PhD Position in Learning PDEs from Data

Updated: 3 months ago
Job Type: Temporary
Deadline: 16 Dec 2022

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:

  • 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.
  • 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).

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