Bifidelity Data-Assisted Neural Networks in Nonintrusive Reduced-Order Modeling

In this paper, we present a new nonintrusive reduced basis method when a cheap low-fidelity model and expensive high-fidelity model are available. The method relies on proper orthogonal decomposition (POD) to generate the high-fidelity reduced basis and a shallow multilayer perceptron to learn the high-fidelity reduced coefficients. In contrast to other methods, one distinct feature of the proposed method is to incorporate the features extracted from the low-fidelity data as the input feature, this approach not only improves the predictive capability of the neural network but also enables the decoupling the high-fidelity simulation from the online stage. Due to its nonintrusive nature, it is applicable to general parameterized problems. We also provide several numerical examples to illustrate the effectiveness and performance of the proposed method.

[1]  P. A. Newman,et al.  Optimization with variable-fidelity models applied to wing design , 1999 .

[2]  Xueyu Zhu,et al.  A bi-fidelity method for the multiscale Boltzmann equation with random parameters , 2019, J. Comput. Phys..

[3]  Ionel M. Navon,et al.  Non-intrusive reduced order modelling of the Navier-Stokes equations , 2015 .

[4]  Dongbin Xiu,et al.  A Stochastic Collocation Algorithm with Multifidelity Models , 2014, SIAM J. Sci. Comput..

[5]  Mats G. Larson,et al.  The Finite Element , 2013 .

[6]  Jing Li,et al.  When Bifidelity Meets CoKriging: An Efficient Physics-Informed Multifidelity Method , 2018, SIAM J. Sci. Comput..

[7]  Jan S. Hesthaven,et al.  Non-intrusive reduced order modeling of nonlinear problems using neural networks , 2018, J. Comput. Phys..

[8]  J. Hesthaven,et al.  Data-driven reduced order modeling for time-dependent problems , 2019, Computer Methods in Applied Mechanics and Engineering.

[9]  Kirthevasan Kandasamy,et al.  Multi-Fidelity Black-Box Optimization with Hierarchical Partitions , 2018, ICML.

[10]  D. Xiao.  Non-intrusive reduced order models and their applications , 2016 .

[11]  Guangyao Li,et al.  Multi-fidelity optimization for sheet metal forming process , 2011 .

[12]  Zhi-Hua Zhou,et al.  Ensemble Methods: Foundations and Algorithms , 2012 .

[13]  A. Patera,et al.  A PRIORI CONVERGENCE OF THE GREEDY ALGORITHM FOR THE PARAMETRIZED REDUCED BASIS METHOD , 2012 .

[14]  Tim Colonius,et al.  Guide to Spectral Proper Orthogonal Decomposition , 2020, AIAA Journal.

[15]  Yvon Maday,et al.  Reduced basis method for the rapid and reliable solution of partial differential equations , 2006 .

[16]  Akil C. Narayan,et al.  Practical error bounds for a non-intrusive bi-fidelity approach to parametric/stochastic model reduction , 2018, J. Comput. Phys..

[17]  A. O'Hagan,et al.  Predicting the output from a complex computer code when fast approximations are available , 2000 .

[18]  Jonathan P. How,et al.  Reinforcement learning with multi-fidelity simulators , 2014, 2014 IEEE International Conference on Robotics and Automation (ICRA).

[19]  Theresa Dawn Robinson,et al.  Surrogate-Based Optimization Using Multifidelity Models with Variable Parameterization and Corrected Space Mapping , 2008 .

[20]  Wolfgang Hörmann,et al.  Quasi Importance Sampling , 2005 .

[21]  P Perdikaris,et al.  Multi-fidelity modelling via recursive co-kriging and Gaussian–Markov random fields , 2015, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[22]  Dongbin Xiu,et al.  Computational Aspects of Stochastic Collocation with Multifidelity Models , 2014, SIAM/ASA J. Uncertain. Quantification.

[23]  A. Quarteroni,et al.  Reduced Basis Methods for Partial Differential Equations: An Introduction , 2015 .

[24]  Qian Wang,et al.  Non-intrusive reduced order modeling of unsteady flows using artificial neural networks with application to a combustion problem , 2019, J. Comput. Phys..

[25]  Sebastian Ruder,et al.  An overview of gradient descent optimization algorithms , 2016, Vestnik komp'iuternykh i informatsionnykh tekhnologii.

[26]  J. Hesthaven,et al.  Certified Reduced Basis Methods for Parametrized Partial Differential Equations , 2015 .

[27]  Kurt Hornik,et al.  Approximation capabilities of multilayer feedforward networks , 1991, Neural Networks.

[28]  Ionel M. Navon,et al.  A parameterized non-intrusive reduced order model and error analysis for general time-dependent nonlinear partial differential equations and its applications , 2017 .

[29]  Tiangang Cui,et al.  Multifidelity importance sampling , 2016 .

[30]  Dongbin Xiu,et al.  Multi-fidelity stochastic collocation method for computation of statistical moments , 2017, J. Comput. Phys..

[31]  J. Hesthaven,et al.  A non-intrusive multifidelity method for the reduced order modeling of nonlinear problems , 2020 .

[32]  George Cybenko,et al.  Approximation by superpositions of a sigmoidal function , 1989, Math. Control. Signals Syst..

[33]  Shun Zhang,et al.  Reduced Basis Multiscale Finite Element Methods for Elliptic Problems , 2015, Multiscale Model. Simul..

[34]  Omer San,et al.  An artificial neural network framework for reduced order modeling of transient flows , 2018, Commun. Nonlinear Sci. Numer. Simul..

[35]  A. Patera,et al.  Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations , 2007 .

[36]  Danny C. Sorensen,et al.  Nonlinear Model Reduction via Discrete Empirical Interpolation , 2010, SIAM J. Sci. Comput..

[37]  A. Chatterjee.  An introduction to the proper orthogonal decomposition , 2000 .

[38]  Peng Chen,et al.  Model order reduction methods in computational uncertainty quantification , 2017 .

[39]  R. Iman.  Latin Hypercube Sampling , 2006 .

[40]  Mohammad Bagher Menhaj,et al.  Training feedforward networks with the Marquardt algorithm , 1994, IEEE Trans. Neural Networks.

[41]  A. Patera,et al.  Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations , 2007 .

[42]  Jan S. Hesthaven,et al.  Reduced order modeling for nonlinear structural analysis using Gaussian process regression , 2018, Computer Methods in Applied Mechanics and Engineering.

[43]  Kurt Hornik,et al.  Multilayer feedforward networks are universal approximators , 1989, Neural Networks.

[44]  S. Pawar,et al.  A deep learning enabler for nonintrusive reduced order modeling of fluid flows , 2019, Physics of Fluids.

[45]  Jan S. Hesthaven,et al.  Certified reduced basis method for electromagnetic scattering and radar cross section estimation , 2012 .

[46]  Kirthevasan Kandasamy,et al.  Gaussian Process Bandit Optimisation with Multi-fidelity Evaluations , 2016, NIPS.

[47]  Slawomir Koziel,et al.  Multi-fidelity design optimization of transonic airfoils using physics-based surrogate modeling and shape-preserving response prediction , 2010, J. Comput. Sci..