Predicting and explaining nonlinear material response using deep
  Physically Guided Neural Networks with Internal Variables
Javier Orera-Echeverria Jacobo Ayensa-Jiménez Manuel Doblare
Nonlinear materials are often difficult to model with classical state model
theory because they have a complex and sometimes inaccurate physical and
mathematical description or we simply do not know how to describe such
materials in terms of relations between external and internal variables. In
many disciplines, Neural Network methods have arisen as powerful tools to
identify very complex and non-linear correlations. In this work, we use the
very recently developed concept of Physically Guided Neural Networks with
Internal Variables (PGNNIV) to discover constitutive laws using a model-free
approach and training solely with measured force-displacement data. PGNNIVs
make a particular use of the physics of the problem to enforce constraints on
specific hidden layers and are able to make predictions without internal
variable data. We demonstrate that PGNNIVs are capable of predicting both
internal and external variables under unseen load scenarios, regardless of the
nature of the material considered (linear, with hardening or softening behavior
and hyperelastic), unravelling the constitutive law of the material hence
explaining its nature altogether, placing the method in what is known as
eXplainable Artificial Intelligence (XAI).