  In this study, it is demonstrated that Recurrent Neural Networks (RNNs),
specifically those utilizing Gated Recurrent Unit (GRU) architecture, possess
the capability to learn the complex dynamics of rate-and-state friction laws
from synthetic data. The data employed for training the network is generated
through the application of traditional rate-and-state friction equations
coupled with the aging law for state evolution. A novel aspect of our approach
is the formulation of a loss function that explicitly accounts for the direct
effect by means of automatic differentiation. It is found that the RNN, with
its GRU architecture, effectively learns to predict changes in the friction
coefficient resulting from velocity jumps (with and without noise in the target
data), thereby showcasing the potential of machine learning models in
understanding and simulating the physics of frictional processes.
