An autoencoder-based reduced-order model for eigenvalue problems with application to neutron diffusion

Using an autoencoder for dimensionality reduction, this paper presents a novel projection-based reduced-order model for eigenvalue problems. Reduced-order modelling relies on finding suitable basis functions which define a low-dimensional space in which a high-dimensional system is approximated. Proper orthogonal decomposition (POD) and singular value decomposition (SVD) are often used for this purpose and yield an optimal linear subspace. Autoencoders provide a nonlinear alternative to POD/SVD, that may capture, more efficiently, features or patterns in the high-fidelity model results. Reduced-order models based on an autoencoder and a novel hybrid SVD-autoencoder are developed. These methods are compared with the standard POD-Galerkin approach and are applied to two test cases taken from the field of nuclear reactor physics.

[1]  Karen Willcox,et al.  Proper orthogonal decomposition extensions for parametric applications in compressible aerodynamics , 2003 .

[2]  Bernd R. Noack,et al.  Cluster-based reduced-order modelling of a mixing layer , 2013, Journal of Fluid Mechanics.

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

[4]  Lalit K. Mestha,et al.  A deep learning framework for model reduction of dynamical systems , 2017, 2017 IEEE Conference on Control Technology and Applications (CCTA).

[5]  A. Ammar,et al.  PGD-Based Computational Vademecum for Efficient Design, Optimization and Control , 2013, Archives of Computational Methods in Engineering.

[6]  Zachary M. Prince,et al.  Application of proper generalized decomposition to multigroup neutron diffusion eigenvalue calculations , 2020 .

[7]  Yu-Bin Yang,et al.  Image Restoration Using Very Deep Convolutional Encoder-Decoder Networks with Symmetric Skip Connections , 2016, NIPS.

[8]  Jing Wang,et al.  A Folded Neural Network Autoencoder for Dimensionality Reduction , 2012, INNS-WC.

[9]  E. Oja.  Simplified neuron model as a principal component analyzer , 1982, Journal of mathematical biology.

[10]  Jean C. Ragusa,et al.  Reduced-order modeling of parameterized multi-group diffusion k-eigenvalue problems , 2019 .

[11]  Fynn Scheben,et al.  Iterative Methods for Criticality Computations in Neutron Transport Theory , 2011 .

[12]  Pascal Vincent,et al.  Stacked Denoising Autoencoders: Learning Useful Representations in a Deep Network with a Local Denoising Criterion , 2010, J. Mach. Learn. Res..

[13]  Nils Thürey,et al.  Latent Space Physics: Towards Learning the Temporal Evolution of Fluid Flow , 2018, Comput. Graph. Forum.

[14]  D. Ryckelynck,et al.  A priori hyperreduction method: an adaptive approach , 2005 .

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

[16]  C. Pain,et al.  Model identification of reduced order fluid dynamics systems using deep learning , 2017, International Journal for Numerical Methods in Fluids.

[17]  H. Bourlard,et al.  Auto-association by multilayer perceptrons and singular value decomposition , 1988, Biological Cybernetics.

[18]  Karen Willcox,et al.  A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems , 2015, SIAM Rev..

[19]  C. Eckart,et al.  The approximation of one matrix by another of lower rank , 1936 .

[20]  Danny C. Sorensen,et al.  A Posteriori Error Estimation for DEIM Reduced Nonlinear Dynamical Systems , 2014, SIAM J. Sci. Comput..

[21]  Charbel Farhat,et al.  Progressive construction of a parametric reduced‐order model for PDE‐constrained optimization , 2014, ArXiv.

[22]  Eleni Chatzi,et al.  Parametrized reduced order modeling for cracked solids , 2020, International Journal for Numerical Methods in Engineering.

[23]  Pablo Salinas,et al.  Non-intrusive model reduction for a 3D unstructured mesh control volume finite element reservoir model and its application to fluvial channels , 2018 .

[24]  Vassilios Theofilis,et al.  Modal Analysis of Fluid Flows: An Overview , 2017, 1702.01453.

[25]  Wolfgang Polifke,et al.  Black-box system identification for reduced order model construction , 2014 .

[26]  Benjamin Peherstorfer,et al.  Projection-based model reduction: Formulations for physics-based machine learning , 2019, Computers & Fluids.

[27]  Francisco J. Gonzalez,et al.  Deep convolutional recurrent autoencoders for learning low-dimensional feature dynamics of fluid systems , 2018, ArXiv.

[28]  Kurt Hornik,et al.  Neural networks and principal component analysis: Learning from examples without local minima , 1989, Neural Networks.

[29]  Jimmy J. Lin,et al.  WTF: the who to follow service at Twitter , 2013, WWW.

[30]  K. Sefiane,et al.  The European Consortium For Mathematics in Industry , 2008 .

[31]  Sepp Hochreiter,et al.  Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs) , 2015, ICLR.

[32]  Jim E. Morel,et al.  A Self-Adjoint Angular Flux Equation , 1999 .

[33]  Michele Milano,et al.  Neural network modeling for near wall turbulent flow , 2002 .

[34]  L. Sirovich.  Turbulence and the dynamics of coherent structures. I. Coherent structures , 1987 .

[35]  Pierre Kerfriden,et al.  Local/global model order reduction strategy for the simulation of quasi‐brittle fracture , 2011, 1108.3167.

[36]  P. Villon,et al.  Multi-fidelity POD surrogate-assisted optimization: Concept and aero-design study , 2017 .

[37]  Gene H. Golub,et al.  Matrix computations , 1983 .

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

[39]  Thomas M. Evans,et al.  Eigenvalue Solvers for Modeling Nuclear Reactors on Leadership Class Machines , 2017, ArXiv.

[40]  Max Kuhn,et al.  Applied Predictive Modeling , 2013 .

[41]  Peter Benner,et al.  Comparison of Methods for Parametric Model Order Reduction of Time-Dependent Problems , 2017 .

[42]  Kenji Kashima,et al.  Nonlinear model reduction by deep autoencoder of noise response data , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[43]  Geoffrey E. Hinton,et al.  Reducing the Dimensionality of Data with Neural Networks , 2006, Science.

[44]  Petros Koumoutsakos,et al.  Machine Learning for Fluid Mechanics , 2019, Annual Review of Fluid Mechanics.

[45]  Christopher C. Pain,et al.  A combined immersed body and adaptive mesh method for simulating neutron transport within complex structures , 2019 .

[46]  K. C. Hoang,et al.  An hp-proper orthogonal decomposition–moving least squares approach for molecular dynamics simulation , 2016 .

[47]  L. Durlofsky,et al.  Use of reduced-order models in well control optimization , 2017 .

[48]  Kookjin Lee,et al.  Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders , 2018, J. Comput. Phys..

[49]  H. V. D. Vorst,et al.  Model Order Reduction: Theory, Research Aspects and Applications , 2008 .

[50]  Gabriel Pinski,et al.  Citation influence for journal aggregates of scientific publications: Theory, with application to the literature of physics , 1976, Inf. Process. Manag..

[51]  Gianluigi Rozza,et al.  Fast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a POD-Galerkin method and a vascular shape parametrization , 2016, J. Comput. Phys..

[52]  Sebastian Schöps,et al.  An algorithmic comparison of the Hyper-Reduction and the Discrete Empirical Interpolation Method for a nonlinear thermal problem , 2016, ArXiv.

[53]  Omer San,et al.  Memory embedded non-intrusive reduced order modeling of non-ergodic flows , 2019 .

[54]  B. R. Noack.  Turbulence, Coherent Structures, Dynamical Systems and Symmetry , 2013 .

[55]  Zhang Chunyu,et al.  Fast solution of neutron diffusion problem by reduced basis finite element method , 2018 .

[56]  Rodney Cannon Schmidt,et al.  COUPLED NEUTRONICS AND THERMAL-HYDRAULIC SOLUTION OF A FULL-CORE PWR USING VERA-CS , 2014 .

[57]  Gianluigi Rozza,et al.  Reduced Basis Approaches in Time-Dependent Non-Coercive Settings for Modelling the Movement of Nuclear Reactor Control Rods , 2016 .

[58]  Yike Guo,et al.  A reduced order model for turbulent flows in the urban environment using machine learning , 2019, Building and Environment.

[59]  S Niroomandi,et al.  Real‐time simulation of surgery by reduced‐order modeling and X‐FEM techniques , 2012, International journal for numerical methods in biomedical engineering.

[61]  G. Rozza,et al.  A multi-physics reduced order model for the analysis of Lead Fast Reactor single channel , 2016 .

[62]  Timothy Dozat,et al.  Incorporating Nesterov Momentum into Adam , 2016 .

[63]  Christopher C. Pain,et al.  A POD reduced‐order model for eigenvalue problems with application to reactor physics , 2013 .