Postdoc Optimal Transport & Machine Learning Methods for Inverse Problems

We are looking for a postdoctoral candidate who will work within the new research group on Data-Driven Scientic Computing led by Olga Mula, located at CASA, the Center for Analysis, Scientic Computing and Applications of TU Eindhoven.

The goal of the group is to develop algorithms mixing the strengths of physics-based PDE methods with the ones offered by data-driven machine learning approaches. Both strategies have classically been considered separately, despite that they often provide complementary descriptions of the same reality. The group will address the growing need to combine them in an optimal way, using strategies that will depend on the application.

The candidate's research project will consist in developing numerical methods for inverse problems where the goal is to recover the state of a physical system based on a limited set of noisy partial observations. The study will focus on physical phenomena that are modeled with high-dimensional PDEs involving strong advection effects. Examples of such equations can be conservation laws, transport or kinetic equations, Fokker-Planck equations, or Mean Field Game Equations. For this type of advection-dominated problems, it is known that classical linear approximation methods are not well suited. The postdoctoral candidate will address this problem by developing nonlinear methods based on optimal transport, and machine learning. In order to address high dimensionality, and many-query evaluations, nonlinear model order reduction of parametric PDEs may be required.

References [1, 2] could serve as a starting point for the intended research project.

References
[1] V. Ehrlacher, D. Lombardi, O. Mula, and F.-X. Vialard. Nonlinear model reduction on metric
spaces. application to one-dimensional conservative pdes in wasserstein spaces. ESAIM M2AN,
54(6):21592197, 2020.
[2] Albert Cohen, Wolfgang Dahmen, Olga Mula, and James Nichols. Nonlinear reduced models for
state and parameter estimation. SIAM/ASA Journal on Uncertainty Quantication, 10(1):227
267, 2022.