# Compensation of Modeling Errors for the Aeroacoustic Inverse Problem with Tools from Deep Learning

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Modeling

## 3. Source Power Reconstruction with FISTA

- A gradient step with respect to the first summand of the objective function;
- Application of the proximal mapping (see Definition 6.1, p. 129, [34]) of the regularization part.

- To ensure convergence, the step size $\tau $ must satisfy$$\tau <{\left(\underset{\mathbf{q}\in {\mathbb{R}}^{N},x\ne 0}{sup}\frac{{\u2225{\mathcal{C}}^{\ast}\mathcal{C}\left(\mathbf{q}\right)\u2225}_{2}}{{\u2225x\u2225}_{2}}\right)}^{-1}\phantom{\rule{4pt}{0ex}},$$
- As the observed CSM ${\mathbf{C}}^{\mathrm{obs}}$ is Hermitian, the upper diagonal part can be neglected. Therefore, $\mathcal{S}$ denotes the index set of the lower triangular part of the CSM. The operation ${\mathrm{tril}}_{\mathcal{S}}\left((\right)\xb7)$ sets all entries to zero that do not belong to $\mathcal{S}$. Moreover, the principle of diagonal removal can be easily incorporated by defining $\mathcal{S}$ as the lower triangular indices without the diagonal.
- The operation ${\mathbf{v}}^{\left(n\right)}\odot {\mathbf{G}}^{H}$ multiplies each column of ${\mathbf{G}}^{H}$ component-wise by the vector ${\mathbf{v}}^{\left(n\right)}$.
- The operation ${(\xb7)}^{+}$ takes the positive part component-wise, i.e., ${\left(x\right)}^{+}=max(x,0)$.

## 4. Optimization of Phase Modeling Parameters

#### 4.1. Unrolled FISTA

- The set of trainable parameters of $\mathcal{F}$ are the entries of the phase matrix $\Phi $. Those are shared for each layer. Note that the set of trainable and non-trainable parameters can be varied but we will restrict our investigations to the case where $\Phi $ is trainable.
- The starting value ${\mathbf{q}}^{\left(0\right)}$ is the data input of the neural network and is transformed to the pair $\left({\mathbf{q}}^{\left(0\right)},{\mathbf{q}}^{\left(0\right)}\right)$ before the first layer.
- The network consists of ${n}_{\mathrm{iter}}$ layers, where each layer represents one FISTA iteration. In each layer, the following two operations are applied to the pair $\left(\mathbf{q},{\mathbf{q}}^{\prime}\right)$:
- 1.
- The current pair is propagated forward by linear operations, encoded in $\mathrm{ags}$ (8), which essentially depend on the trainable parameters $\Phi $ and the non-trainable parameters $\mathbf{R},{\mathbf{C}}^{\mathrm{obs}},\tau ,{\beta}_{n},{\alpha}_{1},{\alpha}_{2}$.
- 2.
- After the propagation step, the activation function $\mathrm{proxAc}$ (9) is applied.

- After the last layer, the output pair $\left({\mathbf{q}}^{\left({n}_{\mathrm{iter}}\right)},\phantom{\rule{4pt}{0ex}}{\mathbf{q}}^{\left({n}_{\mathrm{iter}-1}\right)}\right)$ is transformed to the final output ${\mathbf{q}}^{\left({n}_{\mathrm{iter}}\right)}$.

#### 4.2. Constrained Residual Minimization

## 5. Numerical Examples

#### 5.1. Systematic Modeling Error

#### 5.2. Random Modeling Error

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Sketch of the unrolled FISTA network $\mathcal{F}$. One forward pass through the network is equivalent to the application of ${n}_{\mathrm{iter}}$ FISTA iterations. Green bounding boxes indicate one iteration of the FISTA algorithm, trainable parameters are marked in blue.

**Figure 2.**Reference solutions for numerical example. (

**a**) Exact solution; (

**b**) FISTA solution with exact phase (${\u2225\mathbf{q}-{\mathbf{q}}^{\u2020}\u2225}_{2}=7.49$).

**Figure 3.**Solutions for numerical example with systematic phase perturbation. (

**a**) FISTA solution with perturbed phase (${\u2225\mathbf{q}-{\mathbf{q}}^{\u2020}\u2225}_{2}=106.61$); (

**b**) FISTA solution after phase error compensation (${\u2225\mathbf{q}-{\mathbf{q}}^{\u2020}\u2225}_{2}=104.46$).

**Figure 4.**Solutions for numerical example with random phase perturbation. (

**a**) FISTA solution with perturbed phase (${\u2225\mathbf{q}-{\mathbf{q}}^{\u2020}\u2225}_{2}=106.31$); (

**b**) FISTA solution after phase error compensation (${\u2225\mathbf{q}-{\mathbf{q}}^{\u2020}\u2225}_{2}=6.56$).

**Figure 5.**Cost and error graphs for several Helmholtz numbers. The colored area indicates the interval $\overline{\mathrm{cos}\mathrm{t}\left(n\right)}\pm 2\phantom{\rule{0.166667em}{0ex}}{\mathrm{cos}\mathrm{t}}_{\mathrm{dev}}^{\pm}\left(n\right)$ and $\overline{\mathrm{err}\left(n\right)}\pm 2\phantom{\rule{0.166667em}{0ex}}{\mathrm{err}}_{\mathrm{dev}}^{\pm}\left(n\right)$.

Optimization algorithm | Gradient descent with Nesterov momentum (p. 353, [40]) |
---|---|

Learning rate | lr$\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}{10}^{-3}$ |

Momentum parameter | momentum$\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}0.9$ |

Gradient descent steps | ${n}_{\mathrm{grad}}=200$ |

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**MDPI and ACS Style**

Raumer, H.-G.; Ernst, D.; Spehr, C.
Compensation of Modeling Errors for the Aeroacoustic Inverse Problem with Tools from Deep Learning. *Acoustics* **2022**, *4*, 834-848.
https://doi.org/10.3390/acoustics4040050

**AMA Style**

Raumer H-G, Ernst D, Spehr C.
Compensation of Modeling Errors for the Aeroacoustic Inverse Problem with Tools from Deep Learning. *Acoustics*. 2022; 4(4):834-848.
https://doi.org/10.3390/acoustics4040050

**Chicago/Turabian Style**

Raumer, Hans-Georg, Daniel Ernst, and Carsten Spehr.
2022. "Compensation of Modeling Errors for the Aeroacoustic Inverse Problem with Tools from Deep Learning" *Acoustics* 4, no. 4: 834-848.
https://doi.org/10.3390/acoustics4040050