# Extreme Learning Machines as Encoders for Sparse Reconstruction

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Sparse Reconstruction Problem

#### 2.1. Sparse Reconstruction Theory

#### 2.2. Data-Driven Sparse Basis Computation Using POD

#### 2.3. Data-Driven Sparse Basis Computation Using an ELM Autoencoder

#### 2.4. Measurement Locations, Data Basis, and Incoherence

#### 2.5. Sparse Recovery Framework

Algorithm 1:${l}_{2}$-based algorithm: Sparse reconstruction with known basis, $\mathsf{\Phi}$. |

#### 2.6. Algorithmic Complexity

## 3. Data Generation for Canonical Cylinder Wake

## 4. Sparse Reconstruction of Cylinder Wake Limit-Cycle Dynamics

#### 4.1. Sparse Reconstruction Experiments and Analysis

#### 4.2. Sparsity and Energy Metrics

#### 4.3. Sparse Reconstruction of Limit-Cycle Dynamics in Cylinder Wakes Using the POD Basis

#### 4.4. Sparse Reconstruction Using the ELM Basis

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Comparison of Predicted ELM Features and Snapshot Reconstruction

**Figure A1.**Sparse reconstructed (SR) POD coefficients for different sensor placements compared with full reconstruction (FR) for a given ${K}^{\ast}=3$.

**Figure A2.**Comparison of ELM-based parse reconstruction with full reconstruction flow fields for a single snapshot at T = 48.8, using random sensor placement with seed = 101 (maximum reconstruction error) and seed = 106 (minimum reconstruction error) from the choice of ten different experiments. Blue solid line: FR. Red dashed line: SR. Black stars: sensor locations. (

**a**) Seed = 101 (max error); and (

**b**) Seed = 106 (min error).

**Figure A3.**Comparison of the sparse reconstructed (SR) ELM features with the corresponding full data reconstruction (FR) predicted features. The two SR cases correspond to sensor placement with maximum error (seed = 101) and minimum error (seed = 106).

## Appendix B. Impact of Retaining the Data Mean for Sparse Reconstruction

**Figure A4.**Isocontours of first five POD modes with and without removing the mean from the snapshot data for cylinder wake flow with $Re$ = 100. (

**a**) 1st POD mode (with mean); (

**b**) 1st POD mode (without mean); (

**c**) 2nd POD mode (with mean); (

**d**) 2nd POD mode (without mean); (

**e**) 3rd POD mode (with mean); (

**f**) 3rd POD mode (without mean); (

**g**) 4th POD mode (with mean); (

**h**) 4th POD mode (without mean); (

**i**) 5th POD mode (with mean); and (

**j**) 5th POD mode (without mean).

**Figure A5.**The isocontours of the normalized mean squared POD-based sparse reconstruction errors (${l}_{2}$ norms). The top figures correspond to the case with mean removed for learning the basis. The bottom figures correspond to the case with the mean retained. Left: normalized absolute error metric, ${\u03f5}_{1}$. Right: normalized relative error metric, ${\u03f5}_{2}$. (

**a**) ${\u03f5}_{1}$ (without mean); (

**b**) ${\u03f5}_{2}$ (without mean); (

**c**) ${\u03f5}_{1}$ (with mean); and (

**d**) ${\u03f5}_{2}$ (with mean).

## Appendix C. POD-Based Sparse Reconstruction for Cylinder Wake with Re = 800

**Figure A6.**Isocontours of the normalized mean squared POD-based sparse reconstruction errors (${l}_{2}$ norm) for $Re=800$ case, corresponding to the sensor placement with maximum and minimum errors from the chosen ensemble. The average error across the entire ensemble of ten random sensor placements is also shown. Left: normalized absolute error metric, ${\u03f5}_{1}$. Right: normalized relative error metric, ${\u03f5}_{2}$. (

**a**) ${\u03f5}_{1}$ (maximum); (

**b**) ${\u03f5}_{2}$ (maximum); (

**c**) ${\u03f5}_{1}$ (minimum); (

**d**) ${\u03f5}_{2}$ (minimum); (

**e**) ${\u03f5}_{1}$ (average); and (

**f**) ${\u03f5}_{2}$ (average).

**Figure A7.**Isocontours of the normalized mean squared (${l}_{2}$) ELM-based sparse reconstruction errors of $Re=800$ case for maximum, minimum, and average error using different choices for random sensor placement. Left: normalized absolute error metric, ${\u03f5}_{1}$. Right: normalized relative error metric, ${\u03f5}_{2}$. (

**a**) ${\u03f5}_{1}$ (maximum); (

**b**) ${\u03f5}_{2}$ (maximum); (

**c**) ${\u03f5}_{1}$ (minimum); (

**d**) ${\u03f5}_{2}$ (minimum); (

**e**) ${\u03f5}_{1}$ (average); and (

**f**) ${\u03f5}_{2}$ (average).

**Figure A8.**Isocontours of the normalized mean squared (${l}_{2}$) ELM-based sparse reconstruction errors of $Re=800$ case for maximum, minimum, and average error using different choices for the random input weights in learning the basis. Left: normalized absolute error metric, ${\u03f5}_{1}$. Right: normalized relative error metric, ${\u03f5}_{2}$. (

**a**) ${\u03f5}_{1}$ (maximum); (

**b**) ${\u03f5}_{2}$ (maximum); (

**c**) ${\u03f5}_{1}$ (minimum); (

**d**) ${\u03f5}_{2}$ (minimum); (

**e**) ${\u03f5}_{1}$ (average); and (

**f**) ${\u03f5}_{2}$ (average).

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**Figure 1.**Schematic illustration of ${l}_{2}$ (

**a**) and ${l}_{1}$ (

**b**) minimization reconstruction for sparse recovery using a single-pixel measurement matrix. The numerical values in C are represented by colors: black (1), white (0). The other colors represent numbers that are neither 0 nor 1. In the above schematics $\tilde{\mathbf{x}}\in {\mathbb{R}}^{P}$, $C\in {\mathbb{R}}^{P\times N}$, $\mathsf{\Phi}\in {\mathbb{R}}^{N\times {N}_{b}}$ and $a\in {\mathbb{R}}^{{N}_{b}}$, where ${N}_{b}\le N$. The number of colored cells in a represents the system sparsity K. $K={N}_{b}$ for ${l}_{2}$ and $K<{N}_{b}$ for ${l}_{1}$.

(a) | (b) |

**Figure 2.**Schematic of the extreme learning machine (ELM) autoencoder network. In this architecture, the output features are the same as input features.

**Figure 3.**Schematic of a gappy proper orthogonal decomposition (GPOD)-like sparse reconstruction (SR) formulation. The numerical values represented by the colored blocks are: black (1), white (0), color (other numbers).

**Figure 4.**Isocontour plots of the stream-wise velocity component for cylinder flow at $Re=100$ and $T=25,68,$ and 200, showing the evolution of the flow field. Here, T represents the time non-dimensionalized by the advection time-scale.

**Figure 5.**The temporal evolution of the first three normalized proper orthogonal decomposition (POD) coefficients for limit cycle cylinder flow at $Re=100$. (

**a**) 2D view; (

**b**) 3D view.

**Figure 6.**Error ($er{r}_{K}^{POD},er{r}_{K}^{ELM}$) using different number of basis (K) for both POD reconstruction and ELM prediction.

**Figure 7.**Isocontours of the normalized mean squared POD-based sparse reconstruction errors (${l}_{2}$ norms) using different random seeds (101 and 102) for sensor placements. Left: normalized absolute error metric, ${\u03f5}_{1}$. Right: normalized relative error metric, ${\u03f5}_{2}$. (

**a**) ${\u03f5}_{1}$ (seed = 101); (

**b**) ${\u03f5}_{2}$ (seed = 101); (

**c**) ${\u03f5}_{1}$ (seed = 102); and (

**d**) ${\u03f5}_{2}$ (seed = 102).

**Figure 8.**Isocontours of the normalized mean squared POD-based sparse reconstruction errors (${l}_{2}$ norm) using different random seed (108 and 109) for sensor placements. Left: normalized absolute error metric, ${\u03f5}_{1}$. Right: normalized relative error metric, ${\u03f5}_{2}$. (

**a**) ${\u03f5}_{1}$ (seed = 108); (

**b**) ${\u03f5}_{2}$ (seed = 108); (

**c**) ${\u03f5}_{1}$ (seed = 109); and (

**d**) ${\u03f5}_{2}$ (seed = 109).

**Figure 9.**Comparison of POD-based SR and full reconstruction (FR) for different random sensor placement and ${P}^{\ast}\le {K}^{\ast}$. Blue solid line: FR. Red dashed line: SR. Black stars: sensor locations. (

**a**) Seed = 101 (at ${P}^{\ast}=3$ and ${K}^{\ast}=3$); (

**b**) Seed = 102 (at ${P}^{\ast}=3$ and ${K}^{\ast}=3$); (

**c**) Seed = 108 (at ${P}^{\ast}=2$ and ${K}^{\ast}=3$); and (

**d**) Seed = 109 (at ${P}^{\ast}=2$ and ${K}^{\ast}=3$).

**Figure 10.**Comparison of POD-based SR and FR for different random sensor placement and ${P}^{\ast}>{K}^{\ast}$. Blue solid line: FR. Red dashed line: SR. Black stars: sensor locations. (

**a**) Seed = 101 (at ${P}^{\ast}=10$ and ${K}^{\ast}=5$); (

**b**) Seed = 102 (at ${P}^{\ast}=10$ and ${K}^{\ast}=5$); (

**c**) Seed = 108 (at ${P}^{\ast}=10$ and ${K}^{\ast}=5$); and (

**d**) Seed = 109 (at ${P}^{\ast}=10$ and ${K}^{\ast}=5$).

**Figure 11.**Isocontours of the normalized mean squared POD-based sparse reconstruction errors (${l}_{2}$ norm) corresponding to the sensor placement with maximum and minimum errors from the chosen ensemble of random sensor arrangements. The average error across the entire ensemble of ten random sensor placements is also shown. Left: normalized absolute error metric, ${\u03f5}_{1}$. Right: normalized relative error metric, ${\u03f5}_{2}$. (

**a**) ${\u03f5}_{1}$ (maximum); (

**b**) ${\u03f5}_{2}$ (maximum); (

**c**) ${\u03f5}_{1}$ (minimum); (

**d**) ${\u03f5}_{2}$ (minimum); (

**e**) ${\u03f5}_{1}$ (average); and (

**f**) ${\u03f5}_{2}$ (average).

**Figure 12.**Isocontours of the normalized mean squared (${l}_{2}$) ELM-based sparse reconstruction errors for the maximum, minimum, and average using different choices of the random input weights. Left: normalized absolute error metric, ${\u03f5}_{1}$. Right: normalized relative error metric, ${\u03f5}_{2}$. (

**a**) ${\u03f5}_{1}$ (maximum); (

**b**) ${\u03f5}_{2}$ (maximum); (

**c**) ${\u03f5}_{1}$ (minimum); (

**d**) ${\u03f5}_{2}$ (minimum); (

**e**) ${\u03f5}_{1}$ (average); and (

**f**) ${\u03f5}_{2}$ (average).

**Figure 13.**Isocontours of the normalized mean squared (${l}_{2}$) ELM-based sparse reconstruction error for maximum, minimum, and average error using different choices for the random sensor placement. Left: normalized absolute error metric, ${\u03f5}_{1}$. Right: normalized relative error metric, ${\u03f5}_{2}$. (

**a**) ${\u03f5}_{1}$ (maximum); (

**b**) ${\u03f5}_{2}$ (maximum); (

**c**) ${\u03f5}_{1}$ (minimum); (

**d**) ${\u03f5}_{2}$ (minimum); (

**e**) ${\u03f5}_{1}$ (average); and (

**f**) ${\u03f5}_{2}$ (average).

**Table 1.**The choice of sparse reconstruction algorithm based on problem design using parameters P (sensor sparsity), K (targeted system sparsity), and ${N}_{b}$ (candidate basis dimension).

Case | $\mathit{K}-{\mathit{N}}_{\mathit{b}}$ Relationship | $\mathit{P}-\mathit{K}$ Relationship | Algorithm | Reconstructed Dimension |
---|---|---|---|---|

1 | $K={N}_{b}$ | $P\ge K$ | ${l}_{2}$ | K |

2 | $K={N}_{b}$ | $P<K$ | ${l}_{1}$ | P |

3 | $K<{N}_{b}$ | $P<K<{N}_{b}$ | ${l}_{1}$ | P |

4 | $K<{N}_{b}$ | ${N}_{b}>P\ge K$ | ${l}_{1}$ | K or P |

5 | $K<{N}_{b}$ | $P\ge {N}_{b}>K$ | ${l}_{2}$ | K or ${N}_{b}$ |

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**MDPI and ACS Style**

Al Mamun, S.M.A.; Lu, C.; Jayaraman, B. Extreme Learning Machines as Encoders for Sparse Reconstruction. *Fluids* **2018**, *3*, 88.
https://doi.org/10.3390/fluids3040088

**AMA Style**

Al Mamun SMA, Lu C, Jayaraman B. Extreme Learning Machines as Encoders for Sparse Reconstruction. *Fluids*. 2018; 3(4):88.
https://doi.org/10.3390/fluids3040088

**Chicago/Turabian Style**

Al Mamun, S M Abdullah, Chen Lu, and Balaji Jayaraman. 2018. "Extreme Learning Machines as Encoders for Sparse Reconstruction" *Fluids* 3, no. 4: 88.
https://doi.org/10.3390/fluids3040088