  This work combines state-of-the-art machine learning techniques with
highest-level electronic structure calculations and full-dimensional quantum
tunneling calculations to obtain a quantitative characterization of tunneling
splittings for system sizes that are currently out of reach using traditional
approaches. For intramolecular hydrogen transfer in tropolone, the best
computed splitting including perturbative corrections in the ring-polymer
instanton calculations is 0.94 cm$^{-1}$ and compares with 0.974 cm$^{-1}$ from
experiments. On the other hand, for intermolecular double hydrogen transfer in
the (propiolic acid)-(formic acid) dimer, the computations yield 0.0147
cm$^{-1}$ which is larger by 40 % compared with experiment (0.0097 cm$^{-1}$)
but still in much better agreement than previous attempts (0.63 cm$^{-1}$). The
strategy pursued in the present work is applicable to yet larger systems and
other properties of interest and provides a rational route for highest-accuracy
energy functions for prediction and benchmarking electronic structure methods
vis-a-vis experiments.
