# A05 - Probing scales in equilibrated systems by optimal nonequilibrium forcing

**Head(s):** Prof. Dr. Carsten Hartmann (BTU Cottbus-Senftenberg), Prof. Dr. Susanna Röblitz (ZIB and University of Bergen), Prof. Dr. Christof Schütte (FU Berlin), PD Dr. Marcus Weber (ZIB)**Project member(s):** Dr. Ilja Klebanov, Lorenz Richter, Alexander Sikorski, Dr. Martin Weiser**Participating institution(s):** FU Berlin, ZIB

## Project Summary

This project is devoted to the analysis and simulation of rare statistical fluctuations in multiscale random dynamical systems beyond equilibrium that are driven by external forcing. Stochastic control theory is a key methodology in the project, specifically, we exploit an intimate duality between the cumulant generating functions of certain path functionals and entropy minimisation where the latter is interpreted within the framework of stochastic control theory. As a result we obtain a variational principle that determines a probability measure that we use to obtain precise importance sampling estimates of the quantities under consideration.

For multiscale system, the corresponding optimal controls can be highly oscillatory and thus the primary focus of this project so far has been to develop a thorough theoretical understanding of the variational problem using averaging, homogenisation and model reduction techniques. For large-scale systems having multiple time scales, especially without clear scale separation (such as large biomolecules with multivalent bonds), solving optimal control problems numerically can be notoriously difficult, which underpins the need for suitable data-driven model reduction and stochastic approximation techniques. These will be explored in the second funding period.

One of the key ideas for the second funding period of the project is to exploit the intimate connection of the variational formulation with Bayes’ Theorem and to develop algorithms that scale polynomially with the system dimension. Specifically, we plan to use this connection in both directions and (i) employ non-parametric inference techniques to approximate potentially high-dimensional optimal importance sampling distributions as Bayesian posteriors by minimizing either the corresponding cross-entropy or the relative entropy (a.k.a. Kullback-Leibler divergence), and (ii) use the connection with importance sampling to design efficient numerical strategies for data assimilation of multiscale diffusions.

A second idea is to study iterative methods for solving the variational problem in connection with model reduction techniques. Specifically, we plan to revisit approximate policy iteration and iterative numerical schemes for (decoupled) forward-backward stochastic differential equations. Here the key insight is that the semilinear dynamic programming equation associated with the aforementioned optimal control problem has a rather specific structure, which makes them amenable to data-driven model reduction methods that do not rely on scale separation, such as conditional expectation or Markov state modelling. The aim is to derive computable error bounds for situations, in which the high-dimensional optimal controls are approximated by a numerical discretisation of a reduced-order model.

We, furthermore, plan to study realistic molecular systems, aiming at the (in-silico) design of multivalent drug-like molecules with a specific multiscale dissociation profile. Solving either the corresponding data assimilation problems or the associated stochastic control problem cannot be done without using tailored sampling techniques, and the idea here is to reformulate available enhanced sampling algorithms, such as metadynamics or the adaptive biasing force method, using our optimal control framework. Not only will this lead to optimised molecular dynamics sampling algorithms, but, conversely, it will also allow us to incorporate prior information that is obtained from, e.g., a metadynamics simulation or a Markov state model into the construction of adaptive importance sampling schemes.