  The discovery of novel experimental techniques often lags behind contemporary
theoretical understanding. In particular, it can be difficult to establish
appropriate measurement protocols without analytic descriptions of the
underlying system-of-interest. Here we propose a statistical learning framework
that avoids the need for such descriptions for ergodic systems. We validate
this framework by using Monte Carlo simulation and deep neural networks to
learn a mapping between low-field nuclear magnetic resonance spectra and proton
exchange rates in ethanol-water mixtures. We found that trained networks
exhibited normalized-root-mean-square errors of less than 1% for exchange rates
under 150 s-1 but performed poorly for rates above this range. This
differential performance occurred because low-field measurements are
indistinguishable from one another at fast exchange. Nonetheless, where a
discoverable relationship between indirect measurements and emergent dynamics
exists, we demonstrate the possibility of approximating it without the need for
precise analytic descriptions, allowing experimental science to flourish in the
midst of ongoing theoretical work
