An artificial neural network approach to bifurcating phenomena in computational fluid dynamics

This work deals with the investigation of bifurcating fluid phenomena using a reduced order modelling setting aided by artificial neural networks. We discuss the POD-NN approach dealing with non-smooth solutions set of nonlinear parametrized PDEs. Thus, we study the NavierStokes equations describing: (i) the Coanda effect in a channel, and (ii) the lid driven triangular cavity flow, in a physical/geometrical multi-parametrized setting, considering the effects of the domain’s configuration on the position of the bifurcation points. Finally, we propose a reduced manifold-based bifurcation diagram for a non-intrusive recovery of the critical points evolution. Exploiting such detection tool, we are able to efficiently obtain information about the pattern flow behaviour, from symmetry breaking profiles to attaching/spreading vortices, even at high Reynolds numbers.

[1]  Guigang Zhang,et al.  Deep Learning , 2016, Int. J. Semantic Comput..

[2]  E. Allgower,et al.  Introduction to Numerical Continuation Methods , 1987 .

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

[4]  Annalisa Quaini,et al.  Reduced basis model order reduction for Navier–Stokes equations in domains with walls of varying curvature , 2019, International Journal of Computational Fluid Dynamics.

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

[6]  Gianluigi Rozza,et al.  Supremizer stabilization of POD–Galerkin approximation of parametrized steady incompressible Navier–Stokes equations , 2015 .

[7]  Natalia Gimelshein,et al.  PyTorch: An Imperative Style, High-Performance Deep Learning Library , 2019, NeurIPS.

[8]  Sören Bartels,et al.  Numerical Approximation of Partial Differential Equations , 2016 .

[9]  J. Hesthaven,et al.  Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations , 2007 .

[10]  Gianluigi Rozza,et al.  Driving bifurcating parametrized nonlinear PDEs by optimal control strategies: application to Navier-Stokes equations with model order reduction , 2020, ArXiv.

[11]  Annalisa Quaini,et al.  Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: Applications to Coanda effect in cardiology , 2017, J. Comput. Phys..

[12]  Yvon Maday,et al.  RB (Reduced basis) for RB (Rayleigh–Bénard) , 2013 .

[13]  Gianluigi Rozza,et al.  On the Application of Reduced Basis Methods to Bifurcation Problems in Incompressible Fluid Dynamics , 2017, J. Sci. Comput..

[14]  Annalisa Quaini,et al.  A Reduced Order Modeling Technique to Study Bifurcating Phenomena: Application to the Gross-Pitaevskii Equation , 2020, SIAM J. Sci. Comput..

[15]  Alfio Quarteroni,et al.  Machine learning for fast and reliable solution of time-dependent differential equations , 2019, J. Comput. Phys..

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

[17]  G. Karniadakis,et al.  On the Convergence of Physics Informed Neural Networks for Linear Second-Order Elliptic and Parabolic Type PDEs , 2020, Communications in Computational Physics.

[18]  Annalisa Quaini,et al.  A localized reduced-order modeling approach for PDEs with bifurcating solutions , 2018, Computer Methods in Applied Mechanics and Engineering.

[19]  S. Volkwein,et al.  MODEL REDUCTION USING PROPER ORTHOGONAL DECOMPOSITION , 2008 .

[20]  Jan S. Hesthaven,et al.  Physics-informed machine learning for reduced-order modeling of nonlinear problems , 2021, J. Comput. Phys..

[21]  Vassilios Theofilis,et al.  Three-dimensional flow instability in a lid-driven isosceles triangular cavity , 2011, Journal of Fluid Mechanics.

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

[23]  Paris Perdikaris,et al.  Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations , 2019, J. Comput. Phys..

[24]  R. Molinaro,et al.  Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs , 2020, ArXiv.

[25]  José M. Vega,et al.  On the use of POD-based ROMs to analyze bifurcations in some dissipative systems , 2012 .

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

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

[28]  J. A. Kuznecov.  Elements of applied bifurcation theory , 1998 .

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

[30]  J. Rappaz,et al.  Numerical analysis for nonlinear and bifurcation problems , 1997 .

[31]  S. Brunton,et al.  Discovering governing equations from data by sparse identification of nonlinear dynamical systems , 2015, Proceedings of the National Academy of Sciences.

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

[33]  Jimmy Ba,et al.  Adam: A Method for Stochastic Optimization , 2014, ICLR.

[34]  Gitta Kutyniok,et al.  A Theoretical Analysis of Deep Neural Networks and Parametric PDEs , 2019, Constructive Approximation.

[35]  A. Cohen,et al.  Model Reduction and Approximation: Theory and Algorithms , 2017 .

[36]  Gianluigi Rozza,et al.  Reduced Basis Approaches for Parametrized Bifurcation Problems held by Non-linear Von Kármán Equations , 2018, Journal of Scientific Computing.

[37]  Jacques Rappaz,et al.  Finite Dimensional Approximation of Non-Linear Problems .1. Branches of Nonsingular Solutions , 1980 .

[38]  R. Seydel.  Practical Bifurcation and Stability Analysis , 1994 .

[39]  J. M. Bergadà,et al.  The lid-driven right-angled isosceles triangular cavity flow , 2019, Journal of Fluid Mechanics.

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

[41]  Claudio Canuto,et al.  Efficient computation of bifurcation diagrams with a deflated approach to reduced basis spectral element method , 2021, Adv. Comput. Math..

[42]  Steven L. Brunton,et al.  Learning normal form autoencoders for data-driven discovery of universal, parameter-dependent governing equations , 2021, ArXiv.

[43]  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..

[44]  N. Nguyen,et al.  An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations , 2004 .

[45]  Benjamin Stamm,et al.  EFFICIENT GREEDY ALGORITHMS FOR HIGH-DIMENSIONAL PARAMETER SPACES WITH APPLICATIONS TO EMPIRICAL INTERPOLATION AND REDUCED BASIS METHODS ∗ , 2014 .

[46]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .

[47]  Simon W. Funke,et al.  Deflation Techniques for Finding Distinct Solutions of Nonlinear Partial Differential Equations , 2014, SIAM J. Sci. Comput..

[48]  E. Erturk,et al.  Fine Grid Numerical Solutions of Triangular Cavity Flow , 2005, ArXiv.

[49]  Benjamin Peherstorfer,et al.  Dynamic data-driven reduced-order models , 2015 .

[50]  Y. Kuznetsov.  Elements of applied bifurcation theory (2nd ed.) , 1998 .

[51]  P. G. Ciarlet,et al.  Linear and Nonlinear Functional Analysis with Applications , 2013 .

[52]  Andrea Manzoni,et al.  A Comprehensive Deep Learning-Based Approach to Reduced Order Modeling of Nonlinear Time-Dependent Parametrized PDEs , 2020, Journal of Scientific Computing.

[53]  Annalisa Quaini,et al.  Symmetry breaking and preliminary results about a Hopf bifurcation for incompressible viscous flow in an expansion channel , 2016 .