{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,15]],"date-time":"2026-05-15T14:49:12Z","timestamp":1778856552115,"version":"3.51.4"},"reference-count":63,"publisher":"Springer Science and Business Media LLC","issue":"11","license":[{"start":{"date-parts":[[2021,11,22]],"date-time":"2021-11-22T00:00:00Z","timestamp":1637539200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2021,11,22]],"date-time":"2021-11-22T00:00:00Z","timestamp":1637539200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Nat Comput Sci"],"DOI":"10.1038\/s43588-021-00158-0","type":"journal-article","created":{"date-parts":[[2021,11,22]],"date-time":"2021-11-22T12:03:42Z","timestamp":1637582622000},"page":"744-753","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":131,"title":["Identifiability and predictability of integer- and fractional-order epidemiological models using physics-informed neural networks"],"prefix":"10.1038","volume":"1","author":[{"given":"Ehsan","family":"Kharazmi","sequence":"first","affiliation":[]},{"given":"Min","family":"Cai","sequence":"additional","affiliation":[]},{"given":"Xiaoning","family":"Zheng","sequence":"additional","affiliation":[]},{"given":"Zhen","family":"Zhang","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0976-1987","authenticated-orcid":false,"given":"Guang","family":"Lin","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9713-7120","authenticated-orcid":false,"given":"George Em","family":"Karniadakis","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,11,22]]},"reference":[{"key":"158_CR1","doi-asserted-by":"publisher","first-page":"eabd4563","DOI":"10.1126\/sciadv.abd4563","volume":"6","author":"SE Kreps","year":"2020","unstructured":"Kreps, S. E. & Kriner, D. L. Model uncertainty, political contestation and public trust in science: evidence from the COVID-19 pandemic. Sci. Adv. 6, eabd4563 (2020).","journal-title":"Sci. Adv."},{"key":"158_CR2","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1056\/NEJMp2016822","volume":"383","author":"I Holmdahl","year":"2020","unstructured":"Holmdahl, I. & Buckee, C. Wrong but useful\u2014what COVID-19 epidemiologic models can and cannot tell us. N. Engl. J. Med. 383, 303\u2013305 (2020).","journal-title":"N. Engl. J. Med."},{"key":"158_CR3","unstructured":"Science Brief: Emerging SARS-CoV-2 Variants (CDC, 2019); https:\/\/www.cdc.gov\/coronavirus\/2019-ncov\/more\/science-and-research\/scientific-brief-emerging-variants.html"},{"key":"158_CR4","doi-asserted-by":"publisher","first-page":"713","DOI":"10.1126\/science.abb5659","volume":"368","author":"S Cobey","year":"2020","unstructured":"Cobey, S. Modeling infectious disease dynamics. Science 368, 713\u2013714 (2020).","journal-title":"Science"},{"key":"158_CR5","doi-asserted-by":"publisher","first-page":"128","DOI":"10.1038\/s43588-021-00028-9","volume":"1","author":"W Edeling","year":"2021","unstructured":"Edeling, W. et al. The impact of uncertainty on predictions of the CovidSim epidemiological code. Nat. Comput. Sci. 1, 128\u2013135 (2021).","journal-title":"Nat. Comput. Sci."},{"key":"158_CR6","doi-asserted-by":"publisher","unstructured":"Cramer, E. et al. reichlab\/covid19-forecast-hub: release for Zenodo, 20210816 https:\/\/doi.org\/10.5281\/zenodo.5208210 (2020).","DOI":"10.5281\/zenodo.5208210"},{"key":"158_CR7","doi-asserted-by":"publisher","first-page":"W10415","DOI":"10.1029\/2008WR007577","volume":"45","author":"P Chakraborty","year":"2009","unstructured":"Chakraborty, P., Meerschaert, M. M. & Lim, C. Y. Parameter estimation for fractional transport: a particle-tracking approach. Water Resources Res. 45, W10415 (2009).","journal-title":"Water Resources Res."},{"key":"158_CR8","doi-asserted-by":"publisher","first-page":"36","DOI":"10.1016\/j.mbs.2014.11.008","volume":"262","author":"Y Cho","year":"2015","unstructured":"Cho, Y., Kim, I. & Sheen, D. A fractional-order model for MINMOD millennium. Math. Biosci. 262, 36\u201345 (2015).","journal-title":"Math. Biosci."},{"key":"158_CR9","doi-asserted-by":"publisher","first-page":"2559","DOI":"10.1002\/2016WR019748","volume":"53","author":"JF Kelly","year":"2017","unstructured":"Kelly, J. F., Bolster, D., Meerschaert, M. M., Drummond, J. D. & Packman, A. I. FracFit: a robust parameter estimation tool for fractional calculus models. Water Resources Res. 53, 2559\u20132567 (2017).","journal-title":"Water Resources Res."},{"key":"158_CR10","doi-asserted-by":"publisher","first-page":"172","DOI":"10.1016\/j.jmva.2013.09.010","volume":"123","author":"CY Lim","year":"2014","unstructured":"Lim, C. Y., Meerschaert, M. M. & Scheffler, H. P. Parameter estimation for operator scaling random fields. J. Multivariate Anal. 123, 172\u2013183 (2014).","journal-title":"J. Multivariate Anal."},{"key":"158_CR11","doi-asserted-by":"publisher","first-page":"2095","DOI":"10.1016\/j.ijheatmasstransfer.2011.12.012","volume":"55","author":"HR Ghazizadeh","year":"2012","unstructured":"Ghazizadeh, H. R., Azimi, A. & Maerefat, M. An inverse problem to estimate relaxation parameter and order of fractionality in fractional single-phase-lag heat equation. Int. J. Heat Mass Transfer 55, 2095\u20132101 (2012).","journal-title":"Int. J. Heat Mass Transfer"},{"key":"158_CR12","doi-asserted-by":"publisher","first-page":"1131","DOI":"10.1080\/00207160.2017.1378811","volume":"95","author":"B Yu","year":"2018","unstructured":"Yu, B., Jiang, X. Y. & Qi, H. T. Numerical method for the estimation of the fractional parameters in the fractional mobile\/immobile advection-diffusion model. Int. J. Comput. Math. 95, 1131\u20131150 (2018).","journal-title":"Int. J. Comput. Math."},{"key":"158_CR13","doi-asserted-by":"publisher","first-page":"933","DOI":"10.1080\/00207160.2020.1792453","volume":"98.5","author":"JL Suzuki","year":"2021","unstructured":"Suzuki, J. L. & Zayernouri, M. A self-singularity-capturing scheme for fractional differential equations. Int. J. Comput. Math. 98.5, 933\u2013960 (2021).","journal-title":"Int. J. Comput. Math."},{"key":"158_CR14","doi-asserted-by":"publisher","first-page":"694","DOI":"10.1016\/j.jcp.2017.07.052","volume":"348","author":"GF Pang","year":"2017","unstructured":"Pang, G. F., Perdikaris, P., Cai, W. & Karniadakis, G. E. Discovering variable fractional orders of advection-dispersion equations from field data using multi-fidelity Bayesian optimization. J. Comput. Phys. 348, 694\u2013714 (2017).","journal-title":"J. Comput. Phys."},{"key":"158_CR15","doi-asserted-by":"publisher","first-page":"110","DOI":"10.1007\/s10915-019-00935-0","volume":"80","author":"E Kharazmi","year":"2019","unstructured":"Kharazmi, E. & Zayernouri, M. Fractional sensitivity equation method: application to fractional model construction. J. Sci. Comput. 80, 110\u2013140 (2019).","journal-title":"J. Sci. Comput."},{"key":"158_CR16","doi-asserted-by":"publisher","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. E. 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":"158_CR17","doi-asserted-by":"publisher","first-page":"e1009334","DOI":"10.1371\/journal.pcbi.1009334","volume":"17","author":"S Zhang","year":"2021","unstructured":"Zhang, S., Ponce, J., Zhang, Z., Lin, G. & Karniadakis, G. An integrated framework for building trustworthy data-driven epidemiological models: application to the COVID-19 outbreak in New York City. PLoS Comput. Biol. 17, e1009334 (2021).","journal-title":"PLoS Comput. Biol."},{"key":"158_CR18","doi-asserted-by":"publisher","first-page":"1401","DOI":"10.1080\/17415977.2020.1849182","volume":"29","author":"XC Zheng","year":"2020","unstructured":"Zheng, X. C. & Wang, H. Uniquely identifying the variable order of time-fractional partial differential equations on general multi-dimensional domains. Inverse Problems Sci. Eng. 29, 1401\u20131411 (2020).","journal-title":"Inverse Problems Sci. Eng."},{"key":"158_CR19","doi-asserted-by":"publisher","first-page":"110632","DOI":"10.1016\/j.chaos.2020.110632","volume":"143","author":"H Jahanshahi","year":"2021","unstructured":"Jahanshahi, H., Munoz-Pacheco, J. M., Bekiros, S. & Alotaibi, N. D. A fractional-order SIRD model with time-dependent memory indexes for encompassing the multi-fractional characteristics of the COVID-19. Chaos Solitons Fractals 143, 110632 (2021).","journal-title":"Chaos Solitons Fractals"},{"key":"158_CR20","doi-asserted-by":"publisher","DOI":"10.1038\/s41598-020-77849-7","volume":"10","author":"A Taghvaei","year":"2020","unstructured":"Taghvaei, A., Georgiou, T. T., Norton, L. & Tannenbaum, A. Fractional SIR epidemiological models. Sci. Rep. 10, 20882 (2020).","journal-title":"Sci. Rep."},{"key":"158_CR21","first-page":"101","volume":"3","author":"ZE Ma","year":"2012","unstructured":"Ma, Z. E. & Jin, Z. The stability of an SIR epidemic model with time delays. Math. Biosci. Eng. 3, 101\u2013109 (2012).","journal-title":"Math. Biosci. Eng."},{"key":"158_CR22","doi-asserted-by":"crossref","unstructured":"Ma, Z. E., Zhou, Y. C. & Wu, J. H. Modeling and Dynamics of Infectious Diseases (World Scientific, 2009).","DOI":"10.1142\/7223"},{"key":"158_CR23","doi-asserted-by":"publisher","first-page":"250","DOI":"10.1007\/BF00169563","volume":"33","author":"E Beretta","year":"1995","unstructured":"Beretta, E. & Takeuchi, Y. Global stability of an SIR epidemic model with time delays. J. Math. Biol. 33, 250\u2013260 (1995).","journal-title":"J. Math. Biol."},{"key":"158_CR24","doi-asserted-by":"publisher","first-page":"4107","DOI":"10.1016\/S0362-546X(01)00528-4","volume":"47","author":"E Beretta","year":"2001","unstructured":"Beretta, E., Hara, T., Ma, W. & Takeuchi, Y. Global asymptotic stability of an SIR epidemic model with distributed time delay. Nonlinear Anal. 47, 4107\u20134115 (2001).","journal-title":"Nonlinear Anal."},{"key":"158_CR25","doi-asserted-by":"publisher","first-page":"468","DOI":"10.1007\/s11538-016-0151-7","volume":"78","author":"CN Angstmann","year":"2016","unstructured":"Angstmann, C. N., Henry, B. I. & Mcgann, A. V. A fractional order recovery SIR model from a stochastic process. Bull. Math. Biol. 78, 468\u2013499 (2016).","journal-title":"Bull. Math. Biol."},{"key":"158_CR26","doi-asserted-by":"publisher","first-page":"1055","DOI":"10.1007\/s00466-020-01889-z","volume":"66","author":"PK Jha","year":"2020","unstructured":"Jha, P. K., Cao, L. & Oden, J. T. Bayesian-based predictions of COVID-19 evolution in Texas using multispecies mixture-theoretic continuum models. Comput. Mech. 66, 1055\u20131068 (2020).","journal-title":"Comput. Mech."},{"key":"158_CR27","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1007\/s00477-016-1298-3","volume":"31","author":"G Christakos","year":"2017","unstructured":"Christakos, G., Zhang, C. T. & He, J. Y. A traveling epidemic model of space-time disease spread. Stochastic Environ. Res. Risk Assess. 31, 305\u2013314 (2017).","journal-title":"Stochastic Environ. Res. Risk Assess."},{"key":"158_CR28","first-page":"186437","volume":"2014","author":"EM Lotfi","year":"2014","unstructured":"Lotfi, E. M., Maziane, M., Hattaf, K. & Yousfi, N. Partial differential equations of an epidemic model with spatial diffusion. Int. J. Partial Differ. Eqn. 2014, 186437 (2014).","journal-title":"Int. J. Partial Differ. Eqn."},{"key":"158_CR29","doi-asserted-by":"publisher","unstructured":"Horwitz, L. et al. Trends in COVID-19 risk-adjusted mortality rates in a single health system. Preprint at https:\/\/doi.org\/10.1101\/2020.08.11.20172775 (2020).","DOI":"10.1101\/2020.08.11.20172775"},{"key":"158_CR30","doi-asserted-by":"publisher","unstructured":"Petrilli, C. M. et al. Factors associated with hospitalization and critical illness among 4,103 patients with COVID-19 disease in New York City. Preprint at https:\/\/doi.org\/10.1101\/2020.04.08.20057794 (2020).","DOI":"10.1101\/2020.04.08.20057794"},{"key":"158_CR31","unstructured":"NYC Coronavirus Disease 2019 (COVID-19) Data. https:\/\/github.com\/nychealth\/coronavirus-data"},{"key":"158_CR32","doi-asserted-by":"publisher","first-page":"43","DOI":"10.3390\/math8010043","volume":"8","author":"M Cai","year":"2020","unstructured":"Cai, M. & Li, C. P. Numerical approaches to fractional integrals and derivatives: a review. Mathematics 8, 43 (2020).","journal-title":"Mathematics"},{"key":"158_CR33","doi-asserted-by":"crossref","unstructured":"Li, C. P. & Cai, M. Theory and Numerical Approximations of Fractional Integrals and Derivatives (SIAM, 2019).","DOI":"10.1137\/1.9781611975888"},{"key":"158_CR34","doi-asserted-by":"crossref","unstructured":"Li, C. P. & Zeng, F. H. Numerical Methods for Fractional Calculus (CRC Press, 2015).","DOI":"10.1201\/b18503"},{"key":"158_CR35","unstructured":"NYC Health. COVID-19 data: vaccines. https:\/\/www1.nyc.gov\/site\/doh\/covid\/covid-19-data-vaccines.page"},{"key":"158_CR36","unstructured":"The COVID Tracking Project. Michigan overview. https:\/\/covidtracking.com\/data\/state\/michigan"},{"key":"158_CR37","unstructured":"Michigan.gov. COVID-19 vaccine dashboard. https:\/\/www.michigan.gov\/coronavirus\/0,9753,7-406-98178_103214-547150-,00.html"},{"key":"158_CR38","unstructured":"COVID-19 Rhode Island data."},{"key":"158_CR39","unstructured":"COVID-19 data repository by the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University. GitHub https:\/\/github.com\/CSSEGISandData\/COVID-19"},{"key":"158_CR40","doi-asserted-by":"crossref","unstructured":"Fishman, G. S. Monte Carlo: Concepts, Algorithms and Applications (Springer, 1996).","DOI":"10.1007\/978-1-4757-2553-7"},{"key":"158_CR41","doi-asserted-by":"publisher","first-page":"1118","DOI":"10.1137\/040615201","volume":"27","author":"DB Xiu","year":"2005","unstructured":"Xiu, D. B. & Hesthaven, J. S. High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27, 1118\u20131139 (2005).","journal-title":"SIAM J. Sci. Comput."},{"key":"158_CR42","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1016\/j.apnum.2005.03.003","volume":"56","author":"ZZ Sun","year":"2006","unstructured":"Sun, Z. Z. & Wu, X. N. A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193\u2013209 (2006).","journal-title":"Appl. Numer. Math."},{"key":"158_CR43","doi-asserted-by":"publisher","first-page":"1533","DOI":"10.1016\/j.jcp.2007.02.001","volume":"225","author":"YM Lin","year":"2007","unstructured":"Lin, Y. M. & Xu, C. J. Finite difference\/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533\u20131552 (2007).","journal-title":"J. Comput. Phys."},{"key":"158_CR44","doi-asserted-by":"publisher","first-page":"A2603","DOI":"10.1137\/18M1229845","volume":"41","author":"GF Pang","year":"2019","unstructured":"Pang, G. F., Lu, L. & Karniadakis, G. E. fPINNs: fractional physics-informed neural networks. SIAM J. Sci. Comput. 41, A2603\u2013A2626 (2019).","journal-title":"SIAM J. Sci. Comput."},{"key":"158_CR45","doi-asserted-by":"publisher","first-page":"368","DOI":"10.1126\/science.abd1668","volume":"369","author":"CJE Metcalf","year":"2020","unstructured":"Metcalf, C. J. E., Morris, D. H. & Park, S. W. Mathematical models to guide pandemic response. Science 369, 368\u2013369 (2020).","journal-title":"Science"},{"key":"158_CR46","doi-asserted-by":"publisher","first-page":"420","DOI":"10.1038\/s41586-020-2554-8","volume":"584","author":"XJ Hao","year":"2020","unstructured":"Hao, X. J. et al. Reconstruction of the full transmission dynamics of COVID-19 in Wuhan. Nature 584, 420\u2013424 (2020).","journal-title":"Nature"},{"key":"158_CR47","doi-asserted-by":"publisher","first-page":"911","DOI":"10.1093\/cid\/cir007","volume":"52","author":"P Fine","year":"2011","unstructured":"Fine, P., Eames, K. & Heymann, D. L. \u2018Herd immunity\u2019: a rough guide. Clin. Infect. Dis. 52, 911\u2013916 (2011).","journal-title":"Clin. Infect. Dis."},{"key":"158_CR48","doi-asserted-by":"publisher","first-page":"4023","DOI":"10.1073\/pnas.1616438114","volume":"114","author":"CM Peak","year":"2017","unstructured":"Peak, C. M., Childs, L. M., Grad, Y. H. & Buckee, C. O. Comparing nonpharmaceutical interventions for containing emerging epidemics. Proc. Natl Acad. Sci. USA 114, 4023\u20134028 (2017).","journal-title":"Proc. Natl Acad. Sci. USA"},{"key":"158_CR49","doi-asserted-by":"publisher","first-page":"226","DOI":"10.7326\/M20-1565","volume":"173","author":"NP Jewell","year":"2020","unstructured":"Jewell, N. P., Lewnard, J. A. & Jewell, B. L. Caution warranted: using the institute for health metrics and evaluation model for predicting the course of the COVID-19 pandemic. Ann. Intern. Med. 173, 226\u2013227 (2020).","journal-title":"Ann. Intern. Med."},{"key":"158_CR50","doi-asserted-by":"publisher","first-page":"124501","DOI":"10.1103\/PhysRevFluids.4.124501","volume":"4","author":"M Raissi","year":"2019","unstructured":"Raissi, M., Babaee, H. & Givi, P. Deep learning of turbulent scalar mixing. Phys. Rev. Fluids 4, 124501 (2019).","journal-title":"Phys. Rev. Fluids"},{"key":"158_CR51","doi-asserted-by":"publisher","first-page":"672112789","DOI":"10.1016\/j.cma.2019.112789","volume":"360","author":"ZP Mao","year":"2020","unstructured":"Mao, Z. P., Jagtap, A. D. & Karniadakis, G. E. Physics-informed neural networks for high-speed flows. Comput. Methods Appl. Mech. Eng. 360, 672112789 (2020).","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"158_CR52","doi-asserted-by":"publisher","first-page":"A292","DOI":"10.1137\/18M1225409","volume":"42","author":"L Yang","year":"2020","unstructured":"Yang, L., Zhang, D. K. & Karniadakis, G. E. Physics-informed generative adversarial networks for stochastic differential equations. SIAM J. Sci. Comput. 42, A292\u2013A317 (2020).","journal-title":"SIAM J. Sci. Comput."},{"key":"158_CR53","unstructured":"Kharazmi, E., Zhang, Z. Q. & Karniadakis, G. E. Variational physics-informed neural networks for solving partial differential equations. Preprint at https:\/\/arxiv.org\/abs\/1912.00873v1 (2019)."},{"key":"158_CR54","doi-asserted-by":"publisher","first-page":"113547","DOI":"10.1016\/j.cma.2020.113547","volume":"374","author":"E Kharazmi","year":"2021","unstructured":"Kharazmi, E., Zhang, Z. Q. & 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":"158_CR55","doi-asserted-by":"publisher","first-page":"109136","DOI":"10.1016\/j.jcp.2019.109136","volume":"404","author":"AD Jagtap","year":"2019","unstructured":"Jagtap, A. D., Kawaguchi, K. & Karniadakis, G. E. Adaptive activation functions accelerate convergence in deep and physics-informed neural networks. J. Comput. Phys. 404, 109136 (2019).","journal-title":"J. Comput. Phys."},{"key":"158_CR56","doi-asserted-by":"publisher","unstructured":"Jagtap, A. D., Kawaguchi, K. & Karniadakis, G. E. Locally adaptive activation functions with slope recovery term for deep and physics-informed neural networks. Proc. R. Soc. A 476, https:\/\/doi.org\/10.1098\/rspa.2020.0334 (2020).","DOI":"10.1098\/rspa.2020.0334"},{"key":"158_CR57","doi-asserted-by":"publisher","first-page":"113028","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."},{"key":"158_CR58","unstructured":"Shin, Y., Darbon, J. & Karniadakis, G. E. On the convergence and generalization of physics informed neural networks. Preprint at https:\/\/arxiv.org\/abs\/2004.01806 (2020)."},{"key":"158_CR59","unstructured":"Wang, S. F., Teng, Y. J. & Perdikaris, P. Understanding and mitigating gradient pathologies in physics-informed neural networks. Preprint at https:\/\/arxiv.org\/abs\/2001.04536 (2020)."},{"key":"158_CR60","first-page":"5595","volume":"18","author":"A Baydin","year":"2017","unstructured":"Baydin, A., Pearlmutter, B. A., Radul, A. A. & Siskind, J. M. Automatic differentiation in machine learning: a survey. J. Mach. Learn. Res. 18, 5595\u20135637 (2017).","journal-title":"J. Mach. Learn. Res."},{"key":"158_CR61","unstructured":"Abadi, M. et al. Tensorflow: large-scale machine learning on heterogeneous distributed systems. Preprint at https:\/\/arxiv.org\/abs\/1603.04467 (2016)."},{"key":"158_CR62","doi-asserted-by":"publisher","unstructured":"Kharazmi, E. et al. Identifiability and predictability of integer- and fractional-order epidemiological models using physics-informed neural networks. Zenodo https:\/\/doi.org\/10.5281\/zenodo.5565308 (2021).","DOI":"10.5281\/zenodo.5565308"},{"key":"158_CR63","unstructured":"Kharazmi, E. & Cai, M. PINN-COVID. GitHub https:\/\/github.com\/ehsankharazmi\/PINN-COVID (2021)."}],"container-title":["Nature Computational Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.nature.com\/articles\/s43588-021-00158-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s43588-021-00158-0","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s43588-021-00158-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,2,8]],"date-time":"2023-02-08T18:14:11Z","timestamp":1675880051000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.nature.com\/articles\/s43588-021-00158-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,22]]},"references-count":63,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2021,11]]}},"alternative-id":["158"],"URL":"https:\/\/doi.org\/10.1038\/s43588-021-00158-0","relation":{"has-preprint":[{"id-type":"doi","id":"10.1101\/2021.04.05.21254919","asserted-by":"object"}]},"ISSN":["2662-8457"],"issn-type":[{"value":"2662-8457","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,11,22]]},"assertion":[{"value":"5 April 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 October 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 November 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}