{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,5]],"date-time":"2026-05-05T03:17:56Z","timestamp":1777951076269,"version":"3.51.4"},"reference-count":15,"publisher":"IOP Publishing","issue":"3","license":[{"start":{"date-parts":[[2026,5,4]],"date-time":"2026-05-04T00:00:00Z","timestamp":1777852800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2026,5,4]],"date-time":"2026-05-04T00:00:00Z","timestamp":1777852800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/iopscience.iop.org\/info\/page\/text-and-data-mining"}],"funder":[{"name":"BMW Group"},{"name":"Rice University Wagoner Foreign Study Scholarship"}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"short-container-title":["Mach. Learn.: Sci. Technol."],"published-print":{"date-parts":[[2026,6,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Recent advances in protein structure prediction, such as AlphaFold, have demonstrated the power of deep neural architectures like the Evoformer for capturing complex spatial and evolutionary constraints on protein conformation. However, the depth of the Evoformer, comprising 48 stacked blocks, introduces high computational costs and rigid layerwise discretization. Inspired by neural ordinary differential equations (Neural ODEs), we propose a continuous-depth formulation of the Evoformer, replacing its 48 discrete blocks with a Neural ODE parameterization that preserves its core attention-based operations. This continuous-time Evoformer achieves constant memory cost (in depth) via the adjoint method, while allowing a principled trade-off between runtime and accuracy through adaptive ODE solvers. Benchmarking on protein structure prediction tasks, we find that the Neural ODE-based Evoformer produces structurally plausible predictions and reliably captures certain secondary structure elements, such as\n                    <jats:italic>\u03b1<\/jats:italic>\n                    -helices, though it does not fully replicate the accuracy of the original architecture. However, our model achieves this performance using dramatically fewer resources, just 17.5\u2009h of training on a single GPU, providing a proof of principle that continuous-depth models can serve as a lightweight alternative for biomolecular modeling. This work opens new directions for efficient and adaptive protein structure prediction frameworks.\n                  <\/jats:p>","DOI":"10.1088\/2632-2153\/ae5c55","type":"journal-article","created":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T22:53:30Z","timestamp":1775602410000},"page":"035008","update-policy":"https:\/\/doi.org\/10.1088\/crossmark-policy","source":"Crossref","is-referenced-by-count":1,"title":["Protein folding with neural ordinary differential equations"],"prefix":"10.1088","volume":"7","author":[{"ORCID":"https:\/\/orcid.org\/0009-0002-5237-8972","authenticated-orcid":true,"given":"Arielle","family":"Sanford","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0006-5775-9730","authenticated-orcid":false,"given":"Shuo","family":"Sun","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6386-0230","authenticated-orcid":true,"given":"Christian B","family":"Mendl","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"266","published-online":{"date-parts":[[2026,5,4]]},"reference":[{"key":"mlstae5c55bib1","doi-asserted-by":"publisher","first-page":"583","DOI":"10.1038\/s41586-021-03819-2","type":"journal-article","article-title":"Highly accurate protein structure prediction with AlphaFold","volume":"596","author":"Jumper","year":"2021","journal-title":"Nature"},{"key":"mlstae5c55bib2","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1038\/s41586-024-07487-w","type":"journal-article","article-title":"Accurate structure prediction of biomolecular interactions with AlphaFold 3","volume":"630","author":"Abramson","year":"2024","journal-title":"Nature"},{"key":"mlstae5c55bib3","doi-asserted-by":"publisher","first-page":"D439","DOI":"10.1093\/nar\/gkab1061","type":"journal-article","article-title":"AlphaFold protein structure database: massively expanding the structural coverage of protein-sequence space with high-accuracy models","volume":"50","author":"Varadi","year":"2021","journal-title":"Nucleic Acids Res."},{"key":"mlstae5c55bib4","type":"conference-proceedings","article-title":"Neural ordinary differential equations","volume":"vol 31","author":"Chen","year":"2018"},{"key":"mlstae5c55bib5","first-page":"pp 6696","type":"conference-proceedings","article-title":"Neural controlled differential equations for irregular time series","volume":"vol 33","author":"Kidger","year":"2020"},{"key":"mlstae5c55bib6","article-title":"Neural SDE: stabilizing neural ode networks with stochastic noise","author":"Liu","year":"2019","type":"preprint"},{"key":"mlstae5c55bib7","article-title":"Predicting ordinary differential equations with transformers","author":"Becker","year":"2023","type":"preprint"},{"key":"mlstae5c55bib8","doi-asserted-by":"publisher","first-page":"15278","DOI":"10.1021\/acs.iecr.3c01639","type":"journal-article","article-title":"Fluid-GPT: efficient predictions of particle trajectories and erosion","volume":"62","author":"Yang","year":"2023","journal-title":"Ind. 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