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Reduced Order Modeling of Dynamical Systems Using Artificial Neural Networks Applied to Water Circulation

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 12321)


General circulation models are essential tools in weather and hydrodynamic simulation. They solve discretized, complex physical equations in order to compute evolutionary states of dynamical systems, such as the hydrodynamics of a lake. However, high-resolution numerical solutions using such models are extremely computational and time consuming, often requiring a high performance computing architecture to be executed satisfactorily. Machine learning (ML)-based low-dimensional surrogate models are a promising alternative to speed up these simulations without undermining the quality of predictions. In this work, we develop two examples of fast, reliable, low-dimensional surrogate models to produce a 36 h forecast of the depth-averaged hydrodynamics at Lake George NY, USA. Our ML approach uses two widespread artificial neural network (ANN) architectures: fully connected neural networks and long short-term memory. These ANN architectures are first validated in the deterministic and chaotic regimes of the Lorenz system and then combined with proper orthogonal decomposition (to reduce the dimensionality of the incoming input data) to emulate the depth-averaged hydrodynamics of a flow simulator called SUNTANS. Results show the ANN-based reduced order models have promising accuracy levels (within \(6\%\) of the prediction range) and advocate for further investigation into hydrodynamic applications.


  • Model reduction
  • Dynamical systems
  • Artificial neural networks
  • Water circulation

Supported by The Jefferson Project at Lake George, which is a collaboration of Rensselaer Polytechnic Institute, IBM, and The FUND for Lake George.

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  1. 1.

    Writing the correlation matrix as an integral over the domain \(\varOmega \) means that we are weighting each variable in the vector \(q'\) associated to a lattice point \(\mathbf {x}_{i}\) with the corresponding volume \(V_{i}\) of a fictitious control volume enclosing that lattice point. For fluctuation data \(q'\) collected from unstructured finite element-like grids, the enclosing volume of a point \(\mathbf {x}_{i}\) taken at the centroid of a mesh cell is the cell’s own volume.

  2. 2.

    Relative errors are not suited for the analysis because the time derivative of the reference solution has many values close to zero.

  3. 3.

    The LSTM minimization algorithm finds 264, 690 optimal weights and biases while FCNN requires only 55, 101.


  1. Benjamin, S.G., et al.: A north American hourly assimilation and model forecast cycle: the rapid refresh. Mon. Weather Rev. 144(4), 1669–1694 (2016).

    CrossRef  Google Scholar 

  2. Brunton, S.L., Noack, B.R., Koumoutsakos, P.: Machine learning for fluid mechanics. Annu. Rev. Fluid Mech. 52(1), 477–508 (2020).

    CrossRef  MATH  Google Scholar 

  3. Fringer, O., Gerritsen, M., Street, R.: An unstructured-grid, finite-volumne, nonhydrostatic, parallel coastal ocean simulator. Ocean Model. 14, 139–173 (2006)

    CrossRef  Google Scholar 

  4. Gonzalez, F., Balajewicz, M.: Deep convolutional recurrent autoencoders for learning low-dimensional feature dynamics of fluid systems (2018)

    Google Scholar 

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

  6. Karevan, Z., Suykens, J.: Transductive LSTM for time-series prediction: an application to weather forecasting. Neural Netw. (2020).

  7. Kim, B., Azevedo, V.C., Thuerey, N., Kim, T., Gross, M., Solenthaler, B.: Deep fluids: a generative network for parameterized fluid simulations. Comput. Graph. Forum (Proc. Eurographics) 38(2) (2019)

    Google Scholar 

  8. Lee, K., Carlberg, K.: Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders. J. Comput. Phys. (2019).

    CrossRef  Google Scholar 

  9. Lui, H.F.S.: Construction of reduced order models for fluid flows using deep neural networks. Master’s thesis, State University of Campinas (2019)

    Google Scholar 

  10. Miljanovic, M.: Comparative analysis of recurrent and finite impulse response neural networks in time series prediction. Indian J. Comput. Sci. Eng. 3 (2012).

  11. Mohan, A., Gaitonde, D.: A deep learning based approach to reduced order modeling for turbulent flow control using LSTM neural networks (2018)

    Google Scholar 

  12. Navrátil, J., King, A., Rios, J., Kollias, G., Torrado, R., Codas, A.: Accelerating physics-based simulations using end-to-end neural network proxies: an application in oil reservoir modeling. Front. Big Data 2 (2019).

  13. Nielsen, M.: Neural Networks and Deep Learning. Determination Press (2015).

  14. Sak, H., Senior, A., Beaufays, F.: Long short-term memory recurrent neural network architectures for large scale acoustic modeling (2014)

    Google Scholar 

  15. Schmidhuber, J.: Deep learning in neural networks: an overview. Neural Netw. 61, 85–117 (2015)

    Google Scholar 

  16. Skamarock, W.C., et al.: A description of the advanced research WRF version 3. NCAR technical note -475+STR (2008)

    Google Scholar 

  17. Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. In: Ghahramani, Z., Welling, M., Cortes, C., Lawrence, N.D., Weinberger, K.Q. (eds.) Advances in Neural Information Processing Systems 27, pp. 3104–3112. Curran Associates, Inc. (2014).

  18. Vlachas, P., Byeon, W., Wan, Z., Sapsis, T., Koumoutsakos, P.: Data-driven forecasting of high-dimensional chaotic systems with long short-term memory networks. Proc. Roy. Soc. A: Math. Phys. Eng. Sci. 474(2213), 20170844 (2018)

    CrossRef  MathSciNet  Google Scholar 

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

  20. Wang, J.X., Wu, J.L., Xiao, H.: Physics-informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on DNS data. Phys. Rev. Fluids 2, 034603 (2017).

  21. Wang, Q., Ripamonti, N., Hesthaven, J.: Recurrent neural network closure of parametric POD-Galerkin reduced-order models based on the Mori-Zwanzig formalism (2019)

    Google Scholar 

  22. Watson, C.D., et al.: The application of an internet of things cyber-infrastructure for the study of ecology of lake George in the Jefferson project (2018)

    Google Scholar 

  23. Wynne, T., et al.: Evolution of a cyanobacterial bloom forecast system in western lake Erie: development and initial evaluation. J. Great Lakes Res. 39, 90–99 (2013). Remote Sensing of the Great Lakes and Other Inland Waters

    Google Scholar 

  24. Xiang, Z., Yan, J., Demir, I.: A rainfall-runoff model with LSTM-based sequence-to-sequence learning. Water Resour. Rese. 56 (2020).

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Correspondence to Alberto Costa Nogueira Jr. .

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Costa Nogueira, A., de Sousa Almeida, J.L., Auger, G., Watson, C.D. (2020). Reduced Order Modeling of Dynamical Systems Using Artificial Neural Networks Applied to Water Circulation. In: Jagode, H., Anzt, H., Juckeland, G., Ltaief, H. (eds) High Performance Computing. ISC High Performance 2020. Lecture Notes in Computer Science(), vol 12321. Springer, Cham.

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