  In this work, we provide a mathematical formulation for error propagation in
flow trajectory prediction using data-driven turbulence closure modeling. Under
the assumption that the predicted state of a large eddy simulation prediction
must be close to that of a subsampled direct numerical simulation, we retrieve
an upper bound for the prediction error when utilizing a data-driven closure
model. We also demonstrate that this error is significantly affected by the
time step size and the Jacobian which play a role in amplifying the initial
one-step error made by using the closure. Our analysis also shows that the
error propagates exponentially with rollout time and the upper bound of the
system Jacobian which is itself influenced by the Jacobian of the closure
formulation. These findings could enable the development of new regularization
techniques for ML models based on the identified error-bound terms, improving
their robustness and reducing error propagation.
