{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,13]],"date-time":"2026-10-13T14:51:31Z","timestamp":1791903091956,"version":"4.3.4"},"reference-count":57,"publisher":"American Chemical Society (ACS)","issue":"19","license":[{"start":{"date-parts":[[2026,9,25]],"date-time":"2026-09-25T00:00:00Z","timestamp":1790294400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100006151","name":"Basic Energy Sciences","doi-asserted-by":"publisher","award":["DESC0002128"],"award-info":[{"award-number":["DESC0002128"]}],"id":[{"id":"10.13039\/100006151","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100023581","name":"National Science Foundation Graduate Research Fellowship Program","doi-asserted-by":"publisher","award":["DGE-2039656"],"award-info":[{"award-number":["DGE-2039656"]}],"id":[{"id":"10.13039\/100023581","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,10,13]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Chemical potentials are among the most important properties that can be obtained from a molecular simulation since they define many technologically relevant collective properties such as solubilities and activity coefficients. The chemical potential of a species in solution is typically obtained by computing the free energy change of adding that species into a bulk system, a calculation typically very expensive for systems of technological interest such as electrolytes, due to the lack of phase space overlap between \u201cnot-inserted\u2033 and \u201cinserted\u2033 states. Recently, normalizing flows have been introduced as a way to accelerate free energy computations by learning a bijective function to map the configuration space of one Boltzmann distribution onto another. Currently, these trainable mappings are constructed to be as expressive as possible, theoretically having the ability to create a perfect mapping between states. This expressivity makes them difficult to train, limiting their ability to be generated \u201con-the-fly\u2033 for any new free energy estimation challenge, and in practice these mappings have shown only modest performance improvements for liquid systems. We address these issues by introducing the concept of a \u201cminimal\u2033 normalizing flow (MNF). This is a trainable bijective mapping that is intentionally limited in expressivity and instead applies low-dimensional, physically informed transformations. Useful MNFs can be trained in approximately 1 min of GPU time due to their simplicity and our introduction of a novel loss function based on the Bhattacharyya distance as an alternative to the Kullback\u2013Leibler divergence. We show how calculations of chemical potentials of pure and binary Lennard-Jones particle systems can be accelerated by at least an order of magnitude with a simple radial mapping. For a more complex illustration of the approach, we apply a two-dimensional (radial and orientational) mapping to the solvation of sodium and chloride ions in water, showing that MNFs can increase the effective sample size by as much as 3 times for charging free energy calculations and as much as 8 times for calculating free energy changes due to force field perturbations. This provides the foundation for the development of physically informed normalizing flows that can accelerate complex free energy calculations while retaining low training costs.<\/jats:p>","DOI":"10.1021\/acs.jctc.6c01266","type":"journal-article","created":{"date-parts":[[2026,9,25]],"date-time":"2026-09-25T15:02:43Z","timestamp":1790348563000},"page":"9976-9991","source":"Crossref","is-referenced-by-count":0,"title":["Accelerating Chemical\nPotential Calculations with\nMinimal Normalizing Flows"],"prefix":"10.1021","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-6673-9195","authenticated-orcid":true,"given":"Philippe B.","family":"Baron","sequence":"first","affiliation":[{"name":"Princeton University , , , ,","place":["Princeton, New Jersey, United States, 08540"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8152-6615","authenticated-orcid":true,"given":"Athanassios Z.","family":"Panagiotopoulos","sequence":"additional","affiliation":[{"name":"Princeton University , , , ,","place":["Princeton, New Jersey, United States, 08540"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"316","published-online":{"date-parts":[[2026,9,25]]},"reference":[{"key":"2026101310082063500_cit1","doi-asserted-by":"publisher","first-page":"673","DOI":"10.1038\/s41586-023-05905-z","article-title":"Computational approaches streamlining drug discovery","volume":"616","author":"Sadybekov","year":"2023","journal-title":"Nature"},{"key":"2026101310082063500_cit2","doi-asserted-by":"publisher","first-page":"3137","DOI":"10.1021\/acs.accounts.5c00465","article-title":"On Free Energy Calculations\nin Drug\nDiscovery","volume":"58","author":"Ghidini","year":"2025","journal-title":"Acc. 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Matter"},{"key":"2026101310082063500_cit6","doi-asserted-by":"publisher","first-page":"124504","DOI":"10.1063\/1.4943780","article-title":"Consensus on the solubility of NaCl\nin water from computer simulations using the chemical potential route","volume":"144","author":"Benavides","year":"2016","journal-title":"J. Chem. Phys."},{"key":"2026101310082063500_cit7","doi-asserted-by":"publisher","first-page":"224117","DOI":"10.1063\/5.0304180","article-title":"Efficient\ncalculation of crystal-solution coexistence\nlines for aqueous electrolytes","volume":"163","author":"Baron","year":"2025","journal-title":"J. Chem. Phys."},{"key":"2026101310082063500_cit8","doi-asserted-by":"publisher","first-page":"3677","DOI":"10.1021\/acs.jctc.0c00161","article-title":"Molecular\nForce Field Development\nfor Aqueous Electrolytes: 2. Polarizable Models Incorporating Crystalline\nChemical Potential and Their Accurate Simulations of Halite, Hydrohalite,\nAqueous Solutions of NaCl, and Solubility","volume":"16","author":"Dockal","year":"2020","journal-title":"J.\nChem. Theory Comput."},{"key":"2026101310082063500_cit9","doi-asserted-by":"publisher","first-page":"010903","DOI":"10.1063\/5.0012102","article-title":"Simulations\nof activities, solubilities, transport properties, and nucleation\nrates for aqueous electrolyte solutions","volume":"153","author":"Panagiotopoulos","year":"2020","journal-title":"J.\nChem. Phys."},{"key":"2026101310082063500_cit10","doi-asserted-by":"publisher","first-page":"2460","DOI":"10.1021\/acs.jctc.9b00941","article-title":"Lennard-Jones Parameters Determined\nto Reproduce the Solubility of NaCl and KCl in SPC\/E, TIP3P, and TIP4P\/2005\nWater","volume":"16","author":"Yagasaki","year":"2020","journal-title":"J. Chem. Theory Comput."},{"key":"2026101310082063500_cit11","doi-asserted-by":"publisher","first-page":"214112","DOI":"10.1063\/5.0272640","article-title":"Top-down\noptimization of aqueous electrolyte force\nfields to model chemical potentials and solubilities","volume":"162","author":"Baron","year":"2025","journal-title":"J. Chem. Phys."},{"key":"2026101310082063500_cit12","doi-asserted-by":"publisher","first-page":"1420","DOI":"10.1063\/1.1740409","article-title":"High-temperature\nequation of state by a perturbation method. I. Nonpolar gases","volume":"22","author":"Zwanzig","year":"1954","journal-title":"J. Chem. Phys."},{"key":"2026101310082063500_cit13","doi-asserted-by":"publisher","first-page":"245","DOI":"10.1016\/0021-9991(76)90078-4","article-title":"Efficient\nestimation of free energy differences from Monte Carlo data","volume":"22","author":"Bennett","year":"1976","journal-title":"J. Comput. 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Chem.\nTheory Comput."},{"key":"2026101310082063500_cit33","doi-asserted-by":"publisher","first-page":"184111","DOI":"10.1063\/5.0320214","article-title":"Assessing generative modeling approaches\nfor free energy estimates\nin condensed matter","volume":"164","author":"Schebek","year":"2026","journal-title":"J. Chem. Phys."},{"key":"2026101310082063500_cit34","doi-asserted-by":"publisher","first-page":"124104","DOI":"10.1063\/5.0164662","article-title":"Learned\nmappings for targeted free\nenergy perturbation between peptide conformations","volume":"159","author":"Willow","year":"2023","journal-title":"J. Chem. 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J.; Katariya, Y.; Leary, C.; Maclaurin, D.; Necula, G.; Paszke, A.; VanderPlas, J.; et al., JAX: composable transformations\nof Python+NumPy programs, 2018; http:\/\/github.com\/jax-ml\/jax."},{"key":"2026101310082063500_cit54","first-page":"7511","article-title":"Neural Spline Flows","volume-title":"Proceedings\nof the 33rd International Conference on Neural\nInformation Processing Systems","author":"Durkan","year":"2019"},{"key":"2026101310082063500_cit55","doi-asserted-by":"publisher","first-page":"8511","DOI":"10.1021\/acs.jpcb.1c04019","article-title":"Individual Ion Activity Coefficients\nin Aqueous Electrolytes\nfrom Explicit-Water Molecular Dynamics Simulations","volume":"125","author":"Saravi","year":"2021","journal-title":"J. Phys. Chem. B"},{"key":"2026101310082063500_cit56","doi-asserted-by":"publisher","first-page":"3330","DOI":"10.1021\/acs.jpcb.7b09861","article-title":"System-Size Dependence of Electrolyte\nActivity Coefficients\nin Molecular Simulations","volume":"122","author":"Young","year":"2018","journal-title":"J. Phys. Chem. B"},{"key":"2026101310082063500_cit57","doi-asserted-by":"publisher","first-page":"2079","DOI":"10.1002\/jcc.25520","article-title":"Molecular Enhanced\nSampling with Autoencoders: On-The-Fly\nCollective Variable Discovery and Accelerated Free Energy Landscape\nExploration","volume":"39","author":"Chen","year":"2018","journal-title":"J. Comput. Chem."}],"container-title":["Journal of Chemical Theory\nand Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubs.acs.org\/doi\/pdf\/10.1021\/acs.jctc.6c01266","content-type":"application\/pdf","content-version":"vor","intended-application":"unspecified"},{"URL":"https:\/\/pubs.acs.org\/jctcce\/article-pdf\/doi\/10.1021\/acs.jctc.6c01266\/68656492\/acs.jctc.6c01266.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pubs.acs.org\/jctcce\/article-pdf\/doi\/10.1021\/acs.jctc.6c01266\/68656492\/acs.jctc.6c01266.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,13]],"date-time":"2026-10-13T14:29:47Z","timestamp":1791901787000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.acs.org\/jctcce\/article\/22\/19\/9976\/5438754\/Accelerating-Chemical-Potential-Calculations-with"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9,25]]},"references-count":57,"journal-issue":{"issue":"19","published-online":{"date-parts":[[2026,9,25]]},"published-print":{"date-parts":[[2026,10,13]]}},"URL":"https:\/\/doi.org\/10.1021\/acs.jctc.6c01266","relation":{},"ISSN":["1549-9618","1549-9626"],"issn-type":[{"value":"1549-9618","type":"print"},{"value":"1549-9626","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,9,25]]}}}