Background

Many of the most challenging problems in predictive modelling involve multimodal energy landscapes. Such systems can get trapped in local energy minima on long simulation timescales, preventing efficient exploration of the state space. Polymer models (of systems such as proteins [1]) create particularly significant challenges due to their long structures. Each mode corresponds to a different polymer folding, and the challenge is to unfold and refold the polymer on short timescales to sample all physical configurations.

Many techniques have attempted to tackle the general problem, with equi-energy sampling offering particular promise [2].This biases the system away from visited configurations – and samples are re-weighted once the process has converged. The technique essentially amounts, however, to promoting the temperature to an auxiliary variable – leading to poor convergence when integrated into traditional gradient-based simulation methods (e.g., molecular dynamics) as the optimal time step changes with temperature. Metropolis Monte Carlo (MC) is numerically stable and so circumvents the time-step issue, but its diffusive dynamics mix very slowly.

Recent advances in comp-stats [3] and stat-phys [4] have led to state-of-the-art MC sampling algorithms that drive the system through its state space with ballistic-style dynamics. These piecewise deterministic Markov processes (PDMPs) are the optimal candidate for tackling these issues as they combine full numerical stability with efficient gradient-based dynamics – implying the capacity to surpass the state-of-the-art. This project will develop equi-energy sampling within PDMPs. We will integrate the method into existing PDMPs for atomistic water [5, 6] then move on to polymer models. This will bias polymers away from visited foldings. Future integration of machine-learn potentials will accelerate atomistic sampling of physical foldings of polymers in solution and polymer melts.

References

[1] Dill & MacCallum, Science 338, 6110 (2012)

[2] Kim et al., Phys. Rev. Lett. 97, 050601 (2006)

[3] Bierkens & Roberts, Ann. Appl. Probab. 27, 846 (2017)

[4] Bernard et al., Phys. Rev. E 80, 056704 (2009)

[5] Faulkner et al., J. Chem. Phys. 149, 064113 (2018)

[6] Hoellmer, Qin, Faulkner et al., Comput Phys Commun 253, 107168 (2020)

Project workings

This project brings together expertise from the Warwick Centre for Predictive Modelling, Warwick Statistics and Warwick Physics. The taught HetSys programme will provide the applicant with the advanced skills required to develop the proposed algorithms, allowing us to tackle this exciting and broadly applicable research project.

Large-scale simulations will be performed on Warwick’s high-performance-computing infrastructure, and the applicant will develop a high level of research software engineering skills over the course of the project. There is also the potential for travel to visit related researchers.

Informal enquiries to michael.faulkner@warwick.ac.ukare welcome.