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The algorithm, including \u201cpurification of the basis,\u201d 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 further compact a purified basis and, more significantly, to use the reverse differentiation approach to greatly speed up gradient evaluation. We demonstrate this for our recent four-body water interaction potential. Comparisons of training and testing precision on the MD17 database of energies and gradients (forces) for ethanol against numerous machine-learning 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\u20131000 times faster. Finally, a new PIP potential energy surface (PES) is reported for ethanol based on a more extensive dataset of energies and gradients than in the MD17 database. Diffusion Monte Carlo calculations that fail on MD17-based PESs are successful using the new PES.<\/jats:p>","DOI":"10.1063\/5.0080506","type":"journal-article","created":{"date-parts":[[2022,1,3]],"date-time":"2022-01-03T12:34:23Z","timestamp":1641213263000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":49,"title":["Permutationally invariant polynomial regression for energies and gradients, using reverse differentiation, achieves orders of magnitude speed-up with high precision compared to other machine learning methods"],"prefix":"10.1063","volume":"156","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2566-9539","authenticated-orcid":false,"given":"Paul L.","family":"Houston","sequence":"first","affiliation":[{"name":"Department of Chemistry and Chemical Biology, Cornell University 1 , Ithaca, New York 14853, USA and , Atlanta, Georgia 30332, USA"},{"name":"Department of Chemistry and Biochemistry, Georgia Institute of Technology 1 , Ithaca, New York 14853, USA and , Atlanta, Georgia 30332, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8889-4851","authenticated-orcid":false,"given":"Chen","family":"Qu","sequence":"additional","affiliation":[{"name":"Department of Chemistry and Biochemistry, University of Maryland 2 , College Park, Maryland 20742, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6191-5584","authenticated-orcid":false,"given":"Apurba","family":"Nandi","sequence":"additional","affiliation":[{"name":"Department of Chemistry and Cherry L. 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