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Construction of Reduced Order Models for Fluid Flows Using Deep Feedforward Neural Networks

Hugo F. S. Lui, William R. Wolf

arXiv:1903.05206v1physics.flu-dynphysics.comp-ph

TL;DR

The paper addresses computationally demanding fluid-flow ROM construction and the difficulty of learning complex dynamics with suitable features. It combines SPOD-based modal reduction with deep feedforward neural-network regression in a SINDy-like framework, achieving stable predictions beyond training windows and capturing transient and vortex dynamics across tested flows.

  • Problem

    Large-scale unsteady-flow simulations generate costly datasets, while ROMs may become unstable and regression methods require suitable dynamical features.

  • Method

    The methodology combines SPOD-based flow modal decomposition with deep feedforward neural-network regression in a framework similar to SINDy.

  • Results

    The DNN ROMs capture transient and vortex dynamics, predict beyond the training window with larger time increments, and outperform sparse regression in long-term accuracy and stability.

  • Takeaways & Limitations

    The framework provides reduced-order predictions for nonlinear, compressible, and turbulent flows while retaining key transient and vortex dynamics.

  • Takeaways & Limitations

    Model selection based on coefficient mean absolute error does not assess generalization, and evaluating reconstructed-flow error would be more appropriate but more expensive.

Abstract

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We present a numerical methodology for construction of reduced order models, ROMs, of fluid flows through the combination of flow modal decomposition and regression analysis. Spectral proper orthogonal decomposition, SPOD, is applied to reduce the dimensionality of the model and, at the same time, filter the POD temporal modes. The regression step is performed by a deep feedforward neural network, DNN, and the current framework is implemented in a context similar to the sparse identification of non-linear dynamics algorithm, SINDy. A discussion on the optimization of the DNN hyperparameters is provided for obtaining the best ROMs and an assessment of these models is presented for a canonical nonlinear oscillator and the compressible flow past a cylinder. Then, the method is tested on the reconstruction of a turbulent flow computed by a large eddy simulation of a plunging airfoil under dynamic stall. The reduced order model is able to capture the dynamics of the leading edge stall vortex and the subsequent trailing edge vortex. For the cases analyzed, the numerical framework allows the prediction of the flowfield beyond the training window using larger time increments than those employed by the full order model. We also demonstrate the robustness of the current ROMs constructed via deep feedforward neural networks through a comparison with sparse regression. The DNN approach is able to learn transient features of the flow and presents more accurate and stable long-term predictions compared to sparse regression.

1. Introduction

High-fidelity unsteady-flow simulations generate massive datasets and can be computationally expensive, motivating reduced-order models that retain the main physics. The paper combines modal decomposition with machine-learning regression to address ROM instability and improve applicability to complex flows.

  • Motivation: High-fidelity turbulent-flow simulations can require billions of degrees of freedom, long sampling periods, and small time steps for converged statistics.These requirements make large-scale flow data costly to generate and process.
  • Motivation: Time-resolved simulations and experiments produce large datasets that data-driven algorithms can statistically post-process to investigate complex turbulent-flow mechanisms.Modal decomposition methods such as POD and DMD are among the techniques used for this analysis.
  • Reduced-order modeling: Reduced-order models lower computational costs and can support preliminary design, optimization, and flow-control studies, provided they reproduce key full-scale physical behavior.Their usefulness depends on maintaining the main physical aspects of the full-order models.
  • ROM limitations: ROMs can become unstable because POD basis truncation unbalances the turbulent kinetic-energy budget in Galerkin-projected dynamics.Turbulence models and subspace-rotation methods address this issue, while neural networks have also been used to stabilize POD-Galerkin ROMs.
  • Machine learning for fluid flows: Machine learning has been applied to fluid-flow classification, turbulence-model improvement, Reynolds-stress discrepancy prediction, and flow-feature identification.The cited applications include deep neural networks, random forests, and convolutional neural networks.
  • Present approach: The proposed methodology combines POD-based modal decomposition with feedforward-neural-network regression in a framework similar to SINDy.SPOD is used to obtain a low-dimensional representation while filtering temporal modes, and the method is evaluated on nonlinear dynamics and compressible flows.

2. Theoretical and Numerical Formulation

The formulation reduces fluid-flow dynamics with POD/SPOD modal representations, then learns the temporal evolution of retained coefficients using DNN regression. The resulting ROM reconstructs flowfields beyond training data while balancing predictive performance against interpretability and computational cost.

  • 2.2. Regression via DNN: The regression step learns the nonlinear operator governing coefficient dynamics from temporal derivatives, without requiring spatial derivatives.The setup follows a SINDy-like data-driven identification strategy while using DNNs to learn both weights, biases, and features.
  • 2.1. Flow modal decomposition: The method collects flow snapshots, arranges them into a data matrix, and applies snapshot POD to reduce the high-dimensional dynamical system.POD represents fluctuations with spatial modes and temporal coefficients, while the correlation matrix depends on a selected norm.
  • 2.1. Flow modal decomposition: For compressible flows, pressure- and kinetic-energy-based norms produce different POD modes but similarly stable and accurate DNN ROMs.The correlation matrix is formed over the fluid region using an L2 inner product and has size N × N.
  • 2.1. Flow modal decomposition: SPOD filters high-frequency contributions in turbulent temporal modes while preserving full-order information by redistributing energy toward higher POD modes.The method filters the POD correlation matrix using a windowed function before determining eigenvalues and modes.
  • 2.1. Flow modal decomposition: The retained temporal coefficients form a coupled ODE system whose derivatives are computed numerically before regression.A 10th-order compact scheme is used for high spectral resolution, and the derivatives are assembled into a matrix.
  • 2.2. Regression via DNN: DNN feature learning replaces manually chosen SINDy feature libraries with multilayer representations learned directly from the data.The learned features can capture complex functions through successive processing layers.
  • 2.2. Regression via DNN: After integrating the learned coefficient dynamics, the ROM reconstructs the flowfield using the mean field and spatial modes computed from training data.Because these spatial quantities depend only on spatial coordinates and training data, the framework can predict beyond the training window.
  • 2.3. Data-driven ROMs: DNN-based ROMs cost more than sparse regression but provide better long-term predictive capabilities for the analyzed cases.The trade-off is reduced physical interpretability because DNN features, weights, and biases are nonlinear and non-sparse.

3. Hyperparameter Optimization

The DNN hyperparameter search compares manual, grid, random, and Bayesian strategies, while model selection uses training MAE despite a generalization limitation caused by POD construction.

  • Search strategies: DNN performance depends strongly on network depth, hidden units, regularization, and learning rate, making hyperparameter selection challenging.The large number of free parameters creates a substantial hyperparameter search space.
  • Search strategies: Manual, grid, random, and Bayesian search are the principal procedures considered for hyperparameter optimization.Grid search becomes increasingly time-consuming as the number of hyperparameters grows, while random search can outperform grid search empirically.
  • Model selection: The framework uses random search and Bayesian optimization, ranking candidate ROMs by training mean absolute error (MAE).Candidates with lower MAE are considered more likely to provide the optimal model, although the lowest-MAE model may overfit.
  • Model selection: Training MAE does not assess generalization because POD modes cannot be computed separately without bias when training and validation snapshots jointly determine the correlation matrix.The resulting validation procedure would use POD information from both training and validation sets.
  • Model selection: AIC and BIC avoid requiring a validation set by balancing fit quality against model complexity, but their parameter penalties can be problematic for deep neural networks.These criteria are presented as alternatives to the current model-selection approach.
  • Implementation choices: The study tunes selected DNN hyperparameters, while fixing others through standard procedures or manual choices to reduce the search space.Adam decay rates and numerical-stability constants follow Kingma and Ba, while learning rate and iteration count are chosen manually; activation function is also treated as a hyperparameter.

4. Results

The results evaluate DNN-based reduced-order models on nonlinear oscillators, cylinder flows, and plunging-airfoil dynamics. Across these cases, the models reproduce tested flow dynamics beyond training, while accuracy and long-term stability depend on retained modes and regression method.

  • The study evaluates DNN-ROMs for a nonlinear oscillator, compressible cylinder flow, transient incompressible cylinder flow, and plunging-airfoil dynamic stall.
  • 4.1. Nonlinear oscillator: The nonlinear-oscillator ROM reproduces system dynamics during training and throughout the test set.
  • 4.2.1. Flow past a cylinder: 10 POD modes recover 99% of the energy for all variables and accurately reconstruct the compressible cylinder flow beyond training.
  • 4.2.1. Flow past a cylinder: The cylinder ROM recovers near-field hydrodynamics and far-field acoustics, while two modes preserve stability but show visible discrepancies.
  • 4.2.1. Flow past a cylinder: For transient cylinder flow, DNN reconstruction captures both transient and limit-cycle dynamics, whereas sparse regression fails to predict the long-term limit cycle.
  • 4.2.2. Deep dynamic stall of plunging airfoil: The airfoil ROM reproduces leading- and trailing-edge vortex dynamics beyond training, while higher-wavenumber features require additional SPOD modes and increased cost.
  • 4.2.2. Deep dynamic stall of plunging airfoil: DNN models cost 40 times more computationally than SINDy on average but provide stable and accurate long-time predictions beyond the training window.

5. Conclusions

The paper presents DNN-based reduced order models combining modal decomposition, SPOD filtering, and regression, with hyperparameter optimization and comparisons against sparse regression. Across cylinder and dynamic-stall cases, the ROMs reproduce key flow dynamics, support predictions beyond training, and substantially reduce computational cost.

  • The methodology combines flow modal decomposition with deep feedforward neural-network regression in a framework similar to SINDy.
  • SPOD is used to reduce dimensionality and filter high-frequency content in temporal modes, while hyperparameter optimization is used to improve ROM construction.The study tests random search and Bayesian optimization for DNN hyperparameters.
  • The framework predicts flowfields beyond the training window with larger time increments than the full-order model and reduces the selected dynamic-stall ROM simulation cost to a few seconds.The full-order simulation cost was 100,000 hours, while hyperparameter optimization, training, and evaluation of 700 models took approximately 7 GPU hours.
  • In the cylinder study, 10 POD modes recover 99% of flow energy, and the DNN ROM accurately reconstructs both transient and limit-cycle dynamics.Sparse regression reconstructs the transient regime but not the long-term limit cycle.
  • For dynamic stall, the ROMs capture formation, transport, and ejection of the leading-edge stall vortex and the trailing-edge vortex.The result holds for both full three-dimensional and spanwise-averaged flow solutions.
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