PhD Position in Learning PDEs from Data

Faculteit/Dienst:  Faculteit der Natuurw., Wiskunde & Informatica
Opleidingsniveau:  Master
Functie type:  Promotieplaats
Sluitingsdatum:  16 december 2022
Vacaturenummer:  10860



Simulation-based models, which are often formulated in terms of partial differential equations (PDEs), form the backbone of predictive modelling in the natural sciences and in engineering disciplines. Fundamentally, a scientific model is a simplification of reality that helps us the understand and predict the essential aspects of a system, either to forecast its future behaviour, or to optimize its design according to prescribed requirements. In deep learning research, a fundamentally different approach to predictive modelling has emerged, which employs large models, often with billions of parameters, that are optimized on similarly large datasets. Whilst interpreting these models can be very difficult due to their overparameterized nature, these models are achieving unparalleled predictive accuracy in an ever-increasing range of application domains.


We seek a PhD Candidate that will contribute to this project by carrying out research at the intersection of traditional PDE-based and deep learning methods. You will be positioned between the Amsterdam Machine Learning Lab (AMLab) and the Computational Science Lab (CSL) of the Informatics Institute. You will also be collaborating with Microsoft Research at the Science Park in Amsterdam.


What are you going to do?


One opportunity in this space is to leverage the flexibility of neural networks to learn fast approximate solutions to PDEs. A second frontier, with a huge potential, is learning PDE-based models from observational data. For lumped parameter models (systems of coupled ordinary differential equations) this has already been demonstrated. However, for spatio-temporal systems, which could be modelled by partial differential equations, only very few examples exist. You will work to advance the state of the art in these two domains:

  1. Learning PDEs from Data. Learning PDEs represents an opportunity to learn a parsimonious model that approximates unknown dynamics in a physical system, or to learn a model that describes emergent dynamics are larger length and time scales based on lower-level simulations. In the second stage of the project you will formulate a neural surrogate solver where both the surrogate parameters and the PDE parameters are jointly trained to describe the data. One could view this as a PDE inspired neural network model to describe the time-dynamics of physical systems. By reformulating the PDE solver as a recursive computation, one can use automatic differentiation techniques to learn its basic parameters from observational data. This data will be e.g. time dependent data (videos) from complex fluids or active matter, or microscopic time-lapse data from cell cultures. In a later phase of the project the data could also come from other domains, e.g. tracking data of the movement of individuals in large and dense crowds.
  2. Learning Neural Surrogates for PDE Solvers. PDEs are usually solved using numerical methods. Recently, deep learning methods have made much progress in training surrogate solvers from data produced by these numerical methods. These have the advantage of solving the PDE for new initial or boundary conditions many orders of magnitude faster. Therefore, such surrogates find widespread use in scenarios where the response of the system for many different initial or boundary conditions should be explored (e.g. in design studies) or when uncertainties in both the model parameters and the initial and boundary conditions need to be established (uncertainty quantification).

What do you have to offer?


Your experience and profile:

  • You hold a Master’s degree in Artificial Intelligence, Computational Science, Mathematics, Physics, or a related field;
  • You have experience in solving PDEs (analytically, numerically);
  • You have good working knowledge of machine learning methods, including modern deep learning architectures such as transformers and graph neural networks;
  • Familiarity with probabilistic programming and simulation-based inference is preferred but not a pre-requisite. Similarly, familiarity with methods for incorporating physical symmetries into equivariant networks is a plus, but not required;
  • You have sufficient programming skills in Python or related languages. Familiarity with scientific libraries (e.g. NumPy/SciPy, Stan, TensorFlow, PyTorch, JAX) is preferred;
  • You are interested in interdisciplinary research, crossing AI/Machine Learning, Computational Science, and application domains such as physics;
  • You are fluent in the English language, both in speaking and writing;
  • You are willing to take a light education task, as teaching assistant and in supervising Bachelor’s and Master’s students in their thesis research work.


Our offer


A temporary contract for 38 hours per week for the duration of 4 years (the initial contract will be for a period of 18 months and after satisfactory evaluation it will be extended for a total duration of 4 years). The preferred starting date is as soon as possible. This should lead to a dissertation (PhD thesis). We will draft an educational plan that includes attendance of courses and (international) meetings. We also expect you to assist in teaching undergraduates and master students.


The gross monthly salary, based on 38 hours per week and dependent on relevant experience, ranges between € 2,541 in the first year to € 3,247 in the last year (scale P). This does not include the 8% holiday allowance and the 8,3% year-end allowance the UvA offers. The UFO profile PhD Candidate is applicable. A favourable tax agreement, the ‘30% ruling’, may apply to non-Dutch applicants. The Collective Labour Agreement of Universities of the Netherlands is applicable.


Besides the salary and a vibrant and challenging environment at Science Park we offer you multiple fringe benefits:

  • A thriving research environment. You will have access to a community of students and faculty at the AMLab, which has a tremendous collective expertise in AI research, as well as the students and faculty at the CSL, which has deep expertise in computational methods for modelling physical systems;
  • PhD training courses, both at the Faculty of Science and in our international networks (e.g. ELLIS), with travel support for attendance of international conferences;
  • An excellent placement record for internships and post-graduate employment at top-tier universities and companies;
  • 232 holiday hours per year (based on fulltime) and extra holidays between Christmas and 1 January;
  • Multiple courses to follow from our Teaching and Learning Centre;
  • A complete educational program for PhD students;
  • Multiple courses on topics such as leadership for academic staff;
  • Multiple courses on topics such as time management, handling stress and an online learning platform with 100+ different courses;
  • 7 weeks birth leave (partner leave) with 100% salary;
  • Partly paid parental leave;
  • The possibility to set up a workplace at home;
  • A pension at ABP for which UvA pays two third part of the contribution;
  • The possibility to follow courses to learn Dutch;
  • Help with housing for a studio or small apartment when you’re moving from abroad.

Are you curious to read more about our extensive package of secondary employment benefits, take a look here.


About us


The University of Amsterdam is the Netherlands' largest university, offering the widest range of academic programmes. At the UvA, 30,000 students, 6,000 staff members and 3,000 PhD candidates study and work in a diverse range of fields, connected by a culture of curiosity.


The Faculty of Science has a student body of around 8,000, as well as 1,800 members of staff working in education, research or support services. Researchers and students at the Faculty of Science are fascinated by every aspect of how the world works, be it elementary particles, the birth of the universe or the functioning of the brain.


The mission of the Informatics Institute (IvI) is to perform curiosity-driven and use-inspired fundamental research in Computer Science. The main research themes are Artificial Intelligence, Computational Science and Systems and Network Engineering. Our research involves complex information systems at large, with a focus on collaborative, data driven, computational and intelligent systems, all with a strong interactive component.


The Amsterdam Machine Learning Lab (AMLab) conducts research in the area of large scale modelling of complex data sources. This includes the development of new methods for probabilistic graphical models and non-parametric Bayesian models, the development of faster (approximate) inference and learning methods, deep learning, causal inference, reinforcement learning and multi-agent systems and the application of all of the above to large scale data domains in science and industry ('Big Data problems'). AMLab is co-directed by Max Welling and Jan-Willem van de Meent. AMLab positions itself in the AI research theme, and also with clear links to the Data Science theme of the Informatics Institute.


The Computational Science Lab (CSL) of the Informatics Institute aims to make dynamic complex systems tractable via computational science. We study a broad range of dynamics systems in fields ranging from biomedicine to urban, or socioeconomic systems. We also develop theory of dynamic complex systems based on concepts of information processing. You will be working within a group of researchers highly active on topics ranging from computational heamodynamics, cell suspensions, brain perfusion, immunology, but also simulation platforms and high performance computing for computational (bio)medicine.


Want to know more about our organisation? Read more about working at the University of Amsterdam.


Any questions?


Do you have questions about this vacancy? Or do you want to know more about our organisation? Please contact:


Job application


If you feel the profile fits you, and you are interested in the job, we look forward to receiving your application. You can apply online via the button below. We accept applications until and including 16 December 2022.


Applications should include the following information (all files besides your CV should be submitted in one single pdf file):

  • a detailed CV including the months (not just years) when referring to your education and work experience;
  • a letter of motivation;
  • a list of publications;
  • the names and email addresses of two references who can provide letters of recommendation (please do not include any reference letters in your application).


Please make sure to provide ALL requested documents mentioned above.
You can use the CV field to upload your resume as a separate pdf document. Use the Cover Letter field to upload the other requested documents, including the motivation letter, as one single pdf file.


Only complete applications received within the response period via the link below will be considered.


We will invite potential candidates for interviews soon after the expiration of the vacancy.


Als werkgever handhaaft de UvA een beleid van gelijke kansen. We vinden diversiteit belangrijk en zetten ons volledig in om een plek te zijn waar iedereen zich thuis voelt. We koesteren de onderzoekende geest en het doorzettingsvermogen en laten de ruimte om door te vragen. Bij ons vormen nieuwsgierigheid en creativiteit de heersende cultuur.

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