledge of linear algebra, analysis and probability theory is important. Advanced knowledge is very useful for a more detailed analysis of the procedures. For example, in my master's thesis I was able to use knowledge of ordinary differential equations, concentration inequalities, block matrices and more. This allowed me to prove for a given scenario that neural networks are highly likely to function poorly - even if they are equipped with an arbitrary number of neurons. Fortunately, this scenario is easy to circumvent in practice, but it highlights a problem that should not be neglected.
What do you like about research?
I like the freedom I get from it. I can be creative and come up with many different approaches to solve different problems. I can check my ideas, often quickly, by mathematical calculations or by writing computer programs and get further insights. I like to tinker with tricky questions and find the most elegant solutions. In contrast to solving Sudoku puzzles or the like, you have to deal with a wide variety of problems, which always require new solutions, broaden your own horizon and are often of practical relevance. In addition, I learn a lot about the current state of research by reading new scientific articles. Last but not least, I have the chance to publish new findings and thereby possibly make the world a bit better.
Is it possible to apply a lot of knowledge from mathematics in computer science and vice versa?
Mathematics is useful in many disciplines, including computer science. In theoretical computer science, for example, you ask yourself what problems you can solve in principle with computers and if so, how efficiently. Algebra, number theory, graph theory and combinatorics appear here. The situation is similar in information security. On the other hand, in artificial intelligence, image processing or simulation one often has to deal with linear algebra, analysis and probability theory. And even in more non-mathematical fields such as software engineering or human-computer interaction, basic statistical knowledge is needed to conduct studies.
In the more application-oriented mathematical disciplines, such as numerics or statistics, conversely, programming skills are required, and in some cases advanced computer skills are also helpful - for example, an understanding of the structure of computers helps to make mathematical computer programs faster. But computers can also be useful in other areas of mathematics to gain new insights.
Even though physics probably has a larger overlap with mathematics than computer science, there are still many synergies that I was able to use during my studies.
What can you recommend for studying?
During the course of study there are numerous opportunities to pursue other study-related activities outside the normal curriculum. Even if it is impossible to perceive all of them, I can only recommend to think outside the box and use some of these possibilities. For example, I worked as a student trainee and research assistant, attended lectures in other subjects, participated in competitions or helped with the application of the University of Stuttgart as a university of excellence. I also enjoyed participating in holiday academies at the university. There you can learn about a new topic in a course, go hiking in the mountains, participate in sports activities and exchange ideas with other students and lecturers. I was also lucky enough to meet young researchers from all over the world and some of the best-known researchers in mathematics and computer science at the Heidelberg Laureate Forum. One quickly learns that these award winners are also very human and approachable. All in all, I have learned a great deal through my participation in various events and have also made contacts that are now useful for my research.
Thank you for the interview.
David Holzmüller
Robert Bosch GmbH prizewinner for outstanding B.Sc. degree in mathematics