{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T01:07:09Z","timestamp":1780448829785,"version":"3.54.1"},"reference-count":45,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/100003187","name":"NSF","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100003187","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Journal of Computational Physics"],"published-print":{"date-parts":[[2026,10]]},"DOI":"10.1016\/j.jcp.2026.114986","type":"journal-article","created":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T22:50:27Z","timestamp":1777762227000},"page":"114986","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"C","title":["A parameterized Wasserstein Hamiltonian flow approach to solving nonlinear Schr\u00f6dinger equations"],"prefix":"10.1016","volume":"562","author":[{"given":"Hao","family":"Wu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1009-7726","authenticated-orcid":false,"given":"Shu","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaojing","family":"Ye","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Haomin","family":"Zhou","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"issue":"1","key":"10.1016\/j.jcp.2026.114986_bib0001","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0370-1573(81)90033-8","article-title":"Classical and quantum mechanics of the damped harmonic oscillator","volume":"80","author":"Dekker","year":"1981","journal-title":"Phys. Rep."},{"key":"10.1016\/j.jcp.2026.114986_bib0002","first-page":"714","article-title":"Quantum Mechanics of Many-Electron Systems","volume":"123","author":"Dirac","year":"1929","journal-title":"Proc. Royal Soc. A Math. Phys. Eng. Sci."},{"issue":"5221","key":"10.1016\/j.jcp.2026.114986_bib0003","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1126\/science.269.5221.198","article-title":"Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor","volume":"269","author":"Anderson","year":"1995","journal-title":"Science"},{"key":"10.1016\/j.jcp.2026.114986_bib0004","series-title":"Bose-Einstein condensation","article-title":"Bose-Einstein condensation \/ Lev Pitaevskii, Sandro Stringari","author":"Pitaevskii","year":"2003"},{"key":"10.1016\/j.jcp.2026.114986_bib0005","article-title":"Generative adversarial nets","volume":"27","author":"Goodfellow","year":"2014","journal-title":"Adv. Neural Inf. Process. Syst."},{"issue":"3","key":"10.1016\/j.jcp.2026.114986_bib0006","doi-asserted-by":"crossref","first-page":"1205","DOI":"10.1016\/j.jde.2019.08.046","article-title":"Wasserstein hamiltonian flows","volume":"268","author":"Chow","year":"2020","journal-title":"J. Diff. Eq."},{"key":"10.1016\/j.jcp.2026.114986_bib0007","series-title":"Optimal transport: old and new","volume":"338","author":"Villani","year":"2009"},{"issue":"1","key":"10.1016\/j.jcp.2026.114986_bib0008","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1137\/S1064827594277053","article-title":"Absorbing Boundary Conditions for the Schr\u00f6dinger Equation","volume":"21","author":"Fevens","year":"1999","journal-title":"SIAM J. Sci. Comput."},{"issue":"3","key":"10.1016\/j.jcp.2026.114986_bib0009","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1119\/1.1973991","article-title":"Computer-Generated Motion Pictures of One-Dimensional Quantum-Mechanical Transmission and Reflection Phenomena","volume":"35","author":"Goldberg","year":"1967","journal-title":"Am. J. Phys."},{"key":"10.1016\/j.jcp.2026.114986_bib0010","doi-asserted-by":"crossref","first-page":"6760","DOI":"10.1103\/PhysRevB.43.6760","article-title":"Absorbing boundary conditions for the finite-difference time-domain calculation of the one-dimensional Schr\u00f6dinger equation","volume":"43","author":"Shibata","year":"1991","journal-title":"Phys. Rev. B"},{"issue":"12","key":"10.1016\/j.jcp.2026.114986_bib0011","doi-asserted-by":"crossref","first-page":"8011","DOI":"10.1063\/1.1753661","article-title":"Analysis of the finite difference time domain technique to solve the Schr\u00f6dinger equation for quantum devices","volume":"95","author":"Soriano","year":"2004","journal-title":"J. Appl. Phys."},{"issue":"1","key":"10.1016\/j.jcp.2026.114986_bib0012","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3934\/krm.2013.6.1","article-title":"Mathematical theory and numerical methods for Bose-Einstein condensation","volume":"6","author":"Bao","year":"2013","journal-title":"Kinetic Related Mod."},{"issue":"1","key":"10.1016\/j.jcp.2026.114986_bib0013","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1137\/S1064827501393253","article-title":"Numerical Study of Time-Splitting Spectral Discretizations of Nonlinear Schr\u00f6dinger Equations in the Semiclassical Regimes","volume":"25","author":"Bao","year":"2003","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"10.1016\/j.jcp.2026.114986_bib0014","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1016\/j.jcp.2004.11.008","article-title":"Computing multivalued physical observables for the semiclassical limit of the Schr\u00f6dinger equation","volume":"205","author":"Jin","year":"2005","journal-title":"J. Comput. Phys."},{"issue":"12","key":"10.1016\/j.jcp.2026.114986_bib0015","doi-asserted-by":"crossref","first-page":"2621","DOI":"10.1016\/j.cpc.2013.07.012","article-title":"Computational methods for the dynamics of the nonlinear Schr\u00f6dinger\/Gross\u2013Pitaevskii equations","volume":"184","author":"Antoine","year":"2013","journal-title":"Comput. Phys. Commun."},{"key":"10.1016\/j.jcp.2026.114986_bib0016","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.91.053304","article-title":"Numerical solution of the nonlinear Schr\u00f6dinger equation using smoothed-particle hydrodynamics","volume":"91","author":"Mocz","year":"2015","journal-title":"Phys. Rev. E"},{"key":"10.1016\/j.jcp.2026.114986_bib0017","doi-asserted-by":"crossref","first-page":"647","DOI":"10.1007\/BF02101660","article-title":"On the global existence of Bohmian mechanics","volume":"173","author":"Berndl","year":"1995","journal-title":"Commun. Math. Phys."},{"key":"10.1016\/j.jcp.2026.114986_bib0018","series-title":"Compendium of quantum physics","first-page":"47","article-title":"Bohmian mechanics","author":"D\u00fcrr","year":"2009"},{"key":"10.1016\/j.jcp.2026.114986_bib0019","doi-asserted-by":"crossref","first-page":"3215","DOI":"10.1103\/PhysRevLett.86.3215","article-title":"Quantum Backreaction through the Bohmian Particle","volume":"86","author":"Prezhdo","year":"2001","journal-title":"Phys. Rev. Lett."},{"key":"10.1016\/j.jcp.2026.114986_bib0020","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1016\/j.jcp.2018.10.045","article-title":"Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations","volume":"378","author":"Raissi","year":"2019","journal-title":"J. Comput. phys."},{"issue":"1","key":"10.1016\/j.jcp.2026.114986_bib0021","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s40304-018-0127-z","article-title":"The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems","volume":"6","author":"E","year":"2018","journal-title":"Commun. Math. Stat."},{"key":"10.1016\/j.jcp.2026.114986_bib0022","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2020.109409","article-title":"Weak adversarial networks for high-dimensional partial differential equations","volume":"411","author":"Zang","year":"2020","journal-title":"J. Comput. Phys."},{"issue":"4","key":"10.1016\/j.jcp.2026.114986_bib0023","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1007\/s40304-017-0117-6","article-title":"Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations","volume":"5","author":"Han","year":"2017","journal-title":"Commun. Math. Stat."},{"issue":"17","key":"10.1016\/j.jcp.2026.114986_bib0024","doi-asserted-by":"crossref","first-page":"9183","DOI":"10.1073\/pnas.1922204117","article-title":"A machine learning framework for solving high-dimensional mean field game and mean field control problems","volume":"117","author":"Ruthotto","year":"2020","journal-title":"Proc. Natl. Acad. Sci."},{"key":"10.1016\/j.jcp.2026.114986_bib0025","doi-asserted-by":"crossref","first-page":"1723","DOI":"10.1007\/s11071-021-06554-5","article-title":"Solving localized wave solutions of the derivative nonlinear Schr\u00f6dinger equation using an improved PINN method","volume":"105","author":"Pu","year":"2021","journal-title":"Nonlinear Dynam."},{"key":"10.1016\/j.jcp.2026.114986_bib0026","series-title":"International Conference on Conceptual Structures","article-title":"Physics Informed RNN-DCT Networks for Time-Dependent Partial Differential Equations","author":"Wu","year":"2022"},{"key":"10.1016\/j.jcp.2026.114986_bib0027","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2019.108929","article-title":"Solving many-electron Schr\u00f6dinger equation using deep neural networks","volume":"399","author":"Han","year":"2019","journal-title":"Journal of Computational Physics"},{"issue":"10","key":"10.1016\/j.jcp.2026.114986_bib0028","doi-asserted-by":"crossref","first-page":"891","DOI":"10.1038\/s41557-020-0544-y","article-title":"Deep-neural-network solution of the electronic Schr\u00f6dinger equation","volume":"12","author":"Hermann","year":"2020","journal-title":"Nature Chemistry"},{"issue":"3","key":"10.1016\/j.jcp.2026.114986_bib0029","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevResearch.2.033429","article-title":"Ab initio solution of the many-electron Schr\u00f6dinger equation with deep neural networks","volume":"2","author":"Pfau","year":"2020","journal-title":"Phys. Rev. Res."},{"issue":"6325","key":"10.1016\/j.jcp.2026.114986_bib0030","doi-asserted-by":"crossref","first-page":"602","DOI":"10.1126\/science.aag2302","article-title":"Solving the quantum many-body problem with artificial neural networks","volume":"355","author":"Carleo","year":"2017","journal-title":"Science"},{"key":"10.1016\/j.jcp.2026.114986_bib0031","doi-asserted-by":"crossref","DOI":"10.1016\/j.cpc.2020.107624","article-title":"86 PFLOPS deep potential molecular dynamics simulation of 100 million atoms with ab initio accuracy","volume":"259","author":"Lu","year":"2021","journal-title":"Comput. Phys. Commun."},{"key":"10.1016\/j.jcp.2026.114986_bib0032","unstructured":"E. Orlova, A. Ustimenko, R. Jiang, P.Y. Lu, R. Willett, Deep stochastic mechanics, arXiv: 2305.19685, (2023)."},{"issue":"4","key":"10.1016\/j.jcp.2026.114986_bib0033","doi-asserted-by":"crossref","first-page":"1079","DOI":"10.1103\/PhysRev.150.1079","article-title":"Derivation of the Schr\u00f6dinger equation from Newtonian mechanics","volume":"150","author":"Nelson","year":"1966","journal-title":"Phys. Rev."},{"issue":"8","key":"10.1016\/j.jcp.2026.114986_bib0034","doi-asserted-by":"crossref","first-page":"1774","DOI":"10.1103\/PhysRevD.27.1774","article-title":"Quantization of dynamical systems and stochastic control theory","volume":"27","author":"Guerra","year":"1983","journal-title":"Phys. Rev. D"},{"issue":"1","key":"10.1016\/j.jcp.2026.114986_bib0035","doi-asserted-by":"crossref","first-page":"360","DOI":"10.1137\/23M159281X","article-title":"Parameterized wasserstein hamiltonian flow","volume":"63","author":"Wu","year":"2025","journal-title":"SIAM J. Numer. Anal."},{"key":"10.1016\/j.jcp.2026.114986_bib0036","article-title":"Neural ordinary differential equations","volume":"31","author":"Chen","year":"2018","journal-title":"Adv. Neural Inf. Process. Syst."},{"issue":"19","key":"10.1016\/j.jcp.2026.114986_bib0037","doi-asserted-by":"crossref","DOI":"10.1088\/1751-8121\/acc7db","article-title":"Towards a mathematical theory of the Madelung equations: Takabayasi\u2019s quantization condition, quantum quasi-irrotationality, weak formulations, and the Wallstrom phenomenon","volume":"56","author":"Reddiger","year":"2023","journal-title":"J. Phys. A Math. Theor."},{"issue":"3","key":"10.1016\/j.jcp.2026.114986_bib0038","doi-asserted-by":"crossref","first-page":"1613","DOI":"10.1103\/PhysRevA.49.1613","article-title":"Inequivalence between the Schr\u00f6dinger equation and the Madelung hydrodynamic equations","volume":"49","author":"Wallstrom","year":"1994","journal-title":"Phys. Rev. A"},{"issue":"2","key":"10.1016\/j.jcp.2026.114986_bib0039","doi-asserted-by":"crossref","first-page":"699","DOI":"10.1090\/S0002-9947-1988-0924776-9","article-title":"The density manifold and configuration space quantization","volume":"305","author":"Lafferty","year":"1988","journal-title":"Trans. Am. Math. Soc."},{"issue":"4","key":"10.1016\/j.jcp.2026.114986_bib0040","doi-asserted-by":"crossref","first-page":"858","DOI":"10.4153\/CMB-2011-121-9","article-title":"An Optimal Transport View of Schr\u00f6dinger\u2019s Equation","volume":"55","author":"von Renesse","year":"2012","journal-title":"Canadian Math. Bulletin"},{"issue":"3","key":"10.1016\/j.jcp.2026.114986_bib0041","doi-asserted-by":"crossref","first-page":"1775","DOI":"10.1103\/PhysRevA.58.1775","article-title":"Derivation of the equations of nonrelativistic quantum mechanics using the principle of minimum Fisher information","volume":"58","author":"Reginatto","year":"1998","journal-title":"Phys. Rev. A"},{"key":"10.1016\/j.jcp.2026.114986_bib0042","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1103\/PhysRev.85.180","article-title":"A Suggested Interpretation of the Quantum Theory in Terms of \u201cHidden\u201d Variables. I, II","volume":"85","author":"Bohm","year":"1952","journal-title":"Phys. Rev."},{"key":"10.1016\/j.jcp.2026.114986_bib0043","series-title":"International conference on machine learning","first-page":"214","article-title":"Wasserstein generative adversarial networks","author":"Arjovsky","year":"2017"},{"key":"10.1016\/j.jcp.2026.114986_bib0044","series-title":"International Conference on Learning Representations","article-title":"Score-Based Generative Modeling through Stochastic Differential Equations","author":"Song","year":"2021"},{"key":"10.1016\/j.jcp.2026.114986_bib0045","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/j.cpc.2015.03.012","article-title":"GPELab, a Matlab toolbox to solve Gross-Pitaevskii equations II: dynamics and stochastic simulations","volume":"193","author":"Antoine","year":"2015","journal-title":"Comput. Phys. Commun."}],"container-title":["Journal of Computational Physics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0021999126003396?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0021999126003396?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T00:40:45Z","timestamp":1780447245000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0021999126003396"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,10]]},"references-count":45,"alternative-id":["S0021999126003396"],"URL":"https:\/\/doi.org\/10.1016\/j.jcp.2026.114986","relation":{},"ISSN":["0021-9991"],"issn-type":[{"value":"0021-9991","type":"print"}],"subject":[],"published":{"date-parts":[[2026,10]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"A parameterized Wasserstein Hamiltonian flow approach to solving nonlinear Schr\u00f6dinger equations","name":"articletitle","label":"Article Title"},{"value":"Journal of Computational Physics","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.jcp.2026.114986","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"114986"}}