# Reduced Order Modeling of Nonlinear Vibrating Multiphysics Microstructures with Deep Learning-Based Approaches

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

## 3. Data-Driven Reduced Order Modeling through DL-ROMs

- To map the FOM solutions in a low-dimensional coordinates vector representation (encoding stage), we use the encoder function of a convolutive autoencoder (CAE)$${\tilde{\mathbf{z}}}_{n}(t;\mathbf{\mu},{\mathbf{\theta}}_{E})={\mathbf{f}}_{n}^{E}(\mathbf{U}(t;\mathbf{\mu});{\mathbf{\theta}}_{E});$$
- To describe the system dynamics (reduced dynamics learning), the intrinsic coordinates of the ROM approximation are defined as$${\mathbf{z}}_{n}(t;\mathbf{\mu},{\mathbf{\theta}}_{DF})={\mathbf{\varphi}}_{n}^{DF}(t;\mathbf{\mu},{\mathbf{\theta}}_{DF}),$$
- To model the reduced nonlinear trial manifold ${\mathcal{S}}_{h}^{n}\approx \mathcal{S}$ (decoding stage), we employ the decoder function of a CAE [37,38], that is,$$\begin{array}{c}\hfill {\tilde{\mathcal{S}}}_{h}^{n}=\{{\mathbf{f}}_{h}^{D}({\mathbf{z}}_{n}(t;\mathbf{\mu},{\mathbf{\theta}}_{DF});{\mathbf{\theta}}_{D})\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}{\mathbf{z}}_{n}(t;\mathbf{\mu},{\mathbf{\theta}}_{DF})\in {\mathbb{R}}^{n}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.166667em}{0ex}}t\in [0,T)\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.166667em}{0ex}}\mathbf{\mu}\in \mathcal{P}\}\subset {\mathbb{R}}^{{N}_{h}},\end{array}$$

#### Arc-Length on the FRF as Control Parameter

## 4. Direct Parametrization of Invariant Manifolds

- (1)
- All of the coefficients and vectors in Equations (15) and (16) are computed offline starting from the FOM, in a preliminary phase, that can be seen as the equivalent of the encoding phase in the DL-ROM. The costly offline training can be performed on dedicated platforms and software. Both approaches hence are based on a FOM, which is typically built exploiting a finite element discretization.
- (2)
- (3)
- Equations (13) and (14) represent the parallel of the decoding phase, which reconstructs the global nodal fields starting from the online integration of the ROM. Thus, both ROMs can reproduce the same richness in details of the original FOM since the decoding phase allows one to generate a full field information.

#### Computation of the Steady State Response

`Auto07p`[45],

`Manlab`[46,47];

`Nlvib`[48],

`MATCONT`[49],

`COCO`[50] and

`BifurcationKit`[51], an emerging toolkit for Julia language. All of the continuation solutions discussed in this work have been obtained using

`MATCONT`.

## 5. Applications

#### 5.1. Reconstruction of the Whole Response

#### 5.2. Computation of Manifolds

#### 5.3. Micromirror

#### 5.4. Internal Resonance in a Shallow Arch

#### 5.5. Electromechanical Disk Resonating Gyroscope

^{TM}[62,63,64], a leading tool for the analysis of MEMS. The original model available in the software, inspired by the device proposed in [65], has been slightly modified in our benchmarks, and all of the details are provided in what follows.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**POD-DL-ROM architecture. Starting from the FOM solution $\mathbf{U}(t;\mathbf{\mu})$, the intrinsic coordinates ${\mathbf{V}}_{N}^{T}{\mathbf{u}}_{h}(t;\mathbf{\mu})$ are computed by means of rSVD; the neural network provides their approximation ${\tilde{\mathbf{u}}}_{N}(t;\mathbf{\mu})$ as output. The reconstructed solution ${\tilde{\mathbf{u}}}_{h}(t;\mathbf{\mu})$ is then recovered through the basis matrix.

**Figure 2.**Arch-length abscissa. (

**a**) The arch-length abscissa is first computed along each FRF. (

**b**,

**c**) The new $\omega \left(s\right)$ and $A\left(s\right)$ are single valued functions. Peaks and valleys of the FRF are aligned by rescaling the arc-length abscissa in four different regions, separated by the coloured dots.

**Figure 3.**Schematic comparison between DPIM and DL methods. One can identify a pre-processing stage; an encoding stage; a reduced dynamics solution stage; and, finally, a decoding stage. For DL methods, in this scheme we distinguish between black box approaches for the reduced order dynamics, as in the DL-ROM [24], and model based approaches, as in SINDy proposed by Brunton et al. [28].

**Figure 4.**Micromirror: (

**a**) optical photo, (

**b**) schematic view and (

**c**) detail of the boundary conditions.

**Figure 6.**Micromirror. FRFs of the opening angle for different excitation levels $\beta $. (

**a**) Comparison between the FRFs obtained with the DPIM and the POD-DL-ROM using p = 1:3. (

**b**,

**c**) Enlarged views of the peaks in the response. The training $\beta $ values are labelled in light blue, while the testing ones are in red. The FRFs here reported have been obtained with a testing dataset made of 112,700 instances. The inquiry of the POD-DL-ROM is performed in less than 0.2 s using a Tesla V100 32 GB GPU and an implementation in the Tensorlow DL framework.

**Figure 8.**Micromirror. Invariant manifolds for the modal displacement ${u}_{1}$ in terms of the modal displacement ${u}_{3}$ and velocity ${v}_{3}$ of the master mode, for $\beta =3.0$. The DPIM manifold is the smooth grey surface, while orbits from the POD-DL-ROM are traced as lines. (

**a**,

**d**) Different views for $p=1$. (

**b**,

**e**) Different views for $p=2$. (

**c**,

**f**) Different views for $p=3$. It should be noted that the POD-DL-ROM manifolds for $p=1$ are single curvature surfaces.

**Figure 9.**Micromirror. Invariant manifolds for the modal displacement ${u}_{4}$ in terms of the modal displacement ${u}_{3}$ and velocity ${v}_{3}$ of the master mode, for $\beta =3.0$. The DPIM manifold is the smooth grey surface, while orbits from the POD-DL-ROM are traced as lines. (

**a**,

**d**) Different views for $p=1$. (

**b**,

**e**) Different views for $p=2$. (

**c**,

**f**) Different views for $p=3$. Additionally, in this case the POD-DL-ROM manifolds for $p=1$ are single curvature surfaces.

**Figure 11.**Shallow arch. DLRM FRFs of the mid-span amplitude d and comparison with the reference DPIM FRF. (

**a**) FRFs obtained with 2 and 3 reduced variables. (

**b**,

**c**) Details of the FRFs in (

**a**) showing the excellent accuracy. (

**d**) FRF obtained with 1 reduced variable. Consistently with the presence of an interaction between to modes, one reduced variable in the POD-DL-ROM cannot describe the full evolution. The FRFs here reported have been obtained with a testing dataset made of 241,339 instances. The inquiry of the POD-DL-ROM is performed in less than 0.5 s using a Tesla V100 32 GB GPU and an implementation in the Tensorlow DL framework.

**Figure 12.**Shallow arch. Convergence of the analysis increasing the number of reduced variables. (

**a**,

**b**) Error norms ${E}_{i}^{r}$ and ${E}_{i}$, Equation (18), for $p=1\phantom{\rule{-0.166667em}{0ex}}:\phantom{\rule{-0.166667em}{0ex}}5$. (

**c**,

**d**) FRFs of the master modal amplitudes ${A}_{1},{A}_{6}$, $p=2\phantom{\rule{-0.166667em}{0ex}}:\phantom{\rule{-0.166667em}{0ex}}5$. (

**e**,

**f**) FRFs of the slave modal amplitudes ${A}_{11},{A}_{13}$, $p=2\phantom{\rule{-0.166667em}{0ex}}:\phantom{\rule{-0.166667em}{0ex}}5$. For the sake of clarity only the testing $\beta $ values are considered and the FRFs for $p=1$ are omitted.

**Figure 13.**Shallow arch. Trajectories ${u}_{6},{u}_{1},{v}_{1}$ in a restricted master space. (

**a**–

**d**) POD-DL-ROM orbits are superposed to the DPIM manifolds for $p=2,3$. The orbits for $p=1$ are only presented on (

**b**,

**c**) and elsewhere omitted for clarity. The master orbits are already at convergence with $p=2$ (see Figure 12). The colored dots refer to the portion of FRF targeted, see Figure 14.

**Figure 14.**Shallow arch. (

**a**) Reference FRF split in different color regions. (

**b**) Reference DPIM envelop of orbits where colors denote the envelops for the regions identified in (

**a**). The interaction between the two master modes leads to folding of the surface.

**Figure 15.**Shallow arch. Trajectories ${u}_{13},{u}_{1},{v}_{1}$ in a restricted master space. (

**a**–

**d**) POD-DL-ROM orbits are superposed to the DPIM manifolds for $p=2$ and 4. Other p solutions are omitted for sake of clarity and for $p>4$ orbits almost coincide (see Figure 12). The colored dots refer to the respective FRF region, see Figure 14.

**Figure 17.**DRG: first linear eigenmodes. Due to the symmetry properties, many of the eigenmodes come in degenerate couples sharing the same eigenfrequency and differing only by a rotation around the z axis. Modes 7 and 8 are, respectively, the drive and sense modes.

**Figure 18.**DRG. (

**a**,

**c**) FRFs of the radial displacement of the node indicated by a red circle in (

**b**), which is representative of the drive mode, i.e., mode 7. (

**c**) Comparison of the $p=1$ case and the FOM, clearly highlighting that main dynamic features are not represented adequately with one reduced variable. (

**a**) Comparisons for $p=2,3$. The curves below a threshold ${V}_{\mathrm{AC}}$ correspond to a simple harmonic resonance of the softening drive mode; a plateau starts developing when the sense mode gets autoparametrically activated. Data for the sense mode are collected in (

**d**,

**f**) illustrating the FRFs of the radial displacement of the node indicated by a red circle in (

**e**), which is representative of the sense mode. These FRFs confirm the strong and explosive mode interaction according to which the sense mode reaches abruptly amplitudes of the same order as the drive one. The FRFs here reported have been obtained with a testing dataset made of 1,398,000 instances. The inquiry of the POD-DL-ROM is performed in less than 3 s using a Tesla V100 32 GB GPU and an implementation in the Tensorlow DL framework.

**Figure 19.**DRG. Convergence of modal coordinates when increasing the number of reduced variables. (

**a**,

**b**) Error norms ${E}_{i}^{r}$ and ${E}_{i}$, Equation (18), for $p=1\phantom{\rule{-0.166667em}{0ex}}:\phantom{\rule{-0.166667em}{0ex}}5$. An increase of p over 2 has no effects on the master modes but improves the representation of slave modes. (

**c**,

**d**) FRFs of the master modal amplitudes ${A}_{7},{A}_{8}$, $p=1\phantom{\rule{-0.166667em}{0ex}}:\phantom{\rule{-0.166667em}{0ex}}3$. (

**e**,

**f**) Zoom of selected regions in the FRFs.

**Figure 20.**DRG. Envelop of the computed orbits in the ${u}_{8},{u}_{7},{v}_{7}$ space for the three branches, marked with different colours, of the FRFs highlighted in (

**a**,

**b**) (for mode 7 and mode 8, respectively). The whole Conventor MEMS+ envelop is presented in (

**c**). The comparison is split in three subregions in (

**d**–

**f**) identified by the colored dots.

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## Share and Cite

**MDPI and ACS Style**

Gobat, G.; Fresca, S.; Manzoni, A.; Frangi, A.
Reduced Order Modeling of Nonlinear Vibrating Multiphysics Microstructures with Deep Learning-Based Approaches. *Sensors* **2023**, *23*, 3001.
https://doi.org/10.3390/s23063001

**AMA Style**

Gobat G, Fresca S, Manzoni A, Frangi A.
Reduced Order Modeling of Nonlinear Vibrating Multiphysics Microstructures with Deep Learning-Based Approaches. *Sensors*. 2023; 23(6):3001.
https://doi.org/10.3390/s23063001

**Chicago/Turabian Style**

Gobat, Giorgio, Stefania Fresca, Andrea Manzoni, and Attilio Frangi.
2023. "Reduced Order Modeling of Nonlinear Vibrating Multiphysics Microstructures with Deep Learning-Based Approaches" *Sensors* 23, no. 6: 3001.
https://doi.org/10.3390/s23063001