Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
169 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Data-driven reduced order models using invariant foliations, manifolds and autoencoders (2206.12269v3)

Published 24 Jun 2022 in math.DS, cs.LG, and physics.data-an

Abstract: This paper explores how to identify a reduced order model (ROM) from a physical system. A ROM captures an invariant subset of the observed dynamics. We find that there are four ways a physical system can be related to a mathematical model: invariant foliations, invariant manifolds, autoencoders and equation-free models. Identification of invariant manifolds and equation-free models require closed-loop manipulation of the system. Invariant foliations and autoencoders can also use off-line data. Only invariant foliations and invariant manifolds can identify ROMs, the rest identify complete models. Therefore, the common case of identifying a ROM from existing data can only be achieved using invariant foliations. Finding an invariant foliation requires approximating high-dimensional functions. For function approximation, we use polynomials with compressed tensor coefficients, whose complexity increases linearly with increasing dimensions. An invariant manifold can also be found as the fixed leaf of a foliation. This only requires us to resolve the foliation in a small neighbourhood of the invariant manifold, which greatly simplifies the process. Combining an invariant foliation with the corresponding invariant manifold provides an accurate ROM. We analyse the ROM in case of a focus type equilibrium, typical in mechanical systems. The nonlinear coordinate system defined by the invariant foliation or the invariant manifold distorts instantaneous frequencies and damping ratios, which we correct. Through examples we illustrate the calculation of invariant foliations and manifolds, and at the same time show that Koopman eigenfunctions and autoencoders fail to capture accurate ROMs under the same conditions.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (64)
  1. Optimization Algorithms on Matrix Manifolds. Princeton University Press, 2009.
  2. P. A. Absil and J. Malick. Projection-like retractions on matrix manifolds. SIAM Journal on Optimization, 22(1):135–158, 2012.
  3. D. A. W. Barton. Control-based continuation: Bifurcation and stability analysis for physical experiments. Mechanical Systems and Signal Processing, 84:54–64, 2017.
  4. R. E. Bellman. Adaptive control processes. Princeton university press, 2015.
  5. R. Bergmann. Manopt.jl: Optimization on manifolds in Julia. Journal of Open Source Software, 7(70):3866, 2022.
  6. W.-J. Beyn and V. Thümmler. Phase Conditions, Symmetries and PDE Continuation, pages 301–330. Springer Netherlands, Dordrecht, 2007.
  7. S.A. Billings. Nonlinear System Identification: "NARMAX" Methods in the Time, Frequency, and Spatio-Temporal Domains. Wiley, 2013.
  8. N. Boumal. An introduction to optimization on smooth manifolds. To appear with Cambridge University Press, Mar 2022.
  9. S. Boyd and L. Vandenberghe. Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares. Cambridge University Press, 2018.
  10. T. Breunung and G. Haller. Explicit backbone curves from spectral submanifolds of forced-damped nonlinear mechanical systems. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 474(2213), 2018.
  11. Extracting qualitative dynamics from experimental data. Physica D: Nonlinear Phenomena, 20(2):217–236, 1986.
  12. Compressive sensing and low-rank libraries for classification of bifurcation regimes in nonlinear dynamical systems. SIAM Journal on Applied Dynamical Systems, 13(4):1716–1732, 2014.
  13. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences of the United States of America, 113(15):3932–3937, 2016.
  14. The parameterization method for invariant manifolds I: Manifolds associated to non-resonant subspaces. Indiana Univ. Math. J., 52:283–328, 2003.
  15. The parameterization method for invariant manifolds iii: overview and applications. Journal of Differential Equations, 218(2):444–515, 2005.
  16. M. Casdagli. Nonlinear prediction of chaotic time series. Physica D: Nonlinear Phenomena, 35(3):335–356, 1989.
  17. Data-driven modeling and prediction of non-linearizable dynamics via spectral submanifolds. Nat Commun, 13(872), 2022.
  18. Data-driven discovery of coordinates and governing equations. Proceedings of the National Academy of Sciences of the United States of America, 116(45):22445–22451, 2019.
  19. R. R. Coifman and S. Lafon. Diffusion maps. Applied and Computational Harmonic Analysis, 21(1):5–30, 2006.
  20. Trust Region Methods. MPS-SIAM Series on Optimization. SIAM, 2000.
  21. R. de la Llave. Invariant manifolds associated to nonresonant spectral subspaces. Journal of Statistical Physics, 87(1):211–249, 1997.
  22. D.L. Donoho. Compressed sensing. IEEE Transactions on Information Theory, 52(4):1289–1306, 2006.
  23. Measurement of nonlinear normal modes using multi-harmonic stepped force appropriation and free decay. Mechanical Systems and Signal Processing, 76-77:612 – 633, 2016.
  24. Deep neural network approximation theory. IEEE Transactions on Information Theory, 67(5):2581–2623, 2021.
  25. N. Fenichel. Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J., 21:193–226, 1972.
  26. L. Grasedyck. Hierarchical singular value decomposition of tensors. SIAM Journal on Matrix Analysis and Applications, 31(4):2029–2054, 2010.
  27. A literature survey of low-rank tensor approximation techniques. GAMM-Mitteilungen, 36(1):53–78, 2013.
  28. W. Hackbusch and S. Kühn. A new scheme for the tensor representation. Journal of Fourier Analysis and Applications, 15:706–722, 2009.
  29. G. Haller and S. Ponsioen. Nonlinear normal modes and spectral submanifolds: existence, uniqueness and use in model reduction. Nonlinear Dynamics, 86(3):1493–1534, 2016.
  30. R. Hermann and A. Krener. Nonlinear controllability and observability. IEEE Transactions on Automatic Control, 22(5):728–740, 1977.
  31. K. Hornik. Approximation capabilities of multilayer feedforward networks. Neural Networks, 4(2):251 – 257, 1991.
  32. Identification of instantaneous frequency and damping from transient decay data. Journal of Vibration and Acoustics, Transactions of the ASME, 142(5):051111, 2020.
  33. Learning normal form autoencoders for data-driven discovery of universal,parameter-dependent governing equations, 2021.
  34. Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system-level analysis. Communications in Mathematical Sciences, 1(4):715 – 762, 2003.
  35. Equation-free multiscale computation: Algorithms and applications. Annual Review of Physical Chemistry, 60(1):321–344, 2009.
  36. M. A. Kramer. Nonlinear principal component analysis using autoassociative neural networks. AIChE Journal, 37(2):233–243, 1991.
  37. S. Lang. Fundamentals of Differential Geometry. Graduate Texts in Mathematics. Springer New York, 2012.
  38. H. Blaine Lawson, Jr. Foliations. Bull. Amer. Math. Soc., 80:369–418, 1974.
  39. I. Mezić. Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dynamics, 41(1-3):309–325, 2005.
  40. I. Mezić. Koopman operator, geometry, and learning of dynamical systems. Notices of the American Mathematical Society, 68(7):1087–1105, 2021.
  41. Coordinate descent converges faster with the gauss-southwell rule than random selection. In Proceedings of the 32nd International Conference on International Conference on Machine Learning - Volume 37, ICML’15, page 1632–1641, 2015.
  42. Neural Networks: Tricks of the Trade. Lecture Notes in Computer Science. Springer Berlin Heidelberg, 2003.
  43. Iterative Solution of Nonlinear Equations in Several Variables. Classics in Applied Mathematics. SIAM, 1970.
  44. I. V. Oseledets. Tensor-train decomposition. SIAM J. Sci. Comput., 33(5):2295–2317, 2011.
  45. A unified approach to attractor reconstruction. Chaos: An Interdisciplinary Journal of Nonlinear Science, 17(1):013110, 2007.
  46. Matrix product state representations. Quantum Info. Comput., 7(5):401–430, 2007.
  47. Topological properties of the set of functions generated by neural networks of fixed size. Foundations of Computational Mathematics, 21(2):375–444, 2021.
  48. Automated computation of autonomous spectral submanifolds for nonlinear modal analysis. Journal of Sound and Vibration, 420:269 – 295, 2018.
  49. Application of nonlinear dynamic analysis to the identification and control of nonlinear systems - iii. n-dimensional systems. Journal of Process Control, 8(1):35–46, 1998.
  50. A. J. Roberts. Appropriate initial conditions for asymptotic descriptions of the long-term evolution of dynamical systems. J. Austral. Math. Soc. Ser. B, 31:48–75, 1989.
  51. A. J. Roberts. Boundary conditions for approximate differential equations. J. Austral. Math. Soc. Ser. B, 34:54–80, 1992.
  52. S. W. Shaw and C Pierre. Normal-modes of vibration for nonlinear continuous systems. J. Sound Vibr., 169(3):319–347, 1994.
  53. J. Sieber and B. Krauskopf. Control based bifurcation analysis for experiments. Nonlinear Dynamics, 51:365–377, 2008.
  54. G. Strang and T. Nguyen. Wavelets and Filter Banks. Wellesley-Cambridge Press, 1996.
  55. R. Szalai. Invariant spectral foliations with applications to model order reduction and synthesis. Nonlinear Dynamics, 101(4):2645–2669, 2020.
  56. Nonlinear model identification and spectral submanifolds for multi-degree-of-freedom mechanical vibrations. Proc. R. Soc. A, 473:20160759, 2017.
  57. F. Takens. Detecting strange attractors in turbulence. In D. Rand and Lai-Sang Young, editors, Dynamical Systems and Turbulence, Warwick 1980, pages 366–381. Springer Berlin Heidelberg, 1981.
  58. A global geometric framework for nonlinear dimensionality reduction. Science, 290(5500):2319–2323, 2000.
  59. Impact hammer-based analysis of nonlinear effects in bolted lap joint. In Proceedings of ISMA2016 including USD2016, ISMA2016, page 789–802, 2016.
  60. A. Uschmajew and B. Vandereycken. The geometry of algorithms using hierarchical tensors. Linear Algebra and its Applications, 439(1):133–166, 2013.
  61. Direct computation of nonlinear mapping via normal form for reduced-order models of finite element nonlinear structures. Computer Methods in Applied Mechanics and Engineering, 384:113957, 2021.
  62. H. Whitney. Differentiable manifolds. Annals of Mathematics, 37(3):645–680, 1936.
  63. No free lunch theorems for optimization. IEEE Transactions on Evolutionary Computation, 1(1):67–82, 1997.
  64. Reconstruction of normal forms by learning informed observation geometries from data. Proceedings of the National Academy of Sciences of the United States of America, 114(38):E7865–E7874, 2017.
Citations (7)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now