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

• Given a space that you can only test at points, we introduce a heuristic that can learn what is present.
• The heuristic applies a mesh to the space and refines it around interesting points.
• This heuristic can reduce the work to discover features by an order of magnitude.
• These spaces arise in studies of the phase diagrams of self-organizing systems.

In a complex self-organizing system, small changes in the interactions between the system's components can result in different emergent macrostructures or macrobehavior. In chemical engineering and material science, such spontaneously self-assembling systems, using polymers, nanoscale or 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 correspond 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, however. In general, one would like to learn the most information using the fewest queries. Here we introduce a learning heuristic for issuing queries to map and search a two-dimensional parameter space. Using a method inspired by adaptive mesh refinement, the heuristic iteratively issues batches of queries to be executed in parallel based on past information. By adjusting the search criteria, different types of searches (for example, a uniform search, exploring boundaries, sampling all regions equally) can be flexibly implemented. We show that this method will densely search the space, while preferentially targeting certain features. 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.

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