{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,27]],"date-time":"2026-06-27T16:49:02Z","timestamp":1782578942224,"version":"3.54.5"},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T00:00:00Z","timestamp":1780617600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T00:00:00Z","timestamp":1780617600000},"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":["DMS-2204288"],"award-info":[{"award-number":["DMS-2204288"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2238486, DMS-2511283"],"award-info":[{"award-number":["DMS-2238486, DMS-2511283"]}],"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":[[2026,7]]},"DOI":"10.1007\/s10915-026-03348-y","type":"journal-article","created":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T16:18:47Z","timestamp":1780676327000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Self-Test Loss Functions for Learning Weak-Form Operators and Gradient Flows"],"prefix":"10.1007","volume":"108","author":[{"given":"Yuan","family":"Gao","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Quanjun","family":"Lang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6842-7922","authenticated-orcid":false,"given":"Fei","family":"Lu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,6,5]]},"reference":[{"key":"3348_CR1","doi-asserted-by":"crossref","unstructured":"Bakry, D., Gentil, I., Ledoux, M.: Analysis and Geometry of Markov Diffusion Operators. Grundlehren der Mathematischen Wissenschaften, vol. 348. Springer, Cham (2014)","DOI":"10.1007\/978-3-319-00227-9"},{"issue":"11","key":"3348_CR2","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6420\/abb447","volume":"36","author":"G Bao","year":"2020","unstructured":"Bao, G., Ye, X., Zang, Y., Zhou, H.: Numerical solution of inverse problems by weak adversarial networks. Inverse Prob. 36(11), 115003 (2020)","journal-title":"Inverse Prob."},{"key":"3348_CR3","doi-asserted-by":"crossref","unstructured":"Carrillo, J., Craig, K., Yao, Y.: Aggregation-diffusion equations: dynamics, asymptotics, and singular limits. In Active Particles, Volume 2, pages 65\u2013108. Springer, (2019)","DOI":"10.1007\/978-3-030-20297-2_3"},{"key":"3348_CR4","doi-asserted-by":"crossref","unstructured":"Jose, A., Carrillo, G., Estrada-Rodriguez, L., Mikolas, S., Tang: Sparse identification of nonlocal interaction kernels in nonlinear gradient flow equations via partial inversion, (2024). arXiv preprint arXiv:2402.06355","DOI":"10.1142\/S0218202525500137"},{"key":"3348_CR5","unstructured":"Chada, N.K., Lang, Q., Lu, F., Wang, X.: A data-adaptive RKHS prior for Bayesian learning of kernels in operators. J. Mach. Learn. Res. 25(317), 1\u201337 (2024)"},{"issue":"45","key":"3348_CR6","doi-asserted-by":"publisher","first-page":"22445","DOI":"10.1073\/pnas.1906995116","volume":"116","author":"K Champion","year":"2019","unstructured":"Champion, K., Bethany Lusch, J., Kutz, N., Brunton, S.L.: Data-driven discovery of coordinates and governing equations. Proc. Natl. Acad. Sci. 116(45), 22445\u201322451 (2019)","journal-title":"Proc. Natl. Acad. Sci."},{"issue":"2","key":"3348_CR7","doi-asserted-by":"publisher","first-page":"811","DOI":"10.1137\/22M1522504","volume":"62","author":"T De Ryck","year":"2024","unstructured":"De Ryck, T., Mishra, S., Molinaro, R.: wpinns: Weak physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws. SIAM J. Numer. Anal. 62(2), 811\u2013841 (2024)","journal-title":"SIAM J. Numer. Anal."},{"key":"3348_CR8","unstructured":"Dozat, T.: Incorporating Nesterov momentum into Adam, (2016). (ICLR 2016 workshop)"},{"key":"3348_CR9","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)","DOI":"10.1007\/s40304-018-0127-z"},{"key":"3348_CR10","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2021.114502","volume":"390","author":"H Gao","year":"2022","unstructured":"Gao, H., Zahr, M.J., Wang, J.: X Physics-informed graph neural galerkin networks: A unified framework for solving pde-governed forward and inverse problems. Comput. Methods Appl. Mech. Eng. 390, 114502 (2022)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"8","key":"3348_CR11","doi-asserted-by":"publisher","first-page":"3816","DOI":"10.1088\/1361-6544\/ab853d","volume":"33","author":"Y Gao","year":"2020","unstructured":"Gao, Y., Liu, J.-G., Lu, J., Marzuola, J.L.: Analysis of a continuum theory for broken bond crystal surface models with evaporation and deposition effects. Nonlinearity 33(8), 3816 (2020)","journal-title":"Nonlinearity"},{"key":"3348_CR12","first-page":"49","volume":"25","author":"Y Gao","year":"2019","unstructured":"Gao, Y., Liu, J.-G., Lu, X.Y.: Gradient flow approach to an exponential thin film equation: global existence and latent singularity. ESAIM: Control, Optimisation and Calculus of Variations 25, 49 (2019)","journal-title":"ESAIM: Control, Optimisation and Calculus of Variations"},{"key":"3348_CR13","doi-asserted-by":"publisher","first-page":"445","DOI":"10.1017\/S0962492921000064","volume":"30","author":"O Ghattas","year":"2021","unstructured":"Ghattas, O., Willcox, K.: Learning physics-based models from data: perspectives from inverse problems and model reduction. Acta Numer 30, 445\u2013554 (2021)","journal-title":"Acta Numer"},{"key":"3348_CR14","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.coisb.2020.07.009","volume":"22","author":"W Gilpin","year":"2020","unstructured":"Gilpin, W., Huang, Y., Forger, D.B.: Learning dynamics from large biological data sets: machine learning meets systems biology. Current Opinion in Systems Biology 22, 1\u20137 (2020)","journal-title":"Current Opinion in Systems Biology"},{"key":"3348_CR15","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2020.132817","volume":"421","author":"B Hamzi","year":"2021","unstructured":"Hamzi, B., Owhadi, H.: Learning dynamical systems from data: A simple cross-validation perspective, part i: Parametric kernel flows. Physica D 421, 132817 (2021)","journal-title":"Physica D"},{"key":"3348_CR16","unstructured":"Hansen, P.C.: The L-curve and its use in the numerical treatment of inverse problems. In in Computational Inverse Problems in Electrocardiology, ed. P. Johnston, Advances in Computational Bioengineering, pages 119\u2013142. WIT Press, (2000)"},{"key":"3348_CR17","doi-asserted-by":"crossref","unstructured":"Hu, Z., Liu, C., Wang, Y., Xu, Z. Energetic variational neural network discretizations of gradient flows. SIAM J. Sci. Comput. 46(4), A2528\u2013A2556 (2024)","DOI":"10.1137\/22M1529427"},{"key":"3348_CR18","unstructured":"Isakov, V.: Inverse problems for partial differential equations, volume 127. Springer, (2006)"},{"issue":"12","key":"3348_CR19","doi-asserted-by":"publisher","first-page":"3588","DOI":"10.1016\/j.jfa.2016.09.014","volume":"271","author":"P-E Jabin","year":"2016","unstructured":"Jabin, P.-E., Wang, Z.: Mean field limit and propagation of chaos for Vlasov systems with bounded forces. J. Funct. Anal. 271(12), 3588\u20133627 (2016)","journal-title":"J. Funct. Anal."},{"key":"3348_CR20","doi-asserted-by":"crossref","unstructured":"Jabin, P.E., Wang, Z.: Mean field limit for stochastic particle systems. In Active Particles, Volume 1, pages 379\u2013402. Springer, (2017)","DOI":"10.1007\/978-3-319-49996-3_10"},{"key":"3348_CR21","unstructured":"Kharazmi, E., Zhang, Z., Karniadakis, G.E.: Variational physics-informed neural networks for solving partial differential equations, (2019). arXiv preprint arXiv:1912.00873"},{"key":"3348_CR22","doi-asserted-by":"publisher","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."},{"key":"3348_CR23","doi-asserted-by":"crossref","unstructured":"Lang, Q., Lu, F., Learning interaction kernels in mean-field equations of first-order systems of interacting particles. SIAM J. Sci. Comput. 44(1), A260\u2013A285 (2022)","DOI":"10.1137\/20M1377072"},{"key":"3348_CR24","doi-asserted-by":"crossref","unstructured":"Lang, Q., Lu, F., Identifiability of interaction kernels in mean-field equations of interacting particles. Foundations of Data Science 5(4), 480\u2013502 (2023)","DOI":"10.3934\/fods.2023007"},{"key":"3348_CR25","unstructured":"Long, Z., Lu, Y., Ma, X., Dong, B. PDE-Net: Learning PDEs from Data. In Proceedings of the 35th International Conference on Machine Learning, volume\u00a080, page\u00a09. PMLR, (2018)"},{"key":"3348_CR26","unstructured":"Lu, F., Lang, Q., An, Q.: Data adaptive RKHS Tikhonov regularization for learning kernels in operators. Proceedings of Mathematical and Scientific Machine Learning, PMLR 190, 158\u2013172 (2022)"},{"key":"3348_CR27","doi-asserted-by":"crossref","unstructured":"Lu, F., Maggioni, M., Tang, S.: Learning interaction kernels in heterogeneous systems of agents from multiple trajectories. J. Mach. Learn. Res. 22(32), 1\u201367 (2021)","DOI":"10.1007\/s10208-021-09521-z"},{"key":"3348_CR28","doi-asserted-by":"crossref","unstructured":"Lu, F., Maggioni, M., Tang, S.: Learning interaction kernels in stochastic systems of interacting particles from multiple trajectories. Foundations of Computational Mathematics, pages 1\u201355, (2021)","DOI":"10.1007\/s10208-021-09521-z"},{"issue":"29","key":"3348_CR29","doi-asserted-by":"publisher","first-page":"14424","DOI":"10.1073\/pnas.1822012116","volume":"116","author":"L Fei","year":"2019","unstructured":"Fei, L., Zhong, M., Tang, S., Maggioni, M.: Nonparametric inference of interaction laws in systems of agents from trajectory data. Proc. Natl. Acad. Sci. U.S.A. 116(29), 14424\u201314433 (2019)","journal-title":"Proc. Natl. Acad. Sci. U.S.A."},{"key":"3348_CR30","doi-asserted-by":"crossref","unstructured":"Lu, Y., Li, X., Liu, C., Tang, Q., Wang, Y.: Learning generalized diffusions using an energetic variational approach, (2024)","DOI":"10.2139\/ssrn.5054728"},{"key":"3348_CR31","doi-asserted-by":"crossref","unstructured":"Messenger, D.A., Bortz, D.M.: Bortz: Weak sindy for partial differential equations. J. Comput. Phys. 443, 110525 (2021)","DOI":"10.1016\/j.jcp.2021.110525"},{"issue":"3","key":"3348_CR32","doi-asserted-by":"publisher","first-page":"1474","DOI":"10.1137\/20M1343166","volume":"19","author":"DA Messenger","year":"2021","unstructured":"Messenger, D.A., Bortz, D.M.: Weak sindy: Galerkin-based data-driven model selection. Multiscale Modeling & Simulation 19(3), 1474\u20131497 (2021)","journal-title":"Multiscale Modeling & Simulation"},{"key":"3348_CR33","unstructured":"Messenger, D.A., Tran, A., Dukic, V., Bortz, D.M.: The weak form is stronger than you think, (2024). arXiv preprint arXiv:2409.06751"},{"key":"3348_CR34","volume-title":"Stochastic differential equations: an introduction with applications","author":"B \u00d8ksendal","year":"2013","unstructured":"\u00d8ksendal, B.: Stochastic differential equations: an introduction with applications, 6th edn. Springer Science & Business Media, New York (2013)","edition":"6"},{"key":"3348_CR35","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1016\/j.jcp.2019.03.040","volume":"389","author":"H Owhadi","year":"2019","unstructured":"Owhadi, H., Yoo, G.R.: Kernel flows: From learning kernels from data into the abyss. J. Comput. Phys. 389, 22\u201347 (2019)","journal-title":"J. Comput. Phys."},{"key":"3348_CR36","unstructured":"Ren, K., Zhang, L.: Data-driven joint inversions for PDE models, (2022). arXiv preprint arXiv:2210.09228"},{"issue":"2197","key":"3348_CR37","doi-asserted-by":"publisher","first-page":"20160446","DOI":"10.1098\/rspa.2016.0446","volume":"473","author":"H Schaeffer","year":"2017","unstructured":"Schaeffer, H.: Learning partial differential equations via data discovery and sparse optimization. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473(2197), 20160446 (2017)","journal-title":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences"},{"key":"3348_CR38","doi-asserted-by":"crossref","unstructured":"Schafer, W.R.: Schafer: What is a Savitzky-Golay filter? IEEE Signal Process. Mag. 28(4), 111\u2013117 (2011)","DOI":"10.1109\/MSP.2011.941097"},{"issue":"2","key":"3348_CR39","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1016\/j.cam.2006.11.021","volume":"211","author":"LF Shampine","year":"2008","unstructured":"Shampine, L.F.: Vectorized adaptive quadrature in matlab. J. Comput. Appl. Math. 211(2), 131\u2013140 (2008)","journal-title":"J. Comput. Appl. Math."},{"key":"3348_CR40","doi-asserted-by":"publisher","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":"3348_CR41","doi-asserted-by":"crossref","unstructured":"Song, W., Jiang, S., Camps-Valls, G., Mathew Williams, L., Zhang, M.R., Vereecken, H., He, L., Hu, X., Shi, L.: Towards data-driven discovery of governing equations in geosciences. Communications Earth & Environment 5(1), 589 (2024)","DOI":"10.1038\/s43247-024-01760-6"},{"key":"3348_CR42","unstructured":"Song, Y., Sohl-Dickstein, J., Diederik, P., Kingma, A., Kumar, S., Ermon, B., Poole: Score-based generative modeling through stochastic differential equations. In: International Conference on Learning Representations, (2021)"},{"key":"3348_CR43","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2024.112950","volume":"506","author":"R Stephany","year":"2024","unstructured":"Stephany, R., Earls, C.: Weak-pde-learn: A weak form based approach to discovering pdes from noisy, limited data. J. Comput. Phys. 506, 112950 (2024)","journal-title":"J. Comput. Phys."},{"key":"3348_CR44","doi-asserted-by":"crossref","unstructured":"Villani, C.: Topics in optimal transportation, volume\u00a058. American Mathematical Soc., (2003)","DOI":"10.1090\/gsm\/058"},{"key":"3348_CR45","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s11222-021-10009-7","volume":"31","author":"Y Wang","year":"2021","unstructured":"Wang, Y., Chen, J., Liu, C., Kang, L.: Particle-based energetic variational inference. Stat. Comput. 31, 1\u201317 (2021)","journal-title":"Stat. Comput."},{"issue":"1","key":"3348_CR46","doi-asserted-by":"publisher","first-page":"B80","DOI":"10.1137\/21M1413018","volume":"44","author":"L Yang","year":"2022","unstructured":"Yang, L., Daskalakis, C., Karniadakis, G.E.: Generative ensemble regression: Learning particle dynamics from observations of ensembles with physics-informed deep generative models. SIAM J. Sci. Comput. 44(1), B80\u2013B99 (2022)","journal-title":"SIAM J. Sci. Comput."},{"key":"3348_CR47","first-page":"6876897","volume":"99","author":"R Yao","year":"2022","unstructured":"Yao, R., Chen, X., Yang, Y.: Mean-field nonparametric estimation of interacting particle systems. 99, 6876897 (2022)","journal-title":"Mean-field nonparametric estimation of interacting particle systems."},{"key":"3348_CR48","doi-asserted-by":"publisher","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."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-026-03348-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-026-03348-y","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-026-03348-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,27]],"date-time":"2026-06-27T16:31:41Z","timestamp":1782577901000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-026-03348-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6,5]]},"references-count":48,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,7]]}},"alternative-id":["3348"],"URL":"https:\/\/doi.org\/10.1007\/s10915-026-03348-y","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,6,5]]},"assertion":[{"value":"3 January 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 December 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 May 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 June 2026","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 declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of Interest"}}],"article-number":"28"}}