Numerical Assessment of a Nonintrusive Surrogate Model Based on Recurrent Neural Networks and Proper Orthogonal Decomposition: Rayleigh–Bénard Convection

References

• Abadía-Heredia, R., M. López-Martín, B. Carro, J. I. Arribas, J. M. Pérez, and S. Le Clainche. 2022. “A Predictive Hybrid Reduced Order Model Based on Proper Orthogonal Decomposition Combined with Deep Learning Architectures.” Expert Systems with Applications 187: Article ID 115910. doi:10.1016/j.eswa.2021.115910.
   Web of Science ®Google Scholar
• Ahmed, M., and O. San. 2018. “Stabilized Principal Interval Decomposition Method for Model Reduction of Nonlinear Convective Systems with Moving Shocks.” Computational and Applied Mathematics 37 (5): 6870–6902. doi:10.1007/s40314-018-0718-z.
   Google Scholar
• Ahmed, S. E., S. Pawar, O. San, A. Rasheed, T. Iliescu, and B. R. Noack. 2021. “On Closures for Reduced Order Models—A Spectrum of First-Principle to Machine-Learned Avenues.” Physics of Fluids 33 (9): Article ID 091301. doi:10.1063/5.0061577.
   Web of Science ®Google Scholar
• Ahmed, S. E., and O. San. 2020. “Breaking the Kolmogorov Barrier in Model Reduction of Fluid Flows.” Fluids 5 (1): 26. doi:10.3390/fluids5010026.
   Web of Science ®Google Scholar
• Ahmed, S. E., O. San, A. Rasheed, and T. Iliescu. 2020. “A Long Short-term Memory Embedding for Hybrid Uplifted Reduced Order Models.” Physica D: Nonlinear Phenomena 409: Article ID 132471. doi:10.1016/j.physd.2020.132471.
   Web of Science ®Google Scholar
• Ahmed, S. E., O. San, A. Rasheed, and T. Iliescu. 2021. “Nonlinear Proper Orthogonal Decomposition for Convection-Dominated Flows.” Physics of Fluids 33 (12): Article ID 121702. doi:10.1063/5.0074310.
   Web of Science ®Google Scholar
• Amsallem, D., and B. Haasdonk. 2016. “PEBL-ROM: Projection-Error Based Local Reduced-Order Models.” Advanced Modeling and Simulation in Engineering Sciences 3 (1): 1–25. doi:10.1186/s40323-016-0059-7.
   Google Scholar
• Amsallem, D., M. J. Zahr, and C. Farhat. 2012. “Nonlinear Model Order Reduction Based on Local Reduced-Order Bases.” International Journal for Numerical Methods in Engineering 92 (10): 891–916. doi:10.1002/nme.v92.10.
   Web of Science ®Google Scholar
• Amsallem, D., M. J. Zahr, and K. Washabaugh. 2015. “Fast Local Reduced Basis Updates for the Efficient Reduction of Nonlinear Systems with Hyper-Reduction.” Advances in Computational Mathematics 41 (5): 1187–1230. doi:10.1007/s10444-015-9409-0.
   Web of Science ®Google Scholar
• Aubry, N., W. Y. Lian, and E. S. Titi. 1993. “Preserving Symmetries in the Proper Orthogonal Decomposition.” SIAM Journal on Scientific Computing 14 (2): 483–505. doi:10.1137/0914030.
   Web of Science ®Google Scholar
• Bergmann, M., C. H. Bruneau, and A. Iollo. 2009. “Enablers for Robust POD Models.” Journal of Computational Physics 228 (2): 516–538. doi:10.1016/j.jcp.2008.09.024.
   Web of Science ®Google Scholar
• Berkooz, G., P. Holmes, and J. L. Lumley. 1993. “The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows.” Annual Review of Fluid Mechanics 25 (1): 539–575. doi:10.1146/fluid.1993.25.issue-1.
   Web of Science ®Google Scholar
• Briley, W. R. 1971. “A Numerical Study of Laminar Separation Bubbles Using the Navier–Stokes Equations.” Journal of Fluid Mechanics 47 (4): 713–736. doi:10.1017/S0022112071001332.
   Google Scholar
• Brunton, S. L., B. R. Noack, and P. Koumoutsakos. 2020. “Machine Learning for Fluid Mechanics.” Annual Review of Fluid Mechanics 52: 477–508. doi:10.1146/fluid.2020.52.issue-1.
   Web of Science ®Google Scholar
• Burkardt, J., M. Gunzburger, and H. C. Lee. 2006. “Centroidal Voronoi Tessellation-based Reduced-Order Modeling of Complex Systems.” SIAM Journal on Scientific Computing 28 (2): 459–484. doi:10.1137/5106482750342221x.
   Web of Science ®Google Scholar
• Burkardt, J., M. Gunzburger, and H. C. Lee. 2006. “POD and CVT-based Reduced-Order Modeling of Navier–Stokes Flows.” Computer Methods in Applied Mechanics and Engineering 196 (1–3): 337–355. doi:10.1016/j.cma.2006.04.004.
   Web of Science ®Google Scholar
• Cai, T., Z. Deng, Y. Park, S. Mohammadshahi, Y. Liu, and K. C. Kim. 2021. “Acquisition of kHz-Frequency Two-dimensional Surface Temperature Field Using Phosphor Thermometry and Proper Orthogonal Decomposition Assisted Long Short-Term Memory Neural Networks.” International Journal of Heat and Mass Transfer 165: Article ID 120662. doi:10.1016/j.ijheatmasstransfer.2020.120662.
   Web of Science ®Google Scholar
• Christensen, E. A., M. Brøns, and J. N. Sørensen. 1999. “Evaluation of Proper Orthogonal Decomposition-based Decomposition Techniques Applied to Parameter-Dependent Nonturbulent Flows.” SIAM Journal on Scientific Computing 21 (4): 1419–1434. doi:10.1137/S1064827598333181.
   Web of Science ®Google Scholar
• Deane, A., I. Kevrekidis, G. E. Karniadakis, and S. Orszag. 1991. “Low-dimensional Models for Complex Geometry Flows: Application to Grooved Channels and Circular Cylinders.” Physics of Fluids A: Fluid Dynamics 3 (10): 2337–2354. doi:10.1063/1.857881.
   Web of Science ®Google Scholar
• Deng, Z., Y. Chen, Y. Liu, and K. C. Kim. 2019. “Time-Resolved Turbulent Velocity Field Reconstruction Using a Long Short-term Memory (LSTM)-based Artificial Intelligence Framework.” Physics of Fluids 31 (7): Article ID 075108. doi:10.1063/1.5111558.
   Web of Science ®Google Scholar
• Dihlmann, M., M. Drohmann, and B. Haasdonk. 2011. “Model Reduction of Parametrized Evolution Problems Using the Reduced Basis Method with Adaptive Time-partitioning.” Proceedings of ADMOS 2011: 64.
   Google Scholar
• Drohmann, M., B. Haasdonk, and M Ohlberger. 2011. “Adaptive Reduced Basis Methods for Nonlinear Convection–Diffusion Equations.” In Finite Volumes for Complex Applications VI Problems & Perspectives, 369–377. Springer.
   Google Scholar
• Eftang, J. L., D. J. Knezevic, and A. T. Patera. 2011. “An Hp Certified Reduced Basis Method for Parametrized Parabolic Partial Differential Equations.” Mathematical and Computer Modelling of Dynamical Systems 17 (4): 395–422. doi:10.1080/13873954.2011.547670.
   Web of Science ®Google Scholar
• Eftang, J. L., A. T. Patera, and E. M. Rønquist. 2010. “An ‘hp’ Certified Reduced Basis Method for Parametrized Elliptic Partial Differential Equations.” SIAM Journal on Scientific Computing 32 (6): 3170–3200. doi:10.1137/090780122.
   Web of Science ®Google Scholar
• Eftang, J. L., and B. Stamm. 2012. “Parameter Multi-domain ‘hp’ Empirical Interpolation.” International Journal for Numerical Methods in Engineering 90 (4): 412–428. doi:10.1002/nme.3327.
   Web of Science ®Google Scholar
• Eivazi, H., H. Veisi, M. H. Naderi, and V. Esfahanian. 2020. “Deep Neural Networks for Nonlinear Model Order Reduction of Unsteady Flows.” Physics of Fluids 32 (10): Article ID 105104. doi:10.1063/5.0020526.
   Web of Science ®Google Scholar
• Esfahanian, V., A. B. Ansari, and F. Torabi. 2015. “Simulation of Lead-acid Battery Using Model Order Reduction.” Journal of Power Sources 279: 294–305. doi:10.1016/j.jpowsour.2014.12.149.
   Web of Science ®Google Scholar
• Greif, C., and K. Urban. 2019. “Decay of the Kolmogorov N-width for Wave Problems.” Applied Mathematics Letters 96: 216–222. doi:10.1016/j.aml.2019.05.013.
   Web of Science ®Google Scholar
• Grimberg, S., C. Farhat, R. Tezaur, and C. Bou-Mosleh. 2021. “Mesh Sampling and Weighting for the Hyperreduction of Nonlinear Petrov–Galerkin Reduced-order Models with Local Reduced-order Bases.” International Journal for Numerical Methods in Engineering 122 (7): 1846–1874. doi:10.1002/nme.v122.7.
   Web of Science ®Google Scholar
• Haasdonk, B., M. Dihlmann, and M. Ohlberger. 2011. “A Training Set and Multiple Bases Generation Approach for Parameterized Model Reduction Based on Adaptive Grids in Parameter Space.” Mathematical and Computer Modelling of Dynamical Systems 17 (4): 423–442. doi:10.1080/13873954.2011.547674.
   Web of Science ®Google Scholar
• Hochreiter, S., and J. Schmidhuber. 1997. “Long Short-term Memory.” Neural Computation 9 (8): 1735–1780. doi:10.1162/neco.1997.9.8.1735.
   PubMed Web of Science ®Google Scholar
• Holmes, P., J. L. Lumley, G. Berkooz, and C. W Rowley. 2012. Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge: Cambridge University Press.
   Google Scholar
• Holmes, P. J., J. L. Lumley, G. Berkooz, J. C. Mattingly, and R. W. Wittenberg. 1997. “Low-dimensional Models of Coherent Structures in Turbulence.” Physics Reports 287 (4): 337–384. doi:10.1016/S0370-1573(97)00017-3.
   Google Scholar
• Huang, D., J. N. Fuhg, C. Weißenfels, and P. Wriggers. 2020. “A Machine Learning Based Plasticity Model Using Proper Orthogonal Decomposition.” Computer Methods in Applied Mechanics and Engineering 365: Article ID 113008. doi:10.1016/j.cma.2020.113008.
   Web of Science ®Google Scholar
• Im, S., J. Lee, and M. Cho. 2021. “Surrogate Modeling of Elasto-plastic Problems Via Long Short-term Memory Neural Networks and Proper Orthogonal Decomposition.” Computer Methods in Applied Mechanics and Engineering 385: Article ID 114030. doi:10.1016/j.cma.2021.114030.
   Web of Science ®Google Scholar
• Jacquier, P., A. Abdedou, V. Delmas, and A. Soulaïmani. 2021. “Non-intrusive Reduced-order Modeling Using Uncertainty-Aware Deep Neural Networks and Proper Orthogonal Decomposition: Application to Flood Modeling.” Journal of Computational Physics 424: Article ID 109854. doi:10.1016/j.jcp.2020.109854.
   Web of Science ®Google Scholar
• Kaiser, E., B. R. Noack, L. Cordier, A. Spohn, M. Segond, M. Abel, G. Daviller, and R. K. Niven. 2014. “Cluster-based Reduced-order Modelling of a Mixing Layer.” Journal of Fluid Mechanics 754: 365–414. doi:10.1017/jfm.2014.355.
   Web of Science ®Google Scholar
• Kalb, V. L., and A. E. Deane. 2007. “An Intrinsic Stabilization Scheme for Proper Orthogonal Decomposition Based Low-dimensional Models.” Physics of Fluids 19 (5): Article ID 054106. doi:10.1063/1.2723149.
   Web of Science ®Google Scholar
• Kapteyn, M. G., J. V. Pretorius, and K. E. Willcox. 2021. “A Probabilistic Graphical Model Foundation for Enabling Predictive Digital Twins At Scale.” Nature Computational Science 1 (5): 337–347. doi:10.1038/s43588-021-00069-0.
   Google Scholar
• Karhunen, K. 1946. “Zur Spektraltheorie Stochastischer Prozesse.” Annales Academiae Scientiarum Fennicae, AI 34:
   Google Scholar
• Kherad, M., M. K. Moayyedi, and F. Fotouhi. 2021. “Reduced Order Framework for Convection Dominant and Pure Diffusive Problems Based on Combination of Deep Long Short-term Memory and Proper Orthogonal Decomposition/dynamic Mode Decomposition Methods.” International Journal for Numerical Methods in Fluids 93 (3): 853–873. doi:10.1002/fld.v93.3.
   Web of Science ®Google Scholar
• Kolmogoroff, A. 1936. “Uber Die Beste Annaherung Von Funktionen Einer Gegebenen Funktionenklasse.” The Annals of Mathematics 37: 107–110. doi:10.2307/1968691.
   Google Scholar
• Kosambi, D. 1943. “Statistics in Function Space.” Journal of the Indian Mathematical Society 7: 76–88.
   Google Scholar
• Kunisch, K., and S. Volkwein. 1999. “Control of the Burgers Equation by a Reduced-order Approach Using Proper Orthogonal Decomposition.” Journal of Optimization Theory and Applications 102 (2): 345–371. doi:10.1023/A:1021732508059.
   Web of Science ®Google Scholar
• Kunisch, K., and S. Volkwein. 2001. “Galerkin Proper Orthogonal Decomposition Methods for Parabolic Problems.” Numerische Mathematik 90 (1): 117–148. doi:10.1007/s002110100282.
   Web of Science ®Google Scholar
• Lele, S. K. 1992. “Compact Finite Difference Schemes with Spectral-like Resolution.” Journal of Computational Physics 103 (1): 16–42. doi:10.1016/0021-9991(92)90324-R.
   Web of Science ®Google Scholar
• Loeve, M. 1948. “Functions Aleatoires Du Second Ordre.” Processus Stochastique Et Mouvement Brownien 366–420.
   Google Scholar
• Lumley, J. L. 1967. “The Structure of Inhomogeneous Turbulent Flows.” Atmospheric Turbulence and Radio Wave Propagation 166–178.
   Google Scholar
• Majda, A. 2003. Introduction to PDEs and Waves for the Atmosphere and Ocean. Vol. 9. NY: American Mathematical Society.
   Google Scholar
• Maulik, R., B. Lusch, and P. Balaprakash. 2021. “Reduced-order Modeling of Advection-dominated Systems with Recurrent Neural Networks and Convolutional Autoencoders.” Physics of Fluids 33 (3): Article ID 037106. doi:10.1063/5.0039986.
   Web of Science ®Google Scholar
• Obukhov, A. 1954. “Statistical Description of Continuous Fields.” Transactions of the Geophysical International Academy Nauk USSR 24 (24): 3–42.
   Google Scholar
• Olah, C. 2015. Understanding LSTM Networks. Accessed at 27, December 2022. http://colah.github.io/posts/2015-08-Understanding-LSTMs/.
   Google Scholar
• Ooi, C., Q. T. Le, M. H. Dao, V. B. Nguyen, H. H. Nguyen, and T. Ba. 2021. “Modeling Transient Fluid Simulations with Proper Orthogonal Decomposition and Machine Learning.” International Journal for Numerical Methods in Fluids 93 (2): 396–410. doi:10.1002/fld.v93.2.
   Web of Science ®Google Scholar
• Park, H., and M. Lee. 1998. “An Efficient Method of Solving the Navier–Stokes Equations for Flow Control.” International Journal for Numerical Methods in Engineering 41 (6): 1133–1151. doi:10.1002/(ISSN)1097-0207.
   Web of Science ®Google Scholar
• Pawar, S., S. Rahman, H. Vaddireddy, O. San, A. Rasheed, and P. Vedula. 2019. “A Deep Learning Enabler for Nonintrusive Reduced Order Modeling of Fluid Flows.” Physics of Fluids 31 (8): Article ID 085101. doi:10.1063/1.5113494.
   Web of Science ®Google Scholar
• Pearson, K. 1901. “LIII. On Lines and Planes of Closest Fit to Systems of Points in Space.” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 2 (11): 559–572.
   Google Scholar
• Peherstorfer, B., D. Butnaru, K. Willcox, and H. J. Bungartz. 2014. “Localized Discrete Empirical Interpolation Method.” SIAM Journal on Scientific Computing 36 (1): A168–A192. doi:10.1137/130924408.
   Web of Science ®Google Scholar
• Pugachev, V. S. 1953. “The General Theory of Correlation of Random Functions.” Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 17 (5): 401–420.
   Google Scholar
• Rasheed, A., O. San, and T. Kvamsdal. 2020. “Digital Twin: Values, Challenges and Enablers From a Modeling Perspective.” IEEE Access 8: 21980–22012. doi:10.1109/Access.6287639.
   Web of Science ®Google Scholar
• Rathinam, M., and L. R. Petzold. 2003. “A New Look at Proper Orthogonal Decomposition.” SIAM Journal on Numerical Analysis 41 (5): 1893–1925. doi:10.1137/S0036142901389049.
   Web of Science ®Google Scholar
• Ravindran, S. 2000. “Reduced-order Adaptive Controllers for Fluid Flows Using POD.” Journal of Scientific Computing 15 (4): 457–478. doi:10.1023/A:1011184714898.
   Google Scholar
• Redeker, M., and B. Haasdonk. 2015. “A POD-EIM Reduced Two-scale Model for Crystal Growth.” Advances in Computational Mathematics 41 (5): 987–1013. doi:10.1007/s10444-014-9367-y.
   Web of Science ®Google Scholar
• San, O. 2015. “A Novel High-order Accurate Compact Stencil Poisson Solver: Application to Cavity Flows.” International Journal of Applied Mechanics 7 (01): Article ID 1550006. doi:10.1142/S1758825115400062.
   Google Scholar
• San, O., and J. Borggaard. 2015. “Principal Interval Decomposition Framework for POD Reduced-order Modeling of Convective Boussinesq Flows.” International Journal for Numerical Methods in Fluids 78 (1): 37–62. doi:10.1002/fld.v78.1.
   Web of Science ®Google Scholar
• San, O., and R. Maulik. 2018. “Neural Network Closures for Nonlinear Model Order Reduction.” Advances in Computational Mathematics 44 (6): 1717–1750. doi:10.1007/s10444-018-9590-z.
   Web of Science ®Google Scholar
• San, O., R. Maulik, and M. Ahmed. 2019. “An Artificial Neural Network Framework for Reduced Order Modeling of Transient Flows.” Communications in Nonlinear Science and Numerical Simulation 77: 271–287. doi:10.1016/j.cnsns.2019.04.025.
   Web of Science ®Google Scholar
• San, O., A. Rasheed, and T. Kvamsdal. 2021. “Hybrid Analysis and Modeling, Eclecticism, and Multifidelity Computing Toward Digital Twin Revolution.” GAMM-Mitteilungen – Surveys for Applied Mathematics and Mechanics 44: Article ID e202100007. https://doi.org/10.1002/gamm.202100007
   Google Scholar
• Shahbazi, A. A., and V. Esfahanian. 2019. “Reduced-order Modeling of Lead-acid Battery Using Cluster Analysis and Orthogonal Cluster Analysis Method.” International Journal of Energy Research 43 (13): 6779–6798. https://doi.org/10.1002/er.4645
   Web of Science ®Google Scholar
• Sirovich, L. 1987. “Turbulence and the Dynamics of Coherent Structures. I. Coherent Structures.” Quarterly of Applied Mathematics 45 (3): 561–571.
   Web of Science ®Google Scholar
• Srinivasan, P. A., L. Guastoni, H. Azizpour, P. Schlatter, and R. Vinuesa. 2019. “Predictions of Turbulent Shear Flows Using Deep Neural Networks.” Physical Review Fluids 4 (5): Article ID 054603. doi:10.1103/PhysRevFluids.4.054603.
   Web of Science ®Google Scholar
• Vinuesa, R., H. Azizpour, I. Leite, M. Balaam, V. Dignum, S. Domisch, M. Tegmark, and F. Fuso Nerini. 2020. “The Role of Artificial Intelligence in Achieving the Sustainable Development Goals.” Nature Communications 11 (1): 1–10. doi:10.1038/s41467-019-14108-y.
   PubMed Web of Science ®Google Scholar
• Volkwein, S. 2001. “Optimal Control of a Phase-Field Model Using Proper Orthogonal Decomposition.” ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik: Applied Mathematics and Mechanics 81 (2): 83–97. doi:10.1002/(ISSN)1521-4001.
   Web of Science ®Google Scholar
• Wang, M., H. X. Li, X. Chen, and Y. Chen. 2016. “Deep Learning-based Model Reduction for Distributed Parameter Systems.” IEEE Transactions on Systems, Man, and Cybernetics: Systems 46 (12): 1664–1674. doi:10.1109/TSMC.2016.2605159.
   Web of Science ®Google Scholar
• Wang, Z., D. Xiao, F. Fang, R. Govindan, C. C. Pain, and Y. Guo. 2018. “Model Identification of Reduced Order Fluid Dynamics Systems Using Deep Learning.” International Journal for Numerical Methods in Fluids 86 (4): 255–268. doi:10.1002/fld.v86.4.
   Web of Science ®Google Scholar
• Washabaugh, K., D. Amsallem, M. Zahr, and C Farhat. 2012. “Nonlinear Model Reduction for CFD Problems Using Local Reduced-Order Bases.” In 42nd AIAA Fluid Dynamics Conference and Exhibit, 2686. New Orleans, Louisiana.
   Google Scholar
• Weinan, E., and J. G. Liu. 1996. “Essentially Compact Schemes for Unsteady Viscous Incompressible Flows.” Journal of Computational Physics 126 (1): 122–138. doi:10.1006/jcph.1996.0125.
   Web of Science ®Google Scholar
• Wieland, B. 2015. “Implicit Partitioning Methods for Unknown Parameter Sets.” Advances in Computational Mathematics 41 (5): 1159–1186. doi:10.1007/s10444-015-9404-5.
   Web of Science ®Google Scholar