  Permutationally invariant polynomial (PIP) regression has been used to obtain
machine-learned (ML) potential energy surfaces, including analytical gradients,
for many molecules and chemical reactions. Recently, the approach has been
extended to moderate size molecules and applied to systems up to 15 atoms. The
algorithm, including "purification of the basis", is computationally efficient
for energies; however, we found that the recent extension to obtain analytical
gradients, despite being a remarkable advance over previous methods, could be
further improved. Here we report developments to compact further a purified
basis and, more significantly, to use the reverse gradient approach to greatly
speed up gradient evaluation. We demonstrate this for our recent 4-body water
interaction potential. Comparisons of training and testing precision on the
MD17 database of energies and gradients (forces) for ethanol against GP-SOAP,
ANI, sGDML, PhysNet, pKREG, KRR, and other methods, which were recently
assessed by Dral and co-workers, are given. The PIP fits are as precise as
those using these methods, but the PIP computation time for energy and force
evaluation is shown to be 10 to 1000 times faster. Finally, a new PIP PES is
reported for ethanol based on a more extensive dataset of energies and
gradients than in the MD17 database. Diffusion Monte Carlo calculations which
fail on MD17-based PESs are successful using the new PES.
