Identifying invariant ergodic subsets and barriers to mixing by cutting and shuffling: Study in a birotated hemisphere

Lynn, Thomas F.; Ottino, Julio M.; Umbanhowar, Paul B.; Lueptow, Richard M.

doi:10.1103/physreve.101.012204

Cited by 4 publications

(1 citation statement)

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“…In one dimension, cutting-and-shuffling is described mathematically by interval exchange transforms (IETs) [42][43][44][45][46][47][48][49][50], a subset of broader cut-and-shuffle mixing actions called piecewise isometries (PWIs) [41,[51][52][53]. An IET cuts a one dimensional domain into intervals that are then reordered according to a specific permutation.…”

Section: Introductionmentioning

confidence: 99%

Potentialities and Limitations of Machine Learning to Solve Mixing Problems: A Case Study

Lynn¹,

Ottino²,

Lueptow³

et al. 2022

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Cut-and-shuffle mixing is an instructive candidate system with which to assess the potential of machine learning (ML) as an approach to solve difficult mixing problems. We focus on a specific subset of cut-and-shuffle systems, the one-dimensional interval exchange transform. This class of mixing operations is well studied, and a simple mixing methodology, which we refer to as the longest segment (LS) method, works well under a broad range of situations. We use supervised learning to train a neural network (NN) to emulate the LS mixing algorithm for mixing a one-dimensional domain of two species. We find that a generic deep NN can emulate the LS method with good accuracy but cannot generalize to conditions significantly outside its training repertoire. The challenges in defining the mixing problem and generalizing a ML mixing approach are indicative of those expected for more complex systems where optimal or near optimal mixing methods remain unknown.

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