Learning Markovian Homogenized Models in Viscoelasticity

TL;DR

This paper develops a machine learning approach using recurrent neural networks to explicitly model the homogenized constitutive behavior of viscoelastic materials, capturing memory effects and enabling efficient simulations.

Contribution

It introduces a novel method to approximate homogenized viscoelastic models with RNNs, explicitly capturing memory effects from fine-scale simulations.

Findings

RNNs effectively approximate homogenized models

Memory effects are captured by internal variables

Simulations validate the approach

Abstract

Fully resolving dynamics of materials with rapidly-varying features involves expensive fine-scale computations which need to be conducted on macroscopic scales. The theory of homogenization provides an approach to derive effective macroscopic equations which eliminates the small scales by exploiting scale separation. An accurate homogenized model avoids the computationally-expensive task of numerically solving the underlying balance laws at a fine scale, thereby rendering a numerical solution of the balance laws more computationally tractable. In complex settings, homogenization only defines the constitutive model implicitly, and machine learning can be used to learn the constitutive model explicitly from localized fine-scale simulations. In the case of one-dimensional viscoelasticity, the linearity of the model allows for a complete analysis. We establish that the homogenized constitutive model may be approximated by a recurrent neural network (RNN) that captures the memory. The memory is encapsulated in the evolution of an appropriate finite set of internal variables, discovered through the learning process and dependent on the history of the strain. Simulations are presented which validate the theory. Guidance for the learning of more complex models, such as arise in plasticity, by similar techniques, is given.