Source-linked AI summary
A neural network approach for the blind deconvolution of turbulent flows
Romit Maulik, Omer San
TL;DR
The paper addresses blind recovery of subfilter-scale flow information without assuming a predefined filtering kernel. It develops a supervised single-layer feed-forward ANN and reports strong a-priori performance across two-dimensional and three-dimensional turbulence, with potential for data-driven closure modeling.
Problem
Subfilter-scale recovery typically uses approximate deconvolution with a user-defined low-pass filtering kernel, although complex flows may exhibit more contorted natural filter shapes.
Method
The study trains a single-layer feed-forward ANN through supervised learning to identify nonlinear relationships between filtered or noisy coarse-grained fields and target flow variables, without explicitly specifying a filter kernel.
Results
The data-driven framework achieves similar deconvolution performance to several popular structural closure strategies without explicit filter-kernel specification and performs well in a-priori tests across homogeneous and stratified turbulence cases.
Takeaways & Limitations
The approach shows promise as a data-driven component for physics-augmented turbulence closure and for deconvolution across different flow physics.
Takeaways & Limitations
Performance depends on training data that cover many physical regimes, because the model can reproduce physical behaviors similar to those represented in its training data.
Abstract
from arXiv · showhide
We present a single-layer feedforward artificial neural network architecture trained through a supervised learning approach for the deconvolution of flow variables from their coarse grained computations such as those encountered in large eddy simulations. We stress that the deconvolution procedure proposed in this investigation is blind, i.e. the deconvolved field is computed without any pre-existing information about the filtering procedure or kernel. This may be conceptually contrasted to the celebrated approximate deconvolution approaches where a filter shape is predefined for an iterative deconvolution process. We demonstrate that the proposed blind deconvolution network performs exceptionally well in the a-priori testing of both two-dimensional Kraichnan and three-dimensional Kolmogorov turbulence and shows promise in forming the backbone of a physics-augmented data-driven closure for the Navier-Stokes equations.
1. Introduction
The paper develops a blind, supervised neural-network approach for recovering subfilter-scale flow information without specifying a filtering kernel. It evaluates statistical recovery, cross-flow generalization, noisy-data regularization, and comparisons with structural closures.
- Motivation: The approach is motivated by image-reconstruction methods and by the prospect of augmenting first-principles fluid mechanics with data-driven learning.The paper situates machine learning as an augmentation to established governing laws and turbulence closure modeling.
- Contribution: The study develops a single-layer feed-forward ANN that learns a nonlinear relationship between filtered or coarse-grained inputs and target flow variables.The architecture is designed for 2D and 3D homogeneous isotropic turbulence and a compressible stratified turbulence case.
- Evaluation: Noisy-data recovery is used to assess whether the procedure can stabilize aliasing error relevant to coarse-grained large eddy simulations.The study examines predictive performance when inputs are perturbed by noise and targets remain true counterparts.
- Novelty: Blind deconvolution avoids assuming a Gaussian or other predefined filtering kernel, unlike approximate deconvolution methods.The missing kernel assumption motivates additional regularization for flows with distinctly non-Gaussian distributions.
- Evaluation: The evaluation tests inertial-range scaling, probability-density functions, cross-validation, and transfer between different flow-physics cases.The framework is also examined against structural closures requiring an explicitly specified low-pass filter.
2. Artificial Neural Networks
The paper uses a single-hidden-layer feed-forward ANN with a tan-sigmoid hidden activation and linear output, trained through an extreme learning machine procedure. Random first-layer parameters are fixed, while output weights are obtained through a Moore-Penrose pseudoinverse.
- 2.1. Architecture: Each neuron adds a bias to its input and applies a transfer function, with layer outputs represented through matrix weights and activations.The lth-layer weights connect the preceding layer's outputs to the current layer.
- 2.1. Architecture: The network uses one hidden layer with tan-sigmoid activation between raw inputs and targets, while the output layer uses linear activation.Input nodes perform no biasing or activation, and the outer layer has no bias.
- 2.2. Extreme Learning Machine: Extreme learning machine training targets very fast single-layer ANN fitting through least-squares approximation.The method initializes first-layer weights and biases with small random values and solves for output weights using a generalized pseudoinverse.
- 2.2. Extreme Learning Machine: The ELM output is a linear, no-bias transformation of hidden-layer responses trained against target vectors.This regression formulation contrasts with earlier image-deblurring work using classification in deep convolutional networks.
- 2.2. Extreme Learning Machine: The training algorithm initializes random hidden-layer parameters, computes the hidden-layer matrix pseudoinverse, and obtains optimal output weights.The deployment algorithm then propagates test inputs through the trained hidden layer to produce predictions.
3. Blind Deconvolution
The study develops a supervised single-layer ANN for blind deconvolution, recovering flow variables from filtered or noisy coarse data without prescribing a filtering kernel. It compares this data-driven approach with approximate deconvolution and evaluates its ability to recover subfilter content and stabilize noisy inputs.
- Blind deconvolution: Unlike approximate deconvolution, the method does not assume a predefined Gaussian or other filtering kernel.The data are used to implicitly estimate the underlying blur-kernel shape.
- Blind deconvolution: The proposed blind deconvolution framework uses supervised learning to map perturbed, coarse-grained fields to their unfiltered target fields.Training data include relationships between coarsened true data and versions perturbed by filtering or noise.
- Perturbations: Low-pass spatial filtering tests inertial-range recovery and provides the direct test of blind deconvolution capability.The filtering uses a Gaussian kernel with standard deviation σ held at a default value of 1.0.
- Training procedure: The ANN is trained separately for each perturbation type using an extreme learning machine, with 100 hidden neurons in the reported study.The ELM approach provides tractable training times even with several hundred hidden neurons.
4. Cross-Validation
The cross-validation strategy tests whether the trained ANN generalizes beyond its training data across physically similar fields, perturbations, numerical arrangements, and flow physics. Shifted coarse data sets and independent perturbations support a-priori evaluation, while cross-physics testing probes the universal nature of the learned deconvolution and regularization.
- Cross-validation: Cross-validation assesses whether the data-driven model performs similarly on solution fields not used during training.The goal is to distinguish generalization from localization to the training data.
- Data generation: The experiments use high-fidelity simulations to generate coarse and shifted data for cross-validation, with 512^3 degrees of freedom in 3D and 2048^2 in the 2D case.The coarse shifted sets contain 64^3 degrees of freedom in 3D and 256^2 in 2D.
- A-priori cross-validation: Spatial shifting creates numerically different data sets that retain the physics of the same fine-grained solution field.The procedure can produce up to 63 distinct 3D and 511 distinct 2D data sets.
- A-priori cross-validation: One data set is filtered with Gaussian smoothing for training inputs, while unfiltered data provide network outputs and three similarly generated sets provide testing data.This setup supports a perfectly a-priori analysis because all sets derive from the same high-fidelity solution field.
- Cross-physics validation: A further cross-validation tier trains and tests across different flow physics to examine the universal nature of the learned deconvolution and regularization.This is described as a more challenging test than cross-validation among physically similar numerical arrangements.
5. Results
Across three benchmark turbulence settings, the proposed architecture reconstructs filtered flow statistics and fields, stabilizes aliasing in noisy inputs, and achieves subfilter-stress errors near established closure models without assuming a spatial filter shape.
- Test cases: The study evaluates the framework on two-dimensional homogeneous isotropic turbulence, three-dimensional homogeneous isotropic turbulence, and three-dimensional stratified turbulence.The cases represent distinct cascade physics and include Kraichnan, Kolmogorov, and stratified Kelvin–Helmholtz settings.
- Kraichnan turbulence: The proposed architecture captures substantially more of the Kraichnan inertial range and reconstructs the vorticity PDF trends from filtered inputs.The assessment uses kinetic-energy spectra and vorticity probability density functions, with visual recovery of smaller structures also reported.
- Kraichnan turbulence: Noisy-input recovery provides marginal inertial-range improvement but stabilizes cut-off-scale energy pileup, thereby reducing aliasing error across testing data.The reported regularization effect is attributed to the ELM training procedure and is considered promising for coarse-grained large eddy simulations.
- Kolmogorov turbulence: For filtered inputs in the Kolmogorov case, the ANN enhances inertial-range approximation and captures the peak and tail of the true velocity PDF accurately.Similar qualitative behavior is obtained across three testing-data versions, although performance differs with the amount of subfilter information requiring reconstruction.
- Kolmogorov turbulence: For noisy Kolmogorov inputs, the framework reduces high-wavenumber content and recovers PDF peaks, but accurate PDF-tail capture remains elusive.The same lower tail accuracy is observed in the two-dimensional test case.
- Closure comparison: The ANN performs similarly to widely used structural closures, while AD3 performs better on average because its specified Gaussian filter matches the convolution filter.Despite this comparison, the proposed blind method obtains subfilter-stress mean-squared errors near those of the examined closure models without assuming a spatial filter shape.
- Stratified turbulence: In stratified turbulence, regularization effectively removes aliasing and produces good three-dimensional field recoveries, with quantitative trends similar to the other test cases.The comparison with structural closures reports that the data-driven closure performs quite well, while AD3 remains superior under its prior Gaussian-filter specification.
6. Universality
The framework retains strong reconstruction performance when applied across times, Reynolds numbers, and distinct flow configurations. These tests suggest that learned deconvolution and regularization can transfer across flows sharing relevant turbulence characteristics.
- Universality of closure behavior: Universality is assessed through kinetic-energy spectra, velocity PDFs, velocity-field recovery, and subfilter-stress predictions across deconvolution and regularization datasets.The section includes both low-pass-filtered perturbations and noisy perturbation inputs for stratified turbulence.
- Cross-validation across flow conditions: Training at t = 15 produces exceptional closure recovery at t = 20 for the same Taylor-Green vortex simulation.The reconstructed fields exhibit trends similar to those observed in previous test cases.
- Universality of closure behavior: The stratified-turbulence architecture produces subfilter-stress predictions similar to established structural closures without explicitly defining a low-pass spatial filter.Probability density functions are compared across different subfilter-stress components.
- Cross-validation across flow conditions: Stratified-turbulence tests retain remarkable inertial-range recovery and regularization for both filtered and noisy inputs when trained with Taylor-Green vortex data.The only stated links between training and testing data are the k^-5/3 cascade and perturbation magnitude.
7. Concluding Remarks
The study proposes an ANN framework for blind deconvolution and regularization of filtered or perturbed turbulence fields, evaluated on two- and three-dimensional flows. It performs similarly to established structural closures, while broader deployment requires a-posteriori testing, careful training-data sampling, and architectural evaluation.
- Concluding remarks: The proposed ANN performs data-driven deconvolution and regularization for low-pass-filtered turbulence fields in canonical homogeneous-isotropic and stratified compressible flows.The study examines both two-dimensional and three-dimensional test cases with coarsened and perturbed high-fidelity simulation data.
- Concluding remarks: The ELM training approach provides innate regularization for noisy data and is selected for fast training and generalization that avoids overfitting.The recovered variables are described as smoother under noisy inputs.
- Concluding remarks: The ANN performs similarly to widely used structural subfilter modeling strategies, although AD3 performs better on average when its specified filter matches the convolution filter.The comparison favors AD3 under matched-filter conditions, while the ANN does not assume a filtering kernel.
- Limitations and future work: A fully a-posteriori analysis remains a natural follow-up to the present a-priori investigation.The supplied conclusion identifies this as a future evaluation rather than a completed test.
- Limitations and future work: Training-data sampling must expose the framework to many physical regimes because predictions reproduce behaviors similar to those represented in training data.The authors also identify outlier detection and preprocessing as needed components for noisy data.
- Limitations and future work: Performance should be estimated across network configurations because ANN behavior depends strongly on architecture as well as training data.Relevant configurations include inputs, outputs, neuron counts, and activation functions.
- Concluding remarks: Fast training facilitates generating regularized network weights for extremely large datasets, supporting this preliminary avenue for data-driven turbulence closures.The conclusion characterizes the work as an initial glimpse rather than a completed closure framework.