{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,23]],"date-time":"2026-04-23T14:54:08Z","timestamp":1776956048191,"version":"3.51.4"},"reference-count":77,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,3,18]],"date-time":"2023-03-18T00:00:00Z","timestamp":1679097600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,3,18]],"date-time":"2023-03-18T00:00:00Z","timestamp":1679097600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100006831","name":"United States Department of Defense|U.S. Air Force","doi-asserted-by":"publisher","award":["A9550-18-1-0502"],"award-info":[{"award-number":["A9550-18-1-0502"]}],"id":[{"id":"10.13039\/100006831","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009896","name":"United States Department of Defense|U.S. Navy","doi-asserted-by":"publisher","award":["N00014-18-1-2527"],"award-info":[{"award-number":["N00014-18-1-2527"]}],"id":[{"id":"10.13039\/100009896","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009896","name":"United States Department of Defense|U.S. Navy","doi-asserted-by":"publisher","award":["N00014-20-1-278"],"award-info":[{"award-number":["N00014-20-1-278"]}],"id":[{"id":"10.13039\/100009896","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009896","name":"United States Department of Defense|U.S. Navy","doi-asserted-by":"publisher","award":["N00014-20-1-2093"],"award-info":[{"award-number":["N00014-20-1-2093"]}],"id":[{"id":"10.13039\/100009896","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009896","name":"United States Department of Defense|U.S. Navy","doi-asserted-by":"publisher","award":["A9550-18-1-0502"],"award-info":[{"award-number":["A9550-18-1-0502"]}],"id":[{"id":"10.13039\/100009896","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009896","name":"United States Department of Defense|U.S. Navy","doi-asserted-by":"publisher","award":["N00014-18-1-2527"],"award-info":[{"award-number":["N00014-18-1-2527"]}],"id":[{"id":"10.13039\/100009896","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009896","name":"United States Department of Defense|U.S. Navy","doi-asserted-by":"publisher","award":["N00014-20-1-278"],"award-info":[{"award-number":["N00014-20-1-278"]}],"id":[{"id":"10.13039\/100009896","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009896","name":"United States Department of Defense|U.S. Navy","doi-asserted-by":"publisher","award":["N00014-20-1-2093"],"award-info":[{"award-number":["N00014-20-1-2093"]}],"id":[{"id":"10.13039\/100009896","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Sci Rep"],"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A normalizing flow (NF) is a mapping that transforms a chosen probability distribution to a normal distribution. Such flows are a common technique used for data generation and density estimation in machine learning and data science. The density estimate obtained with a NF requires a change of variables formula that involves the computation of the Jacobian determinant of the NF transformation. In order to tractably compute this determinant, continuous normalizing flows (CNF) estimate the mapping and its Jacobian determinant using a neural ODE. Optimal transport (OT) theory has been successfully used to assist in finding CNFs by formulating them as OT problems with a soft penalty for enforcing the standard normal distribution as a target measure. A drawback of OT-based CNFs is the addition of a hyperparameter,<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03b1<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, that controls the strength of the soft penalty and requires significant tuning. We present JKO-Flow, an algorithm to solve OT-based CNF without the need of tuning<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03b1<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. This is achieved by integrating the OT CNF framework into a Wasserstein gradient flow framework, also known as the JKO scheme. Instead of tuning<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03b1<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, we repeatedly solve the optimization problem for a fixed<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03b1<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>effectively performing a JKO update with a time-step<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03b1<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Hence we obtain a \u201ddivide and conquer\u201d algorithm by repeatedly solving simpler problems instead of solving a potentially harder problem with large<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03b1<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1038\/s41598-023-31521-y","type":"journal-article","created":{"date-parts":[[2023,3,26]],"date-time":"2023-03-26T20:03:17Z","timestamp":1679860997000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Taming hyperparameter tuning in continuous normalizing flows using the JKO scheme"],"prefix":"10.1038","volume":"13","author":[{"given":"Alexander","family":"Vidal","sequence":"first","affiliation":[]},{"given":"Samy","family":"Wu Fung","sequence":"additional","affiliation":[]},{"given":"Luis","family":"Tenorio","sequence":"additional","affiliation":[]},{"given":"Stanley","family":"Osher","sequence":"additional","affiliation":[]},{"given":"Levon","family":"Nurbekyan","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,3,18]]},"reference":[{"issue":"1","key":"31521_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s41781-020-0035-2","volume":"4","author":"J Brehmer","year":"2020","unstructured":"Brehmer, J., Kling, F., Espejo, I. & Cranmer, K. Madminer: Machine learning-based inference for particle physics. Comput. Softw. Big Sci. 4(1), 1\u201325 (2020).","journal-title":"Comput. Softw. Big Sci."},{"issue":"4","key":"31521_CR2","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.91.045002","volume":"91","author":"G Carleo","year":"2019","unstructured":"Carleo, G. et al. Machine learning and the physical sciences. Rev. Modern Phys. 91(4), 045002 (2019).","journal-title":"Rev. Modern Phys."},{"issue":"6457","key":"31521_CR3","doi-asserted-by":"publisher","DOI":"10.1126\/science.aaw1147","volume":"365","author":"F No\u00e9","year":"2019","unstructured":"F.\u00a0No\u00e9, S.\u00a0Olsson, J.\u00a0K\u00f6hler, & H.\u00a0Wu. Boltzmann generators: Sampling equilibrium states of many-body systems with deep learning. Science. 365(6457), eaaw1147 (2019).","journal-title":"Science"},{"key":"31521_CR4","doi-asserted-by":"crossref","unstructured":"I.\u00a0Kobyzev, S.\u00a0Prince, & M.\u00a0Brubaker. Normalizing flows: An introduction and review of current methods. in IEEE Transactions on Pattern Analysis and Machine Intelligence (2020).","DOI":"10.1109\/TPAMI.2020.2992934"},{"key":"31521_CR5","unstructured":"G.\u00a0Papamakarios, E.\u00a0Nalisnick, D.\u00a0J. Rezende, S.\u00a0Mohamed, & B.\u00a0Lakshminarayanan. Normalizing flows for probabilistic modeling and inference. arXiv:1912.02762 (2019)."},{"key":"31521_CR6","unstructured":"D.\u00a0J. Rezende & S.\u00a0Mohamed. Variational inference with normalizing flows. in International Conference on Machine Learning (ICML), 1530\u20131538 (2015)."},{"key":"31521_CR7","doi-asserted-by":"crossref","unstructured":"G.\u00a0Peyr\u00e9 & M.\u00a0Cuturi. Computational optimal transport. (2018).","DOI":"10.1561\/9781680835519"},{"key":"31521_CR8","doi-asserted-by":"crossref","unstructured":"Villani, C. in Topics in Optimal Transportation Vol. 58 (American Mathematical Society, Providence, RI, 2003).","DOI":"10.1090\/gsm\/058\/03"},{"key":"31521_CR9","unstructured":"R.\u00a0Baptista, Y.\u00a0Marzouk, R.\u00a0E. Morrison, & O.\u00a0Zahm. Learning non-Gaussian graphical models via Hessian scores and triangular transport. arXiv preprint arXiv:2101.03093 (2021)."},{"key":"31521_CR10","unstructured":"G.\u00a0Papamakarios, T.\u00a0Pavlakou, & I.\u00a0Murray. Masked autoregressive flow for density estimation. in Advances in Neural Information Processing Systems (NeurIPS), 2338\u20132347 (2017)."},{"issue":"3","key":"31521_CR11","doi-asserted-by":"publisher","first-page":"987","DOI":"10.1007\/s00365-022-09570-9","volume":"55","author":"J Zech","year":"2022","unstructured":"Zech, J. & Marzouk, Y. Sparse approximation of triangular transports, part II: The infinite-dimensional case. Construct. Approximation 55(3), 987\u20131036 (2022).","journal-title":"Construct. Approximation"},{"key":"31521_CR12","unstructured":"C.\u00a0Chen, C.\u00a0Li, L.\u00a0Chen, W.\u00a0Wang, Y.\u00a0Pu, & L.\u00a0C. Duke. Continuous-time flows for efficient inference and density estimation. in International Conference on Machine Learning (ICML), 824\u2013833 (2018)."},{"key":"31521_CR13","unstructured":"W.\u00a0Grathwohl, R.\u00a0T. Chen, J.\u00a0Betterncourt, I.\u00a0Sutskever, & D.\u00a0Duvenaud. FFJORD: Free-form continuous dynamics for scalable reversible generative models. in International Conference on Learning Representations (ICLR) (2019)."},{"key":"31521_CR14","unstructured":"C.\u00a0Finlay, J.-H. Jacobsen, L.\u00a0Nurbekyan, & A.\u00a0M. Oberman. How to train your neural ODE: The world of Jacobian and kinetic regularization. in International Conference on Machine Learning (ICML), 3154\u20133164 (2020)."},{"key":"31521_CR15","doi-asserted-by":"crossref","unstructured":"D.\u00a0Onken, S.\u00a0Wu\u00a0Fung, X.\u00a0Li, & L.\u00a0Ruthotto. Ot-flow: Fast and accurate continuous normalizing flows via optimal transport. in Proceedings of the AAAI Conference on Artificial Intelligence, Vol.\u00a035 (2021).","DOI":"10.1609\/aaai.v35i10.17113"},{"key":"31521_CR16","volume-title":"Optimal Transport: Old and New","author":"C Villani","year":"2008","unstructured":"Villani, C. Optimal Transport: Old and New Vol. 338 (Springer Science & Business Media, New York, 2008)."},{"key":"31521_CR17","unstructured":"L.\u00a0Yang and G.\u00a0E. Karniadakis. Potential flow generator with $${L}_2$$ optimal transport regularity for generative models. in IEEE Transactions on Neural Networks and Learning Systems (2020)."},{"key":"31521_CR18","unstructured":"L.\u00a0Zhang, W.\u00a0E, & L.\u00a0Wang. Monge-Amp\u00e8re flow for generative modeling. arXiv:1809.10188 (2018)."},{"issue":"17","key":"31521_CR19","doi-asserted-by":"publisher","first-page":"9183","DOI":"10.1073\/pnas.1922204117","volume":"117","author":"L Ruthotto","year":"2020","unstructured":"Ruthotto, L., Osher, S. J., Li, W., Nurbekyan, L. & Fung, S. W. A machine learning framework for solving high-dimensional mean field game and mean field control problems. Proc. Natl. Acad. Sci. 117(17), 9183\u20139193 (2020).","journal-title":"Proc. Natl. Acad. Sci."},{"issue":"1","key":"31521_CR20","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1137\/S0036141096303359","volume":"29","author":"R Jordan","year":"1998","unstructured":"Jordan, R., Kinderlehrer, D. & Otto, F. The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29(1), 1\u201317 (1998).","journal-title":"SIAM J. Math. Anal."},{"key":"31521_CR21","unstructured":"A.\u00a0Salim, A.\u00a0Korba, & G.\u00a0Luise. The Wasserstein proximal gradient algorithm. in Advances in Neural Information Processing Systems (H.\u00a0Larochelle, M.\u00a0Ranzato, R.\u00a0Hadsell, M.\u00a0Balcan, & H.\u00a0Lin, eds.), Vol.\u00a033, 12356\u201312366. (Curran Associates, Inc., 2020)."},{"key":"31521_CR22","doi-asserted-by":"crossref","unstructured":"L.\u00a0Tenorio. An Introduction to Data Analysis and Uncertainty Quantification for Inverse Problems. (SIAM, 2017).","DOI":"10.1137\/1.9781611974928"},{"key":"31521_CR23","doi-asserted-by":"crossref","unstructured":"M.\u00a0Burger, S.\u00a0Osher, J.\u00a0Xu, & G.\u00a0Gilboa. Nonlinear inverse scale space methods for image restoration. in Variational, Geometric, and Level Set Methods in Computer Vision (N.\u00a0Paragios, O.\u00a0Faugeras, T.\u00a0Chan, & C.\u00a0Schn\u00f6rr, eds.), 25\u201336 (Springer, 2005).","DOI":"10.1007\/11567646_3"},{"issue":"2","key":"31521_CR24","doi-asserted-by":"publisher","first-page":"460","DOI":"10.1137\/040605412","volume":"4","author":"S Osher","year":"2005","unstructured":"Osher, S., Burger, M., Goldfarb, D., Xu, J. & Yin, W. An iterative regularization method for total variation-based image restoration. Multisc. Model. Simulat. 4(2), 460\u2013489 (2005).","journal-title":"Multisc. Model. Simulat."},{"issue":"5\u20136","key":"31521_CR25","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1561\/2200000073","volume":"11","author":"G Peyr\u00e9","year":"2019","unstructured":"Peyr\u00e9, G. & Cuturi, M. Computational optimal transport. Foundations Trends Mach. Learn. 11(5\u20136), 355\u2013607 (2019).","journal-title":"Foundations Trends Mach. Learn."},{"key":"31521_CR26","doi-asserted-by":"crossref","unstructured":"F.\u00a0Santambrogio. Optimal Transport for aAplied Mathematicians, Vol.\u00a087 of Progress in Nonlinear Differential Equations and their Applications. Birkh\u00e4user\/Springer, Cham, 2015. Calculus of variations, PDEs, and modeling.","DOI":"10.1007\/978-3-319-20828-2"},{"key":"31521_CR27","unstructured":"M.\u00a0Welling & Y.\u00a0W. Teh. Bayesian learning via stochastic gradient Langevin dynamics. in International Conference on Machine Learning (ICML), 681\u2013688 (2011)."},{"issue":"3","key":"31521_CR28","doi-asserted-by":"publisher","first-page":"375","DOI":"10.1007\/s002110050002","volume":"84","author":"J-D Benamou","year":"2000","unstructured":"Benamou, J.-D. & Brenier, Y. A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numerische Mathematik 84(3), 375\u2013393 (2000).","journal-title":"Numerische Mathematik"},{"issue":"2","key":"31521_CR29","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/1944345.1944349","volume":"58","author":"H Avron","year":"2011","unstructured":"Avron, H. & Toledo, S. Randomized algorithms for estimating the trace of an implicit symmetric positive semi-definite matrix. J. ACM (JACM) 58(2), 1\u201334 (2011).","journal-title":"J. ACM (JACM)"},{"issue":"2","key":"31521_CR30","doi-asserted-by":"publisher","first-page":"433","DOI":"10.1080\/03610919008812866","volume":"19","author":"MF Hutchinson","year":"1990","unstructured":"Hutchinson, M. F. A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines. Commun. Stat.-Simulat. Comput. 19(2), 433\u2013450 (1990).","journal-title":"Commun. Stat.-Simulat. Comput."},{"key":"31521_CR31","doi-asserted-by":"crossref","DOI":"10.1002\/9781118575574","volume-title":"Multivariate Density Estimation: Theory, Practice, and Visualization","author":"DW Scott","year":"2015","unstructured":"Scott, D. W. Multivariate Density Estimation: Theory, Practice, and Visualization (Wiley, New York, 2015)."},{"key":"31521_CR32","volume-title":"Density Estimation for Statistics and Data Analysis","author":"BW Silverman","year":"1986","unstructured":"Silverman, B. W. Density Estimation for Statistics and Data Analysis (CRC Press, New York, 1986)."},{"issue":"3","key":"31521_CR33","doi-asserted-by":"publisher","first-page":"198","DOI":"10.1016\/S0010-4655(00)00243-5","volume":"136","author":"K Cranmer","year":"2001","unstructured":"Cranmer, K. Kernel estimation in high-energy physics. Comput. Phys. Commun. 136(3), 198\u2013207 (2001).","journal-title":"Comput. Phys. Commun."},{"issue":"4","key":"31521_CR34","first-page":"567","volume":"12","author":"O Collaboration","year":"2000","unstructured":"Collaboration, O. & Abbiendi, G. Search for neutral Higgs bosons in collisions at 189 gev. Eur. Phys. J. C-Particles Fields. 12(4), 567\u2013586 (2000).","journal-title":"Eur. Phys. J. C-Particles Fields."},{"issue":"11","key":"31521_CR35","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1145\/3422622","volume":"63","author":"I Goodfellow","year":"2020","unstructured":"Goodfellow, I. et al. Generative adversarial networks. Commun. ACM 63(11), 139\u2013144 (2020).","journal-title":"Commun. ACM"},{"issue":"2","key":"31521_CR36","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1002\/cpa.21423","volume":"66","author":"EG Tabak","year":"2013","unstructured":"Tabak, E. G. & Turner, C. V. A family of nonparametric density estimation algorithms. Commun. Pure Appl. Math. 66(2), 145\u2013164 (2013).","journal-title":"Commun. Pure Appl. Math."},{"key":"31521_CR37","unstructured":"L.\u00a0Dinh, D.\u00a0Krueger, & Y.\u00a0Bengio. NICE: Non-linear independent components estimation. in International Conference on Learning Representations (ICLR) (Y.\u00a0Bengio & Y.\u00a0LeCun, eds.) (2015)."},{"key":"31521_CR38","unstructured":"L.\u00a0Dinh, J.\u00a0Sohl-Dickstein, & S.\u00a0Bengio. Density estimation using real NVP. in International Conference on Learning Representations (ICLR) (2017)."},{"key":"31521_CR39","unstructured":"D.\u00a0P. Kingma, T.\u00a0Salimans, R.\u00a0Jozefowicz, X.\u00a0Chen, I.\u00a0Sutskever, & M.\u00a0Welling. Improved variational inference with inverse autoregressive flow. in Advances in Neural Information Processing Systems (NeurIPS), 4743\u20134751 (2016)."},{"key":"31521_CR40","unstructured":"D.\u00a0P. Kingma & P.\u00a0Dhariwal. Glow: Generative flow with invertible 1x1 convolutions. in Advances in Neural Information Processing Systems (NeurIPS), 10215\u201310224 (2018)."},{"key":"31521_CR41","unstructured":"C.\u00a0Durkan, A.\u00a0Bekasov, I.\u00a0Murray, & G.\u00a0Papamakarios. Neural spline flows. in Advances in Neural Information Processing Systems (NeurIPS), 7509\u20137520 (2019)."},{"key":"31521_CR42","unstructured":"C.-W. Huang, D.\u00a0Krueger, A.\u00a0Lacoste, & A.\u00a0Courville. Neural autoregressive flows. in International Conference on Machine Learning (ICML) 2078\u20132087 (2018)."},{"key":"31521_CR43","unstructured":"A.\u00a0Wehenkel & G.\u00a0Louppe. Unconstrained monotonic neural networks. in Advances in Neural Information Processing Systems (NeurIPS), 1543\u20131553 (2019)."},{"key":"31521_CR44","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2022.111202","volume":"461","author":"L Guo","year":"2022","unstructured":"Guo, L., Wu, H. & Zhou, T. Normalizing field flows: Solving forward and inverse stochastic differential equations using physics-informed flow models. J. Comput. Phys. 461, 111202 (2022).","journal-title":"J. Comput. Phys."},{"key":"31521_CR45","unstructured":"C.-W. Huang, R.\u00a0T. Chen, C.\u00a0Tsirigotis, & A.\u00a0Courville. Convex potential flows: Universal probability distributions with optimal transport and convex optimization. arXiv preprint arXiv:2012.05942 (2020)."},{"issue":"11","key":"31521_CR46","first-page":"2","volume":"2","author":"RM Neal","year":"2011","unstructured":"Neal, R. M. MCMC using Hamiltonian dynamics. Handbook Markov Chain Monte Carlo 2(11), 2 (2011).","journal-title":"Handbook Markov Chain Monte Carlo"},{"key":"31521_CR47","unstructured":"Y.\u00a0Park, D.\u00a0Maddix, F.-X. Aubet, K.\u00a0Kan, J.\u00a0Gasthaus, & Y.\u00a0Wang. Learning quantile functions without quantile crossing for distribution-free time series forecasting. in International Conference on Artificial Intelligence and Statistics, 8127\u20138150. (PMLR, 2022)."},{"issue":"2","key":"31521_CR48","doi-asserted-by":"publisher","DOI":"10.1002\/gamm.202100008","volume":"44","author":"L Ruthotto","year":"2021","unstructured":"Ruthotto, L. & Haber, E. An introduction to deep generative modeling. GAMM-Mitteilungen 44(2), e202100008 (2021).","journal-title":"GAMM-Mitteilungen"},{"key":"31521_CR49","unstructured":"T.\u00a0Salimans, D.\u00a0Kingma, & M.\u00a0Welling. Markov chain Monte Carlo and variational inference: Bridging the gap. in International Conference on Machine Learning (ICML), 1218\u20131226 (2015)."},{"key":"31521_CR50","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1023\/A:1009632428145","volume":"7","author":"J Suykens","year":"1998","unstructured":"Suykens, J., Verrelst, H. & Vandewalle, J. On-line learning Fokker-Planck machine. Neural Process. Lett. 7, 81\u201389 (1998).","journal-title":"Neural Process. Lett."},{"key":"31521_CR51","unstructured":"T.\u00a0Q. Chen, Y.\u00a0Rubanova, J.\u00a0Bettencourt, & D.\u00a0K. Duvenaud. Neural ordinary differential equations. in Advances in Neural Information Processing Systems (NeurIPS), 6571\u20136583, (2018)."},{"key":"31521_CR52","doi-asserted-by":"crossref","unstructured":"G.\u00a0Avraham, Y.\u00a0Zuo, & T.\u00a0Drummond. Parallel optimal transport GAN. in IEEE\/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 4406\u20134415 (2019).","DOI":"10.1109\/CVPR.2019.00454"},{"key":"31521_CR53","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.cagd.2018.10.005","volume":"68","author":"N Lei","year":"2019","unstructured":"Lei, N., Su, K., Cui, L., Yau, S.-T. & Gu, X. D. A geometric view of optimal transportation and generative model. Comput. Aided Geometr. Design 68, 1\u201321 (2019).","journal-title":"Comput. Aided Geometr. Design"},{"key":"31521_CR54","unstructured":"J.\u00a0Lin, K.\u00a0Lensink, & E.\u00a0Haber. Fluid flow mass transport for generative networks. arXiv:1910.01694 (2019)."},{"key":"31521_CR55","unstructured":"T.\u00a0Salimans, H.\u00a0Zhang, A.\u00a0Radford, & D.\u00a0N. Metaxas. Improving GANs using optimal transport. in International Conference on Learning Representations (ICLR) (2018)."},{"key":"31521_CR56","unstructured":"M.\u00a0Sanjabi, J.\u00a0Ba, M.\u00a0Razaviyayn, & J.\u00a0D. Lee. On the convergence and robustness of training gans with regularized optimal transport. in Advances in Neural Information Processing Systems (NeurIPS), 7091\u20137101 (2018)."},{"key":"31521_CR57","unstructured":"A.\u00a0Tanaka. Discriminator optimal transport. in Advances in Neural Information Processing Systems (NeurIPS), 6816\u20136826 (2019)."},{"key":"31521_CR58","unstructured":"D.\u00a0Onken & L.\u00a0Ruthotto. Discretize-optimize vs. optimize-discretize for time-series regression and continuous normalizing flows. arXiv:2005.13420 (2020)."},{"key":"31521_CR59","unstructured":"J.\u00a0Fan, Q.\u00a0Zhang, A.\u00a0Taghvaei, & Y.\u00a0Chen. Variational Wasserstein gradient flow. in Proceedings of the 39th International Conference on Machine Learning, volume 162 of Proceedings of Machine Learning Research (K.\u00a0Chaudhuri, S.\u00a0Jegelka, L.\u00a0Song, C.\u00a0Szepesvari, G.\u00a0Niu, & S.\u00a0Sabato, eds.), 6185\u20136215. PMLR, 17\u201323 Jul 2022."},{"key":"31521_CR60","unstructured":"D.\u00a0Alvarez-Melis, Y.\u00a0Schiff, & Y.\u00a0Mroueh. Optimizing functionals on the space of probabilities with input convex neural networks. arXiv preprint arXiv:2106.00774 (2021)."},{"key":"31521_CR61","unstructured":"C.\u00a0Bunne, L.\u00a0Papaxanthos, A.\u00a0Krause, & M.\u00a0Cuturi. Proximal optimal transport modeling of population dynamics. in International Conference on Artificial Intelligence and Statistics, 6511\u20136528. (PMLR, 2022)."},{"key":"31521_CR62","first-page":"15243","volume":"34","author":"P Mokrov","year":"2021","unstructured":"Mokrov, P. et al. Large-scale Wasserstein gradient flows. Adv. Neural Inform. Process. Syst. 34, 15243\u201315256 (2021).","journal-title":"Adv. Neural Inform. Process. Syst."},{"key":"31521_CR63","doi-asserted-by":"crossref","unstructured":"L.\u00a0Nurbekyan, W.\u00a0Lei, & Y.\u00a0Yang. Efficient natural gradient descent methods for large-scale optimization problems. arXiv preprint arXiv:2202.06236 (2022).","DOI":"10.1137\/22M1477805"},{"key":"31521_CR64","unstructured":"H.\u00a0Heaton, S.\u00a0W. Fung, A.\u00a0T. Lin, S.\u00a0Osher, & W.\u00a0Yin. Wasserstein-based projections with applications to inverse problems. arXiv preprint arXiv:2008.02200 (2020)."},{"issue":"2","key":"31521_CR65","doi-asserted-by":"publisher","first-page":"429","DOI":"10.2307\/1428011","volume":"29","author":"A M\u00fcller","year":"1997","unstructured":"M\u00fcller, A. Integral probability metrics and their generating classes of functions. Adv. Appl. Probability 29(2), 429\u2013443 (1997).","journal-title":"Adv. Appl. Probability"},{"key":"31521_CR66","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-4869-3","volume-title":"The Methods of Distances in the Theory of Probability and Statistics","author":"ST Rachev","year":"2013","unstructured":"Rachev, S. T., Klebanov, L. B., Stoyanov, S. V. & Fabozzi, F. The Methods of Distances in the Theory of Probability and Statistics Vol. 10 (Springer, New York, 2013)."},{"issue":"3","key":"31521_CR67","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1070\/SM1976v030n03ABEH002280","volume":"30","author":"VM Zolotarev","year":"1976","unstructured":"Zolotarev, V. M. Metric distances in spaces of random variables and their distributions. Math. USSR-Sbornik 30(3), 373 (1976).","journal-title":"Math. USSR-Sbornik"},{"issue":"25","key":"31521_CR68","first-page":"723","volume":"13","author":"A Gretton","year":"2012","unstructured":"Gretton, A., Borgwardt, K. M., Rasch, M. J., Sch\u00f6lkopf, B. & Smola, A. A kernel two-sample test. J. Mach. Learn. Res. (JMLR) 13(25), 723\u2013773 (2012).","journal-title":"J. Mach. Learn. Res. (JMLR)"},{"key":"31521_CR69","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781316219232","volume-title":"An Introduction to the Theory of Reproducing Kernel Hilbert Spaces","author":"VI Paulsen","year":"2016","unstructured":"Paulsen, V. I. & Raghupathi, M. An Introduction to the Theory of Reproducing Kernel Hilbert Spaces Vol. 152 (Cambridge University Press, Cambridge, 2016)."},{"key":"31521_CR70","unstructured":"K.\u00a0Fukumizu, A.\u00a0Gretton, X.\u00a0Sun, & B.\u00a0Sch\u00f6lkopf. Kernel measures of conditional dependence. Adv. Neural Inform. Process. Syst. 20 (2007)."},{"key":"31521_CR71","doi-asserted-by":"crossref","unstructured":"K.\u00a0He, X.\u00a0Zhang, S.\u00a0Ren, & J.\u00a0Sun. Deep residual learning for image recognition. in IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 770\u2013778 (2016).","DOI":"10.1109\/CVPR.2016.90"},{"key":"31521_CR72","unstructured":"B.\u00a0Roe. MiniBooNE particle identification. UCI Machine Learning Repository (2010)."},{"key":"31521_CR73","unstructured":"W.\u00a0H. Fleming & H.\u00a0M. Soner. Controlled Markov Processes and Viscosity Solutions, Vol.\u00a025 of Stochastic Modelling and Applied Probability, 2nd Edn. (Springer, 2006)."},{"key":"31521_CR74","doi-asserted-by":"crossref","unstructured":"D.\u00a0Onken, L.\u00a0Nurbekyan, X.\u00a0Li, S.\u00a0W. Fung, S.\u00a0Osher, & L.\u00a0Ruthotto. A neural network approach applied to multi-agent optimal control. in 2021 European Control Conference (ECC), 1036\u20131041. (IEEE, 2021).","DOI":"10.23919\/ECC54610.2021.9655103"},{"key":"31521_CR75","doi-asserted-by":"crossref","unstructured":"D.\u00a0Onken, L.\u00a0Nurbekyan, X.\u00a0Li, S.\u00a0W. Fung, S.\u00a0Osher, & L.\u00a0Ruthotto. A neural network approach for high-dimensional optimal control applied to multiagent path finding. in IEEE Transactions on Control Systems Technology (2022).","DOI":"10.23919\/ECC54610.2021.9655103"},{"key":"31521_CR76","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2022.111136","volume":"459","author":"S Agrawal","year":"2022","unstructured":"Agrawal, S., Lee, W., Fung, S. W. & Nurbekyan, L. Random features for high-dimensional nonlocal mean-field games. J. Comput. Phys. 459, 111136 (2022).","journal-title":"J. Comput. Phys."},{"issue":"31","key":"31521_CR77","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.2024713118","volume":"118","author":"AT Lin","year":"2021","unstructured":"Lin, A. T., Fung, S. W., Li, W., Nurbekyan, L. & Osher, S. J. Alternating the population and control neural networks to solve high-dimensional stochastic mean-field games. Proc. Natl. Acad. Sci. 118(31), e2024713118 (2021).","journal-title":"Proc. Natl. Acad. Sci."}],"container-title":["Scientific Reports"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.nature.com\/articles\/s41598-023-31521-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s41598-023-31521-y","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s41598-023-31521-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,9]],"date-time":"2023-12-09T16:52:03Z","timestamp":1702140723000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.nature.com\/articles\/s41598-023-31521-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,18]]},"references-count":77,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,12]]}},"alternative-id":["31521"],"URL":"https:\/\/doi.org\/10.1038\/s41598-023-31521-y","relation":{},"ISSN":["2045-2322"],"issn-type":[{"value":"2045-2322","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,3,18]]},"assertion":[{"value":"13 December 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 March 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 March 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing Interests"}}],"article-number":"4501"}}