  Neural networks are increasingly used to estimate parameters in quantitative
MRI, in particular in magnetic resonance fingerprinting. Their advantages over
the gold standard non-linear least square fitting are their superior speed and
their immunity to the non-convexity of many fitting problems. We find, however,
that in heterogeneous parameter spaces, i.e. in spaces in which the variance of
the estimated parameters varies considerably, good performance is hard to
achieve and requires arduous tweaking of the loss function, hyper parameters,
and the distribution of the training data in parameter space. Here, we address
these issues with a theoretically well-founded loss function: the Cram\'er-Rao
bound (CRB) provides a theoretical lower bound for the variance of an unbiased
estimator and we propose to normalize the squared error with respective CRB.
With this normalization, we balance the contributions of hard-to-estimate and
not-so-hard-to-estimate parameters and areas in parameter space, and avoid a
dominance of the former in the overall training loss. Further, the CRB-based
loss function equals one for a maximally-efficient unbiased estimator, which we
consider the ideal estimator. Hence, the proposed CRB-based loss function
provides an absolute evaluation metric. We compare a network trained with the
CRB-based loss with a network trained with the commonly used means squared
error loss and demonstrate the advantages of the former in numerical, phantom,
and in vivo experiments.
