메뉴 건너뛰기




Volumn 170, Issue 1, 2004, Pages 27-58

Modelling the dynamics of nonlinear partial differential equations using neural networks

Author keywords

Karhunen Loeve decomposition; Neural networks; One dimensional Kuramoto Sivashinsky equation; Two dimensional Navier Stokes equations

Indexed keywords

DATA ACQUISITION; MATHEMATICAL MODELS; NEURAL NETWORKS; NUMERICAL METHODS; PARAMETER ESTIMATION; PARTIAL DIFFERENTIAL EQUATIONS;

EID: 3042634773     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2003.12.045     Document Type: Article
Times cited : (43)

References (67)
  • 1
    • 0002826159 scopus 로고
    • Neural networks in control system
    • Antsaklis P. Neural networks in control system IEEE Control Systems Mag. 10 3 1990 3-5
    • (1990) IEEE Control Systems Mag. , vol.10 , Issue.3 , pp. 3-5
    • Antsaklis, P.1
  • 2
    • 0001188213 scopus 로고
    • Kuramoto-Sivashinsky dynamics on the center-unstable manifold
    • Armbruster D. Guckenheimer J. Holmes P.J. Kuramoto-Sivashinsky dynamics on the center-unstable manifold SIAM J. Appl. Math. 49 3 1989 676-691
    • (1989) SIAM J. Appl. Math. , vol.49 , Issue.3 , pp. 676-691
    • Armbruster, D.1    Guckenheimer, J.2    Holmes, P.J.3
  • 6
    • 0001208814 scopus 로고
    • Kolmogorov's seminar on selected problems of analysis
    • Arnold V.I. Meshalkin L.D. Kolmogorov's seminar on selected problems of analysis Usprkhi Mat. Nauk. 15 1960 247-250
    • (1960) Usprkhi Mat. Nauk. , vol.15 , pp. 247-250
    • Arnold, V.I.1    Meshalkin, L.D.2
  • 7
    • 0001108405 scopus 로고
    • Preserving symmetries in the proper orthogonal decomposition
    • Aubry N. Lian W-Y. Titi E.S. Preserving symmetries in the proper orthogonal decomposition SIAM J. Sci. Comput. 14 1993 483-505
    • (1993) SIAM J. Sci. Comput. , vol.14 , pp. 483-505
    • Aubry, N.1    Lian, W-Y.2    Titi, E.S.3
  • 8
    • 0003014302 scopus 로고
    • Attractors of partial differential equations and estimate of their dimension
    • Babin A.V. Vishik M.I. Attractors of partial differential equations and estimate of their dimension Uspeki Mat. Nauk. 38 1983 133-187
    • (1983) Uspeki Mat. Nauk. , vol.38 , pp. 133-187
    • Babin, A.V.1    Vishik, M.I.2
  • 10
    • 0009632228 scopus 로고
    • Galerkin projections and the proper orthogonal decomposition for equivariant equations
    • Berkooz G. Titi E.S. Galerkin projections and the proper orthogonal decomposition for equivariant equations Phys. Lett. A 174 1993 94-102
    • (1993) Phys. Lett. A , vol.174 , pp. 94-102
    • Berkooz, G.1    Titi, E.S.2
  • 11
    • 0024928281 scopus 로고
    • Speech dynamics and recurrent neural networks
    • Glasgow, UK
    • H. Bourlard, C.J. Wellekens, Speech dynamics and recurrent neural networks, Proceedings of ICASSP, Glasgow, UK, 1989, pp. 33-36.
    • (1989) Proceedings of ICASSP , pp. 33-36
    • Bourlard, H.1    Wellekens, C.J.2
  • 13
    • 0040927711 scopus 로고    scopus 로고
    • The Whitney reduction network: A method for computing autoassociative graphs
    • Broomhead D.S. Kirby M. The Whitney reduction network: a method for computing autoassociative graphs Neural Comput. 30 2001 2595-2616
    • (2001) Neural Comput. , vol.30 , pp. 2595-2616
    • Broomhead, D.S.1    Kirby, M.2
  • 16
    • 0003241758 scopus 로고
    • Integral manifold and inertial manifolds for dissipative partial differential equations
    • New York: Springer
    • Constantin P. Foias C. Nicolaenko B. Teman R. Integral manifold and inertial manifolds for dissipative partial differential equations, Applied Mathematical Sciences 1989 Springer New York
    • (1989) Applied Mathematical Sciences
    • Constantin, P.1    Foias, C.2    Nicolaenko, B.3    Teman, R.4
  • 17
    • 0023999086 scopus 로고
    • On the dimension of the attractors in two-dimensional turbulence
    • Constantin P. Foias C. Teman R. On the dimension of the attractors in two-dimensional turbulence Physica D 30 3 1988 284-296
    • (1988) Physica D , vol.30 , Issue.3 , pp. 284-296
    • Constantin, P.1    Foias, C.2    Teman, R.3
  • 19
    • 0025404590 scopus 로고
    • On the relations between modelling approaches for speech recognition
    • Ephrain Y. Rabiner L. On the relations between modelling approaches for speech recognition IEEE Trans. Acoust. Speech Signal Process. 36 1990 372-379
    • (1990) IEEE Trans. Acoust. Speech Signal Process , vol.36 , pp. 372-379
    • Ephrain, Y.1    Rabiner, L.2
  • 21
    • 0013326768 scopus 로고
    • On the Hausdorff dimension of an attractor for the two-dimensional Navier-Stokes equations
    • Foias C. Teman R. On the Hausdorff dimension of an attractor for the two-dimensional Navier-Stokes equations Phys. Lett. 93 9 1983 451-454
    • (1983) Phys. Lett. , vol.93 , Issue.9 , pp. 451-454
    • Foias, C.1    Teman, R.2
  • 24
    • 0041040359 scopus 로고    scopus 로고
    • Application of artificial neural network for estimating tight gas sand intrinsic permeability
    • Garrouch A.A. Smaoui N. Application of artificial neural network for estimating tight gas sand intrinsic permeability Energ. Fuel. 10 1996 1053-1059
    • (1996) Energ. Fuel. , vol.10 , pp. 1053-1059
    • Garrouch, A.A.1    Smaoui, N.2
  • 28
    • 0003304963 scopus 로고
    • Geometric theory of semilinear parabolic equations
    • New York: Springer
    • Henry D. Geometric theory of semilinear parabolic equations Lecture Notes in Mathematics Vol. 840 1981 Springer New York
    • (1981) Lecture Notes in Mathematics , vol.840
    • Henry, D.1
  • 29
    • 0021835689 scopus 로고
    • Neural computation of decisions optimization problems
    • Hopfield J.J. Tank D.W. Neural computation of decisions optimization problems Biol. Cybernet. 52 1985 141-144
    • (1985) Biol. Cybernet. , vol.52 , pp. 141-144
    • Hopfield, J.J.1    Tank, D.W.2
  • 30
    • 0001710505 scopus 로고
    • Analysis of a complex statistical variables into principal components
    • Hotelling H. Analysis of a complex statistical variables into principal components J. Ed. Psychol. 24 1933 498-520
    • (1933) J. Ed. Psychol. , vol.24 , pp. 498-520
    • Hotelling, H.1
  • 31
    • 0000618153 scopus 로고
    • Order and complexity in the Kuramoto-Sivashinsky model of weakly turbulent interface
    • Hyman J.M. Nicolaenko B. Zaleski S. Order and complexity in the Kuramoto-Sivashinsky model of weakly turbulent interface Physica D 23 1986 265-292
    • (1986) Physica D , vol.23 , pp. 265-292
    • Hyman, J.M.1    Nicolaenko, B.2    Zaleski, S.3
  • 33
    • 0002351604 scopus 로고
    • Bifurcation computations on an approximate inertial manifold for the 2-D Navier-Stokes equations
    • Jolly M.S. Bifurcation computations on an approximate inertial manifold for the 2-D Navier-Stokes equations Physica D 36 1993 8-20
    • (1993) Physica D , vol.36 , pp. 8-20
    • Jolly, M.S.1
  • 34
    • 34047228827 scopus 로고
    • Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: Analysis and computations
    • Jolly M.S. Kevrekidis I.G. Titi E.S. Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations Physica D 44 1990 38-60
    • (1990) Physica D , vol.44 , pp. 38-60
    • Jolly, M.S.1    Kevrekidis, I.G.2    Titi, E.S.3
  • 35
    • 0025448183 scopus 로고
    • Back in the saddle again: A computer assisted study of the Kuramoto-Sivashinsky equation
    • Kevrekidis I.G. Nicolaenko B. Scovel C. Back in the saddle again: a computer assisted study of the Kuramoto-Sivashinsky equation SIAM J. Appl. Math. 50 3 1990 760-790
    • (1990) SIAM J. Appl. Math. , vol.50 , Issue.3 , pp. 760-790
    • Kevrekidis, I.G.1    Nicolaenko, B.2    Scovel, C.3
  • 36
    • 0005906830 scopus 로고
    • The nonlinear reduction of high-dimensional dynamical systems via neural networks
    • Kirby M. Miranda R. The nonlinear reduction of high-dimensional dynamical systems via neural networks Phys. Rev. Lett. 72 12 1994 1822
    • (1994) Phys. Rev. Lett. , vol.72 , Issue.12 , pp. 1822
    • Kirby, M.1    Miranda, R.2
  • 37
    • 0024213260 scopus 로고
    • Associative neural network model for the generation of temporal patterns
    • Kleifeld D. Sompolinsky H. Associative neural network model for the generation of temporal patterns Biophysics 54 1988 1039-1051
    • (1988) Biophysics , vol.54 , pp. 1039-1051
    • Kleifeld, D.1    Sompolinsky, H.2
  • 38
    • 0000507179 scopus 로고
    • Persistent propagation of concentration waves in dissipative media far from thermal equilibrium
    • Kuramoto Y. Tsuzuki T. Persistent propagation of concentration waves in dissipative media far from thermal equilibrium Prog. Theoret. Phys. 55 1976 356-369
    • (1976) Prog. Theoret. Phys. , vol.55 , pp. 356-369
    • Kuramoto, Y.1    Tsuzuki, T.2
  • 39
    • 0000241853 scopus 로고
    • Deterministic nonperiodic flow
    • Lorenz E.N. Deterministic nonperiodic flow J. Atmos. Sci. 20 1963 130-141
    • (1963) J. Atmos. Sci. , vol.20 , pp. 130-141
    • Lorenz, E.N.1
  • 41
    • 0005750037 scopus 로고
    • Negatively invariant sets of compact maps and an extension of a theorem of Cartwright
    • Mallet-Paret J. Negatively invariant sets of compact maps and an extension of a theorem of Cartwright J. Differential Equations 22 1976 331
    • (1976) J. Differential Equations , vol.22 , pp. 331
    • Mallet-Paret, J.1
  • 42
    • 0003290175 scopus 로고
    • On the dimension of the compact invariant sets of certain nonlinear maps
    • New York: Springer
    • Mané R. On the dimension of the compact invariant sets of certain nonlinear maps Lecture Notes in Mathematics Vol. 898 1981 Springer New York
    • (1981) Lecture Notes in Mathematics , vol.898
    • Mané, R.1
  • 43
    • 0001883464 scopus 로고
    • An example of absence of turbulence for any Reynolds number
    • Marchioro C. An example of absence of turbulence for any Reynolds number Comm. Math. Phys. 105 1986 99-106
    • (1986) Comm. Math. Phys. , vol.105 , pp. 99-106
    • Marchioro, C.1
  • 44
    • 0027063388 scopus 로고
    • Discrete vs. continuous-time nonlinear signal processing of Cu electrodissolution data
    • Martinez R. Krisher R.K. Kevrekidis I.G. Kube M. Hudson J.L. Discrete vs. continuous-time nonlinear signal processing of Cu electrodissolution data Chem. Eng. Comm. 118 1992 25-48
    • (1992) Chem. Eng. Comm. , vol.118 , pp. 25-48
    • Martinez, R.1    Krisher, R.K.2    Kevrekidis, I.G.3    Kube, M.4    Hudson, J.L.5
  • 45
    • 0001491794 scopus 로고
    • Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous fluid
    • Meshalkin L.D. Sinal A.G. Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous fluid J. Appl. Math. Mech. 25 1961 1700-1705
    • (1961) J. Appl. Math. Mech. , vol.25 , pp. 1700-1705
    • Meshalkin, L.D.1    Sinal, A.G.2
  • 46
    • 0009576141 scopus 로고
    • Symmetry breaking homoclinic chaos in the Kolmogorov flows, in nonlinear world
    • V. G. Baryakhtar et al. (Ed.), Singapore: World Scientific Kiev, (1989)
    • Nicolaenko B. She Z.S. Symmetry breaking homoclinic chaos in the Kolmogorov flows, in nonlinear world Baryakhtar V.G., et-al. International workshop on Nonlinear and Turbulent Processes in Physics, Kiev, (1989) 1990 602-618 World Scientific Singapore
    • (1990) International Workshop on Nonlinear and Turbulent Processes in Physics , pp. 602-618
    • Nicolaenko, B.1    She, Z.S.2
  • 47
    • 0007454831 scopus 로고
    • Temporal intermittency and turbulence production in the Kolmogorov flow
    • H. K. Moffatt (Ed.), Cambridge, UK: Cambridge University press
    • Nicolaenko B. She Z.S. Temporal intermittency and turbulence production in the Kolmogorov flow Moffatt H.K. Topological Fluid Mechanics 1990 256-277 Cambridge University press Cambridge, UK
    • (1990) Topological Fluid Mechanics , pp. 256-277
    • Nicolaenko, B.1    She, Z.S.2
  • 48
    • 0009633753 scopus 로고
    • Symmetry breaking homoclinic chaos and vorticity bursts in periodic Navier-Stokes flows
    • Nicolaenko B. She Z.S. Symmetry breaking homoclinic chaos and vorticity bursts in periodic Navier-Stokes flows European J. Mech. B 10 1991 67-74
    • (1991) European J. Mech. B , vol.10 , pp. 67-74
    • Nicolaenko, B.1    She, Z.S.2
  • 50
    • 0001202597 scopus 로고
    • Learning state space trajectories in recurrent neural networks
    • Pearlmutter B.A. Learning state space trajectories in recurrent neural networks Neural Comput. 1 1989 263-269
    • (1989) Neural Comput. , vol.1 , pp. 263-269
    • Pearlmutter, B.A.1
  • 53
    • 0000806354 scopus 로고
    • Metastability and vortex pairing in the Kolmogorov flow
    • She Z.S. Metastability and vortex pairing in the Kolmogorov flow Phys. Lett. A 124 1987
    • (1987) Phys. Lett. A , vol.124
    • She, Z.S.1
  • 54
    • 0009576281 scopus 로고
    • Large-scale dynamical and transition to turbulence in the two-dimensional Kolmogorov flow
    • H. Branover, M. Mond, & Y. Unger (Eds.), Washington, DC: AIAA
    • She Z.S. Large-scale dynamical and transition to turbulence in the two-dimensional Kolmogorov flow Branover H. Mond M. Unger Y. Current Trends in Turbulence Research 1988 374-396 AIAA Washington, DC
    • (1988) Current Trends in Turbulence Research , pp. 374-396
    • She, Z.S.1
  • 55
    • 0000246863 scopus 로고
    • Turbulence and dynamics of coherent structures. I. coherent structures
    • Sirovich L. Turbulence and dynamics of coherent structures. I. coherent structures Quart. Appl. Math. 45 1987 561-571
    • (1987) Quart. Appl. Math. , vol.45 , pp. 561-571
    • Sirovich, L.1
  • 56
    • 0017554878 scopus 로고
    • Nonlinear analysis of hydrodynamic instability in laminar flames. I. derivation of basic equations
    • Sivashinsky G. Nonlinear analysis of hydrodynamic instability in laminar flames. I. derivation of basic equations Acta Astronaut. 4 1977 1177-1206
    • (1977) Acta Astronaut. , vol.4 , pp. 1177-1206
    • Sivashinsky, G.1
  • 57
    • 0031389364 scopus 로고    scopus 로고
    • Artificial neural network-based low-dimensional model for spatio-temporally varying cellular flames
    • Smaoui N. Artificial neural network-based low-dimensional model for spatio-temporally varying cellular flames Appl. Math. Modeling 21 1997 739-748
    • (1997) Appl. Math. Modeling , vol.21 , pp. 739-748
    • Smaoui, N.1
  • 58
    • 0033731429 scopus 로고    scopus 로고
    • An artificial neural network noise reduction method for chaotic attractors
    • Smaoui N. An artificial neural network noise reduction method for chaotic attractors Internat. J. Comput. Math. 73 2000 417-431
    • (2000) Internat. J. Comput. Math. , vol.73 , pp. 417-431
    • Smaoui, N.1
  • 59
    • 0036289618 scopus 로고    scopus 로고
    • A model for the unstable manifold of the bursting behavior in the 2d Navier-Stokes flow
    • Smaoui N. A model for the unstable manifold of the bursting behavior in the 2d Navier-Stokes flow SIAM J. Sci. Comput. 23 2001 824-840
    • (2001) SIAM J. Sci. Comput. , vol.23 , pp. 824-840
    • Smaoui, N.1
  • 60
    • 3042518384 scopus 로고    scopus 로고
    • Linear vs. nonlinear dimensionality reduction of high dimensional dynamical system
    • to appear
    • N. Smaoui, Linear vs. nonlinear dimensionality reduction of high dimensional dynamical system, SIAM J. Sci. Comput. (2004), to appear.
    • (2004) SIAM J. Sci. Comput.
    • Smaoui, N.1
  • 61
    • 0141906256 scopus 로고    scopus 로고
    • Analyzing the dynamics of cellular flames using Karhunen-Loéve decomposition and autoassociative neural networks
    • Smaoui N. Al-Yakoob S. Analyzing the dynamics of cellular flames using Karhunen-Loéve decomposition and autoassociative neural networks SIAM J. Sci. Comput. 24 5 2003 1790-1808
    • (2003) SIAM J. Sci. Comput. , vol.24 , Issue.5 , pp. 1790-1808
    • Smaoui, N.1    Al-Yakoob, S.2
  • 62
    • 0031215981 scopus 로고    scopus 로고
    • Symmetry and the Karhunen-Loéve analysis
    • Smaoui N. Armbruster D. Symmetry and the Karhunen-Loéve analysis SIAM J. Sci. Comput. 18 1997 1526-1532
    • (1997) SIAM J. Sci. Comput. , vol.18 , pp. 1526-1532
    • Smaoui, N.1    Armbruster, D.2
  • 63
    • 0031179927 scopus 로고    scopus 로고
    • A new approach combining Karhunen-Loéve analysis and artificial neural network for estimating gas sand permeability
    • Smaoui N. Garrouch A.A. A new approach combining Karhunen-Loéve analysis and artificial neural network for estimating gas sand permeability J. Petrol. Sci. Eng. 18 1997 101-112
    • (1997) J. Petrol. Sci. Eng. , vol.18 , pp. 101-112
    • Smaoui, N.1    Garrouch, A.A.2
  • 64
    • 0034086515 scopus 로고    scopus 로고
    • Using Karhunen-Loéve decommission and artificial neural network to model fluid flow in porous media
    • Smaoui N. Gharbi R. Using Karhunen-Loéve decommission and artificial neural network to model fluid flow in porous media Appl. Math. Modeling 24 2000 657-675
    • (2000) Appl. Math. Modeling , vol.24 , pp. 657-675
    • Smaoui, N.1    Gharbi, R.2
  • 65
    • 3042604028 scopus 로고    scopus 로고
    • Classifications of human faces using K-L decomposition and radial basis function neural networks
    • Smaoui N. Matar I.M. Classifications of human faces using K-L decomposition and radial basis function neural networks Internat. J. Comput. Math. 80 3 2003 325-345
    • (2003) Internat. J. Comput. Math. , vol.80 , Issue.3 , pp. 325-345
    • Smaoui, N.1    Matar, I.M.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.