Data-driven model reduction, Wiener projections, and the Koopman-Mori-Zwanzig formalism

[1]  George Em Karniadakis,et al.  Incorporation of memory effects in coarse-grained modeling via the Mori-Zwanzig formalism. , 2015, The Journal of chemical physics.

[2]  Lai-Sang Young,et al.  Entropy formula for random transformations , 1988 .

[3]  Ilan Koren,et al.  Efficient reduction for diagnosing Hopf bifurcation in delay differential systems: Applications to cloud-rain models. , 2020, Chaos.

[4]  Eric Darve,et al.  Computing the non-Markovian coarse-grained interactions derived from the Mori-Zwanzig formalism in molecular systems: Application to polymer melts. , 2017, The Journal of chemical physics.

[5]  M. Powell.  The BOBYQA algorithm for bound constrained optimization without derivatives , 2009 .

[6]  Alan Edelman,et al.  Julia: A Fresh Approach to Numerical Computing , 2014, SIAM Rev..

[7]  I. Kevrekidis,et al.  Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations , 1990 .

[8]  Clarence W. Rowley,et al.  A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition , 2014, Journal of Nonlinear Science.

[9]  Robert P. Gilbert,et al.  Closure method for spatially averaged dynamics of particle chains , 2010, 1010.4832.

[10]  E. J. Hannan,et al.  Multiple time series , 1970 .

[11]  Dieter Forster,et al.  Hydrodynamic fluctuations, broken symmetry, and correlation functions , 1975 .

[12]  Alexandre J. Chorin,et al.  Data-based stochastic model reduction for the Kuramoto--Sivashinsky equation , 2015, 1509.09279.

[13]  H. Grabert,et al.  Projection Operator Techniques in Nonequilibrium Statistical Mechanics , 1982 .

[14]  D. Giannakis,et al.  Nonparametric forecasting of low-dimensional dynamical systems. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  Michael Ghil,et al.  Data-driven non-Markovian closure models , 2014, 1411.4700.

[16]  Giovanni Samaey,et al.  Equation-free multiscale computation: algorithms and applications. , 2009, Annual review of physical chemistry.

[17]  Nathan A. Baker,et al.  Data-driven parameterization of the generalized Langevin equation , 2016, Proceedings of the National Academy of Sciences.

[18]  Dmitri Kondrashov,et al.  Data-adaptive harmonic spectra and multilayer Stuart-Landau models. , 2017, Chaos.

[19]  E Weinan,et al.  Model Reduction with Memory and the Machine Learning of Dynamical Systems , 2018, Communications in Computational Physics.

[20]  Lina Ma,et al.  Coarse-graining Langevin dynamics using reduced-order techniques , 2018, J. Comput. Phys..

[21]  T. Kailath.  Lectures on Wiener and Kalman Filtering , 2003 .

[22]  Alexandre J. Chorin,et al.  Comparison of continuous and discrete-time data-based modeling for hypoelliptic systems , 2016, 1605.02273.

[23]  Shu Wang,et al.  Implicit-solvent coarse-grained modeling for polymer solutions via Mori-Zwanzig formalism. , 2019, Soft matter.

[24]  Traian Iliescu,et al.  Data-Driven Filtered Reduced Order Modeling of Fluid Flows , 2017, SIAM J. Sci. Comput..

[25]  P. Walters.  Introduction to Ergodic Theory , 1977 .

[26]  Wei Wang,et al.  Effective Dynamics of Stochastic Partial Differential Equations , 2014 .

[27]  A. Yaglom.  Correlation Theory of Stationary and Related Random Functions I: Basic Results , 1987 .

[28]  William H. Press,et al.  Numerical recipes , 1990 .

[29]  L. Arnold.  Random Dynamical Systems , 2003 .

[30]  Mtw,et al.  Stochastic flows and stochastic differential equations , 1990 .

[31]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[32]  Y. Kifer.  Ergodic theory of random transformations , 1986 .

[33]  Peter H. Baxendale,et al.  The Lyapunov spectrum of a stochastic flow of diffeomorphisms , 1986 .

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

[35]  John Harlim,et al.  Parametric reduced models for the nonlinear Schrödinger equation. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[36]  K. Duraisamy,et al.  Non-Markovian Closure Models for Large Eddy Simulations using the Mori-Zwanzig Formalism , 2016, 1611.03311.

[37]  Jaideep Pathak,et al.  Model-Free Prediction of Large Spatiotemporally Chaotic Systems from Data: A Reservoir Computing Approach. , 2018, Physical review letters.

[38]  A. Chorin,et al.  Stochastic Tools in Mathematics and Science , 2005 .

[39]  Timothy D. Sauer,et al.  Time-Scale Separation from Diffusion-Mapped Delay Coordinates , 2013, SIAM J. Appl. Dyn. Syst..

[40]  Jianqing Fan,et al.  Nonlinear Time Series : Nonparametric and Parametric Methods , 2005 .

[41]  E Weinan,et al.  Invariant measures for Burgers equation with stochastic forcing , 2000, math/0005306.

[42]  Panagiotis Stinis.  Stochastic Optimal Prediction for the Kuramoto-Sivashinsky Equation , 2004, Multiscale Model. Simul..

[43]  A. Yaglom,et al.  An Introduction to the Theory of Stationary Random Functions , 1963 .

[44]  Steven L. Brunton,et al.  Dynamic mode decomposition - data-driven modeling of complex systems , 2016 .

[45]  Grigorios A. Pavliotis,et al.  Multiscale Methods: Averaging and Homogenization , 2008 .

[46]  Richard A. Davis,et al.  Introduction to time series and forecasting , 1998 .

[47]  D. Giannakis,et al.  Bridging Data Science and Dynamical Systems Theory , 2020, 2002.07928.

[48]  E Weinan,et al.  The heterogeneous multiscale method* , 2012, Acta Numerica.

[49]  Brian A. Freno,et al.  Machine-learning error models for approximate solutions to parameterized systems of nonlinear equations , 2018, Computer Methods in Applied Mechanics and Engineering.

[50]  H. L. Dryden,et al.  Investigations on the Theory of the Brownian Movement , 1957 .

[51]  Anthony J. Roberts,et al.  Resolution of subgrid microscale interactions enhances the discretisation of nonautonomous partial differential equations , 2013, Appl. Math. Comput..

[52]  Eric Darve,et al.  Computing generalized Langevin equations and generalized Fokker–Planck equations , 2009, Proceedings of the National Academy of Sciences.

[53]  S. Cox,et al.  Exponential Time Differencing for Stiff Systems , 2002 .

[54]  I. Mezić.  Spectral Properties of Dynamical Systems, Model Reduction and Decompositions , 2005 .

[55]  Alexandre J. Chorin,et al.  Optimal prediction with memory , 2002 .

[56]  Andrey Gavrilov,et al.  Principal nonlinear dynamical modes of climate variability , 2015, Scientific Reports.

[57]  Juan M. Restrepo,et al.  Dimension Reduction for Systems with Slow Relaxation , 2016, 1609.09222.

[58]  Shixiao W. Jiang,et al.  Modeling of missing dynamical systems: deriving parametric models using a nonparametric framework , 2019, 1905.08082.

[59]  S. Billings.  Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains , 2013 .

[60]  Hao Wu,et al.  Data-Driven Model Reduction and Transfer Operator Approximation , 2017, J. Nonlinear Sci..

[61]  Chris Cameron.  Relative efficiency of Gaussian stochastic process sampling procedures , 2003 .

[62]  Norbert Wiener,et al.  Extrapolation, Interpolation, and Smoothing of Stationary Time Series, with Engineering Applications , 1949 .

[63]  G. P. King,et al.  Extracting qualitative dynamics from experimental data , 1986 .

[64]  J. Hyman,et al.  The Kuramoto-Sivashinsky equation: a bridge between PDE's and dynamical systems , 1986 .

[65]  G. Froyland,et al.  Almost-invariant sets and invariant manifolds — Connecting probabilistic and geometric descriptions of coherent structures in flows , 2009 .

[66]  Steven L. Brunton,et al.  On dynamic mode decomposition: Theory and applications , 2013, 1312.0041.

[67]  P. Schmid,et al.  Dynamic mode decomposition of numerical and experimental data , 2008, Journal of Fluid Mechanics.

[68]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[69]  Eric Vanden-Eijnden,et al.  Subgrid-Scale Parameterization with Conditional Markov Chains , 2008 .

[70]  Lloyd N. Trefethen,et al.  Fourth-Order Time-Stepping for Stiff PDEs , 2005, SIAM J. Sci. Comput..

[71]  R. Zwanzig.  Nonequilibrium statistical mechanics , 2001, Physics Subject Headings (PhySH).

[72]  James D. Hamilton.  Time Series Analysis , 1994 .

[73]  Heyrim Cho,et al.  Statistical analysis and simulation of random shocks in stochastic Burgers equation , 2014, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[74]  William H. Press,et al.  Numerical Recipes 3rd Edition: The Art of Scientific Computing , 2007 .

[75]  Alexandre J. Chorin,et al.  Discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics , 2015, Proceedings of the National Academy of Sciences.

[76]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .