A05 - Probing scales in equilibrated systems by optimal nonequilibrium forcing
Head(s): Prof. Dr. Carsten Hartmann (BTU Cottbus-Senftenberg), Prof. Dr. Christof Schütte (ZIB), PD Dr. Marcus Weber (ZIB)
Project member(s): Enric Ribera Borrell, Dr. Lara Neureither, Dr. Lorenz Richter, Alexander Sikorski, Dr. Martin Weiser
Participating institution(s): FU Berlin, ZIB, (BTU Cottbus-Senftenberg)
The project is concerned with the analysis and simulation of rare statistical fluctuations of stochastic processes with multiple scales. More specifically, we study methods for the efficient sampling of rare events in molecular systems – addressing one of the main computational bottlenecks in molecular dynamics simulation. One of the key conceptual insights from the previous funding period was that sampling of path-dependent properties associated with rare events (e.g., transition probabilities or pathways) can be recast as a variational problem that boils down to a stochastic optimal control problem, for which several equivalent formulations
exist. We have studied the connection between the associated Hamilton–Jacobi–Bellman (HJB) equations, backward stochastic differential equations (BSDE) and variational inference, all of which can be expressed in form of minimisation problems that involve divergences between probability measures on the space of trajectories (path space). The latter include a novel logvariance divergence that has been proved to have favourable scaling properties in high dimensions and give rise to iterative machine-learning (ML) based methods to approximate optimal controls.
In the next funding period, these results shall be generalised and applied to a high-dimensional multiscale system from molecular dynamics that addresses both fundamental research questions and biological applications. Specifically, we will study a receptor binding process, with the aim of obtaining precise estimates of binding rates and, if possible, dominant pathways. Computing binding rates or pathways by direct numerical simulation is hardly possible without using tailored sampling methods that can speed up the simulation of rare events and that are applicable to high-dimensional systems. Therefore we will study variational approximations of the optimal controls that can be obtained from low-dimensional representations of the molecular systems (e.g., low-dimensional stochastic differential equations). A central question here is whether low-dimensional representations in terms of molecular reaction coordinates (that are often available for molecular systems in equilibrium) provide suitable representations also for controlled non-equilibrium systems.
Regarding the computational aspects of the targeted application, we will generalise our variational formulation of importance sampling to problems that involve random stopping times that appear in connection with reaction rate and reaction path computations. This requires to devise robust algorithms to solve BSDEs with random (i.e., indefinite) terminal time and possibly non-smooth terminal conditions that represent elliptic boundary value problems of HJB type. Moreover, we will extend our various results on suboptimal importance sampling (e.g., for slow-fast systems) to the random stopping time setting, which will then provide the
theoretical basis to handle large-scale systems from biomolecular dynamics.