{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,6]],"date-time":"2026-08-06T19:44:20Z","timestamp":1786045460413,"version":"3.56.0"},"reference-count":71,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2021,4,17]],"date-time":"2021-04-17T00:00:00Z","timestamp":1618617600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2021,4,17]],"date-time":"2021-04-17T00:00:00Z","timestamp":1618617600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/100000121","name":"Division of Mathematical Sciences","doi-asserted-by":"publisher","award":["1759536"],"award-info":[{"award-number":["1759536"]}],"id":[{"id":"10.13039\/100000121","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000121","name":"Division of Mathematical Sciences","doi-asserted-by":"publisher","award":["1916467"],"award-info":[{"award-number":["1916467"]}],"id":[{"id":"10.13039\/100000121","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Stat Comput"],"published-print":{"date-parts":[[2021,5]]},"DOI":"10.1007\/s11222-021-10009-7","type":"journal-article","created":{"date-parts":[[2021,4,17]],"date-time":"2021-04-17T07:02:55Z","timestamp":1618642975000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Particle-based energetic variational inference"],"prefix":"10.1007","volume":"31","author":[{"given":"Yiwei","family":"Wang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jiuhai","family":"Chen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chun","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6000-3436","authenticated-orcid":false,"given":"Lulu","family":"Kang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2021,4,17]]},"reference":[{"issue":"1","key":"10009_CR1","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1111\/j.2517-6161.1966.tb00626.x","volume":"28","author":"SM Ali","year":"1966","unstructured":"Ali, S.M., Silvey, S.D.: A general class of coefficients of divergence of one distribution from another. J. R. Stat. Soc. Ser. B 28(1), 131\u2013142 (1966)","journal-title":"J. R. Stat. Soc. Ser. B"},{"issue":"1","key":"10009_CR2","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00229-006-0003-0","volume":"121","author":"L Ambrosio","year":"2006","unstructured":"Ambrosio, L., Lisini, S., Savar\u00e9, G.: Stability of flows associated to gradient vector fields and convergence of iterated transport maps. Manuscr. Math. 121(1), 1\u201350 (2006)","journal-title":"Manuscr. Math."},{"key":"10009_CR3","unstructured":"Arbel, M., Korba, A., Salim, A., Gretton, A.: Maximum mean discrepancy gradient flow. In: Advances in Neural Information Processing Systems, pp. 6484\u20136494 (2019)"},{"issue":"1","key":"10009_CR4","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1093\/imanum\/8.1.141","volume":"8","author":"J Barzilai","year":"1988","unstructured":"Barzilai, J., Borwein, J.M.: Two-point step size gradient methods. IMA J. Numer. Anal. 8(1), 141\u2013148 (1988)","journal-title":"IMA J. Numer. Anal."},{"key":"10009_CR5","volume-title":"Pattern Recognition and Machine Learning","author":"CM Bishop","year":"2006","unstructured":"Bishop, C.M.: Pattern Recognition and Machine Learning. Springer, New York (2006)"},{"issue":"518","key":"10009_CR6","doi-asserted-by":"publisher","first-page":"859","DOI":"10.1080\/01621459.2017.1285773","volume":"112","author":"DM Blei","year":"2017","unstructured":"Blei, D.M., Kucukelbir, A., McAuliffe, J.D.: Variational inference: a review for statisticians. J. Am. Stat. Assoc. 112(518), 859\u2013877 (2017)","journal-title":"J. Am. Stat. Assoc."},{"key":"10009_CR7","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1090\/conm\/526\/10376","volume":"526","author":"JA Carrillo","year":"2010","unstructured":"Carrillo, J.A., Lisini, S.: On the asymptotic behavior of the gradient flow of a polyconvex functional. Nonlinear Partial Differ. Equ. Hyperbolic Wave Phenom. 526, 37\u201351 (2010)","journal-title":"Nonlinear Partial Differ. Equ. Hyperbolic Wave Phenom."},{"issue":"3","key":"10009_CR8","doi-asserted-by":"publisher","first-page":"1463","DOI":"10.1007\/s10915-017-0594-5","volume":"75","author":"JA Carrillo","year":"2018","unstructured":"Carrillo, J.A., D\u00fcring, B., Matthes, D., McCormick, D.S.: A Lagrangian scheme for the solution of nonlinear diffusion equations using moving simplex meshes. J. Sci. Comput. 75(3), 1463\u20131499 (2018)","journal-title":"J. Sci. Comput."},{"issue":"2","key":"10009_CR9","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1007\/s00526-019-1486-3","volume":"58","author":"JA Carrillo","year":"2019","unstructured":"Carrillo, J.A., Craig, K., Patacchini, F.S.: A blob method for diffusion. Calc. Var. Partial. Differ. Equ. 58(2), 53 (2019)","journal-title":"Calc. Var. Partial. Differ. Equ."},{"issue":"3","key":"10009_CR10","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1080\/00031305.1992.10475878","volume":"46","author":"G Casella","year":"1992","unstructured":"Casella, G., George, E.I.: Explaining the Gibbs sampler. Am. Stat. 46(3), 167\u2013174 (1992)","journal-title":"Am. Stat."},{"key":"10009_CR11","unstructured":"Chen, C., Zhang, R., Wang, W., Li, B., Chen, L.: A unified particle-optimization framework for scalable Bayesian sampling (2018). arXiv preprint arXiv:1805.11659"},{"key":"10009_CR12","unstructured":"Chen, P., Wu, K., Chen, J., O\u2019Leary-Roseberry, T., Ghattas, O.: Projected stein variational Newton: a fast and scalable Bayesian inference method in high dimensions (2019). arXiv preprint arXiv:1901.08659"},{"key":"10009_CR13","unstructured":"Dai, B., He, N., Dai, H., Song, L.: Provable Bayesian inference via particle mirror descent. In: Artificial Intelligence and Statistics, pp. 985\u2013994 (2016)"},{"issue":"2","key":"10009_CR14","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1137\/0911018","volume":"11","author":"P Degond","year":"1990","unstructured":"Degond, P., Mustieles, F.J.: A deterministic approximation of diffusion equations using particles. SIAM J. Sci. Comput. 11(2), 293\u2013310 (1990)","journal-title":"SIAM J. Sci. Comput."},{"key":"10009_CR15","unstructured":"Detommaso, G., Cui, T., Marzouk, Y., Spantini, A., Scheichl, R.: A Stein variational Newton method. In: Advances in Neural Information Processing Systems, pp. 9169\u20139179 (2018)"},{"key":"10009_CR16","doi-asserted-by":"crossref","unstructured":"Du, Q., Feng, X.: The phase field method for geometric moving interfaces and their numerical approximations (2019). arXiv preprint arXiv:1902.04924","DOI":"10.1016\/bs.hna.2019.05.001"},{"issue":"2","key":"10009_CR17","doi-asserted-by":"publisher","first-page":"216","DOI":"10.1016\/0370-2693(87)91197-X","volume":"195","author":"S Duane","year":"1987","unstructured":"Duane, S., Kennedy, A.D., Pendleton, B.J., Roweth, D.: Hybrid Monte Carlo. Phys. Lett. B 195(2), 216\u2013222 (1987)","journal-title":"Phys. Lett. B"},{"key":"10009_CR18","first-page":"2121","volume":"12","author":"J Duchi","year":"2011","unstructured":"Duchi, J., Hazan, E., Singer, Y.: Adaptive subgradient methods for online learning and stochastic optimization. J. Mach. Learn. Res. 12, 2121\u20132159 (2011)","journal-title":"J. Mach. Learn. Res."},{"issue":"23","key":"10009_CR19","doi-asserted-by":"publisher","first-page":"7815","DOI":"10.1016\/j.jcp.2012.07.022","volume":"231","author":"TA El Moselhy","year":"2012","unstructured":"El Moselhy, T.A., Marzouk, Y.M.: Bayesian inference with optimal maps. J. Comput. Phys. 231(23), 7815\u20137850 (2012)","journal-title":"J. Comput. Phys."},{"issue":"3","key":"10009_CR20","doi-asserted-by":"publisher","first-page":"737","DOI":"10.1137\/04061386X","volume":"37","author":"LC Evans","year":"2005","unstructured":"Evans, L.C., Savin, O., Gangbo, W.: Diffeomorphisms and nonlinear heat flows. SIAM J. Math. Anal. 37(3), 737\u2013751 (2005)","journal-title":"SIAM J. Math. Anal."},{"key":"10009_CR21","unstructured":"Francois, D., Wertz, V., Verleysen, M., et\u00a0al.: About the locality of kernels in high-dimensional spaces. In: International Symposium on Applied Stochastic Models and Data Analysis, pp. 238\u2013245. Citeseer (2005)"},{"key":"10009_CR22","unstructured":"Frogner, C., Poggio, T.: Approximate inference with Wasserstein gradient flows (2018). arXiv preprint arXiv:1806.04542"},{"key":"10009_CR23","doi-asserted-by":"publisher","DOI":"10.1201\/b16018","volume-title":"Bayesian Data Analysis","author":"A Gelman","year":"2013","unstructured":"Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A., Rubin, D.B.: Bayesian Data Analysis. Chapman and Hall\/CRC, Boca Raton (2013)"},{"key":"10009_CR24","doi-asserted-by":"publisher","first-page":"721","DOI":"10.1109\/TPAMI.1984.4767596","volume":"6","author":"S Geman","year":"1984","unstructured":"Geman, S., Geman, D.: Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. 6, 721\u2013741 (1984)","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"10009_CR25","unstructured":"Gershman, S.J., Hoffman, M.D., Blei, D.M.: Nonparametric variational inference. In: Proceedings of the 29th International Conference on Machine Learning, pp. 235\u2013242 (2012)"},{"key":"10009_CR26","doi-asserted-by":"crossref","unstructured":"Giga, M.H., Kirshtein, A., Liu, C.: Variational modeling and complex fluids. Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, pp. 1\u201341 (2017)","DOI":"10.1007\/978-3-319-10151-4_2-1"},{"key":"10009_CR27","volume-title":"A First Course in Continuum Mechanics","author":"O Gonzalez","year":"2008","unstructured":"Gonzalez, O., Stuart, A.M.: A First Course in Continuum Mechanics. Cambridge University Press, Cambridge (2008)"},{"issue":"3","key":"10009_CR28","doi-asserted-by":"publisher","first-page":"375","DOI":"10.1007\/s001800050022","volume":"14","author":"H Haario","year":"1999","unstructured":"Haario, H., Saksman, E., Tamminen, J.: Adaptive proposal distribution for random walk Metropolis algorithm. Comput. Stat. 14(3), 375\u2013396 (1999)","journal-title":"Comput. Stat."},{"issue":"1","key":"10009_CR29","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1093\/biomet\/57.1.97","volume":"57","author":"WK Hastings","year":"1970","unstructured":"Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57(1), 97\u2013109 (1970)","journal-title":"Biometrika"},{"issue":"3","key":"10009_CR30","doi-asserted-by":"publisher","first-page":"435","DOI":"10.1103\/RevModPhys.49.435","volume":"49","author":"PC Hohenberg","year":"1977","unstructured":"Hohenberg, P.C., Halperin, B.I.: Theory of dynamic critical phenomena. Rev. Mod. Phys. 49(3), 435 (1977)","journal-title":"Rev. Mod. Phys."},{"key":"10009_CR31","unstructured":"Iserles, A.: A first course in the numerical analysis of differential equations. No.\u00a044 in Cambridge Texts in Applied Mathematics. Cambridge University Press, New York (2009)"},{"issue":"1","key":"10009_CR32","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\u2013Planck equation. SIAM J. Math. Anal. 29(1), 1\u201317 (1998)","journal-title":"SIAM J. Math. Anal."},{"issue":"2","key":"10009_CR33","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1023\/A:1007665907178","volume":"37","author":"MI Jordan","year":"1999","unstructured":"Jordan, M.I., Ghahramani, Z., Jaakkola, T.S., Saul, L.K.: An introduction to variational methods for graphical models. Mach. Learn. 37(2), 183\u2013233 (1999)","journal-title":"Mach. Learn."},{"key":"10009_CR34","unstructured":"Kingma, D.P., Salimans, T., Jozefowicz, R., Chen, X., Sutskever, I., Welling, M.: Improved variational inference with inverse autoregressive flow. In: Advances in Neural Information Processing Systems, pp. 4743\u20134751 (2016)"},{"key":"10009_CR35","doi-asserted-by":"crossref","unstructured":"Lacombe, G., Mas-Gallic, S.: Presentation and analysis of a diffusion-velocity method. In: ESAIM: Proceedings, vol.\u00a07, pp. 225\u2013233. EDP Sciences (1999)","DOI":"10.1051\/proc:1999021"},{"key":"10009_CR36","unstructured":"Li, L., Liu, J.G., Liu, Z., Lu, J.: A stochastic version of Stein variational gradient descent for efficient sampling (2019). arXiv preprint arXiv:1902.03394"},{"key":"10009_CR37","doi-asserted-by":"crossref","unstructured":"Liu, C.: An introduction of elastic complex fluids: an energetic variational approach. In: Multi-Scale Phenomena in Complex Fluids: Modeling, Analysis and Numerical Simulation, pp. 286\u2013337. World Scientific (2009)","DOI":"10.1142\/9789814273268_0004"},{"key":"10009_CR38","unstructured":"Liu, Q.: Stein variational gradient descent as gradient flow. In: Advances in Neural Information Processing Systems, pp. 3115\u20133123 (2017)"},{"key":"10009_CR39","unstructured":"Liu, Q., Wang, D.: Stein variational gradient descent: a general purpose Bayesian inference algorithm. In: Advances in Neural Information Processing Systems, pp. 2378\u20132386 (2016)"},{"key":"10009_CR40","doi-asserted-by":"publisher","first-page":"109566","DOI":"10.1016\/j.jcp.2020.109566","volume":"417","author":"C Liu","year":"2020","unstructured":"Liu, C., Wang, Y.: On Lagrangian schemes for porous medium type generalized diffusion equations: a discrete energetic variational approach. J. Comput. Phys. 417, 109566 (2020a)","journal-title":"J. Comput. Phys."},{"issue":"6","key":"10009_CR41","doi-asserted-by":"publisher","first-page":"B1541","DOI":"10.1137\/20M1326684","volume":"42","author":"C Liu","year":"2020","unstructured":"Liu, C., Wang, Y.: A variational Lagrangian scheme for a phase field model: a discrete energetic variational approach. SIAM J. Sci. Comput. 42(6), B1541\u2013B1569 (2020b)","journal-title":"SIAM J. Sci. Comput."},{"key":"10009_CR42","doi-asserted-by":"crossref","unstructured":"Liu, C., Zhu, J.: Riemannian Stein variational gradient descent for Bayesian inference. In: 32nd AAAI Conference on Artificial Intelligence (2018)","DOI":"10.1609\/aaai.v32i1.11810"},{"key":"10009_CR43","unstructured":"Liu, C., Zhuo, J., Cheng, P., Zhang, R., Zhu, J.: Understanding and accelerating particle-based variational inference. In: International Conference on Machine Learning, pp. 4082\u20134092 (2019)"},{"issue":"2","key":"10009_CR44","doi-asserted-by":"publisher","first-page":"648","DOI":"10.1137\/18M1187611","volume":"51","author":"J Lu","year":"2019","unstructured":"Lu, J., Lu, Y., Nolen, J.: Scaling limit of the Stein variational gradient descent: the mean field regime. SIAM J. Math. Anal. 51(2), 648\u2013671 (2019)","journal-title":"SIAM J. Math. Anal."},{"key":"10009_CR45","volume-title":"Information Theory, Inference and Learning Algorithms","author":"DJ MacKay","year":"2003","unstructured":"MacKay, D.J., Mac Kay, D.J.: Information Theory, Inference and Learning Algorithms. Cambridge University Press, Cambridge (2003)"},{"issue":"1","key":"10009_CR46","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1051\/m2an\/2018045","volume":"53","author":"D Matthes","year":"2019","unstructured":"Matthes, D., Plazotta, S.: A variational formulation of the BDF2 method for metric gradient flows. ESAIM: Math. Model. Numer. Anal. 53(1), 145\u2013172 (2019)","journal-title":"ESAIM: Math. Model. Numer. Anal."},{"issue":"6","key":"10009_CR47","doi-asserted-by":"publisher","first-page":"1087","DOI":"10.1063\/1.1699114","volume":"21","author":"N Metropolis","year":"1953","unstructured":"Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equation of state calculations by fast computing machines. J. Chem. Phys. 21(6), 1087\u20131092 (1953)","journal-title":"J. Chem. Phys."},{"key":"10009_CR48","unstructured":"Mika, S., Ratsch, G., Weston, J., Scholkopf, B., Mullers, K.R.: Fisher discriminant analysis with kernels. In: Neural networks for signal processing IX: Proceedings of the 1999 IEEE Signal Processing Society Workshop, pp. 41\u201348. IEEE (1999)"},{"key":"10009_CR49","volume-title":"Machine Learning: A Probabilistic Perspective","author":"KP Murphy","year":"2012","unstructured":"Murphy, K.P.: Machine Learning: A Probabilistic Perspective. MIT Press, Cambridge (2012)"},{"key":"10009_CR50","unstructured":"Neal, R.M.: Probabilistic inference using Markov chain Monte Carlo methods. Department of Computer Science, University of Toronto Toronto, Ontario, Canada (1993)"},{"key":"10009_CR51","doi-asserted-by":"crossref","unstructured":"Neal, R.M., Hinton, G.E.: A view of the EM algorithm that justifies incremental, sparse, and other variants. In: Learning in Graphical Models, pp. 355\u2013368. Springer (1998)","DOI":"10.1007\/978-94-011-5014-9_12"},{"issue":"4","key":"10009_CR52","doi-asserted-by":"publisher","first-page":"405","DOI":"10.1103\/PhysRev.37.405","volume":"37","author":"L Onsager","year":"1931","unstructured":"Onsager, L.: Reciprocal relations in irreversible processes. I. Phys. Rev. 37(4), 405 (1931a)","journal-title":"Phys. Rev."},{"issue":"12","key":"10009_CR53","doi-asserted-by":"publisher","first-page":"2265","DOI":"10.1103\/PhysRev.38.2265","volume":"38","author":"L Onsager","year":"1931","unstructured":"Onsager, L.: Reciprocal relations in irreversible processes. II. Phys. Rev. 38(12), 2265 (1931b)","journal-title":"Phys. Rev."},{"key":"10009_CR54","unstructured":"Papamakarios, G., Nalisnick, E., Rezende, D.J., Mohamed, S., Lakshminarayanan, B.: Normalizing flows for probabilistic modeling and inference (2019). arXiv preprint arXiv:1912.02762"},{"issue":"3","key":"10009_CR55","doi-asserted-by":"publisher","first-page":"378","DOI":"10.1016\/0550-3213(81)90056-0","volume":"180","author":"G Parisi","year":"1981","unstructured":"Parisi, G.: Correlation functions and computer simulations. Nucl. Phys. B 180(3), 378\u2013384 (1981)","journal-title":"Nucl. Phys. B"},{"issue":"1","key":"10009_CR56","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1112\/plms\/s1-5.1.119","volume":"1","author":"L Rayleigh","year":"1873","unstructured":"Rayleigh, L.: Note on the numerical calculation of the roots of fluctuating functions. Proc. Lond. Math. Soc. 1(1), 119\u2013124 (1873)","journal-title":"Proc. Lond. Math. Soc."},{"key":"10009_CR57","unstructured":"Rezende, D.J., Mohamed, S.: Variational inference with normalizing flows (2015). arXiv preprint arXiv:1505.05770"},{"issue":"4","key":"10009_CR58","doi-asserted-by":"publisher","first-page":"341","DOI":"10.2307\/3318418","volume":"2","author":"GO Roberts","year":"1996","unstructured":"Roberts, G.O., Tweedie, R.L., et al.: Exponential convergence of Langevin distributions and their discrete approximations. Bernoulli 2(4), 341\u2013363 (1996)","journal-title":"Bernoulli"},{"issue":"5","key":"10009_CR59","doi-asserted-by":"publisher","first-page":"877","DOI":"10.1137\/0314056","volume":"14","author":"RT Rockafellar","year":"1976","unstructured":"Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 877\u2013898 (1976)","journal-title":"SIAM J. Control Optim."},{"issue":"10","key":"10009_CR60","doi-asserted-by":"publisher","first-page":"4628","DOI":"10.1063\/1.436415","volume":"69","author":"PJ Rossky","year":"1978","unstructured":"Rossky, P.J., Doll, J.D., Friedman, H.L.: Brownian dynamics as smart Monte Carlo simulation. J. Chem. Phys. 69(10), 4628\u20134633 (1978)","journal-title":"J. Chem. Phys."},{"key":"10009_CR61","unstructured":"Salimans, T., Kingma, D., Welling, M.: Markov chain Monte Carlo and variational inference: bridging the gap. In: International Conference on Machine Learning, pp. 1218\u20131226 (2015)"},{"key":"10009_CR62","unstructured":"Salman, H., Yadollahpour, P., Fletcher, T., Batmanghelich, K.: Deep diffeomorphic normalizing flows (2018). arXiv preprint arXiv:1810.03256"},{"issue":"1","key":"10009_CR63","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1007\/s13373-017-0101-1","volume":"7","author":"F Santambrogio","year":"2017","unstructured":"Santambrogio, F.: $$\\{$$Euclidean, metric, and Wasserstein$$\\}$$ gradient flows: an overview. Bull. Math. Sci 7(1), 87\u2013154 (2017)","journal-title":"Bull. Math. Sci"},{"issue":"1","key":"10009_CR64","first-page":"31","volume":"20","author":"S Sonoda","year":"2019","unstructured":"Sonoda, S., Murata, N.: Transport analysis of infinitely deep neural network. J. Mach. Learn. Res. 20(1), 31\u201382 (2019)","journal-title":"J. Mach. Learn. Res."},{"key":"10009_CR65","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1017\/S0962492910000061","volume":"19","author":"AM Stuart","year":"2010","unstructured":"Stuart, A.M.: Inverse problems: a Bayesian perspective. Acta Numer. 19, 451\u2013559 (2010)","journal-title":"Acta Numer."},{"issue":"1","key":"10009_CR66","doi-asserted-by":"publisher","first-page":"217","DOI":"10.4310\/CMS.2010.v8.n1.a11","volume":"8","author":"EG Tabak","year":"2010","unstructured":"Tabak, E.G., Vanden-Eijnden, E., et al.: Density estimation by dual ascent of the log-likelihood. Commun. Math. Sci. 8(1), 217\u2013233 (2010)","journal-title":"Commun. Math. Sci."},{"key":"10009_CR67","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511755422","volume-title":"Mathematical Modeling in Continuum Mechanics","author":"R Temam","year":"2005","unstructured":"Temam, R., Miranville, A.: Mathematical Modeling in Continuum Mechanics. Cambridge University Press, Cambridge (2005)"},{"key":"10009_CR68","volume-title":"Optimal Transport: Old and New","author":"C Villani","year":"2008","unstructured":"Villani, C.: Optimal Transport: Old and New, vol. 338. Springer, Berlin (2008)"},{"issue":"1\u20132","key":"10009_CR69","first-page":"1","volume":"1","author":"MJ Wainwright","year":"2008","unstructured":"Wainwright, M.J., Jordan, M.I., et al.: Graphical models, exponential families, and variational inference. Found. Trends\u00ae Mach. Learn. 1(1\u20132), 1\u2013305 (2008)","journal-title":"Found. Trends\u00ae Mach. Learn."},{"key":"10009_CR70","unstructured":"Wang, D., Tang, Z., Bajaj, C., Liu, Q.: Stein variational gradient descent with matrix-valued kernels. In: Advances in Neural Information Processing Systems, pp. 7834\u20137844 (2019)"},{"key":"10009_CR71","unstructured":"Welling, M., Teh, Y.W.: Bayesian learning via stochastic gradient Langevin dynamics. In: Proceedings of the 28th International Conference on Machine Learning, pp. 681\u2013688 (2011)"}],"container-title":["Statistics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-021-10009-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11222-021-10009-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-021-10009-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,28]],"date-time":"2024-08-28T16:29:10Z","timestamp":1724862550000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11222-021-10009-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,17]]},"references-count":71,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,5]]}},"alternative-id":["10009"],"URL":"https:\/\/doi.org\/10.1007\/s11222-021-10009-7","relation":{},"ISSN":["0960-3174","1573-1375"],"issn-type":[{"value":"0960-3174","type":"print"},{"value":"1573-1375","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,17]]},"assertion":[{"value":"2 June 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 March 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 April 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"34"}}