In stochastic optimal control and conditional generative modelling, a central computational task is to modify a reference diffusion process to maximise a given terminal-time reward. Most existing methods require this reward to be differentiable, using gradients to steer the diffusion towards favourable outcomes. However, in many practical settings, like diffusion bridges, the reward is singular, taking an infinite value if the target is hit and zero otherwise. We introduce a novel framework, based on Malliavin calculus and path-space integration by parts, that enables the development of methods robust to such singular rewards. This allows our approach to handle a broad range of applications, including classification, diffusion bridges, and conditioning without the need for artificial observational noise. We demonstrate that our approach offers stable and reliable training, outperforming existing techniques. 

This talk will focus on systems-theoretic and control theory tools that help characterize the responses of nonlinear systems to external inputs, with an emphasis on how network structure “motifs” introduce constraints on finite-time, transient behaviors.  Of interest are qualitative features that are unique to nonlinear systems, such as non-harmonic responses to periodic inputs or the invariance to input symmetries. These properties play a key role as tools for model discrimination and reverse engineering in systems biology, as well as in characterizing robustness to disturbances. Our research has been largely motivated by biological problems at all scales, from the molecular (e.g., extracellular ligands affecting signaling and gene networks), to cell populations (e.g., resistance to chemotherapy due to systemic interactions between the immune system and tumors; drug-induced mutations; sensed external molecules triggering activations of specific neurons in worms), to interactions of individuals (e.g., periodic or single-shot non-pharmaceutical “social distancing'” interventions for epidemic control). Subject to time constraints, we'll briefly discuss some of these applications.