Skip to main content
SpringerLink
Log in

Operator Inference of Non-Markovian Terms for Learning Reduced Models from Partially Observed State Trajectories

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

This work introduces a non-intrusive model reduction approach for learning reduced models from partially observed state trajectories of high-dimensional dynamical systems. The proposed approach compensates for the loss of information due to the partially observed states by constructing non-Markovian reduced models that make future-state predictions based on a history of reduced states, in contrast to traditional Markovian reduced models that rely on the current reduced state alone to predict the next state. The core contributions of this work are a data sampling scheme to sample partially observed states from high-dimensional dynamical systems and a formulation of a regression problem to fit the non-Markovian reduced terms to the sampled states. Under certain conditions, the proposed approach recovers from data the very same non-Markovian terms that one obtains with intrusive methods that require the governing equations and discrete operators of the high-dimensional dynamical system. Numerical results demonstrate that the proposed approach leads to non-Markovian reduced models that are predictive far beyond the training regime. Additionally, in the numerical experiments, the proposed approach learns non-Markovian reduced models from trajectories with only 20% observed state components that are about as accurate as traditional Markovian reduced models fitted to trajectories with 99% observed components.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

Availability of data and material

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Code availability Code is available from the corresponding author on reasonable request. Code used in some simulations is also available at https://github.com/wayneisaacuy/OpInfPartialObs.

Notes

  1. https://github.com/wayneisaacuy/OpInfPartialObs

References

  1. Antoulas, A.C., Anderson, B.D.Q.: On the scalar rational interpolation problem. IMA J. Math. Control Inf. 3(2–3), 61–88 (1986)

    Article  MATH  Google Scholar 

  2. Antoulas, A.C., Gosea, I.V., Ionita, A.C.: Model reduction of bilinear systems in the Loewner framework. SIAM J. Sci. Comput. 38(5), B889–B916 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ascher, U.M., Ruuth, S.J., Spiteri, R.J.: Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25(2–3), 151–167 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beattie, C., Gugercin, S.: Interpolatory projection methods for structure-preserving model reduction. Syst. Control Lett. 58(3), 225–232 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Beattie, C., Gugercin, S.: Realization-independent \({\cal{H}}_2\)-approximation. In: Proceedings of the 36th IEEE Conference on Decision and Control, pp. 4953–4958, Maui, HI, USA (2012)

  6. Benner, P., Goyal, P., Kramer, B., Peherstorfer, B., Willcox. K.: Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms. Comput. Methods Appl. Mech. Eng. (accepted) (2020)

  7. Benner, P., Gugercin, S., Willcox, K.: A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev. 57(4), 483–531 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Billings, S.A.: Nonlinear system identification : NARMAX methods in the time, frequency, and spatio-temporal domains. Wiley, Chichester (2013)

    Book  MATH  Google Scholar 

  9. Brunton, S.L., Proctor, J.L., Kutz, J.N.: Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl. Acad. Sci. 113(15), 3932–3937 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Champion, K.P., Brunton, S.L., Kutz, J.N.: Discovery of nonlinear multiscale systems: Sampling strategies and embeddings. SIAM J. Appl. Dyn. Syst. 18(1), 312–333 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chorin, A., Hald, O.H.: Stochastic Tools in Mathematics and Science. Springer, New York (2009)

    Book  MATH  Google Scholar 

  12. Chorin, A., Stinis, P.: Problem reduction, renormalization, and memory. Commun. Appl. Math. Comput. Sci. 1(1), 1–27 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chorin, A.J., Hald, O.H., Kupferman, R.: Optimal prediction with memory. Physica D 166(3–4), 239–257 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Clainche, S.L., Vega, J.M.: Higher order dynamic mode decomposition. SIAM J. Appl. Dyn. Syst. 16(2), 882–925 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Degroote, J., Vierendeels, J., Willcox, K.: Interpolation among reduced-order matrices to obtain parameterized models for design, optimization and probabilistic analysis. Int. J. Numer. Meth. Fluids 63, 207–230 (2009)

    MathSciNet  MATH  Google Scholar 

  16. Drmač, Z., Gugercin, S., Beattie, C.: Vector fitting for matrix-valued rational approximation. SIAM J. Sci. Comput. 37(5), A2346–A2379 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Givon, D., Kupferman, R., Stuart, A.: Extracting macroscopic dynamics: model problems and algorithms. Nonlinearity 17(6), R55–R127 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gosea, I.V., Antoulas, A.C.: Data-driven model order reduction of quadratic-bilinear systems. Numer. Linear Algebra Appl. 25(6), e2200 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gugercin, S., Antoulas, A.C.: A survey of model reduction by balanced truncation and some new results. Int. J. Control 77(8), 748–766 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gustavsen, B., Semlyen, A.: Rational approximation of frequency domain responses by vector fitting. IEEE Trans. Power Delivery 14(3), 1052–1061 (1999)

    Article  Google Scholar 

  21. Hesthaven, J.S., Rozza, G., Stamm, B.: Certified Reduced Basis Methods for Parametrized Partial Differential Equations. Springer International Publishing, Berlin (2016)

    Book  MATH  Google Scholar 

  22. Hochreiter, S., Schmidhuber, J.: Long short-term memory. Neural Comput. 9(8), 1735–1780 (1997)

    Article  Google Scholar 

  23. Ionita, A.C., Antoulas, A.C.: Data-driven parametrized model reduction in the Loewner framework. SIAM J. Sci. Comput. 36(3), A984–A1007 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Juang, J.-N., Pappa, R.S.: An eigensystem realization algorithm for modal parameter identification and model reduction. J. Guid. Control. Dyn. 8(5), 620–627 (1985)

    Article  MATH  Google Scholar 

  25. Kramer, B., Gugercin, S.: The eigensystem realization algorithm from tangentially interpolated data. Mathematical and Computer Modelling of Dynamical Systems (2016) to appear

  26. Kutz, J.N., Brunton, S.L., Brunton, B.W., Proctor, J.L.: Dynamic mode decomposition: Data-driven modeling of complex systems. SIAM, New Delhi (2016)

    Book  MATH  Google Scholar 

  27. Lefteriu, S., Antoulas, A.: A new approach to modeling multiport systems from frequency-domain data. Computer-Aided Des. Integr. Circuits Syst. IEEE Trans. 29(1), 14–27 (2010)

    Article  Google Scholar 

  28. Lin, K.K., Lu, F.: Data-driven model reduction, Wiener projections, and the Koopman–Mori–Zwanzig formalism. J. Comput. Phys. 424, 109864 (2021)

    Article  MathSciNet  Google Scholar 

  29. Maulik, R., Mohan, A., Lusch, B., Madireddy, S., Balaprakash, P., Livescu, D.: Time-series learning of latent-space dynamics for reduced-order model closure. Physica D 405, 132368 (2020)

    Article  MathSciNet  Google Scholar 

  30. Mayo, A., Antoulas, A.: A framework for the solution of the generalized realization problem. Linear Algebra Appl. 425(2–3), 634–662 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. McQuarrie, S.A., Huang, C., Willcox, K.: Data-driven reduced-order models via regularized operator inference for a single-injector combustion process. arXiv e-prints, page arXiv:2008.02862 (2020)

  32. Mou, C., Liu, H., Wells, D.R., Iliescu, T.: Data-driven correction reduced order models for the quasi-geostrophic equations: a numerical investigation. Int. J. Comput. Fluid Dynam. 34(2), 147–159 (2020)

    Article  MathSciNet  Google Scholar 

  33. Pan, S., Duraisamy, K.: Data-driven discovery of closure models. SIAM J. Appl. Dyn. Syst. 17(4), 2381–2413 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  34. Peherstorfer, B.: Sampling low-dimensional Markovian dynamics for pre-asymptotically recovering reduced models from data with operator inference. SIAM J. Sci. Comput. 42(5), A3489–A3515 (2020)

    Article  MATH  Google Scholar 

  35. Peherstorfer, B., Willcox, K.: Dynamic data-driven reduced-order models. Comput. Methods Appl. Mech. Eng. 291, 21–41 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  36. Peherstorfer, B., Willcox, K.: Data-driven operator inference for non-intrusive projection-based model reduction. Comput. Methods Appl. Mech. Eng. 306, 196–215 (2016)

    Article  MATH  Google Scholar 

  37. Qian, E., Kramer, B., Peherstorfer, B., Willcox, K.: Lift & learn: Physics-informed machine learning for large-scale nonlinear dynamical systems. Physica D 406, 132401 (2020)

    Article  MathSciNet  Google Scholar 

  38. Quarteroni, A., Rozza, G., Manzoni, A.: Certified reduced basis approximation for parametrized partial differential equations and applications. J. Math. Ind. 1(1), 3 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. Rowley, C., Mezić, I., Bagheri, S., Schlatter, P., Henningson, D.: Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115–127 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  40. Rozza, G., Huynh, D., Patera, A.T.: Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Eng. 15(3), 1–47 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  41. Sagaut, P.: Large Eddy Simulation for Incompressible Flows : An Introduction. Springer-Verlag, Berlin New York (2006)

    MATH  Google Scholar 

  42. Schaeffer, H., Caflisch, R., Hauck, C.D., Osher, S.: Sparse dynamics for partial differential equations. Proc. Natl. Acad. Sci. 110(17), 6634–6639 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  43. Schaeffer, H., Tran, G., Ward, R.: Extracting sparse high-dimensional dynamics from limited data. SIAM J. Appl. Math. 78(6), 3279–3295 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  44. Schmid, P.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  45. Schmid, P., Sesterhenn, J.: Dynamic mode decomposition of numerical and experimental data. In: Bulletin of the American Physical Society, 61st APS meeting, page 208. American Physical Society, (2008)

  46. Schulze, P., Unger, B.: Data-driven interpolation of dynamical systems with delay. Syst. Control Lett. 97, 125–131 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  47. Schulze, P., Unger, B., Beattie, C., Gugercin, S.: Data-driven structured realization. Linear Algebra Appl. 537, 250–286 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  48. Shumway, R., Stoffer, D.S.: Time Series Analysis and Its Applications : With R Examples. Springer, New York (2011)

    Book  MATH  Google Scholar 

  49. Stinis, P.: Renormalized Mori–Zwanzig-reduced models for systems without scale separation. Proc. Royal Soc. A: Math. Phys. Eng. Sci. 471(2176), 20140446 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  50. Suárez, E., Adelman, J.L., Zuckerman, D.M.: Accurate estimation of protein folding and unfolding times: Beyond markov state models. J. Chem. Theory Comput. 12(8), 3473–3481 (2016)

    Article  Google Scholar 

  51. Swischuk, R., Kramer, B., Huang, C., Willcox, K.: Learning physics-based reduced-order models for a single-injector combustion process. AIAA J. 58(6), 2658–2672 (2020)

    Article  Google Scholar 

  52. Tu, J.H., Rowley, C.W., Luchtenburg, D.M., Brunton, S.L., Kutz, J.N.: On dynamic mode decomposition: Theory and applications. J. Comput. Dyn. 1(2), 391–421 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  53. Uy, W.I.T., Peherstorfer, B.: Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations. arXiv e-prints, page arXiv:2005.05890, (2020)

  54. Wan, Z.Y., Vlachas, P., Koumoutsakos, P., Sapsis, T.: Data-assisted reduced-order modeling of extreme events in complex dynamical systems. PLoS ONE 13(5), e0197704 (2018)

    Article  Google Scholar 

  55. Wang, J., Ferguson, A.L.: Nonlinear reconstruction of single-molecule free-energy surfaces from univariate time series. Phys. Rev. E 93(3), 1–28 (2016)

  56. Wang, J., Ferguson, A.L.: Recovery of protein folding funnels from single-molecule time series by delay embeddings and manifold learning. J. Phys. Chem. B 122(50), 11931–11952 (2018)

    Article  Google Scholar 

  57. Wang, Q., Ripamonti, N., Hesthaven, J.S.: Recurrent neural network closure of parametric POD-galerkin reduced-order models based on the Mori–Zwanzig formalism. J. Comput. Phys. 410, 109402 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  58. Wang, Z., Akhtar, I., Borggaard, J., Iliescu, T.: Proper orthogonal decomposition closure models for turbulent flows: A numerical comparison. Comput. Methods Appl. Mech. Eng. 237–240, 10–26 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  59. Williams, M.O., Kevrekidis, I.G., Rowley, C.W.: A data–driven approximation of the Koopman operator: Extending dynamic mode decomposition. J. Nonlinear Sci. 25(6), 1307–1346 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  60. Xie, X., Mohebujjaman, M., Rebholz, L.G., Iliescu, T.: Data-driven filtered reduced order modeling of fluid flows. SIAM J. Sci. Comput. 40(3), B834–B857 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  61. Zucchini, W.: Hidden Markov Models for Time Series : An Introduction Using R. CRC Press, Boca Raton, FL (2016)

    MATH  Google Scholar 

Download references

Funding

This work was partially supported by US Department of Energy, Office of Advanced Scientific Computing Research, Applied Mathematics Program (Program Manager Dr. Steven Lee), DOE Award DESC0019334, and by the National Science Foundation under Grant No. 1901091 and under Grant No. 1761068.

Author information

Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York, NY, 10012, USA

    Wayne Isaac Tan Uy & Benjamin Peherstorfer

Authors
  1. Wayne Isaac Tan Uy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Benjamin Peherstorfer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wayne Isaac Tan Uy.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was partially supported by US Department of Energy, Office of Advanced Scientific Computing Research, Applied Mathematics Program (Program Manager Dr. Steven Lee), DOE Award DESC0019334, and by the National Science Foundation under Grant No. 1901091 and under Grant No. 1761068.

Appendix A Error Analysis of the Non-Markovian Reduced Model

Appendix A Error Analysis of the Non-Markovian Reduced Model

We build on the analysis in Sect. 3.1.3 and demonstrate numerically that Markovian reduced models for linear autonomous full systems can achieve lower errors than the proposed non-Markovian reduced models (16). To motivate the numerical experiments that follow, consider a full model with dimension \(N = 2\) and with observed state of dimension \(r = 1\) and reduced dimension \(n=1\). A sufficient condition for which the proposed reduced model with non-Markovian term (16) yields a lower error than a Markovian reduced model is when \({\varvec{A}}_1\) is symmetric positive definite. To see this, observe that for positive integers l (\(l \in {\mathbb {Z}}^+\)), \({\tilde{{\varvec{A}}}}_1, {\varvec{E}}_l \in {\mathbb {R}}\) and that \({\tilde{{\varvec{A}}}}_1 >0\),

$$\begin{aligned} {\varvec{E}}_l = ({\varvec{Q}}^T {\varvec{A}}_1 {\varvec{Q}}^{\perp })^2 (({\varvec{Q}}^{\perp })^T {\varvec{A}}_1 {\varvec{Q}}^{\perp })^{l-1} > 0. \end{aligned}$$

Provided that \({\tilde{{\varvec{z}}}}_0 = {\tilde{{\varvec{z}}}}_0^{(0)}= {\tilde{{\varvec{z}}}}_0^{(L)}\), for fixed \(k \in {\mathbb {Z}}^+\), if \({\tilde{{\varvec{z}}}}_k^{(0)}\) and \({\tilde{{\varvec{z}}}}_k^{(L)}\) are expressed in terms of the initial condition \({\tilde{{\varvec{z}}}}_0\), algebraic calculations show that

$$\begin{aligned} \Vert {\tilde{{\varvec{z}}}}_k - {\tilde{{\varvec{z}}}}_k^{(L)}\Vert _2 \le \Vert {\tilde{{\varvec{z}}}}_k - {\tilde{{\varvec{z}}}}_k^{(0)}\Vert _2 \end{aligned}$$

since \({\tilde{{\varvec{A}}}}_1, {\varvec{E}}_l\) are positive for all \(l \in {\mathbb {Z}}^+.\) Therefore, since

$$\begin{aligned} \Vert {\varvec{z}}_k - {\varvec{V}}{\tilde{{\varvec{z}}}}_k^{(0)}\Vert _2&= \Vert {\varvec{z}}_k - {\varvec{V}}{\tilde{{\varvec{z}}}}_k\Vert _2 + \Vert {\varvec{V}}({\tilde{{\varvec{z}}}}_k - {\tilde{{\varvec{z}}}}_k^{(0)})\Vert _2, \\ \Vert {\varvec{z}}_k - {\varvec{V}}{\tilde{{\varvec{z}}}}_k^{(L)}\Vert _2&= \Vert {\varvec{z}}_k - {\varvec{V}}{\tilde{{\varvec{z}}}}_k\Vert _2 + \Vert {\varvec{V}}({\tilde{{\varvec{z}}}}_k - {\tilde{{\varvec{z}}}}_k^{(L)})\Vert _2, \end{aligned}$$

we conclude that

$$\begin{aligned} \Vert {\varvec{z}}_k - {\varvec{V}}{\tilde{{\varvec{z}}}}_k^{(L)}\Vert _2 \le \Vert {\varvec{z}}_k - {\varvec{V}}{\tilde{{\varvec{z}}}}_k^{(0)}\Vert _2, \end{aligned}$$

i.e., the reduced model with non-Markovian term achieves a lower error than its Markovian counterpart.

Fig. 17
figure 17

The Markovian model yields a more accurate approximation of the observed state dynamics at certain time points than the model with truncated non-Markovian term in this example

Full size image

However, the symmetric positive definiteness of the matrix \({\varvec{A}}_1\) is insufficient when \(N> 2, n > 1\). To see this, consider the following two examples with lag \(L=1\). A numerical implementation is available in PythonFootnote 1 which reproduces Fig. 17 below. We set \(N = 10, n=2\) and consider 30% observed state components for the first example while for the second, we use \(N = 50, n = 40\) and consider 95% observed state components. In both cases, the initial condition \({\tilde{{\varvec{z}}}}_0\) is chosen such that its components are realizations of independent standard normal random variables. The initial condition for the full system is then \({\varvec{x}}_0 = {\varvec{Q}}{\tilde{{\varvec{z}}}}_0\) so that \({\varvec{x}}_0\) satisfies \(({\varvec{Q}}^{\perp })^T {\varvec{x}}_0 = {\varvec{0}}_{N-n}\).

The symmetric positive definite matrix \({\varvec{A}}_1\) is constructed as follows. Its eigenvalues are sampled from a uniform distribution on (0, 1) to ensure that the system is stable. Its orthonormal eigenvectors are then chosen to be the eigenvectors of \(({\varvec{R}}+ {\varvec{R}}^T)/2\) where \({\varvec{R}}^{N \times N}\) is a matrix whose entries are independently sampled from a uniform distribution on (0, 10). The components with indices 1,6,10 of the full state are observed in the first example with the initial condition and basis and system matrices given by

$$\begin{aligned} {\varvec{x}}_0&= \begin{bmatrix} -0.5960&0&0&0&0&1.0333&0&0&0&0.8346 \end{bmatrix}^T, \\ {\varvec{V}}&= \begin{bmatrix} -0.9889 &{} 0.0294\\ 0.0767 &{} -0.7374\\ -0.1269 &{} -0.6748 \end{bmatrix},\\ {\varvec{V}}^{\perp }&= \begin{bmatrix} -0.1453\\ -0.6710\\ 0.7270\\ \end{bmatrix},\\ {\varvec{A}}_1&= \begin{bmatrix} 0.3603 &{} 0.0184 &{} -0.2192 &{} 0.0435 &{} -0.1624 &{} -0.0602 &{} 0.0758 &{} -0.0872 &{} 0.0634 &{} -0.0252\\ 0.0184 &{} 0.2907 &{} -0.1049 &{} 0.1334 &{} 0.0087 &{} 0.0951 &{} -0.0594 &{} -0.0602 &{} -0.0717 &{} 0.1366\\ -0.2192 &{} -0.1049 &{} 0.2978 &{} -0.1695 &{} 0.0887 &{} 0.0648 &{} -0.0924 &{} 0.0624 &{} -0.0213 &{} 0.0079\\ 0.0435 &{} 0.1334 &{} -0.1695 &{} 0.3700 &{} 0.0529 &{} -0.0074 &{} 0.1284 &{} 0.0196 &{} -0.0115 &{} 0.0273\\ -0.1624 &{} 0.0087 &{} 0.0887 &{} 0.0529 &{} 0.4582 &{} 0.0913 &{} 0.1194 &{} -0.0375 &{} 0.0449 &{} 0.1615 \\ -0.0602 &{} 0.0951 &{} 0.0648 &{} -0.0074 &{} 0.0913 &{} 0.4311 &{} -0.0781 &{} -0.0263 &{} 0.2070 &{} 0.1714\\ 0.0758 &{} -0.0594 &{} -0.0924 &{} 0.1284 &{} 0.1194 &{} -0.0781 &{} 0.3804 &{} 0.0296 &{} 0.1548 &{} -0.1197\\ -0.0872 &{} -0.0602 &{} 0.0624 &{} 0.0196 &{} -0.0375 &{} -0.0263 &{} 0.0296 &{} 0.3470 &{} 0.1123 &{} -0.1761\\ 0.0634 &{} -0.0717 &{} -0.0213 &{} -0.0115 &{} 0.0449 &{} 0.2070 &{} 0.1548 &{} 0.1123 &{} 0.5707 &{} -0.1059\\ -0.0252 &{} 0.1366 &{} 0.0079 &{} 0.0273 &{} 0.1615 &{} 0.1714 &{} -0.1197 &{} -0.1761 &{} -0.1059 &{} 0.3255 \end{bmatrix}. \end{aligned}$$

The details of the second example are provided in the repository\(^{1}\).

Figure 17 shows the difference in the relative error

$$\begin{aligned} \frac{1}{\Vert {\varvec{z}}_k\Vert _2} (\Vert {\varvec{z}}_k - {\varvec{V}}{\tilde{{\varvec{z}}}}_k^{(0)}\Vert _2 - \Vert {\varvec{z}}_k - {\varvec{V}}{\tilde{{\varvec{z}}}}_k^{(L)}\Vert _2) \end{aligned}$$

against the time step k. At certain time instances, the Markovian reduced model has a smaller error (negative values on the y-axis) than the model with non-Markovian term of lag \(L=1\). Thus, the conclusion we derived for \(N = 2,n=1\) does not generalize and these examples show that it is possible that the Markovian model gives a more accurate approximation than the truncated non-Markovian model even if the matrix \({\varvec{A}}_1\) is symmetric positive definite. A more rigorous analysis is warranted but is beyond the scope of this work.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Uy, W.I.T., Peherstorfer, B. Operator Inference of Non-Markovian Terms for Learning Reduced Models from Partially Observed State Trajectories. J Sci Comput 88, 91 (2021). https://doi.org/10.1007/s10915-021-01580-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-021-01580-2

Keywords

Search

Navigation