A learning heuristic for space mapping and searching self-organizing systems using adaptive mesh refinement

In a complex self-organizing system, small changes in the interactions between the system’s components can result in different emergent macrostructures or macro-behaviors. In chemical engineering and material science, such spontaneously self-assembling systems, using polymers, nanoor colloidal scale particles, DNA, or other precursors, are an attractive way to create materials that are precisely engineered at a fine scale. Changes to the interactions can often be described by a set of parameters. Different contiguous regions in this parameter space corresponds to different ordered states. Since these ordered states are emergent, often experiment, not analysis, is necessary to create a diagram of ordered states over the parameter space. By issuing queries to points in the parameter space, e.g performing a computational or physical experiment, ordered states can be discovered and mapped . Queries can be costly in terms of resources or time. In general, one would like to learn the most about a space for the fewest total number of queries. Here we introduce a learning heuristic for issuing queries to map and search a two-dimensional parameter space. Using a method inspired by the adaptive mesh refinement method, the heuristic iteratively issues batches of queries to be executed in parallel, based on what has been learned from previous iterations. By adjusting the search criteria of the heuristic, different types of searches (e.g. a uniform search, exploring boundaries, sampling all regions equally) can be flexibly implemented. We show that this method will densely search the space in the limit of infinite queries while preferentially targeting certain features of space. Using numerical examples, including a study simulating the self assembly of complex crystals, we show how this heuristic can discover new regions and map boundaries more accurately than a uniformly distributed set of queries.