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A Deep Learning based Approach to Reduced Order Modeling for Turbulent Flow Control using LSTM Neural Networks

Arvind T. Mohan, Datta V. Gaitonde

arXiv:1804.09269v1physics.comp-phphysics.flu-dyn

TL;DR

High-fidelity turbulence simulations provide insight but make direct modeling computationally and data intensive, motivating reduced-order models for flow control. This paper replaces the usual Galerkin-based temporal modeling with LSTM networks operating on POD coefficients, evaluates LSTM and BiLSTM variants, and uses the Hurst exponent to study memory. The authors report that LSTM shows potential for turbulence dynamics, while BiLSTM performs worse and long-horizon accuracy is constrained by dynamical-system divergence.

  • Problem

    The paper addresses how to model turbulent-flow dynamics and retain useful flow-control capability while avoiding the computational and storage costs of high-fidelity LES/DNS.

  • Method

    The paper builds a non-Galerkin ROM that uses POD temporal coefficients as inputs to LSTM or BiLSTM models, and applies the Hurst exponent to study sequence memory.

  • Results

    LSTM showed potential for modeling turbulence temporal dynamics; BiLSTM consistently performed worse than traditional LSTM, while accuracy deteriorated with increasing prediction horizon.

  • Takeaways & Limitations

    The LSTM-ROM approach is suited to specialized flow-control applications requiring narrow regime generalization and low-cost inference on resource-constrained hardware.

  • Takeaways & Limitations

    Pure data-driven long-horizon prediction is limited because trajectories diverge over time, causing the likelihood of accurate predictions to drop exponentially.

Abstract

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Reduced Order Modeling (ROM) for engineering applications has been a major research focus in the past few decades due to the unprecedented physical insight into turbulence offered by high-fidelity CFD. The primary goal of a ROM is to model the key physics/features of a flow-field without computing the full Navier-Stokes (NS) equations. This is accomplished by projecting the high-dimensional dynamics to a low-dimensional subspace, typically utilizing dimensionality reduction techniques like Proper Orthogonal Decomposition (POD), coupled with Galerkin projection. In this work, we demonstrate a deep learning based approach to build a ROM using the POD basis of canonical DNS datasets, for turbulent flow control applications. We find that a type of Recurrent Neural Network, the Long Short Term Memory (LSTM) which has been primarily utilized for problems like speech modeling and language translation, shows attractive potential in modeling temporal dynamics of turbulence. Additionally, we introduce the Hurst Exponent as a tool to study LSTM behavior for non-stationary data, and uncover useful characteristics that may aid ROM development for a variety of applications.

1 Introduction

Reduced Order Models represent turbulent-flow dynamics in a low-dimensional space to preserve key features while reducing the computational and data costs of high-fidelity simulations. This paper explores deep learning, especially LSTM networks, as a non-Galerkin alternative for turbulence ROMs and flow-control applications.

  • Reduced Order Modeling: ROMs model key turbulent-flow dynamics and coherent features while compressing LES/DNS data.Their low computational cost makes them suitable for designing flow-control logic for actuators.
  • Reduced Order Modeling: POD is widely used to extract mathematically optimal reduced bases from high-fidelity flow data.The reduced representation targets dominant spatio-temporal dynamics computed by the Navier–Stokes equations.
  • Deep Learning Approach: Deep learning offers a non-Galerkin approach that learns dataset patterns and predicts outputs from input variables.The paper motivates this approach because reduced-order modeling is inherently data-driven.
  • Motivation for LSTM: LSTMs are explored because they model sequential data, explicitly accommodate memory, and avoid the vanishing-gradient problem affecting traditional RNNs.Turbulent-flow data may exhibit memory effects that accurate predictive models must accommodate.
  • Motivation for LSTM: The work investigates LSTM feasibility for building reduced-order models of complex turbulent systems.The stated application is turbulent-flow control.

2 Long Short-Term Memory Neural Networks

LSTMs extend recurrent networks with memory and gated information flow, making them suitable for predicting ordered temporal sequences. The paper describes their cell architecture, compares traditional and bidirectional variants, and motivates their use for POD-coefficient prediction in turbulence ROMs.

  • Sequence Modeling: Sequence prediction requires preserving observation order, so RNNs are designed to approximate functions with sequence as an additional dimension.POD temporal-coefficient prediction is framed as a sequence-modeling problem.
  • LSTM Networks: LSTMs use internal memory and recent context to learn temporal dependence while overcoming traditional RNN vanishing-gradient instability.Predictions condition on sequence context rather than only the current input.
  • LSTM Cell Architecture: An LSTM cell regulates information through input, forget, and output gates that selectively add, remove, or transmit information.These gates control the flow of training information between recurrent cells.
  • LSTM Cell Architecture: Selective information control and the forget gate help prevent vanishing gradients and reduce over-fitting by discarding retained information.The cell state and output are propagated through the recurrent network and trained by backpropagation through time.
  • LSTM Variants: The study examines traditional LSTM and bidirectional LSTM architectures to test whether bidirectional processing improves turbulence-ROM performance.Each LSTM hidden cell receives information dependent on the previous time instant, whereas BiLSTM uses two-way information flow.

3 Methodology

The LSTM-ROM uses POD to represent dominant flow dynamics and deep recurrent networks to predict their temporal coefficients from DNS data. Training uses high-fidelity isotropic and magnetohydrodynamic turbulence datasets, including multiple two-dimensional planes.

  • Datasets: The study demonstrates LSTM-ROM using two DNS databases: forced isotropic turbulence and magnetohydrodynamic turbulence.Both datasets come from the Johns Hopkins turbulence database and use U-velocity fields for the reported results.
  • Datasets: The ISO and MHD datasets are three-dimensional DNS simulations on 1024^3 grids, with 5023 and 1024 saved timesteps, respectively.The ISO data represent 10 seconds of flow data, while the MHD dataset represents 2.056 seconds.
  • Data preparation: The methodology decomposes three-dimensional datasets into multiple two-dimensional planes while retaining the temporal snapshots for each plane.This creates many training datasets and exploits qualitative similarities among planes in homogeneous isotropic and MHD turbulence.
  • ROM formulation: POD extracts dominant modes that capture much of the flow energy, while their temporal coefficients describe the evolution of the key flow dynamics.The approach retains this POD-based ROM framework but replaces Galerkin projection with neural-network prediction.
  • ROM formulation: The LSTM-ROM predicts future POD temporal coefficients instead of solving the full Navier–Stokes equations.Predicted coefficients are combined with POD modes to reconstruct the evolving flow field.

1. Select the number of 2-D planes to be used as training datasets (Figure 4)

The workflow selects plane-based training and test datasets, extracts dominant POD modes and coefficients, trains recurrent models, validates future coefficient predictions, and reconstructs the flow field.

  • Dataset selection: A test dataset is selected from the decomposed planes for modeling after the neural network learns from the training datasets.The test dataset is a single equidistant two-dimensional plane.
  • POD extraction: The workflow extracts usually 5–10 highest-energy POD modes and their temporal coefficients from every training and test dataset.Test-set coefficients validate the LSTM-ROM predictions.
  • Network training: LSTM or BiLSTM networks are trained for the selected dominant POD modes.The initial strategy trains models using the temporal coefficients associated with those modes.
  • Validation: Short histories of test-set POD coefficients are used to predict coefficients at the next time instants, which are compared with the true values.Validation is repeated for every selected dominant mode.
  • Flow reconstruction: Predicted temporal coefficients and POD modes are combined to compute the predicted flow field.This reconstruction uses the same POD representation underlying the reduced-order model.
  • Dataset selection: The ISO training and test planes exhibit significant qualitative similarity in their dominant POD modes.This similarity reflects an assumption commonly made in ROM applications.

4 Results

The results show accurate LSTM predictions for dominant POD coefficients in isotropic and MHD turbulence, with unified models addressing limitations of separate mode-specific models. BiLSTM generally performs worse on the isotropic tests, while the Hurst Exponent is used to assess memory in turbulent signals.

  • Experimental setup: A 10-step time window and 10-step prediction horizon were used to keep training computationally manageable while demonstrating LSTM-ROM efficacy.The choices can significantly affect accuracy and require design-level tuning based on the flow physics.
  • 4.1 Isotropic Turbulence: The initial isotropic-turbulence strategy trains separate LSTM or BiLSTM models for each of the five dominant POD temporal coefficients.This produces five trained models, one for each mode.
  • 4.1 Isotropic Turbulence: BiLSTM has higher average MASE than LSTM across all five isotropic POD modes.The paper attributes this to possible overfitting when long-range correlations are not guaranteed in chaotic turbulence signals.
  • 4.1 Isotropic Turbulence: Prediction errors in lower-energy POD modes affect reconstructed flow accuracy less than errors in the first and second modes.The comparison uses a five-mode LSTM-ROM reconstruction at t = 8.10.
  • 4.2 Magnetohydrodynamic Turbulence: The unified model trains one network on samples from POD modes 1–5 and can predict coefficients for all modes.This strategy addresses the memory bottleneck and the possibility that flow features span multiple POD modes.
  • Memory analysis: The Hurst Exponent approaches 0.5 for random-walk-like behavior, indicating weaker discernible trends and less persistent memory.This provides a tool for relating signal memory to LSTM predictive modeling.
  • 4.2 Magnetohydrodynamic Turbulence: The unified model generally performs well across samples, and LSTM predicts each MHD mode with considerable accuracy.The results suggest that the unified model may learn statistics common to different POD modes.

5 Memory effects and LSTM Model Accuracy

The paper uses the Hurst Exponent to quantify temporal memory and examines how persistence and prediction horizon affect LSTM-ROM accuracy. Accuracy generally declines with longer horizons, but the effect depends on persistence and POD mode.

  • Memory quantification: The Hurst Exponent quantifies persistence in POD-coefficient signals, with H≈0.5 indicating randomness and little memory.H→0 indicates anti-persistence, whereas H→1 indicates persistent trends.
  • Memory quantification: LSTM-ROM accuracy depends on the interaction between signal persistence and prediction horizon.Persistence is characterized by the Hurst Exponent, while horizon is the number of future steps predicted.
  • Experimental design: The analysis varies the horizon and tunes the LSTM separately for each POD mode to assess best-case accuracy across persistence levels.Earlier experiments used a fixed horizon and shared tuning, whereas this analysis varies L and retunes each mode.
  • Accuracy trends: Mean MASE generally increases with horizon, with the largest error increases for POD modes 7 and 15 and weaker variation for modes 50 and 100.Modes 400 and 800 maintain consistently high prediction error across horizons.
  • Accuracy trends: Strongly persistent low-rank modes are modeled accurately at short horizons but lose accuracy sharply at longer horizons.These modes contain more energy, making their prediction quality especially important for the resulting ROM.

2. Conversely, LSTM has considerable difficulties modeling strongly anti-persistent

LSTM has difficulty modeling strongly anti-persistent modes, whose prediction error remains high across horizons. Increasing horizon usually reduces accuracy, although weakly persistent modes can sometimes improve.

  • Horizon effects: Increasing the horizon generally reduces accuracy for a given mode, with a less pronounced effect for strongly anti-persistent modes.The horizon effect therefore varies with the mode’s persistence characteristics.
  • ROM implications: Because persistent low-rank modes carry more energy, their poor long-horizon predictions can produce inferior turbulence ROMs.Short-horizon ROM quality benefits from accurate modeling of these high-energy modes.

6 Discussion & Conclusions

The paper finds that LSTM-ROMs can model non-stationary turbulence for specialized flow-control settings, while accuracy declines with prediction horizon and theoretical and modeling assumptions constrain applicability.

  • Findings: LSTM was more accurate than Bidirectional LSTM, which consistently performed worse, likely because it overfits by assuming nonexistent long-range memory.The authors characterize this explanation as a surmise rather than a demonstrated cause.
  • Findings: Prediction accuracy deteriorated as the forecasting horizon increased, consistent with theoretical restrictions associated with Lyapunov-exponent dynamics.For trajectories initially close together, divergence increases farther from the initial state, making long-horizon prediction exponentially less likely to remain accurate.
  • Future directions: Purely data-driven approaches may face difficulties at long horizons, whereas combining LSTM with Navier–Stokes equations is proposed as a future direction.The proposed hybrid approach would use governing equations to enforce spatial dynamics and symmetries.
  • Limitations: The approach assumes that dominant POD spatial modes remain consistent within a flow regime, an assumption that can be restrictive outside simplified, closely related geometries and Reynolds numbers.The paper notes that this consistency generally holds for cylinder-wake shedding at closely related Reynolds numbers when bifurcations do not occur.
  • Applications: The LSTM-ROM is positioned for specialized engineering flow-control applications requiring narrow regime generalization, low inference cost, and no necessary physics insight.Low inference cost is identified as useful for resource-scarce onboard hardware, including microcontrollers and devices with limited memory.
  • Applications: Initial results on two validated DNS datasets suggest that the Hurst exponent can provide an a priori quantitative estimate of ROM success before training.The dependence between Hurst exponent and horizon length requires study across a larger variety of datasets.

7 Appendix

The appendix describes the software implementation and repeated-run procedure used to obtain the reported results.

  • Implementation: The LSTM networks were implemented with TensorFlow and Keras, chosen for scalable, robust, user-friendly, and concise interfaces.Keras served as a simplified Python wrapper for TensorFlow.
  • Implementation: Results were obtained from an ensemble of 10 runs for each case, with 75 epochs per run, to account for neural-network stochasticity.The Hurst-exponent code was obtained from Reference 50.
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