Within the recently approved NWO-ENW (Science domain) Groot project UNRAVEL there are 6 vacancies for PhD candidates:

- The Department of Mathematics and Computer Science of Eindhoven University of Technology has a vacancy for 3 PhD students in its Centre for Analysis, Scientific computing and Applications (CASA). CASA comprises the chairs Scientific Computing (SC) and Applied Analysis (TA). CASA’s major research objective is to develop new and to improve existing mathematical (both analytical and numerical) methods for a wide range of applications in science and engineering.
- The Department of Physics of Eindhoven University of Technology has a vacancy for 1 PhD student, and combines this with another PhD position at SISSA in Trieste (Italy). This project will be carried out within the Fluids and Flows (F&F) group at the Department of Applied Physics (https://www.tue.nl/en/research/research-groups/fluids-and-flows/ ) of Eindhoven University of Technology and within the PhD program in Mathematical Analysis, Modelling and Applications at SISSA (Trieste, Italy).
- Leiden Observatory has a vacancy for 1 PhD student in the Computational Astrophysics Leiden group of prof. Portegies Zwart. Leiden Observatory is part of the Faculty of Science at Leiden University.
- Centrum Wiskunde & Informatica (CWI, Amsterdam) has a vacancy for 1 PhD student in the Scientific Computing group. CWI is the Dutch national research institute for mathematics and computer science.

**The UNRAVEL project**

Our understanding of processes and phenomena in nature and society is being radically transformed by machine learning and the availability of data. This is evident also from the large numbers of researchers embracing deep learning as a tool. At the same time, obstacles and challenges are becoming apparent: most deep-learning approaches require large amounts of data, but in many domains such massive datasets are not available. Furthermore, the emergent behaviour of deep neural networks is usually difficult to interpret. To overcome these drawbacks, the effective use of prior knowledge is key. The main objective of this project: revealing how neural networks can be made much more effective by incorporating mathematical and physical understanding in their design. The project aims to build a mimetic theory of neural networks that will enable their data-efficient and understandable use for scientific discovery in physics, astronomy and beyond.

To achieve this objective, it is necessary to approach the challenge from different angles. For this reason, our team consists of experts from mathematics, computer science, machine learning, physics and astronomy. The work has been organised in 6 individual projects that will work closely together (more information available from the supervisors):

- Algorithms for the discovery of interpretable latent variables. This project is about the design and mathematical analysis of algorithms for the discovery of interpretable latent variables (disentangled representations). The PhD project will be on the one hand about the design of such algorithms, taking state-of-the art candidates called Variational Autoencoders as a basis, and on the other hand about mathematically analyzing why given algorithms do or do not provide the desired results.
- Mimetic, hierarchical training algorithms for neural networks. The goal of this subproject is to improve the efficiency and effectiveness of training algorithms in neural networks (algorithms for optimizing neuron weights). Problem-dependent constraints will be incorporated into these training algorithms. The convergence of the mimetic, hierarchical training algorithms will be studied through mathematical analyses. The resulting neural networks will be tested on the basis of discriminating test cases from fluid mechanics (project 5) and astronomy (project 6).
- Dynamic neural networks and their relation to state-space methods. Dynamic neural networks are networks where the action of neurons is described by scalar ordinary differential equations, enabling the simulation of time-dependent phenomena. Previous work on this type of dynamic neural networks revealed an intimate relationship between the structure and parameters of the network with state space models used to describe the underlying system. It implies that the state space system can be used to predict the topology of the neural network. This relation will be exploited to develop an entirely new theory of model reduction techniques directly for neural networks, including structure preservation methods.
- Machine learning using constrained neural networks and differential equations. In this project mathematical techniques from the fields of ordinary and partial differential equations will be used to better understand and design neural networks. By considering neural networks as the discrete version of a continuous dynamical system, mathematical tools from the field of time integration can be used to analyse the properties of the network, such as its robustness to varying inputs (using nonlinear stability analysis) and the incorporation of constraints in the network (using differential-algebraic equation analysis).
- Machine learning for analysis and control of complex fluid flows. In this project we will combine and further develop the scientific and technological know-how of TU/e and SISSA research groups towards the use of Machine Learning techniques. The aim of the project is to develop tools that will allow a deeper understanding, modelling and control capabilities for turbulent flows.
- Neural networks for N-body simulations. In this project, we will solve the gravitational few-body problem using neural networks. This approach is possible because the underlying equations of motion are chaotic. Consequently, solutions obtained using traditional methods on computers only provide a statistical answer, which can as well be obtained by a neural network. We will train deep artificial neural-networks on ensembles of converged solutions to the few-body problem. The trained network will subsequently be replacing the expensive few-body calculations in large simulations of dense star clusters. This should lead to considerable speed-up compared to more traditional direct integration.

Each of the topics in itself is of a ground-breaking character. The mathematically inclined projects concentrate on fundamental properties of neural networks that potentially have a big influence on future methodologies for constructing networks and for conducting scientific computational research. The fluid flows and astronomy projects concentrate on specific challenges which serve as test cases for potentially more general strategies.

**As a PhD student your tasks are the following**:

- Perform scientific research in the described domain;
- Present results at international conferences;
- Publish results in scientific journals;
- Participate in activities of the group and the department;
- At the universities: assist staff in teaching undergraduate and graduate courses (at most 20% of the time).

**Requirements**

We are looking for talented, enthusiastic PhD candidates who meet the following requirements:

- An MSc in mathematics, physics, astronomy, computer science or a related discipline with a strong background in applications; knowledge of numerical analysis, ordinary and partial differential equations, discretization techniques, machine learning, neural networks, will be beneficial
- Experience with scientific programming, e.g. Matlab, Python, C, C++, Python;
- Creative, pro-active team player with good analytical skills;
- Good communicative skills in English, both written and oral.

**Appointment and salary**:

We offer:

- A full-time appointment for a period of four years, with an intermediate evaluation after nine months;
- A gross salary of € 2395 per month in the first year increasing up to € 3061 per month in the fourth year;
- Support for your personal development and career planning including courses, summer schools, conference visits, etc.;
- A research position in an enthusiastic and internationally renowned research group, and embedding of the research in the UNRAVEL team with strong national and international collaborations;
- A broad package of fringe benefits (e.g. excellent technical infrastructure, saving schemes, excellent sport facilities, and child daycare).

The terms of employment are in accordance with the Collective Labour Agreement for Dutch Universities (CAO Nederlandse Universiteiten) and the Collective Labour Agreement for Research Institutions (CAO Onderzoekinstellingen), respectively. The initial labour agreement will be for a period of 9 or 18 months, depending on the institute. After a positive evaluation, the agreement will be extended. Employees are also entitled to a holiday allowance of 8% of the gross annual salary and a year-end bonus of 8.33%. All participating institutions offer attractive working conditions, including flexible scheduling and help with housing for expat employees.

The application should consist of the following parts:

- A motivation letter;
- A Curriculum Vitae;
- Copies of diplomas and a list of grades of your studies;
- A copy of your master’s thesis;
- Names and addresses of two referees;
- Proof of English language skills (if applicable).

**Information on the various projects**

More information about the project as a whole can be obtained from the coordinator, Prof.dr. W.H.A. (Wil) Schilders, e-mail: w.h.a.schilders[at]tue.nl, phone: +31402475518, mobile: +31651892525.

For the 6 projects, more detailed information can be obtained as follows:

- Subproject 1: dr. Jim Portegies, TU Eindhoven, e-mail: j.w.portegies[at]tue.nl
- Subproject 2: Prof.dr. Barry Koren, TU Eindhoven, e-mail: b.koren[at]tue.nl
- Subproject 3: Prof.dr. Wil Schilders, TU Eindhoven, e-mail: w.h.a.schilders[at]tue.nl
- Subproject 4: dr. Benjamin Sanderse, CWI, e-mail: b.sanderse[at]cwi.nl
- Subproject 5: Prof.dr. Federico Toschi, TU Eindhoven, e-mail: f.toschi[at]tue.nl
- Subproject 6: Prof.dr. Simon Portegies Zwart, Leiden University, e-mail: spz[at]strw.leidenuniv.nl, applications should be entered in the Job Application System

Information about employment conditions:

- For subprojects 1, 2 and 3: please contact mrs. Karin Wels-Noordermeer, email: k.h.wels[at]tue.nl
- For subproject 4: please contact pd[at]cwi.nl
- For subproject 5: please contact f.toschi[at]tue.nl
- For subproject 6: inquiries can be made to gerstel[at]strw.leidenuniv.nl