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Published May 19, 2023 | Published
Journal Article Open

Learning Markovian Homogenized Models in Viscoelasticity


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 for deriving 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 that captures the memory. The memory is encapsulated in the evolution of an appropriate finite set of hidden variables, which are 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, using similar techniques, is given.

Additional Information

The U.S. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes. Copyright is owned by SIAM to the extent not limited by these rights. The authors are grateful to Matt Levine for helpful discussions about training RNN models and to Pierre Suquet for helpful discussions about homogenization theory. The work of the first, second, and third authors was sponsored by the U.S. Army Research Laboratory and was accomplished under Cooperative Agreement W911NF-12-2-0022. The work of the fourth author was funded by the Department of Energy Computational Science Graduate Fellowship under award DE-SC002111.

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August 20, 2023
October 20, 2023