  Neural Network Differential Equation (NN DE) solvers have surged in
popularity due to a combination of factors: computational advances making their
optimization more tractable, their capacity to handle high dimensional
problems, easy interpret-ability of their models, etc. However, almost all NN
DE solvers suffer from a fundamental limitation: they are trained using loss
functions that depend only implicitly on the error associated with the
estimate. As such, validation and error analysis of solution estimates requires
knowledge of the true solution. Indeed, if the true solution is unknown, we are
often reduced to simply hoping that a "low enough" loss implies "small enough"
errors, since explicit relationships between the two are not available/well
defined. In this work, we describe a general strategy for efficiently
constructing error estimates and corrections for Neural Network Differential
Equation solvers. Our methods do not require advance knowledge of the true
solutions and obtain explicit relationships between loss functions and the
error associated with solution estimates. In turn, these explicit relationships
directly allow us to estimate and correct for the errors.
