Following the 2012 breakthrough in deep learning for classification and visions problems, the last decade has seen tremendous raise of interest in machine learning in a wider mathematical research community from foundational research through field specific analysis to applications. 

As data is at the core of any inverse problem, it was a natural direction for the field to investigate how machine learning could aid various aspects of inversion yielding numerous approaches from somewhat ad-hoc but very effective like learned unrolled methods to provably convergent learned regularisers with everything in between. In this talk I will review some on these developments through a lens of the research of our group.   

We introduce a new methodology based on the multirevolution idea for constructing integrators for stochastic differential equations in the situation where the fast oscillations themselves are driven by a Stratonovich noise. Applications include in particular highly-oscillatory Kubo oscillators and spatial discretizations of the nonlinear Schrödinger equation with fast white noise dispersion. We construct a method of weak order two with computational cost and accuracy both independent of the stiffness of the oscillations. A geometric modification that conserves exactly quadratic invariants is also presented. If time allows, we will discuss ongoing work on uniformly accurate methods for such systems. This is a joint work with Gilles Vilmart.

We introduce a novel approach to global optimization  via continuous-time dynamic programming and Hamilton-Jacobi-Bellman (HJB) PDEs. For non-convex, non-smooth objective functions,  we reformulate global optimization as an infinite horizon, optimal asymptotic stabilization control problem. The solution to the associated HJB PDE provides a value function which corresponds to a (quasi)convexification of the original objective.  Using the gradient of the value function, we obtain a  feedback law driving any initial guess towards the global optimizer without requiring derivatives of the original objective. We then demonstrate that this HJB control law can be integrated into other global optimization frameworks to improve its performance and robustness.