{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T11:17:18Z","timestamp":1762255038714,"version":"3.37.3"},"reference-count":62,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,1,20]],"date-time":"2025-01-20T00:00:00Z","timestamp":1737331200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,1,20]],"date-time":"2025-01-20T00:00:00Z","timestamp":1737331200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS1913163"],"award-info":[{"award-number":["DMS1913163"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2025,3]]},"DOI":"10.1007\/s10915-025-02791-7","type":"journal-article","created":{"date-parts":[[2025,1,20]],"date-time":"2025-01-20T08:14:49Z","timestamp":1737360889000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Adaptive Deep Density Approximation for Stochastic Dynamical Systems"],"prefix":"10.1007","volume":"102","author":[{"given":"Junjie","family":"He","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2033-6356","authenticated-orcid":false,"given":"Qifeng","family":"Liao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaoliang","family":"Wan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,1,20]]},"reference":[{"unstructured":"Ben-Hamu, H., Cohen, S., Bose, J., Amos, B., Nickel, M., Grover, A., Chen, R.T.Q., Lipman, Y.: Matching normalizing flows and probability paths on manifolds. In: Proceedings of the 39th International Conference on Machine Learning, pp. 1749\u20131763 (2022)","key":"2791_CR1"},{"key":"2791_CR2","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1016\/j.jcp.2018.06.038","volume":"372","author":"C Brennan","year":"2018","unstructured":"Brennan, C., Venturi, D.: Data-driven closures for stochastic dynamical systems. J. Comput. Phys. 372, 281\u2013298 (2018)","journal-title":"J. Comput. Phys."},{"issue":"6","key":"2791_CR3","doi-asserted-by":"crossref","first-page":"2554","DOI":"10.1137\/080740647","volume":"41","author":"G Carlier","year":"2010","unstructured":"Carlier, G., Galichon, A., Santambrogio, F.: From Knothe\u2019s transport to Brenier\u2019s map and a continuation method for optimal transport. SIAM J. Math. Anal. 41(6), 2554\u20132576 (2010)","journal-title":"SIAM J. Math. Anal."},{"unstructured":"Chen, T.Q., Rubanova, Y., Bettencourt, J., Duvenaud, D.: Neural ordinary differential equations. In: Advances in Neural Information Processing Systems 31, NeurIPS 2018, pp. 6572\u20136583 (2018)","key":"2791_CR4"},{"key":"2791_CR5","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2021.110782","volume":"449","author":"Z Chen","year":"2022","unstructured":"Chen, Z., Churchill, V., Wu, K., Xiu, D.: Deep neural network modeling of unknown partial differential equations in nodal space. J. Comput. Phys. 449, 110782 (2022)","journal-title":"J. Comput. Phys."},{"unstructured":"Chewi, S.: Log-concave sampling. https:\/\/chewisinho.github.io\/main.pdf (2024)","key":"2791_CR6"},{"issue":"4","key":"2791_CR7","doi-asserted-by":"crossref","first-page":"B890","DOI":"10.1137\/12088896X","volume":"35","author":"H Cho","year":"2013","unstructured":"Cho, H., Venturi, D., Karniadakis, G.E.: Adaptive discontinuous Galerkin method for response-excitation PDF equations. SIAM J. Sci. Comput. 35(4), B890\u2013B911 (2013)","journal-title":"SIAM J. Sci. Comput."},{"key":"2791_CR8","doi-asserted-by":"crossref","first-page":"817","DOI":"10.1016\/j.jcp.2015.10.030","volume":"305","author":"H Cho","year":"2016","unstructured":"Cho, H., Venturi, D., Karniadakis, G.E.: Numerical methods for high-dimensional probability density function equations. J. Comput. Phys. 305, 817\u2013837 (2016)","journal-title":"J. Comput. Phys."},{"unstructured":"Dinh, L., Sohl-Dickstein, J., Bengio, S.: Density estimation using Real NVP. In: 5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24\u201326, 2017, Conference Track Proceedings. OpenReview.net (2017)","key":"2791_CR9"},{"key":"2791_CR10","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2021.114129","volume":"387","author":"S Dong","year":"2021","unstructured":"Dong, S., Li, Z.: Local extreme learning machines and domain decomposition for solving linear and nonlinear partial differential equations. Comput. Methods Appl. Mech. Eng. 387, 114129 (2021)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"doi-asserted-by":"crossref","unstructured":"E, W.: A proposal on machine learning via dynamical systems. Commun. Math. Stat. 1(5), 1\u201311 (2017)","key":"2791_CR11","DOI":"10.1007\/s40304-017-0103-z"},{"doi-asserted-by":"crossref","unstructured":"E, W., Yu, B.: The Deep Ritz Method: a deep learning-based numerical algorithm for solving variational problems. Commun. Math. Stat. 6(1), 1\u201312 (2018)","key":"2791_CR12","DOI":"10.1007\/s40304-018-0127-z"},{"key":"2791_CR13","doi-asserted-by":"crossref","DOI":"10.1093\/acprof:oso\/9780199678792.001.0001","volume-title":"Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics","author":"HC Elman","year":"2014","unstructured":"Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics. Oxford University Press, Oxford (2014)"},{"issue":"2","key":"2791_CR14","doi-asserted-by":"crossref","first-page":"401","DOI":"10.4208\/cicp.OA-2022-0090","volume":"32","author":"X Feng","year":"2022","unstructured":"Feng, X., Zeng, L., Zhou, T.: Solving time dependent Fokker-Planck equations via temporal normalizing flow. Commun. Comput. Phys. 32(2), 401\u2013423 (2022)","journal-title":"Commun. Comput. Phys."},{"key":"2791_CR15","volume":"428","author":"H Gao","year":"2021","unstructured":"Gao, H., Sun, L., Wang, J.X.: PhyGeoNet: physics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state pdes on irregular domain. J. Comput. Phys. 428, 110079 (2021)","journal-title":"J. Comput. Phys."},{"doi-asserted-by":"crossref","unstructured":"He, K., Zhang, X., Ren, S., Sun, J.: Delving deep into rectifiers: surpassing human-level performance on imagenet classification. In: Proceedings of the IEEE international conference on computer vision, pp. 1026\u20131034 (2015)","key":"2791_CR16","DOI":"10.1109\/ICCV.2015.123"},{"doi-asserted-by":"crossref","unstructured":"Heinlein, A., Klawonn, A., Lanser, M., Weber, J.: Combining machine learning and domain decomposition methods for the solution of partial differential equations-a review. GAMM-Mitteilungen 44(1) (2021)","key":"2791_CR17","DOI":"10.1002\/gamm.202100001"},{"unstructured":"Hendrycks, D., Gimpel, K.: Gaussian error linear units (gelus). arXiv preprint arXiv:1606.08415 (2016)","key":"2791_CR18"},{"key":"2791_CR19","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-17146-8","volume-title":"Nonparametric and Semiparametric Models","author":"W H\u00e4rdle","year":"2004","unstructured":"H\u00e4rdle, W., Werwatz, A., M\u00fcller, M., Sperlich, S.: Nonparametric and Semiparametric Models. Springer, Berlin, Heidelberg (2004)"},{"key":"2791_CR20","first-page":"1","volume":"24","author":"I Ishikawa","year":"2023","unstructured":"Ishikawa, I., Teshima, T., Tojo, K., Oono, K., Ikeda, M., Sugiyama, M.: Universal approximation property of invertible neural networks. J. Mach. Learn. Res. 24, 1\u201368 (2023)","journal-title":"J. Mach. Learn. Res."},{"key":"2791_CR21","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2020.113028","volume":"365","author":"AD Jagtap","year":"2020","unstructured":"Jagtap, A.D., Kharazmi, E., Karniadakis, G.E.: Conservative physics-informed neural networks on discrete domains for conservation laws: applications to forward and inverse problems. Comput. Methods Appl. Mech. Eng. 365, 113028 (2020)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"4","key":"2791_CR22","doi-asserted-by":"crossref","DOI":"10.1063\/1.3496397","volume":"20","author":"A Karimi","year":"2010","unstructured":"Karimi, A., Paul, M.R.: Extensive chaos in the Lorenz-96 model. Chaos: Interdiscip. J. Nonlinear Sci. 20(4), 043105 (2010)","journal-title":"Chaos: Interdiscip. J. Nonlinear Sci."},{"issue":"6","key":"2791_CR23","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1038\/s42254-021-00314-5","volume":"3","author":"GE Karniadakis","year":"2021","unstructured":"Karniadakis, G.E., Kevrekidis, I.G., Lu, L., Perdikaris, P., Wang, S., Yang, L.: Physics-informed machine learning. Nat. Rev. Phys. 3(6), 422\u2013440 (2021)","journal-title":"Nat. Rev. Phys."},{"key":"2791_CR24","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2020.113547","volume":"374","author":"E Kharazmi","year":"2021","unstructured":"Kharazmi, E., Zhang, Z., Karniadakis, G.E.: hp-VPINNs: variational physics-informed neural networks with domain decomposition. Comput. Methods Appl. Mech. Eng. 374, 113547 (2021)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"unstructured":"Kingma, D.P., Dhariwal, P.: Glow: Generative flow with invertible 1x1 convolutions. In: Advances in Neural Information Processing Systems 31, NeurIPS 2018, pp. 10236\u201310245 (2018)","key":"2791_CR25"},{"key":"2791_CR26","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/B978-044451796-8\/50004-5","volume-title":"Dynamics of Stochastic Systems","author":"V Klyatskin","year":"2005","unstructured":"Klyatskin, V.: Chapter 3 - indicator function and Liouville equation. In: Klyatskin, V. (ed.) Dynamics of Stochastic Systems, pp. 42\u201348. Elsevier Science, Amsterdam (2005)"},{"issue":"11","key":"2791_CR27","doi-asserted-by":"crossref","first-page":"3964","DOI":"10.1109\/TPAMI.2020.2992934","volume":"43","author":"I Kobyzev","year":"2021","unstructured":"Kobyzev, I., Prince, S.J., Brubaker, M.A.: Normalizing flows: an introduction and review of current methods. IEEE Trans. Pattern Anal. Mach. Intell. 43(11), 3964\u20133979 (2021)","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"2791_CR28","doi-asserted-by":"crossref","DOI":"10.1002\/9780470824269","volume-title":"Stochastic Dynamics of Structures","author":"J Li","year":"2009","unstructured":"Li, J., Chen, J.: Stochastic Dynamics of Structures. Wiley, Hoboken (2009)"},{"key":"2791_CR29","doi-asserted-by":"crossref","first-page":"5283","DOI":"10.1109\/ACCESS.2019.2957200","volume":"8","author":"K Li","year":"2020","unstructured":"Li, K., Tang, K., Wu, T., Liao, Q.: D3M: a deep domain decomposition method for partial differential equations. IEEE Access 8, 5283\u20135294 (2020)","journal-title":"IEEE Access"},{"unstructured":"Li, W., Xiang, X., Xu, Y.: Deep domain decomposition method: Elliptic problems. In: Mathematical and Scientific Machine Learning, pp. 269\u2013286. PMLR (2020)","key":"2791_CR30"},{"key":"2791_CR31","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9781139017329","volume-title":"An Introduction to Computational Stochastic PDEs","author":"GJ Lord","year":"2014","unstructured":"Lord, G.J., Powell, C.E., Shardlow, T.: An Introduction to Computational Stochastic PDEs, vol. 50. Cambridge University Press, Cambridge (2014)"},{"unstructured":"Loshchilov, I., Hutter, F.: SGDR: Stochastic gradient descent with warm restarts. In: International Conference on Learning Representations (2017)","key":"2791_CR32"},{"unstructured":"Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. In: International Conference on Learning Representations (2019)","key":"2791_CR33"},{"key":"2791_CR34","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1016\/j.jcp.2014.06.029","volume":"274","author":"DM Luchtenburg","year":"2014","unstructured":"Luchtenburg, D.M., Brunton, S.L., Rowley, C.W.: Long-time uncertainty propagation using generalized polynomial chaos and flow map composition. J. Comput. Phys. 274, 783\u2013802 (2014)","journal-title":"J. Comput. Phys."},{"key":"2791_CR35","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2020.113250","volume":"370","author":"X Meng","year":"2020","unstructured":"Meng, X., Li, Z., Zhang, D., Karniadakis, G.E.: PPINN: parareal physics-informed neural network for time-dependent PDEs. Comput. Methods Appl. Mech. Eng. 370, 113250 (2020)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"2791_CR36","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511812248","volume-title":"Numerical Solution of Partial Differential Equations: An Introduction","author":"KW Morton","year":"2005","unstructured":"Morton, K.W., Mayers, D.F.: Numerical Solution of Partial Differential Equations: An Introduction. Cambridge University Press, Cambridge (2005)"},{"key":"2791_CR37","volume-title":"Noise in Nonlinear Dynamical Systems","author":"F Moss","year":"1989","unstructured":"Moss, F., McClintock, P.V.: Noise in Nonlinear Dynamical Systems, vol. 1. Cambridge University Press, Cambridge (1989)"},{"issue":"57","key":"2791_CR38","first-page":"1","volume":"22","author":"G Papamakarios","year":"2021","unstructured":"Papamakarios, G., Nalisnick, E., Rezende, D.J., Mohamed, S., Lakshminarayanan, B.: Normalizing flows for probabilistic modeling and inference. J. Mach. Learn. Res. 22(57), 1\u201364 (2021)","journal-title":"J. Mach. Learn. Res."},{"key":"2791_CR39","first-page":"2825","volume":"12","author":"F Pedregosa","year":"2011","unstructured":"Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., Vanderplas, J., Passos, A., Cournapeau, D., Brucher, M., Perrot, M., Duchesnay, E.: Scikit-learn: machine learning in python. J. Mach. Learn. Res. 12, 2825\u20132830 (2011)","journal-title":"J. Mach. Learn. Res."},{"key":"2791_CR40","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2023.112464","volume":"493","author":"M Penwarden","year":"2023","unstructured":"Penwarden, M., Jagtap, A.D., Zhe, S., Karniadakis, G.E., Kirby, R.M.: A unified scalable framework for causal sweeping strategies for physics-informed neural networks (PINNs) and their temporal decompositions. J. Comput. Phys. 493, 112464 (2023)","journal-title":"J. Comput. Phys."},{"key":"2791_CR41","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1016\/j.jcp.2018.10.045","volume":"378","author":"M Raissi","year":"2019","unstructured":"Raissi, M., Perdikaris, P., Karniadakis, G.: Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378, 686\u2013707 (2019)","journal-title":"J. Comput. Phys."},{"key":"2791_CR42","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-61544-3","volume-title":"The Fokker\u2013Planck Equation: Methods of Solution and Applications","author":"H Risken","year":"1996","unstructured":"Risken, H.: The Fokker\u2013Planck Equation: Methods of Solution and Applications, vol. 18. Springer, Berlin (1996)"},{"key":"2791_CR43","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, Hoboken (2015)"},{"key":"2791_CR44","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2020.110085","volume":"428","author":"H Sheng","year":"2021","unstructured":"Sheng, H., Yang, C.: PFNN: a penalty-free neural network method for solving a class of second-order boundary-value problems on complex geometries. J. Comput. Phys. 428, 110085 (2021)","journal-title":"J. Comput. Phys."},{"key":"2791_CR45","doi-asserted-by":"crossref","first-page":"1339","DOI":"10.1016\/j.jcp.2018.08.029","volume":"375","author":"J Sirignano","year":"2018","unstructured":"Sirignano, J., Spiliopoulos, K.: DGM: a deep learning algorithm for solving partial differential equations. J. Comput. Phys. 375, 1339\u20131364 (2018)","journal-title":"J. Comput. Phys."},{"key":"2791_CR46","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-011-3712-6","volume-title":"Stochastic Differential Equations: With Applications to Physics and Engineering","author":"K Sobczyk","year":"1991","unstructured":"Sobczyk, K.: Stochastic Differential Equations: With Applications to Physics and Engineering. Springer, Dordrecht (1991)"},{"issue":"3","key":"2791_CR47","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1016\/j.taml.2020.01.023","volume":"10","author":"K Tang","year":"2020","unstructured":"Tang, K., Wan, X., Liao, Q.: Deep density estimation via invertible block-triangular mapping. Theor. Appl. Mech. Lett. 10(3), 143\u2013148 (2020)","journal-title":"Theor. Appl. Mech. Lett."},{"key":"2791_CR48","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2022.111080","volume":"457","author":"K Tang","year":"2022","unstructured":"Tang, K., Wan, X., Liao, Q.: Adaptive deep density approximation for Fokker-Planck equations. J. Comput. Phys. 457, 111080 (2022)","journal-title":"J. Comput. Phys."},{"key":"2791_CR49","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2022.111868","volume":"476","author":"K Tang","year":"2023","unstructured":"Tang, K., Wan, X., Yang, C.: DAS-PINNs: a deep adaptive sampling method for solving high-dimensional partial differential equations. J. Comput. Phys. 476, 111868 (2023)","journal-title":"J. Comput. Phys."},{"doi-asserted-by":"crossref","unstructured":"Tartakovsky, D.M., Gremaud, P.A.: Method of Distributions for Uncertainty Quantification, pp. 763\u2013783. Springer, Cham (2017)","key":"2791_CR50","DOI":"10.1007\/978-3-319-12385-1_27"},{"key":"2791_CR51","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-71050-9","volume-title":"Optimal Transport: Old and New","author":"C Villani","year":"2009","unstructured":"Villani, C.: Optimal Transport: Old and New. Springer, Berlin, Heidelberg (2009)"},{"issue":"2","key":"2791_CR52","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1016\/j.jcp.2005.03.023","volume":"209","author":"X Wan","year":"2005","unstructured":"Wan, X., Karniadakis, G.E.: An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. J. Comput. Phys. 209(2), 617\u2013642 (2005)","journal-title":"J. Comput. Phys."},{"key":"2791_CR53","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2022.111855","volume":"475","author":"S Wang","year":"2023","unstructured":"Wang, S., Perdikaris, P.: Long-time integration of parametric evolution equations with physics-informed DeepONets. J. Comput. Phys. 475, 111855 (2023)","journal-title":"J. Comput. Phys."},{"key":"2791_CR54","volume":"384","author":"S Wang","year":"2021","unstructured":"Wang, S., Wang, H., Perdikaris, P.: On the eigenvector bias of Fourier feature networks: from regression to solving multi-scale PDEs with physics-informed neural networks. Comput. Methods Appl. Mech. Eng. 384, 113938 (2021)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"2791_CR55","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2019.109071","volume":"406","author":"Y Wang","year":"2020","unstructured":"Wang, Y., Cheung, S.W., Chung, E.T., Efendiev, Y., Wang, M.: Deep multiscale model learning. J. Comput. Phys. 406, 109071 (2020)","journal-title":"J. Comput. Phys."},{"key":"2791_CR56","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2019.108963","volume":"400","author":"Z Wang","year":"2020","unstructured":"Wang, Z., Zhang, Z.: A mesh-free method for interface problems using the deep learning approach. J. Comput. Phys. 400, 108963 (2020)","journal-title":"J. Comput. Phys."},{"key":"2791_CR57","volume":"408","author":"K Wu","year":"2020","unstructured":"Wu, K., Xiu, D.: Data-driven deep learning of partial differential equations in modal space. J. Comput. Phys. 408, 109307 (2020)","journal-title":"J. Comput. Phys."},{"doi-asserted-by":"crossref","unstructured":"Xu, Z., Liao, Q., Li, J.: Domain-decomposed Bayesian inversion based on local Karhunen\u2013Lo\u00e8ve expansions. J. Comput. Phys. 112856 (2024)","key":"2791_CR58","DOI":"10.1016\/j.jcp.2024.112856"},{"key":"2791_CR59","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2020.109409","volume":"411","author":"Y Zang","year":"2020","unstructured":"Zang, Y., Bao, G., Ye, X., Zhou, H.: Weak adversarial networks for high-dimensional partial differential equations. J. Comput. Phys. 411, 109409 (2020)","journal-title":"J. Comput. Phys."},{"key":"2791_CR60","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.neunet.2021.12.007","volume":"147","author":"A Zhu","year":"2022","unstructured":"Zhu, A., Jin, P., Tang, Y.: Approximation capabilities of measure-preserving neural networks. Neural Netw. 147, 72\u201380 (2022)","journal-title":"Neural Netw."},{"key":"2791_CR61","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1016\/j.jcp.2018.04.018","volume":"366","author":"Y Zhu","year":"2018","unstructured":"Zhu, Y., Zabaras, N.: Bayesian deep convolutional encoder-decoder networks for surrogate modeling and uncertainty quantification. J. Comput. Phys. 366, 415\u2013447 (2018)","journal-title":"J. Comput. Phys."},{"key":"2791_CR62","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1016\/j.jcp.2019.05.024","volume":"394","author":"Y Zhu","year":"2019","unstructured":"Zhu, Y., Zabaras, N., Koutsourelakis, P.S., Perdikaris, P.: Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data. J. Comput. Phys. 394, 56\u201381 (2019)","journal-title":"J. Comput. Phys."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-025-02791-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-025-02791-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-025-02791-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,2,18]],"date-time":"2025-02-18T17:37:13Z","timestamp":1739900233000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-025-02791-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,20]]},"references-count":62,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,3]]}},"alternative-id":["2791"],"URL":"https:\/\/doi.org\/10.1007\/s10915-025-02791-7","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"type":"print","value":"0885-7474"},{"type":"electronic","value":"1573-7691"}],"subject":[],"published":{"date-parts":[[2025,1,20]]},"assertion":[{"value":"28 June 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 November 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 January 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 January 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have not disclosed any conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"57"}}