{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,4]],"date-time":"2026-07-04T21:09:35Z","timestamp":1783199375023,"version":"3.54.6"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,11,8]],"date-time":"2025-11-08T00:00:00Z","timestamp":1762560000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,11,8]],"date-time":"2025-11-08T00:00:00Z","timestamp":1762560000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Sectorplan Beta"},{"DOI":"10.13039\/501100003246","name":"Nederlandse Organisatie voor Wetenschappelijk Onderzoek","doi-asserted-by":"crossref","award":["NGF.1609.241.020"],"award-info":[{"award-number":["NGF.1609.241.020"]}],"id":[{"id":"10.13039\/501100003246","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100002860","name":"China Sponsorship Council","doi-asserted-by":"publisher","award":["202107650017"],"award-info":[{"award-number":["202107650017"]}],"id":[{"id":"10.13039\/501100002860","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv. Model. and Simul. in Eng. Sci."],"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Gaussian process regression is widely applied in computational science and engineering for surrogate modeling owning to its kernel-based and probabilistic nature. In this work, we propose a Bayesian approach that integrates the variability of input data into the Gaussian process regression for function and partial differential equation approximation. Leveraging two types of observables\u2014noise-corrupted outputs with certain inputs and those with prior-distribution-defined uncertain inputs, a posterior distribution of uncertain inputs is estimated via Bayesian inference. Thereafter, such quantified uncertainties of inputs are incorporated into Gaussian process predictions by means of marginalization. The setting of two types of data aligns with common scenarios of constructing surrogate models for the solutions of partial differential equations, where the data of boundary conditions and initial conditions are typically known while the data of solution may involve uncertainties due to the measurement or stochasticity. The effectiveness of the proposed method is demonstrated through several numerical examples including multiple one-dimensional functions, the heat equation and Allen\u2013Cahn equation. A consistently good performance of generalization is observed, and a substantial reduction in the predictive uncertainties is achieved by the Bayesian inference of uncertain inputs.<\/jats:p>","DOI":"10.1186\/s40323-025-00308-3","type":"journal-article","created":{"date-parts":[[2025,11,8]],"date-time":"2025-11-08T05:29:34Z","timestamp":1762579774000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["PDE-constrained Gaussian process surrogate modeling with uncertain data locations"],"prefix":"10.1186","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4903-1487","authenticated-orcid":false,"given":"Dongwei","family":"Ye","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Weihao","family":"Yan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christoph","family":"Brune","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mengwu","family":"Guo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,11,8]]},"reference":[{"issue":"3","key":"308_CR1","doi-asserted-by":"publisher","first-page":"88","DOI":"10.1007\/s10915-022-01939-z","volume":"92","author":"S Cuomo","year":"2022","unstructured":"Cuomo S, Di Cola VS, Giampaolo F, Rozza G, Raissi M, Piccialli F. Scientific machine learning through physics-informed neural networks: where we are and what\u2019s next. J Sci Comput. 2022;92(3):88.","journal-title":"J Sci Comput"},{"issue":"175","key":"308_CR2","doi-asserted-by":"publisher","first-page":"20200802","DOI":"10.1098\/rsif.2020.0802","volume":"18","author":"A Arzani","year":"2021","unstructured":"Arzani A, Dawson ST. Data-driven cardiovascular flow modelling: examples and opportunities. J R Soc Interface. 2021;18(175):20200802.","journal-title":"J R Soc Interface"},{"key":"308_CR3","doi-asserted-by":"crossref","unstructured":"Sanderse B, Stinis P, Maulik R, Ahmed SE. Scientific machine learning for closure models in multiscale problems: a review; 2024. arXiv preprint arXiv:2403.02913.","DOI":"10.3934\/fods.2024043"},{"key":"308_CR4","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s43588-024-00643-2","volume":"4","author":"SL Brunton","year":"2024","unstructured":"Brunton SL, Kutz JN. Promising directions of machine learning for partial differential equations. Nat Comput Sci. 2024;4:1\u201312.","journal-title":"Nat Comput Sci"},{"issue":"1","key":"308_CR5","doi-asserted-by":"publisher","first-page":"7","DOI":"10.1002\/mp.14602","volume":"48","author":"R Phellan","year":"2021","unstructured":"Phellan R, Hachem B, Clin J, Mac-Thiong JM, Duong L. Real-time biomechanics using the finite element method and machine learning: review and perspective. Med Phys. 2021;48(1):7\u201318.","journal-title":"Med Phys"},{"key":"308_CR6","doi-asserted-by":"publisher","first-page":"110","DOI":"10.3389\/fmats.2019.00110","volume":"6","author":"FE Bock","year":"2019","unstructured":"Bock FE, Aydin RC, Cyron CJ, Huber N, Kalidindi SR, Klusemann B. A review of the application of machine learning and data mining approaches in continuum materials mechanics. Front Mater. 2019;6:110.","journal-title":"Front Mater"},{"issue":"1","key":"308_CR7","doi-asserted-by":"publisher","first-page":"477","DOI":"10.1146\/annurev-fluid-010719-060214","volume":"52","author":"SL Brunton","year":"2020","unstructured":"Brunton SL, Noack BR, Koumoutsakos P. Machine learning for fluid mechanics. Annu Rev Fluid Mech. 2020;52(1):477\u2013508.","journal-title":"Annu Rev Fluid Mech"},{"key":"308_CR8","volume-title":"Gaussian processes for machine learning","author":"CE Rasmussen","year":"2006","unstructured":"Rasmussen CE, Williams CKI. Gaussian processes for machine learning. Cambridge: The MIT Press; 2006."},{"issue":"6","key":"308_CR9","first-page":"119","volume":"52","author":"DG Krige","year":"1951","unstructured":"Krige DG. A statistical approach to some basic mine valuation problems on the Witwatersrand. J South Afr Inst Min Metall. 1951;52(6):119\u201339.","journal-title":"J South Afr Inst Min Metall"},{"issue":"6","key":"308_CR10","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1371\/journal.pone.0130252","volume":"10","author":"ETY Chang","year":"2015","unstructured":"Chang ETY, Strong M, Clayton RH. Bayesian sensitivity analysis of a cardiac cell model using a Gaussian process emulator. PLoS ONE. 2015;10(6):1\u201320.","journal-title":"PLoS ONE"},{"issue":"4","key":"308_CR11","doi-asserted-by":"publisher","first-page":"A2485","DOI":"10.1137\/18M1204991","volume":"41","author":"M Gulian","year":"2019","unstructured":"Gulian M, Raissi M, Perdikaris P, Karniadakis G. Machine learning of space-fractional differential equations. SIAM J Sci Comput. 2019;41(4):A2485\u2013509.","journal-title":"SIAM J Sci Comput"},{"key":"308_CR12","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2019.112602","volume":"357","author":"FS Costabal","year":"2019","unstructured":"Costabal FS, Perdikaris P, Kuhl E, Hurtado DE. Multi-fidelity classification using Gaussian processes: accelerating the prediction of large-scale computational models. Comput Methods Appl Mech Eng. 2019;357: 112602.","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"187","key":"308_CR13","doi-asserted-by":"publisher","first-page":"20210864","DOI":"10.1098\/rsif.2021.0864","volume":"19","author":"D Ye","year":"2022","unstructured":"Ye D, Zun P, Krzhizhanovskaya V, Hoekstra AG. Uncertainty quantification of a three-dimensional in-stent restenosis model with surrogate modelling. J R Soc Interface. 2022;19(187):20210864.","journal-title":"J R Soc Interface"},{"key":"308_CR14","unstructured":"Frigola R, Lindsten F, Sch\u00f6n TB, Rasmussen CE. Bayesian inference and learning in Gaussian process state-space models with particle MCMC. In: Burges CJ, Bottou L, Welling M, Ghahramani Z, Weinberger KQ, editors. Advances in neural information processing systems. vol.\u00a026; 2013."},{"key":"308_CR15","unstructured":"Alaa AM, van\u00a0der Schaar M. Bayesian inference of individualized treatment effects using multi-task Gaussian processes. In: Guyon I, Luxburg UV, Bengio S, Wallach H, Fergus R, Vishwanathan S, et\u00a0al., editors. Advances in neural information processing systems. vol.\u00a030; 2017."},{"key":"308_CR16","unstructured":"Alvarez M, Luengo D, Lawrence ND. Latent force models. In: Artificial intelligence and statistics. PMLR; 2009. p. 9\u201316."},{"key":"308_CR17","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1016\/j.jcp.2017.11.039","volume":"357","author":"M Raissi","year":"2018","unstructured":"Raissi M, Karniadakis GE. Hidden physics models: machine learning of nonlinear partial differential equations. J Comput Phys. 2018;357:125\u201341.","journal-title":"J Comput Phys"},{"key":"308_CR18","first-page":"29386","volume":"35","author":"A Besginow","year":"2022","unstructured":"Besginow A, Lange-Hegermann M. Constraining Gaussian processes to systems of linear ordinary differential equations. Adv Neural Inf Process Syst. 2022;35:29386\u201399.","journal-title":"Adv Neural Inf Process Syst"},{"key":"308_CR19","doi-asserted-by":"crossref","unstructured":"S\u00e4rkk\u00e4 S. Linear operators and stochastic partial differential equations in Gaussian process regression. In: Artificial neural networks and machine learning\u2013ICANN 2011: 21st international conference on artificial neural networks, Espoo, Finland, June 14\u201317, 2011, Proceedings, Part II 21. Springer; 2011. p. 151\u20138.","DOI":"10.1007\/978-3-642-21738-8_20"},{"key":"308_CR20","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2024.108184","volume":"138","author":"D Ye","year":"2024","unstructured":"Ye D, Guo M. Gaussian process learning of nonlinear dynamics. Commun Nonlinear Sci Numer Simul. 2024;138: 108184.","journal-title":"Commun Nonlinear Sci Numer Simul"},{"issue":"1","key":"308_CR21","doi-asserted-by":"publisher","first-page":"A172","DOI":"10.1137\/17M1120762","volume":"40","author":"M Raissi","year":"2018","unstructured":"Raissi M, Perdikaris P, Karniadakis GE. Numerical Gaussian processes for time-dependent and nonlinear partial differential equations. SIAM J Sci Comput. 2018;40(1):A172\u201398.","journal-title":"SIAM J Sci Comput"},{"key":"308_CR22","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2021.110668","volume":"447","author":"Y Chen","year":"2021","unstructured":"Chen Y, Hosseini B, Owhadi H, Stuart AM. Solving and learning nonlinear PDEs with Gaussian processes. J Comput Phys. 2021;447: 110668.","journal-title":"J Comput Phys"},{"key":"308_CR23","doi-asserted-by":"crossref","unstructured":"Batlle P, Chen Y, Hosseini B, Owhadi H, Stuart AM. Error analysis of kernel\/GP methods for nonlinear and parametric PDEs; 2023. arXiv preprint arXiv:2305.04962.","DOI":"10.2139\/ssrn.4831053"},{"issue":"2","key":"308_CR24","doi-asserted-by":"publisher","first-page":"680","DOI":"10.1109\/LRA.2017.2651154","volume":"2","author":"M Ghaffari Jadidi","year":"2017","unstructured":"Ghaffari Jadidi M, Miro JV, Dissanayake G. Warped Gaussian processes occupancy mapping with uncertain inputs. IEEE Robot Autom Lett. 2017;2(2):680\u20137.","journal-title":"IEEE Robot Autom Lett"},{"key":"308_CR25","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1016\/j.enbuild.2014.01.048","volume":"75","author":"MC Burkhart","year":"2014","unstructured":"Burkhart MC, Heo Y, Zavala VM. Measurement and verification of building systems under uncertain data: a Gaussian process modeling approach. Energy Build. 2014;75:189\u201398.","journal-title":"Energy Build"},{"key":"308_CR26","unstructured":"Girard A. Approximate methods for propagation of uncertainty with Gaussian process models. PQDT\u2014global; 2004. p. 168."},{"key":"308_CR27","doi-asserted-by":"publisher","first-page":"433","DOI":"10.1007\/978-3-642-10677-4_49","volume-title":"Neural information processing","author":"P Dallaire","year":"2009","unstructured":"Dallaire P, Besse C, Chaib-draa B. Learning Gaussian process models from uncertain data. In: Leung CS, Lee M, Chan JH, editors. Neural information processing. Berlin: Springer; 2009. p. 433\u201340."},{"issue":"11","key":"308_CR28","doi-asserted-by":"publisher","first-page":"1945","DOI":"10.1016\/j.neucom.2010.09.024","volume":"74","author":"P Dallaire","year":"2011","unstructured":"Dallaire P, Besse C, Chaib-draa B. An approximate inference with Gaussian process to latent functions from uncertain data. Neurocomputing. 2011;74(11):1945\u201355.","journal-title":"Neurocomputing"},{"key":"308_CR29","unstructured":"Mchutchon A, Rasmussen C. Gaussian process training with input noise. In: Shawe-Taylor J, Zemel R, Bartlett P, Pereira F, Weinberger KQ, editors. Advances in neural information processing systems. vol.\u00a024; 2011."},{"key":"308_CR30","unstructured":"Qui\u00f1onero-Candela J, Roweis ST. Data imputation and robust training with Gaussian processes. NIPS; 2003."},{"issue":"4\u20135","key":"308_CR31","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1016\/S0898-1221(99)00057-7","volume":"37","author":"IM Sobol","year":"1999","unstructured":"Sobol IM, Levitan YL. A pseudo-random number generator for personal computers. Comput Math Appl. 1999;37(4\u20135):33\u201340.","journal-title":"Comput Math Appl"},{"key":"308_CR32","unstructured":"GPy.: GPy: a Gaussian process framework in python. http:\/\/github.com\/SheffieldML\/GPy."},{"issue":"2","key":"308_CR33","first-page":"65","volume":"9","author":"M Rudemo","year":"1982","unstructured":"Rudemo M. Empirical choice of histograms and kernel density estimators. Scand J Stat. 1982;9(2):65\u201378.","journal-title":"Scand J Stat"},{"key":"308_CR34","unstructured":"Kingma DP, Ba J. Adam: A method for stochastic optimization; 2014. arXiv preprint arXiv:1412.6980."},{"key":"308_CR35","unstructured":"Bradbury J, Frostig R, Hawkins P, Johnson MJ, Leary C, Maclaurin D, et\u00a0al.: JAX: composable transformations of Python+NumPy programs. http:\/\/github.com\/jax-ml\/jax."},{"issue":"9","key":"308_CR36","doi-asserted-by":"publisher","first-page":"1017","DOI":"10.1016\/0001-6160(75)90106-6","volume":"23","author":"SM Allen","year":"1975","unstructured":"Allen SM, Cahn JW. Coherent and incoherent equilibria in iron-rich iron-aluminum alloys. Acta Metall. 1975;23(9):1017\u201326.","journal-title":"Acta Metall"},{"issue":"2","key":"308_CR37","doi-asserted-by":"publisher","first-page":"490","DOI":"10.1006\/jmaa.2001.7470","volume":"260","author":"S Tersian","year":"2001","unstructured":"Tersian S, Chaparova J. Periodic and homoclinic solutions of extended Fisher-Kolmogorov equations. J Math Anal Appl. 2001;260(2):490\u2013506.","journal-title":"J Math Anal Appl"},{"issue":"5999","key":"308_CR38","doi-asserted-by":"publisher","first-page":"1616","DOI":"10.1126\/science.1179047","volume":"329","author":"S Kondo","year":"2010","unstructured":"Kondo S, Miura T. Reaction-diffusion model as a framework for understanding biological pattern formation. Science. 2010;329(5999):1616\u201320.","journal-title":"Science"}],"container-title":["Advanced Modeling and Simulation in Engineering Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1186\/s40323-025-00308-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1186\/s40323-025-00308-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1186\/s40323-025-00308-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,8]],"date-time":"2025-11-08T05:29:36Z","timestamp":1762579776000},"score":1,"resource":{"primary":{"URL":"https:\/\/amses-journal.springeropen.com\/articles\/10.1186\/s40323-025-00308-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,8]]},"references-count":38,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,12]]}},"alternative-id":["308"],"URL":"https:\/\/doi.org\/10.1186\/s40323-025-00308-3","relation":{},"ISSN":["2213-7467"],"issn-type":[{"value":"2213-7467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,8]]},"assertion":[{"value":"31 December 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 August 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 November 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"All authors declare that they have no Competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"33"}}