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Uncertainty Quantification"],"published-print":{"date-parts":[[2022,12,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>Due to the importance of uncertainty quantification (UQ), the Bayesian approach to inverse problems has recently gained popularity in applied mathematics, physics, and engineering. However, traditional Bayesian inference methods based on Markov chain Monte Carlo (MCMC) tend to be computationally intensive and inefficient for such high-dimensional problems. To address this issue, several methods based on surrogate models have been proposed to speed up the inference process. More specifically, the calibration-emulation-sampling (CES) scheme has been proven to be successful in large dimensional UQ problems. In this work, we propose a novel CES approach for Bayesian inference based on deep neural network models for the emulation phase. The resulting algorithm is computationally more efficient and more robust against variations in the training set. Further, by using an autoencoder (AE) for dimension reduction, we have been able to speed up our Bayesian inference method up to three orders of magnitude. Overall, our method, henceforth called the dimension-reduced emulative autoencoder Monte Carlo (DREAMC) algorithm, is able to scale Bayesian UQ up to thousands of dimensions for inverse problems. Using two low-dimensional (linear and nonlinear) inverse problems, we illustrate the validity of this approach. Next, we apply our method to two high-dimensional numerical examples (elliptic and advection-diffusion) to demonstrate its computational advantages over existing algorithms.<\/jats:p>","DOI":"10.1137\/21m1439456","type":"journal-article","created":{"date-parts":[[2022,12,20]],"date-time":"2022-12-20T09:41:53Z","timestamp":1671529313000},"page":"1684-1713","source":"Crossref","is-referenced-by-count":18,"title":["Scaling Up Bayesian Uncertainty Quantification for Inverse Problems Using Deep Neural Networks"],"prefix":"10.1137","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9167-3715","authenticated-orcid":true,"given":"Shiwei","family":"Lan","sequence":"first","affiliation":[{"name":"School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shuyi","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Babak","family":"Shahbaba","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of California, Irvine, CA 92697-1250 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2022,12,20]]},"reference":[{"key":"ref1","first-page":"SPE-117274-PA","volume":"14","author":"Aanonsen S. I.","year":"2009","journal-title":"Soc. Pet. Eng."},{"key":"ref2","volume-title":"7th Yale Workshop on Adaptive and Learning Systems","author":"Barron A. R.","year":"1992"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1561\/2200000006"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1016\/j.spl.2014.03.016"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2016.12.041"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1016\/j.spa.2011.06.003"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1214\/08-AAP563"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1142\/S0219493708002378"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511569920.003"},{"key":"ref10","volume-title":"Advances in Neural Information Processing Systems","volume":"16","author":"Bottou L.","year":"2004"},{"key":"ref11","doi-asserted-by":"crossref","unstructured":"N. K. Chada , \nA. M. Stuart , and \nX. T. Tong . Tikhonov Regularization within Ensemble Kalman Inversion, preprint, arXiv:1901.10382, 2019.","DOI":"10.1137\/19M1242331"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1007\/s11004-011-9376-z"},{"key":"ref13","doi-asserted-by":"crossref","unstructured":"D. Chicco , \nP. Sadowski , and \nP. Baldi , Deep autoencoder neural networks for gene ontology annotation predictions, in Proceedings of the 5th ACM Conference on BIoinformatics, Computational Biology, and Health Informatics, ACM, New York, 2014, https:\/\/doi.org\/10.1145\/2649387.2649442.","DOI":"10.1145\/2649387.2649442"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1016\/j.neunet.2012.02.023"},{"key":"ref15","doi-asserted-by":"crossref","unstructured":"E. Cleary , \nA. Garbuno-Inigo , \nS. Lan , \nT. Schneider , and \nA. M. Stuart , Calibrate, Emulate, Sample, preprint, arXiv:2001.03689, 2020.","DOI":"10.1016\/j.jcp.2020.109716"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1137\/15M1042127"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1214\/13-STS421"},{"key":"ref18","series-title":"CourseSmart Series","volume-title":"Statistics for Spatio-Temporal Data","author":"Cressie N.","year":"2011"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2015.10.008"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1007\/BF02551274"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-12385-1_7"},{"key":"ref22","doi-asserted-by":"publisher","DOI":"10.1137\/17M1119056"},{"key":"ref23","doi-asserted-by":"publisher","DOI":"10.1007\/s10596-012-9333-z"},{"key":"ref24","doi-asserted-by":"publisher","DOI":"10.1029\/94JC00572"},{"key":"ref25","doi-asserted-by":"publisher","DOI":"10.1007\/s10236-003-0036-9"},{"key":"ref26","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-03711-5"},{"key":"ref27","doi-asserted-by":"publisher","DOI":"10.1007\/s10596-018-9731-y"},{"key":"ref28","doi-asserted-by":"publisher","DOI":"10.1175\/1520-0493(1996)124<0085:AOGADF>2.0.CO;2"},{"key":"ref29","doi-asserted-by":"publisher","DOI":"10.1007\/BF00344251"},{"key":"ref30","doi-asserted-by":"publisher","DOI":"10.1137\/19M1251655"},{"key":"ref31","doi-asserted-by":"publisher","DOI":"10.1137\/19M1304891"},{"key":"ref32","doi-asserted-by":"publisher","DOI":"10.1109\/72.963769"},{"key":"ref33","volume-title":"Deep Learning","author":"Goodfellow I.","year":"2016"},{"key":"ref34","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-27835-1_10"},{"key":"ref35","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-70096-0_39"},{"key":"ref36","doi-asserted-by":"publisher","DOI":"10.1145\/3306346.3322959"},{"key":"ref37","unstructured":"J. Hensman , \nA. Matthews , and \nZ. Ghahramani , Scalable variational Gaussian process classification, in Proceedings of the Eighteenth International Conference on Articial Intelligence and Statistics (San Diego, CA), \nG. Lebanon  and \nS. V. N. Vishwanathan , eds. Proc. Mach. Learn. Res. 38, PMLR, 2015, pp. 351\u2013360, https:\/\/proceedings.mlr.press\/v38\/hensman15.html."},{"key":"ref38","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827503426693"},{"key":"ref39","doi-asserted-by":"publisher","DOI":"10.1126\/science.1127647"},{"key":"ref40","doi-asserted-by":"publisher","DOI":"10.1162\/neco.2006.18.7.1527"},{"key":"ref41","doi-asserted-by":"publisher","DOI":"10.1162\/neco.1997.9.8.1735"},{"key":"ref42","doi-asserted-by":"publisher","DOI":"10.1175\/1520-0493(2001)129<0123:ASEKFF>2.0.CO;2"},{"key":"ref43","doi-asserted-by":"publisher","DOI":"10.1175\/MWR-D-15-0440.1"},{"key":"ref44","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/32\/2\/025002"},{"key":"ref45","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/29\/4\/045001"},{"key":"ref46","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2017.2713099"},{"key":"ref47","doi-asserted-by":"publisher","DOI":"10.1080\/10618600.2021.1886938"},{"key":"ref48","doi-asserted-by":"publisher","DOI":"10.1115\/1.3662552"},{"key":"ref49","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511802270"},{"key":"ref50","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.2019.1592753"},{"key":"ref51","doi-asserted-by":"publisher","DOI":"10.1111\/1467-9868.00294"},{"key":"ref52","unstructured":"D. P. Kingma  and \nM. Welling . Auto-encoding variational Bayes, in Proceedings of the 2nd International Conference on Learning Representations, ICLR 2014, Banff, AB, Canada, 2014."},{"key":"ref53","doi-asserted-by":"publisher","DOI":"10.1561\/2200000056"},{"key":"ref54","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-20325-6"},{"key":"ref55","first-page":"1097","volume-title":"Advances in Neural Information Processing Systems 25","author":"Krizhevsky A.","year":"2012"},{"key":"ref56","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2019.04.043"},{"key":"ref57","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2015.12.032"},{"key":"ref58","doi-asserted-by":"publisher","DOI":"10.1080\/10618600.2014.902764"},{"key":"ref59","unstructured":"S. Lan , \nB. Zhou , and \nB. Shahbaba , Spherical Hamiltonian Monte Carlo for constrained target distributions, in Proceedings of the 31st International Conference on Machine Learning, Beijing, China, 2014, pp. 629\u2013637."},{"key":"ref60","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2013.07.026"},{"key":"ref61","doi-asserted-by":"publisher","DOI":"10.1038\/nature14539"},{"key":"ref62","doi-asserted-by":"publisher","DOI":"10.1162\/neco.1989.1.4.541"},{"key":"ref63","unstructured":"J. Lee , \nJ. Sohl-dickstein , \nJ. Pennington , \nR. Novak , \nS. Schoenholz , and \nY. Bahri . Deep neural networks as Gaussian processes, in Proceedings of the International Conference on Learning Representations, 2018, https:\/\/openreview.net\/forum?id=B1EA-M-0Z."},{"key":"ref64","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-70139-4"},{"key":"ref65","doi-asserted-by":"publisher","DOI":"10.1109\/MSP.2017.2739299"},{"key":"ref66","doi-asserted-by":"publisher","DOI":"10.1109\/TASLP.2014.2383614"},{"key":"ref67","unstructured":"E. T. Nalisnick , On Priors for Bayesian Neural Networks, PhD thesis. University of California-Irvine, Irvine, CA, 2018."},{"key":"ref68","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0745-0"},{"key":"ref69","volume-title":"Handbook of Markov Chain Monte Carlo","author":"Neal R. M.","year":"2010"},{"key":"ref70","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/89.4.769"},{"key":"ref71","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9868.2004.05304.x"},{"key":"ref72","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2005.11.025"},{"key":"ref73","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511535642"},{"key":"ref74","doi-asserted-by":"publisher","DOI":"10.1162\/neco.2008.08-07-592"},{"key":"ref75","doi-asserted-by":"crossref","unstructured":"N. Petra  and \nG. Stadler , Model Variational Inverse Problems Governed by Partial Differential Equations, Technical report, The Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX, 2011.","DOI":"10.21236\/ADA555315"},{"key":"ref76","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492900002919"},{"key":"ref77","doi-asserted-by":"publisher","DOI":"10.1016\/j.patrec.2017.07.016"},{"key":"ref78","doi-asserted-by":"publisher","DOI":"10.1214\/aoap\/1034625254"},{"key":"ref79","doi-asserted-by":"publisher","DOI":"10.1111\/1467-9868.00123"},{"key":"ref80","doi-asserted-by":"publisher","DOI":"10.1038\/323533a0"},{"key":"ref81","doi-asserted-by":"publisher","DOI":"10.1214\/ss\/1177012413"},{"key":"ref82","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-3799-8"},{"key":"ref83","doi-asserted-by":"publisher","DOI":"10.1137\/16M105959X"},{"key":"ref84","doi-asserted-by":"publisher","DOI":"10.1080\/00036811.2017.1386784"},{"key":"ref85","doi-asserted-by":"publisher","DOI":"10.1016\/j.neunet.2014.09.003"},{"key":"ref86","doi-asserted-by":"publisher","DOI":"10.1038\/s41586-019-1923-7"},{"key":"ref87","unstructured":"B. Shahbaba , \nL. M. Lomeli , \nT. Chen , and \nS. Lan , Deep Markov Chain Monte Carlo, preprint, arXiv:1910.05692, 2019."},{"key":"ref88","doi-asserted-by":"publisher","DOI":"10.1038\/nature16961"},{"key":"ref89","doi-asserted-by":"publisher","DOI":"10.1038\/nature24270"},{"key":"ref90","unstructured":"G. Stephenson , Using Derivative Information in the Statistical Analysis of Computer Models, PhD thesis, University of Southampton, Southampton, UK, 2010."},{"key":"ref91","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492910000061"},{"key":"ref92","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRev.159.98"},{"key":"ref93","doi-asserted-by":"crossref","unstructured":"U. Villa , \nN. Petra , and \nO. Ghattas , hIPPYlib: An Extensible Software Framework for Large-Scale Inverse Problems Governed by PDEs; Part I: Deterministic Inversion and Linearized Bayesian Inference, preprint, arXiv:1909.03948, 2020.","DOI":"10.1145\/3428447"},{"key":"ref94","doi-asserted-by":"publisher","DOI":"10.1038\/s41598-020-62484-z"},{"key":"ref95","unstructured":"Y. Wang  and \nW. Li , Information Newton\u2019s Flow: Second-Order Optimization Method in Probability Space, preprint, arXiv:2001.04341, 2020."},{"key":"ref96","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-021-01709-3"},{"key":"ref97","unstructured":"M. Welling  and \nY. W. Teh . Bayesian learning via stochastic gradient Langevin dynamics, in Proceedings of the International Conference on Machine Learning, Bellevue, WA, 2011, pp. 681\u2013688."},{"key":"ref98","doi-asserted-by":"publisher","DOI":"10.1016\/j.spasta.2020.100408"},{"key":"ref99","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2019.06.004"}],"container-title":["SIAM\/ASA Journal on Uncertainty Quantification"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/21M1439456","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:32:10Z","timestamp":1787337130000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/21M1439456"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,20]]},"references-count":99,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2022,12,31]]}},"alternative-id":["10.1137\/21M1439456"],"URL":"https:\/\/doi.org\/10.1137\/21m1439456","relation":{},"ISSN":["2166-2525"],"issn-type":[{"value":"2166-2525","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,12,20]]}}}