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

Arvind Mohan, Datta V. Gaitonde|arXiv (Cornell University)|Apr 24, 2018

Model Reduction and Neural Networks41 references131 citations

TL;DR

This paper develops a deep learning ROM for turbulent flows by modeling POD mode coefficients with LSTM and BiLSTM networks, enabling flow prediction and control applications. It also introduces the Hurst exponent to analyze memory in the data.

ABSTRACT

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.

Motivation & Objective

• Demonstrate a deep learning-based reduced order modeling approach for turbulent flows using POD modes as the reduced basis.
• Investigate the feasibility of LSTM and BiLSTM networks to predict temporal coefficients of POD modes for future flow evolution.
• Assess the stability and accuracy of LSTM-based ROMs for isotropic turbulence and magnetohydrodynamic turbulence datasets.
• Explore unified vs. multiple-model training strategies to improve predictive performance across POD modes.
• Introduce the Hurst exponent as a tool to quantify memory in non-stationary turbulent data and relate it to model performance.

Proposed method

• Use POD to extract dominant flow features (POD modes and temporal coefficients) from DNS databases.
• Train LSTM and Bidirectional LSTM networks to predict future POD coefficients from short history windows (input-output framing).
• Evaluate on two JHTB datasets: Forced Isotropic Turbulence (ISO) and Magnetohydrodynamic Turbulence (MHD) by decomposing 3D data into multiple 2D planes.
• Compare a set of training paradigms: separate models per POD mode vs unified model across modes.
• Analyze memory effects with the Hurst exponent to relate persistence/anti-persistence to prediction horizon performance.

Experimental results

Research questions

• RQ1 Can LSTM (and BiLSTM) predict the temporal coefficients of dominant POD modes well enough to reconstruct accurate future flow fields?
• RQ2 Does a unified neural network model across POD modes outperform separate models for turbulence ROMs?
• RQ3 How do memory characteristics (as captured by the Hurst exponent) influence LSTM-ROM predictive accuracy across horizons?
• RQ4 What is the impact of prediction horizon and mode persistence on ROM performance for isotropic and MHD turbulence datasets?

Key findings

• LSTM predictions generally follow the true temporal trends for the dominant POD modes, with BiLSTM often underperforming compared to LSTM in this turbulent context.
• A unified model across POD modes can achieve good predictive performance, suggesting shared statistical structure among modes.
• LSTM-ROM accuracy deteriorates with longer prediction horizons, especially for highly persistent modes, while highly anti-persistent modes are difficult to predict across horizons.
• The MASE metric shows low errors for many modes, but higher errors occur at samples with sudden dynamics, and lower energy (higher-rank) modes contribute less to overall flow accuracy.
• In isotropic turbulence, lower POD modes dominate energy and display stronger persistence, influencing short-horizon prediction success; in MHD, unified modeling also shows substantial predictive capability.
• BiLSTM generally yields higher errors than LSTM in these chaotic turbulence datasets, likely due to long-range dependency assumptions not aligning with turbulence signals.