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Five architectures are considered: a multilayer perceptron (MLP), an autoregressive neural network (ARNN), a long short-term memory (LSTM), a source-term surrogate for breakage and coalescence (RHS), and a Neural ordinary differential equation (Neural ODE). Surrogates are trained on thousands of QMOM simulations that span a wide range of initial droplet size, dispersed-phase holdup, and turbulent dissipation rate and are evaluated on 12,000 unseen operating conditions. The MLP, ARNN and LSTM accurately reproduce the decay of low-order moments, the nonlinear evolution of higher orders, and average diameters, with typical relative errors of a few percent and mean relative errors down to\n                    <jats:inline-formula>\n                      <jats:tex-math>$$10^{-3}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    , while delivering speed-ups of about three orders of magnitude relative to the direct QMOM numerical solution. The RHS and Neural ODE models are less data-efficient and exhibit larger errors; the former is also slower than QMOM because it requires an additional ODE integration. The proposed methodology is transferable to other homogeneous PBEs and provides a basis for extensions to spatially resolved CFD\u2013PBE frameworks.\n                  <\/jats:p>","DOI":"10.1007\/s00366-026-02335-z","type":"journal-article","created":{"date-parts":[[2026,5,26]],"date-time":"2026-05-26T03:54:12Z","timestamp":1779767652000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Deep learning for modeling the evolution of droplet size distribution in liquid\u2013liquid dispersed systems"],"prefix":"10.1007","volume":"42","author":[{"given":"Mazhar","family":"Bayazidi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Daniele","family":"Marchisio","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Agnese","family":"Marcato","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Antonio","family":"Buffo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,5,26]]},"reference":[{"issue":"9","key":"2335_CR1","doi-asserted-by":"publisher","first-page":"4109","DOI":"10.1021\/acs.iecr.2c03695","volume":"62","author":"G Tan","year":"2023","unstructured":"Tan G, Qian K, Jiang S, Wang J, Wang J (2023) Cfd-pbm investigation on droplet size distribution in a liquid\u2013liquid stirred tank: effect of impeller type. 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