Hierarchical Pattern Discovery in Stochastic Lattice Systems

Abstract. We develop a hierarchical approach for pattern discovery in many-body stochastic systems, motivated by challenges in guiding engineering tasks for nanopattern formation in heteroepitaxial processes. Patterns in such systems have rich morphologies at mesoscales that change dramatically as control parameters vary; typically they form as a result of microscopic particle dynamics in a complex energy landscape, in the presence of stochastic fluctuations. Studying pattern formation mechanisms as functions of the control parameters of the system poses a significant computational challenge which is currently intractable with conventional Kinetic Monte Carlo (kMC) methods. We present hierarchical strategies towards this systems’ task goal by combining mesoscopic PDE and Coarse-Grained Monte Carlo (CGMC) approximations of kMC algorithms that we have developed in our earlier work. More precisely, (i) we employ deterministic mesoscopic PDE as means to obtain an approximate (and in principle rather crude) phase diagram of the system; subsequently, (ii) we employ adaptive CGMC at selected regions of the approximate phase diagram in order to refine it by including interactions and fluctuations properly. Our adaptivity framework allows us to obtain accurate and near-optimal coarse-grainings for each parameter regime, ensuring proper “knowledge representation”– in the information theory sense – of the complex system for the desired observables, e.g., spatial correlations, power spectra or scaling laws. In turn such tools can be also used in model reduction and control of the underlying complex systems.