{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,18]],"date-time":"2026-09-18T08:09:27Z","timestamp":1789718967430,"version":"4.0.1"},"reference-count":30,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"name":"PhILMS","award":["DE-SC0019453"],"award-info":[{"award-number":["DE-SC0019453"]}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-20-1-0358"],"award-info":[{"award-number":["FA9550-20-1-0358"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2022,2]]},"abstract":"<jats:p>We propose a new method for inferring the governing stochastic ordinary differential equations (SODEs) by observing particle ensembles at discrete and sparse time instants, i.e., multiple \u201csnapshots.\u201d Particle coordinates at a single time instant, possibly noisy or truncated, are recorded in each snapshot but are unpaired across the snapshots. By training a physics-informed generative model that generates \u201cfake\u201d sample paths, we aim to fit the observed particle ensemble distributions with a curve in the probability measure space, which is induced from the inferred particle dynamics. We employ different metrics to quantify the differences between distributions, e.g., the sliced Wasserstein distances and the adversarial losses in generative adversarial networks. We refer to this method as generative \u201censemble-regression\u201d (GER), in analogy to the classic \u201cpoint-regression,\u201d where we infer the dynamics by performing regression in the Euclidean space. We illustrate the GER by learning the drift and diffusion terms of particle ensembles governed by SODEs with Brownian motions and L\u00e9vy processes up to 100 dimensions. We also discuss how to treat cases with noisy or truncated observations. Apart from systems consisting of independent particles, we also tackle nonlocal interacting particle systems with unknown interaction potential parameters by constructing a physics-informed loss function. Finally, we investigate scenarios of paired observations and discuss how to reduce the dimensionality in such cases by proving a convergence theorem that provides theoretical support.<\/jats:p>","DOI":"10.1137\/21m1413018","type":"journal-article","created":{"date-parts":[[2022,1,13]],"date-time":"2022-01-13T11:18:27Z","timestamp":1642072707000},"page":"B80-B99","source":"Crossref","is-referenced-by-count":27,"title":["Generative Ensemble Regression: Learning Particle Dynamics from Observations of Ensembles with Physics-informed Deep Generative Models"],"prefix":"10.1137","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7476-9168","authenticated-orcid":true,"given":"Liu","family":"Yang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Constantinos","family":"Daskalakis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9713-7120","authenticated-orcid":true,"given":"George E.","family":"Karniadakis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2022,1,13]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.20188"},{"key":"atypb2","volume-title":"Gradient Flows: In Metric Spaces and in the Space of Probability Measures","author":"Ambrosio L.","year":"2008"},{"key":"atypb3","first-page":"1","volume":"1","author":"Archambeau C.","year":"2007","journal-title":"J. Mach. Learn. Res."},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2021.110219"},{"key":"atypb5","first-page":"6571","volume-title":"Proceedings of the Conference on Advances in Neural Information Processing Systems","author":"Chen T. Q.","year":"2018"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2007.895842"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR.2018.00367"},{"key":"atypb8","volume-title":"Subadditivity of Probability Divergences on Bayes-Nets with Applications to Time Series GANs, preprint, arXiv:2003:00652","author":"Ding M.","year":"2020"},{"key":"atypb9","first-page":"32","volume":"1","author":"Donoho D. L.","year":"2000","journal-title":"AMS Math Challenges Lecture"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1111\/1468-0262.00226"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1198\/073500101316970403"},{"key":"atypb12","first-page":"2672","volume-title":"Proceedings of the Conference on Advances in Neural Information Processing Systems","author":"Goodfellow I.","year":"2014"},{"key":"atypb13","first-page":"5767","volume-title":"Proceedings of the Conference on Advances in Neural Information Processing Systems","author":"Gulrajani I.","year":"2017"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1162\/neco.1997.9.8.1735"},{"key":"atypb15","first-page":"9843","volume-title":"Proceedings of the Conference on Advances in Neural Information Processing Systems","author":"Jia J.","year":"2019"},{"key":"atypb16","volume-title":"Adam: A Method for Stochastic Optimization, preprint, arXiv:1412.6980","author":"Kingma D. P.","year":"2014"},{"key":"atypb17","volume-title":"Scalable Gradients for Stochastic Differential Equations, preprint, arXiv:2001.01328","author":"Li X.","year":"2020"},{"key":"atypb18","volume-title":"RODE-Net: Learning Ordinary Differential Equations with Randomness from Data, preprint, arXiv:2006.02377","author":"Liu J.","year":"2020"},{"key":"atypb19","volume-title":"Neural SDE: Stabilizing Neural ODE Networks with Stochastic Noise, preprint, arXiv:1906.02355","author":"Liu X.","year":"2019"},{"key":"atypb20","first-page":"3","volume-title":"Proceedings of the International Conference on Machine Learning","volume":"30","author":"Maas A. L.","year":"2013"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1007\/s42967-019-00031-y"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1109\/CDC42340.2020.9304193"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1922204117"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1007\/s13373-017-0101-1"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1007\/s11222-013-9441-1"},{"key":"atypb26","volume-title":"Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit, preprint, arXiv:1905.09883","author":"Tzen B.","year":"2019"},{"key":"atypb27","volume-title":"Theoretical Guarantees for Sampling and Inference in Generative Models with Latent Diffusions, preprint, arXiv:1903.01608","author":"Tzen B.","year":"2019"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.91.012148"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1137\/18M1225409"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2019.05.027"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/21M1413018","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:13:48Z","timestamp":1787325228000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/21M1413018"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,13]]},"references-count":30,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,2]]}},"alternative-id":["10.1137\/21M1413018"],"URL":"https:\/\/doi.org\/10.1137\/21m1413018","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,1,13]]}}}