Source-linked AI summary
Machine learning based spatio-temporal super resolution reconstruction of turbulent flows
Kai Fukami, Koji Fukagata, Kunihiko Taira
TL;DR
The paper addresses reconstruction of turbulent flow fields from data that are grossly coarse in both space and time. It combines supervised spatial super resolution with temporal inbetweening using a DSC/MS convolutional neural network. The method reconstructs multiple turbulent-flow cases when temporal spacing is appropriately chosen, but has identifiable limitations near boundaries and at low temporal correlation.
Problem
The study targets the limited applicability of earlier machine-learning super resolution methods, which had been restricted to two-dimensional spatial reconstruction.
Method
The method combines supervised spatial super resolution and temporal inbetweening with a DSC/MS convolutional neural network.
Results
The method reconstructs turbulent flows across two- and three-dimensional cases, with temporal evolution recovered when the interval between frames is appropriately chosen.
Takeaways & Limitations
The approach supports flow reconstruction from grossly coarse space-time data and may support computational and experimental fluid-dynamics applications.
Takeaways & Limitations
Temporal reconstruction fails when the temporal two-point correlation coefficient is R+ ⩽ 0.25, while errors are also elevated near channel-flow boundaries and walls.
Abstract
from arXiv · showhide
We present a new turbulent data reconstruction method with supervised machine learning techniques inspired by super resolution and inbetweening, which can recover high-resolution turbulent flows from grossly coarse flow data in space and time. For the present machine learning based data reconstruction, we use the downsampled skip-connection/multi-scale model based on a convolutional neural network to incorporate the multi-scale nature of fluid flows into its network structure. As an initial example, the model is applied to a two-dimensional cylinder wake at $Re_D$ = 100. The reconstructed flow fields by the proposed method show great agreement with the reference data obtained by direct numerical simulation. Next, we examine the capability of the proposed model for a two-dimensional decaying homogeneous isotropic turbulence. The machine-learned models can follow the decaying evolution from coarse input data in space and time, according to the assessment with the turbulence statistics. The proposed concept is further investigated for a complex turbulent channel flow over a three-dimensional domain at $Re_τ$ =180. The present model can reconstruct high-resolved turbulent flows from very coarse input data in space, and it can also reproduce the temporal evolution when the time interval is appropriately chosen. The dependence on the amount of training snapshots and duration between the first and last frames based on a temporal two-point correlation coefficient are also assessed to reveal the capability and robustness of spatio-temporal super resolution reconstruction. These results suggest that the present method can meet a range of flow reconstructions for supporting computational and experimental efforts.
1. Introduction
Machine learning super resolution had shown promise for reconstructing high-resolution flow fields, but prior applications were limited to two-dimensional spatial reconstruction. This study extends the idea to reconstruct turbulent data in both space and time.
- Machine learning has been applied to fluid dynamics, including turbulence modeling and reduced-order modeling.
- Earlier super resolution work reconstructed 128 × 128 turbulent fields from 4 × 4 inputs using machine learning.
- Prior super resolution applications were limited to two-dimensional spatial reconstruction despite their potential for handling high-resolution flow data.
- Temporal inbetweening estimates intermediate frames between the first and last frames in image and video sequences.
- The present study combines spatial super resolution and temporal inbetweening to reconstruct high-resolution turbulent data from extremely low-resolution inputs in space and time.
2. Approach
The approach trains separate supervised CNN-based models for spatial super resolution and temporal inbetweening, then applies them in alternative orders to reconstruct high-resolution flow fields. Cylinder-wake tests show excellent agreement with DNS and indicate that sequential reconstruction is preferable to a directly combined model.
- The method reconstructs q(xHR, tHR) from q(xLR, tLR) by combining spatial super resolution with temporal inbetweening.
- Separate supervised models Fx and Ft are trained as nonlinear mappings for spatial and temporal reconstruction.
- The DSC/MS architecture combines skip connections with multi-scale processing in a convolutional neural network.
- Two sequential orders are evaluated: spatial reconstruction followed by inbetweening, or inbetweening followed by spatial reconstruction.
- The combined model Fcomb directly reconstructs high-resolution space-time fields from low-resolution inputs but cannot reconstruct the cylinder flow well.
- Approximately 5% L2 error was achieved for the cylinder wake, with reconstructed fields practically indistinguishable from reference DNS data.
3. Results
The method reconstructs turbulent flows from coarse spatial and temporal data, with accuracy depending on input resolution, temporal correlation, training-set size, and flow-region characteristics.
- Two-dimensional turbulence: The reconstructed two-dimensional turbulence captures complex vortex evolution and the decaying nature of the flow across coarse spatio-temporal inputs.Time-ensemble errors and total kinetic energy trends support this assessment.
- Two-dimensional turbulence: 0.198 is the peak L2 error norm for Regime II with low-resolution input and a wide time step, occurring at t = (n+7)∆t.The shifted peak is attributed to absent boundary-condition information, evolution of a large structure over a bi-periodic domain, and accumulated spatial-reconstruction error.
- Two-dimensional turbulence: The reconstructed turbulence statistics follow reference DNS trends, but high-wavenumber kinetic-energy spectra do not match because low- and high-wavenumber components lack correlation.The model nevertheless reconstructs high-resolution two-dimensional turbulence from spatio-temporally coarse data.
- Three-dimensional channel flow: Three-dimensional channel-flow reconstruction is more sensitive to temporal than spatial resolution, with wide and super-wide time steps failing because temporal correlation is insufficient.A medium time step yields reasonable agreement in Q isosurfaces and L2 error, whereas wider steps produce smoothed fields.
- Three-dimensional channel flow: 100 training snapshots for both spatial and temporal models are the minimum reported requirement for recovering turbulent channel-flow fields, while training cost rises substantially with dataset size.Training takes approximately 3 days for {100,100} snapshots and 15 days for {1000,1000} on an NVIDIA Tesla V100 GPU.
4. Conclusions
The supervised spatio-temporal super-resolution models reconstruct turbulent flow fields from highly coarse spatial and temporal data across two- and three-dimensional test cases. Their robustness depends on temporal sampling, training data, and architecture choices, while the method also supports substantial data compression.
- The model reconstructed a two-dimensional cylinder wake spatially and reproduced its temporal evolution using only the first and last frames.
- The decaying homogeneous isotropic turbulence reconstructions reasonably matched reference data in L2 error norm, kinetic energy spectrum, and vorticity probability density.Models were substantially affected by the temporal range of training data.
- For turbulent channel flow at Re_τ = 180, reconstruction succeeded from grossly coarse spatial and temporal data when the interval between frames was appropriately chosen.
- When the temporal two-point correlation coefficient was R+ ⩽0.25, the model was unable to recover the turbulent flow fields in time.
- The models preferentially extract low-wavenumber features to minimize the loss function, motivating physics-informed architectures, loss functions, feature engineering, and better training-data design.
- The model showed reasonable capability with up to 10% noisy input and reconstructed flow fields using as few as 100 training data sets.
- The method recovered a three-dimensional turbulent channel flow containing 7.9 million spatio-temporal elements from 3072 elements, corresponding to 0.04% compression.
Appendix
The appendix evaluates whether non-uniform wall-normal grids alter convolutional-network reconstruction results. Visual differences in inputs did not produce significant differences in contours or L2 error norms.
- The main analysis uses flow fields interpolated from non-uniform wall-normal data onto a uniform grid for convolutional-network inputs and outputs.
- Non-uniform-grid inputs produced no significant differences in streamwise velocity contours or L2 error norm compared with uniform-grid inputs.