The project is inspired by the increasing development and interest in scientific machine learning applied to multi-fidelity data assimilation and inverse problems.  Methods to approximate probability measures have flourished recently also due to the implementation of efficient generative models in machine learning and more principled mathematical methods like diffusion models and transport maps.The target of our studies are inverse problems that concern the environmental sciences, possibly but not necessarily linked to computational fluid dynamics: from the dynamics of polar and continental ice sheets to the oceans acidification and circulation models, to cite a few. SISSA mathLab has a consistent expertise in optimal control and data assimilation for the environmental sciences. The MIT group has several ongoing projects in this area, including an ONR MURI focused on data assimilation for sea ice modeling, and a DOE SciDAC project focused on optimal sensor network design to inform aspects of earth system models. In particular, multi-scale models are sometimes constituted by hybrid systems featuring a coarse scale modeled by a partial differential equation (PDE) and a fine scale modeled by a stochastic differential equation (SDE) expanded on the whole spatial domain. The aim is to develop surrogate models also for this type of multi-scale coupled PDE-SDE systems.The main goals of this project are: (1) leveraging reduced order models in Bayesian inversion or in sequential Bayesian data assimilation; (2) the speed-up of the forward uncertainty propagation with our surrogate and generative models; (3) randomization through PDE-SDE coupled systems; (4) the application of our methodologies to tackle complex and large-scale inverse problems from the environmental sciences.

mathLab members involved: Gianluigi Rozza, Francesco Romor, Marco Tezzele