Stochastic upscaling in solid mechanics: An excercise in machine learning

Abstract

This paper presents a consistent theoretical and computational framework for upscaling in random microstructures. We adopt an information theoretic approach in order to quantify the informational content of the microstructural details and find ways to condense it while assessing quantitatively the approximation introduced. In particular, we substitute the high-dimensional microscale description by a lower-dimensional representation corresponding for example to an equivalent homogeneous medium. The probabilistic characteristics of the latter are determined by minimizing the distortion between actual macroscale predictions and the predictions made using the coarse model. A machine learning framework is essentially adopted in which a vector quantizer is trained using data generated computationally or collected experimentally. Several parallels and differences with similar problems in source coding theory are pointed out and an efficient computational tool is employed. Various applications in linear and non-linear problems in solid mechanics are examined.