# 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

- Vizzaccaro, A.; Givois, A.; Longobardi, P.; Shen, Y.; Deü, J.F.; Salles, L.; Touzé, C.; Thomas, O. Non-intrusive reduced order modeling for the dynamics of geometrically nonlinear flat structures using three-dimensional finite elements. Comput. Mech.
**2020**, 66, 1293–1319. [Google Scholar] [CrossRef] - Touzé, C.; Vizzaccaro, A.; Thomas, O. Model order reduction methods for geometrically nonlinear structures: A review of nonlinear techniques. Nonlinear Dyn.
**2021**, 105, 1141–1190. [Google Scholar] [CrossRef] - Kerschen, G.; Golinval, J.C.; Vakakis, A.F.; Bergman, L.A. The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: An overview. Nonlinear Dyn.
**2005**, 41, 147–169. [Google Scholar] [CrossRef] - Amabili, M.; Touzé, C. Reduced-order models for nonlinear vibrations of fluid-filled circular cylindrical shells: Comparison of POD and asymptotic nonlinear normal modes methods. J. Fluids Struct.
**2007**, 23, 885–903. [Google Scholar] [CrossRef][Green Version] - Amabili, M.; Sarkar, A.; Paıdoussis, M. Reduced-order models for nonlinear vibrations of cylindrical shells via the proper orthogonal decomposition method. J. Fluids Struct.
**2003**, 18, 227–250. [Google Scholar] [CrossRef] - Gobat, G.; Opreni, A.; Fresca, S.; Manzoni, A.; Frangi, A. Reduced order modeling of nonlinear microstructures through Proper Orthogonal Decomposition. Mech. Syst. Signal Process.
**2022**, 171, 108864. [Google Scholar] [CrossRef] - Frangi, A.; Gobat, G. Reduced order modeling of the non-linear stiffness in MEMS resonators. Int. J. Non-Linear Mech.
**2019**, 116, 211–218. [Google Scholar] [CrossRef] - Vizzaccaro, A.; Salles, L.; Touzé, C. Comparison of nonlinear mappings for reduced-order modeling of vibrating structures: Normal form theory and quadratic manifold method with modal derivatives. Nonlinear Dyn.
**2021**, 103, 3335–3370. [Google Scholar] [CrossRef] - Shaw, S.; Pierre, C. Non-linear normal modes and invariant manifolds. J. Sound Vib.
**1991**, 150, 170–173. [Google Scholar] [CrossRef][Green Version] - Shaw, S.W.; Pierre, C. Normal modes for non-linear vibratory systems. J. Sound Vib.
**1993**, 164, 85–124. [Google Scholar] [CrossRef][Green Version] - Ponsioen, S.; Pedergnana, T.; Haller, G. Automated computation of autonomous spectral submanifolds for nonlinear modal analysis. J. Sound Vib.
**2018**, 420, 269–295. [Google Scholar] [CrossRef][Green Version] - Vizzaccaro, A.; Shen, Y.; Salles, L.; Blahoš, J.; Touzé, C. Direct computation of nonlinear mapping via normal form for reduced-order models of finite element nonlinear structures. Comput. Methods Appl. Mech. Eng.
**2021**, 384, 113957. [Google Scholar] [CrossRef] - Opreni, A.; Vizzaccaro, A.; Frangi, A.; Touzé, C. Model Order Reduction based on Direct Normal Form: Application to Large Finite Element MEMS Structures Featuring Internal Resonance. Nonlinear Dyn.
**2021**, 105, 1237–1272. [Google Scholar] [CrossRef] - Vizzaccaro, A.; Opreni, A.; Salles, L.; Frangi, A.; Touzé, C. High order direct parametrisation of invariant manifolds for model order reduction of finite element structures: Application to large amplitude vibrations and uncovering of a folding point. Nonlinear Dyn.
**2022**, 110, 525–571. [Google Scholar] [CrossRef] - Jain, S.; Haller, G. How to compute invariant manifolds and their reduced dynamics in high-dimensional finite element models. Nonlinear Dyn.
**2022**, 107, 1417–1450. [Google Scholar] [CrossRef] - Opreni, A.; Vizzaccaro, A.; Touzé, C.; Frangi, A. High order direct parametrisation of invariant manifolds for model order reduction of finite element structures: Application to generic forcing terms and parametrically excited systems. Nonlinear Dyn. 2022; accepted for publication. [Google Scholar]
- Haghighat, E.; Raissi, M.; Moure, A.; Gomez, H.; Juanes, R. A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics. Comput. Methods Appl. Mech. Eng.
**2021**, 379, 113741. [Google Scholar] [CrossRef] - Kim, Y.; Choi, Y.; Widemann, D.; Zohdi, T. A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder. J. Comput. Phys.
**2022**, 451, 110841. [Google Scholar] [CrossRef] - Raissi, M.; Perdikaris, P.; Karniadakis, G.E. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys.
**2019**, 378, 686–707. [Google Scholar] [CrossRef] - Guo, M.; Hesthaven, J.S. Reduced order modeling for nonlinear structural analysis using Gaussian process regression. Comput. Methods Appl. Mech. Eng.
**2018**, 341, 807–826. [Google Scholar] [CrossRef] - Guo, M.; Hesthaven, J.S. Data-driven reduced order modeling for time-dependent problems. Comput. Methods Appl. Mech. Eng.
**2019**, 345, 75–99. [Google Scholar] [CrossRef] - Lee, K.; Carlberg, K.T. Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders. J. Comput. Phys.
**2020**, 404, 108973. [Google Scholar] [CrossRef][Green Version] - Fries, W.D.; He, X.; Choi, Y. Lasdi: Parametric latent space dynamics identification. Comput. Methods Appl. Mech. Eng.
**2022**, 399, 115436. [Google Scholar] [CrossRef] - Fresca, S.; Dede, L.; Manzoni, A. A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs. J. Sci. Comput.
**2021**, 87, 61. [Google Scholar] [CrossRef] - Fresca, S.; Manzoni, A. POD-DL-ROM: Enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition. Comput. Methods Appl. Mech. Eng.
**2022**, 388, 114181. [Google Scholar] [CrossRef] - Fresca, S.; Gobat, G.; Fedeli, P.; Frangi, A.; Manzoni, A. Deep learning-based reduced order models for the real-time simulation of the nonlinear dynamics of microstructures. Int. J. Numer. Methods Eng.
**2022**, 123, 4749–4777. [Google Scholar] [CrossRef] - Kaiser, E.; Kutz, J.N.; Brunton, S.L. Sparse identification of nonlinear dynamics for model predictive control in the low-data limit. Proc. R. Soc. A
**2018**, 474, 20180335. [Google Scholar] [CrossRef] [PubMed][Green Version] - Brunton, S.L.; Proctor, J.L.; Kutz, J.N. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl. Acad. Sci. USA
**2016**, 113, 3932–3937. [Google Scholar] [CrossRef][Green Version] - Brunton, S.L.; Proctor, J.L.; Kutz, J.N. Sparse identification of nonlinear dynamics with control (SINDYc). IFAC-PapersOnLine
**2016**, 49, 710–715. [Google Scholar] [CrossRef] - Champion, K.; Lusch, B.; Kutz, J.N.; Brunton, S.L. Data-driven discovery of coordinates and governing equations. Proc. Natl. Acad. Sci. USA
**2019**, 116, 22445–22451. [Google Scholar] [CrossRef][Green Version] - Simpson, T.; Dervilis, N.; Chatzi, E. Machine Learning Approach to Model Order Reduction of Nonlinear Systems via Autoencoder and LSTM Networks. J. Eng. Mech.
**2021**, 147, 04021061. [Google Scholar] [CrossRef] - Li, S.; Yang, Y. Data-driven identification of nonlinear normal modes via physics-integrated deep learning. Nonlinear Dyn.
**2021**, 106, 3231–3246. [Google Scholar] [CrossRef] - Fresca, S.; Manzoni, A.; Dedè, L.; Quarteroni, A. Deep learning-based reduced order models in cardiac electrophysiology. PLoS ONE
**2020**, 15, e0239416. [Google Scholar] [CrossRef] - Opreni, A.; Boni, N.; Carminati, R.; Frangi, A. Analysis of the nonlinear response of piezo-micromirrors with the harmonic balance method. Actuators
**2021**, 10, 21. [Google Scholar] [CrossRef] - Detroux, T.; Renson, L.; Masset, L.; Kerschen, G. The harmonic balance method for bifurcation analysis of large-scale nonlinear mechanical systems. Comput. Methods Appl. Mech. Eng.
**2015**, 296, 18–38. [Google Scholar] [CrossRef][Green Version] - Goodfellow, I.; Bengio, Y.; Courville, A. Deep Learning; MIT Press: Cambridge, MA, USA, 2016. [Google Scholar]
- LeCun, Y.; Bottou, L.; Bengio, Y.; Haffner, P. Gradient-based learning applied to document recognition. Proc. IEEE
**1998**, 86, 2278–2324. [Google Scholar] [CrossRef][Green Version] - Hinton, G.E.; Zemel, R. Autoencoders, minimum description length and Helmholtz free energy. In Advances in Neural Information Processing Systems; MIT Press: Cambridge, MA, USA, 1993; Volume 6. [Google Scholar]
- Halko, N.; Martinsson, P.; Tropp, J.A. Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev.
**2011**, 53, 217–288. [Google Scholar] [CrossRef] - Cabré, X.; Fontich, E.; de la Llave, R. The parameterization method for invariant manifolds I: Manifolds associated to non-resonant subspaces. Indiana Univ. Math. J.
**2003**, 52, 283–328. [Google Scholar] [CrossRef][Green Version] - Cabré, X.; Fontich, E.; de la Llave, R. The parameterization method for invariant manifolds II: Regularity with respect to parameters. Indiana Univ. Math. J.
**2003**, 52, 329–360. [Google Scholar] [CrossRef][Green Version] - Cabré, X.; Fontich, E.; De La Llave, R. The parameterization method for invariant manifolds III: Overview and applications. J. Differ. Equ.
**2005**, 218, 444–515. [Google Scholar] [CrossRef][Green Version] - Haro, A.; Canadell, M.; Figueras, J.L.; Luque, A.; Mondelo, J.M. The parameterization method for invariant manifolds. Appl. Math. Sci.
**2016**, 195. [Google Scholar] - Opreni, A.; Vizzaccaro, A.; Touzé, C.; Frangi, A. MORFEInvariantManifold. 2022. Available online: https://github.com/aopreni/MORFEInvariantManifold.jl (accessed on 3 March 2023).
- Doedel, E.J.; Champneys, A.R.; Dercole, F.; Fairgrieve, T.F.; Kuznetsov, Y.A.; Oldeman, B.; Paffenroth, R.; Sandstede, B.; Wang, X.; Zhang, C. AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations; Concordia University: Montreal, QC, Canada, 2007. [Google Scholar]
- Guillot, L.; Cochelin, B.; Vergez, C. A Taylor series-based continuation method for solutions of dynamical systems. Nonlinear Dyn.
**2019**, 98, 2827–2845. [Google Scholar] [CrossRef][Green Version] - Guillot, L.; Lazarus, A.; Thomas, O.; Vergez, C.; Cochelin, B. A purely frequency based Floquet-Hill formulation for the efficient stability computation of periodic solutions of ordinary differential systems. J. Comput. Phys.
**2020**, 416, 109477. [Google Scholar] [CrossRef] - Krack, M.; Gross, J. Harmonic Balance for Nonlinear Vibration Problems; Springer Nature: Cham, Switzerland, 2019; Volume 1. [Google Scholar]
- Dhooge, A.; Govaerts, W.; Kuznetsov, Y.A.; Mestrom, W.; Riet, A.; Sautois, B. MATCONT and CL MATCONT: Continuation Toolboxes in Matlab; Universiteit Gent: Gent, Belgium; University of Twente: Enschede, The Netherlands; Utrecht University: Utrecht, The Netherlands, 2006. [Google Scholar]
- Dankowicz, H.; Schilder, F. Recipes for Continuation; SIAM (Society for Industrial and Applied Mathematics): Philadelphia, PA, USA, 2013. [Google Scholar]
- Veltz, R. BifurcationKit. jl. HAL. 2020. Available online: https://hal.inria.fr/hal-02902346 (accessed on 3 March 2023).
- Brent, S.; James, A.H.; Nicholas, M.I.; Michael, D.K. Lidar Sensor. US20200033449A1. Available online: https://patents.google.com/patent/US20200033449A1/en?q=Lidar+sensor&oq=Lidar+sensor+ (accessed on 3 March 2023).
- Laser Beam Scanning. Available online: https://www.st.com/content/dam/AME/2019/developers-conference-2019/presentations/STDevCon19_2.4-6-Laser-Beam-Scanners-ST.pdf (accessed on 3 March 2023).
- Microsoft. Microsoft Hololens. Available online: https://www.microsoft.com/it-it/hololens (accessed on 3 March 2023).
- Corigliano, A.; De Masi, B.; Frangi, A.; Comi, C.; Villa, A.; Marchi, M. Mechanical characterization of polysilicon through on-chip tensile tests. J. Microelectromech. Syst.
**2004**, 13, 200–219. [Google Scholar] [CrossRef] - Gobat, G.; Guillot, L.; Frangi, A.; Cochelin, B.; Touzé, C. Backbone curves, Neimark–Sacker boundaries and appearance of quasi-periodicity in nonlinear oscillators: Application to 1: 2 internal resonance and frequency combs in MEMS. Meccanica
**2021**, 56, 1937–1969. [Google Scholar] [CrossRef] - Opreni, A.; Vizzaccaro, A.; Boni, N.; Carminati, R.; Mendicino, G.; Touzé, C.; Frangi, A. Fast and Accurate Predictions of MEMS Micromirrors Nonlinear Dynamic Response Using Direct Computation of Invariant Manifolds. In Proceedings of the 2022 IEEE 35th International Conference on Micro Electro Mechanical Systems Conference (MEMS), Tokyo, Japan, 9–13 January 2022; pp. 491–494. [Google Scholar]
- Frangi, A.; De Masi, B.; Confalonieri, F.; Zerbini, S. Threshold shock sensor based on a bistable mechanism: Design, modeling, and measurements. J. Microelectromech. Syst.
**2015**, 24, 2019–2026. [Google Scholar] [CrossRef] - Zega, V.; Gobat, G.; Fedeli, P.; Carulli, P.; Frangi, A.A. Reduced Order Modeling in a Mems Arch Resonator Exhibiting 1: 2 Internal Resonance. In Proceedings of the 2022 IEEE 35th International Conference on Micro Electro Mechanical Systems Conference (MEMS), Tokyo, Japan, 9–13 January 2022; pp. 499–502. [Google Scholar]
- Sharpe, W.N.; Yuan, B.; Vaidyanathan, R.; Edwards, R.L. Measurements of Young’s modulus, Poisson’s ratio, and tensile strength of polysilicon. In Proceedings of the IEEE the Tenth Annual International Workshop on Micro Electro Mechanical Systems. An Investigation of Micro Structures, Sensors, Actuators, Machines and Robots, Nagoya, Japan, 26–30 January 1997; pp. 424–429. [Google Scholar]
- Fresca, S.; Manzoni, A. Real-time simulation of parameter-dependent fluid flows through deep learning-based reduced order models. Fluids
**2021**, 6, 259. [Google Scholar] [CrossRef] - Coventor Inc., A Lam Research Company. Coventor MEMS+
^{TM}. Available online: https://www.coventor.com/ (accessed on 3 March 2023). - Parent, A.; Krust, A.; Lorenz, G.; Favorskiy, I.; Piirainen, T. Efficient nonlinear simulink models of MEMS gyroscopes generated with a novel model order reduction method. In Proceedings of the 2015 Transducers-2015 18th International Conference on Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS), Anchorage, AK, USA, 21–25 June 2015; pp. 2184–2187. [Google Scholar]
- Parent, A.; Krust, A.; Lorenz, G.; Piirainen, T. A novel model order reduction approach for generating efficient nonlinear verilog-a models of mems gyroscopes. In Proceedings of the 2015 IEEE International Symposium on Inertial Sensors and Systems (ISISS) Proceedings, Hapuna Beach, HI, USA, 23–26 March 2015; pp. 1–4. [Google Scholar]
- Ayazi, F.; Najafi, K. A HARPSS polysilicon vibrating ring gyroscope. J. Microelectromech. Syst.
**2001**, 10, 169–179. [Google Scholar] [CrossRef][Green Version] - Nitzan, S.H.; Zega, V.; Li, M.; Ahn, C.H.; Corigliano, A.; Kenny, T.W.; Horsley, D.A. Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope. Sci. Rep.
**2015**, 5, 9036. [Google Scholar] [CrossRef][Green Version] - Polunin, P.M.; Shaw, S.W. Self-induced parametric amplification in ring resonating gyroscopes. Int. J. Non-Linear Mech.
**2017**, 94, 300–308. [Google Scholar] [CrossRef] - Li, D.; Shaw, S.W.; Polunin, P.M. Computational modeling of nonlinear dynamics and its utility in MEMS gyroscopes. J. Struct. Dyn
**2022**, 1, 217–235. [Google Scholar] [CrossRef] - Nayfeh, A.H.; Mook, D.T.; Holmes, P. Nonlinear Oscillations; WILEY-VCH Verlag GmbH & Co. kgaa: Weinheim, Germany, 1980. [Google Scholar]
- Thomsen, J.J.; Thomsen, J.J.; Thomsen, J. Vibrations and Stability; Springer: Berlin/Heidelberg, Germany, 2003. [Google Scholar]
- Van der Avoort, C.; Van der Hout, R.; Bontemps, J.; Steeneken, P.; Le Phan, K.; Fey, R.; Hulshof, J.; Van Beek, J. Amplitude saturation of MEMS resonators explained by autoparametric resonance. J. Micromech. Microeng.
**2010**, 20, 105012. [Google Scholar] [CrossRef] - Gallacher, B.; Burdess, J.; Harish, K. A control scheme for a MEMS electrostatic resonant gyroscope excited using combined parametric excitation and harmonic forcing. J. Micromech. Microeng.
**2006**, 16, 320. [Google Scholar] [CrossRef]

**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.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## 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