Bayesian Nonlocal Operator Regression: A Data-Driven Learning Framework of Nonlocal Models with Uncertainty Quantification
Publication: Journal of Engineering Mechanics
Volume 149, Issue 8
Abstract
We consider the problem of modeling heterogeneous materials where microscale dynamics and interactions affect global behavior. In the presence of heterogeneities in material microstructure it is often impractical, if not impossible, to provide quantitative characterization of material response. The goal of this work is to develop a Bayesian framework for uncertainty quantification (UQ) in material response prediction when using nonlocal models. Our approach combines the nonlocal operator regression (NOR) technique and Bayesian inference. Specifically, additive independent identically distributed Gaussian noise is employed to model the discrepancy between the nonlocal model and the data. Then, we use a Markov chain Monte Carlo (MCMC) method to sample the posterior probability distribution on parameters involved in the nonlocal constitutive law and associated modeling discrepancies relative to higher-fidelity computations. As an application, we consider the propagation of stress waves through a one-dimensional heterogeneous bar with randomly generated microstructure. Several numerical tests illustrate the construction, enabling UQ in nonlocal model predictions. Although nonlocal models have become popular means for homogenization, their statistical calibration with respect to high-fidelity models has not been presented before. This work is a first step in this direction, focused on Bayesian parameter calibration.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
All data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.
Acknowledgments
Y. Fan and Y. Yu would like to acknowledge support by the National Science Foundation under Award DMS-1753031 and the AFOSR Grant FA9550-22-1-0197. Portions of this research were conducted on Lehigh University’s Research Computing infrastructure partially supported by NSF Award 2019035. S. Silling and M. D’Elia would like to acknowledge the support of the Sandia National Laboratories Laboratory-directed Research and Development program and by the US Department of Energy (DOE), Office of Advanced Scientific Computing Research (ASCR) under the Collaboratory on Mathematics and Physics-Informed Learning Machines for Multiscale and Multiphysics Problems project. H. Najm acknowledges support by the US DOE ASCR office, Scientific Discovery through Advanced Computing program. This article has been co-authored by employees of National Technology and Engineering Solutions of Sandia, LLC under Contract No. DE-NA0003525 with the US DOE. The employees own all rights, title, and interest in and to the article and is solely responsible for its contents.
References
Andrieu, C., N. De Freitas, A. Doucet, and M. I. Jordan. 2003. “An introduction to mcmc for machine learning.” Mach. Learn. 50 (1): 5–43. https://doi.org/10.1023/A:1020281327116.
Andrieu, C., and J. Thoms. 2008. “A tutorial on adaptive mcmc.” Stat. Comput. 18 (4): 343–373. https://doi.org/10.1007/s11222-008-9110-y.
Bensoussan, A., J.-L. Lions, and G. Papanicolaou. 2011. Asymptotic analysis for periodic structures. Boston: American Mathematical Society.
Beran, M., and J. McCoy. 1970. “Mean field variations in a statistical sample of heterogeneous linearly elastic solids.” Int. J. Solids Struct. 6 (8): 1035–1054. https://doi.org/10.1016/0020-7683(70)90046-6.
Bobaru, F., J. T. Foster, P. H. Geubelle, and S. A. Silling. 2016. Handbook of peridynamic modeling. London: CRC Press.
Carlin, B. P., and T. A. Louis. 2011. Bayesian methods for data analysis. Boca Raton, FL: Chapman and Hall.
Casella, G., and R. L. Berger. 1990. Statistical inference. Belmont, CA: Duxbury Press.
Cherednichenko, K., V. P. Smyshlyaev, and V. Zhikov. 2006. “Non-local homogenised limits for composite media with highly anisotropic periodic fibres.” Proc. R. Soc. Edinburgh Section A: Math. 136 (1): 87–114. https://doi.org/10.1017/S0308210500004455.
Cui, T., Y. Marzouk, and K. Willcox. 2016. “Scalable posterior approximations for large-scale Bayesian inverse problems via likelihood-informed parameter and state reduction.” J. Comput. Phys. 315 (Apr): 363–387. https://doi.org/10.1016/j.jcp.2016.03.055.
Debusschere, B., H. Najm, P. Pébay, O. Knio, R. Ghanem, and O. Le Maître. 2004. “Numerical challenges in the use of polynomial chaos representations for stochastic processes.” J. Sci. Comput. 26 (2): 698–719. https://doi.org/10.1137/S1064827503427741.
Debusschere, B., K. Sargsyan, C. Safta, and K. Chowdhary. 2017. “The uncertainty quantification toolkit (uqtk).” In Handbook of uncertainty quantification, edited by R. Ghanem, D. Higdon, and H. Owhadi, 1807–1827. Berlin: Springer.
de Moraes, E. A. B., M. D’Elia, and M. Zayernouri. 2022. “Nonlocal machine learning of micro-structural defect evolutions in crystalline materials.” Preprint, submitted May 11, 2022. http://arxiv.org/abs/2205.05729.
Dobson, M., M. Luskin, and C. Ortner. 2010. “Sharp stability estimates for the force-based quasicontinuum approximation of homogeneous tensile deformation.” Multiscale Model. Simul. 8 (3): 782–802. https://doi.org/10.1137/090767005.
Du, Q., B. Engquist, and X. Tian. 2020. “Multiscale modeling, homogenization and nonlocal effects: Mathematical and computational issues.” Contemp. Math. 754 (Apr): 20–26.
Du, Q., M. Gunzburger, R. B. Lehoucq, and K. Zhou. 2013. “A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws.” Math. Models Methods Appl. Sci. 23 (3): 493–540. https://doi.org/10.1142/S0218202512500546.
Du, Q., Y. Tao, and X. Tian. 2017. “A peridynamic model of fracture mechanics with bond-breaking.” J. Elast. 2017 (1): 1–22.
Du, Q., and K. Zhou. 2011. “Mathematical analysis for the peridynamic nonlocal continuum theory.” Math. Modell. Numer. Anal. 45 (2): 217–234. https://doi.org/10.1051/m2an/2010040.
Efendiev, Y., J. Galvis, and T. Y. Hou. 2013. “Generalized multiscale finite element methods (gmsfem).” J. Comput. Phys. 251 (Aug): 116–135. https://doi.org/10.1016/j.jcp.2013.04.045.
Eringen, A. C., and D. G. B. Edelen. 1972. “On nonlocal elasticity.” Int. J. Eng. Sci. 10 (3): 233–248. https://doi.org/10.1016/0020-7225(72)90039-0.
Geers, M. G. D., V. G. Kouznetsova, K. Matouš, and J. Yvonnet. 2017. “Homogenization methods and multiscale modeling.” In Encyclopedia of computational mechanics. 2nd ed., 1–34. Hoboken, NJ: Wiley.
Haario, H., E. Saksman, and J. Tamminen. 2001. “An adaptive Metropolis algorithm.” Bernoulli 7 (Jun): 223–242. https://doi.org/10.2307/3318737.
Hakim, L., G. Lacaze, M. Khalil, K. Sargsyan, H. Najm, and J. Oefelein. 2018. “Probabilistic parameter estimation in a 2-step chemical kinetics model for n-dodecane jet autoignition.” Combust. Theory Model. 22 (3): 446–466. https://doi.org/10.1080/13647830.2017.1403653.
Hansen, P. C. 2000. The L-curve and its use in the numerical treatment of inverse problems. Billerica, MA: WIT Press.
Higdon, D., H. Lee, and C. Holloman. 2003. “Markov chain Monte Carlo-based approaches for inference in computationally intensive inverse problems.” Bayesian Stat. 7 (Aug): 181–197.
Huan, X., C. Safta, K. Sargsyan, G. Geraci, M. S. Eldred, Z. P. Vane, G. Lacaze, J. C. Oefelein, and H. N. Najm. 2018. “Global sensitivity analysis and estimation of model error, toward uncertainty quantification in scramjet computations.” AIAA J. 56 (3): 1170–1184. https://doi.org/10.2514/1.J056278.
Hughes, T. J., G. N. Wells, and A. A. Wray. 2004. “Energy transfers and spectral eddy viscosity in large-eddy simulations of homogeneous isotropic turbulence: Comparison of dynamic smagorinsky and multiscale models over a range of discretizations.” Phys. Fluids 16 (11): 4044–4052. https://doi.org/10.1063/1.1789157.
Jaynes, E. 2003. Probability theory: The logic of science. Edited by G. L. Bretthorst. Cambridge, UK: Cambridge University Press.
Junghans, C., M. Praprotnik, and K. Kremer. 2008. “Transport properties controlled by a thermostat: An extended dissipative particle dynamics thermostat.” Soft Matter 4 (1): 156–161. https://doi.org/10.1039/B713568H.
Karal, F. C., Jr., and J. B. Keller. 1964. “Elastic, electromagnetic, and other waves in a random medium.” J. Math. Phys. 5 (4): 537–547. https://doi.org/10.1063/1.1704145.
Kennedy, M., and A. O’Hagan. 2000. “Predicting the output from a complex computer code when fast approximations are available.” Biometrika 87 (1): 1–13. https://doi.org/10.1093/biomet/87.1.1.
Kennedy, M. C., and A. O’Hagan. 2001. “Bayesian calibration of computer models.” J. R. Stat. Soc. B 63 (3): 425–464. https://doi.org/10.1111/1467-9868.00294.
Kubo, R. 1966. “The fluctuation-dissipation theorem.” Rep. Prog. Phys. 29 (1): 255. https://doi.org/10.1088/0034-4885/29/1/306.
Lang, Q., and F. Lu. 2022. “Learning interaction kernels in mean-field equations of first-order systems of interacting particles.” J. Sci. Comput. 44 (1): 260–285. https://doi.org/10.1137/20M1377072.
Laplace, M. D. 1814. Essai philosophique sur les probabilités. New York: Dover Publications.
Lu, F., Q. An, and Y. Yu. 2022. “Nonparametric learning of kernels in nonlocal operators.” Preprint, submitted May 23, 2022. https://arxiv.org/abs/2205.11006.
Moës, N., J. T. Oden, K. Vemaganti, and J.-F. Remacle. 1999. “Simplified methods and a posteriori error estimation for the homogenization of representative volume elements (rve).” Comput. Methods Appl. Mech. Eng. 176 (1–4): 265–278. https://doi.org/10.1016/S0045-7825(98)00341-7.
Oliver, T. A., and R. D. Moser. 2011. “Bayesian uncertainty quantification applied to RANS turbulence models.” J. Phys.: Conf. Ser. 318 (Aug): 12–16. https://doi.org/10.1088/1742-6596/318/4/042032.
Ortiz, M. 1987. “A method of homogenization of elastic media.” Int. J. Eng. Sci. 25 (7): 923–934. https://doi.org/10.1016/0020-7225(87)90125-X.
Rahali, Y., I. Giorgio, J. Ganghoffer, and F. Dell’isola. 2015. “Homogenization à la piola produces second gradient continuum models for linear pantographic lattices.” Int. J. Eng. Sci. 97 (Jun): 148–172. https://doi.org/10.1016/j.ijengsci.2015.10.003.
Robert, C., and G. Casella. 2004. Monte Carlo statistical methods. Berlin: Springer.
Santosa, F., and W. W. Symes. 1991. “A dispersive effective medium for wave propagation in periodic composites.” SIAM J. Appl. Math. 51 (4): 984–1005. https://doi.org/10.1137/0151049.
Sargsyan, K., X. Huan, and H. Najm. 2019. “Embedded model error representation for Bayesian model calibration.” Int. J. UQ 9 (4): 365–394.
Sargsyan, K., H. Najm, and R. Ghanem. 2015. “On the statistical calibration of physical models.” Int. J. Chem. Kinet. 47 (4): 246–276. https://doi.org/10.1002/kin.20906.
Silling, S. A. 2000. “Reformulation of elasticity theory for discontinuities and long-range forces.” J. Mech. Phys. Solids 48 (1): 175–209. https://doi.org/10.1016/S0022-5096(99)00029-0.
Silling, S. A. 2021. “Propagation of a stress pulse in a heterogeneous elastic bar.” J. Peridyn. Nonlocal Model. 3 (3): 255–275. https://doi.org/10.1007/s42102-020-00048-5.
Silling, S. A., M. D’Elia, Y. Yu, H. You, and M. Fermen-Coker. 2022. “Peridynamic model for single-layer graphene obtained from coarse-grained bond forces.” J. Peridyn. Nonlocal Model. 2022 (1): 1–22. https://doi.org/10.1007/s42102-021-00075-w.
Sivia, D. S., and J. Skilling. 2006. Data analysis: A Bayesian tutorial. 2nd ed. Oxford, UK: Oxford University Press.
Smyshlyaev, V. P., and K. D. Cherednichenko. 2000. “On rigorous derivation of strain gradient effects in the overall behaviour of periodic heterogeneous media.” J. Mech. Phys. Solids 48 (6–7): 1325–1357. https://doi.org/10.1016/S0022-5096(99)00090-3.
Suzuki, J., M. Gulian, M. Zayernouri, and M. D’Elia. 2022. “Fractional modeling in action: A survey of nonlocal models for subsurface transport, turbulent flows, and anomalous materials.” J. Peridyn. Nonlocal Model. 2022 (Oct): 1–68. https://doi.org/10.1007/s42102-022-00085-2.
Tauchert, T. R., and A. Guzelsu. 1972. “An experimental study of dispersion of stress waves in a fiber-reinforced composite.” J. Appl. Mech. 39 (1): 98–102. https://doi.org/10.1115/1.3422677.
Trillos, N. G., and D. Sanz-Alonso. 2017. “The bayesian formulation and well-posedness of fractional elliptic inverse problems.” Inverse Prob. 33 (6): 065006. https://doi.org/10.1088/1361-6420/aa711e.
Vats, D., J. M. Flegal, and G. L. Jones. 2019. “Multivariate output analysis for markov chain Monte Carlo.” Biometrika 106 (2): 321–337. https://doi.org/10.1093/biomet/asz002.
Weinan, E., and B. Engquist. 2003. “Multiscale modeling and computation.” Notices AMS 50 (9): 1062–1070.
Willis, J. R. 1985. “The nonlocal influence of density variations in a composite.” Int. J. Solids Struct. 21 (7): 805–817. https://doi.org/10.1016/0020-7683(85)90084-8.
Xu, X., M. D’Elia, and J. T. Foster. 2021. “A machine-learning framework for peridynamic material models with physical constraints.” Comput. Methods Appl. Mech. Eng. 386 (Apr): 114062. https://doi.org/10.1016/j.cma.2021.114062.
Xu, X., M. D’Elia, C. Glusa, and J. T. Foster. 2022. “Machine-learning of nonlocal kernels for anomalous subsurface transport from breakthrough curves.” Preprint, submitted January 26, 2022. https://arxiv.org/abs/2201.11146.
Xu, X., and J. T. Foster. 2020. “Deriving peridynamic influence functions for one-dimensional elastic materials with periodic microstructure.” J. Peridyn. Nonlocal Model. 2 (4): 337–351. https://doi.org/10.1007/s42102-020-00037-8.
You, H., Y. Yu, S. Silling, and M. D’Elia. 2021a. “Data-driven learning of nonlocal models: From high-fidelity simulations to constitutive laws.” In Proc., AAAI Spring Symp. on Combining Artificial Intelligence and Machine Learning with Physical Sciences. Washington, DC: Association for the Advancement of Artificial Intelligence.
You, H., Y. Yu, S. Silling, and M. D’Elia. 2022. “A data-driven peridynamic continuum model for upscaling molecular dynamics.” Comput. Methods Appl. Mech. Eng. 389 (1): 114400. https://doi.org/10.1016/j.cma.2021.114400.
You, H., Y. Yu, N. Trask, M. Gulian, and M. D’Elia. 2021b. “Data-driven learning of nonlocal physics from high-fidelity synthetic data.” Comput. Methods Appl. Mech. Eng. 374 (Jun): 113553. https://doi.org/10.1016/j.cma.2020.113553.
Zhang, L., H. You, and Y. Yu. 2022. “MetaNOR: A meta-learnt nonlocal operator regression approach for metamaterial modeling.” MRS Commun. 12 (5): 662–677. https://doi.org/10.1557/s43579-022-00250-0.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 149 • Issue 8 • August 2023
Copyright
© 2023 American Society of Civil Engineers.
History
Received: Oct 1, 2022
Accepted: Mar 6, 2023
Published online: May 26, 2023
Published in print: Aug 1, 2023
Discussion open until: Oct 26, 2023
ASCE Technical Topics:
- Analysis (by type)
- Bayesian analysis
- Chemical properties
- Chemistry
- Composite materials
- Continuum mechanics
- Dynamics (solid mechanics)
- Engineering fundamentals
- Engineering materials (by type)
- Engineering mechanics
- Environmental engineering
- Heterogeneity
- Material mechanics
- Material properties
- Materials characterization
- Materials engineering
- Microstructure
- Motion (dynamics)
- Regression analysis
- Solid mechanics
- Statistical analysis (by type)
- Uncertainty principles
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.