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Download the full whitepaper “Reduced Order Modeling of Nonlinear Vibrating Multiphysics Microstructures with Deep Learning-Based Approaches” to learn more.

When designing a MEMS-based gyroscope, device behavior in the nonlinear regime must be predicted and controlled to ensure that the gyroscope works as expected and can be produced at a reasonable cost. To predict and control non-linear gyroscope behavior, the design must be optimized, and online control mechanisms need to be implemented. Two key capabilities must be included in a MEMS model and its simulation environment to complete these tasks: the ability to obtain real-time feedback and the ability to parameterize the MEMS model. This is a major challenge, since predicting behavior or obtaining real-time feedback on a nonlinear system is not a trivial task. Non-linear models cannot be easily described using a simple set of rules and tend to have highly complex behavior. Machine learning, and more specifically deep learning-based modeling, represents a potentially novel technique to predict and control MEMS gyroscope behavior in the non-linear regime.

We apply a deep learning-based model to a MEMS gyroscope. This model fits in the class of models known as “reduced order models” (or “ROMs”). ROMs are simplified models of a complex system that retain their model accuracy but are more computationally efficient (the original, more complex models are referred to as Full Order Models or “FOMs”). We initially created a model of a Disk Resonating Gyroscope (or “DRG”) from Ayazi et al. [10] using Coventor *MEMS+*™. The parametrized *MEMS+* model was developed to mimic the real-time, nonlinear behavior of the selected DRG.

Disk Resonating Gyroscopes are normally configured as concentric or external rings with connecting spokes and anchors (Figure 1). DRGs exhibit inherent mode matching, high thermal stability, low anchor loss, high quality factors (Q), and insensitivity to environmental issues, making them highly accurate and desirable in many navigation applications [12]. In our application, the DRG external ring is modelled in *MEMS+* with 32 Euler-Bernoulli beams, while each arch suspension is comprised of 2 Euler-Bernoulli beams. The center support is a fixed rigid body. A series of parallel-plate electrodes are placed around the ring. The main nonlinearities in DRG device behavior originate in the electromechanical coupling. In Figure 1, each gyroscope component is highlighted using a different color.

This gyroscope is sensitive to angular velocities around the z-axis. The drive mode of the gyroscope is excited by imposing an oscillating bias voltage to the blue electrodes and an opposite phase bias voltage to the yellow electrodes. A constant potential bias voltage of 𝑉𝐷𝐶 = 1𝑉 is imposed on the gyroscope “spokes” (red structure). All the remaining electrodes are grounded. When the device is subjected to an external angular velocity, the Coriolis effect exerts in-plane forces on the ring and activates the sense mode which provides navigation data. Unfortunately, the electrostatic nonlinearities of the parallel plate excitation induce a coupling of the sense and drive modes , which causes an auto-parametric resonance behavior. This non-linear behavior is a major challenge, since the device can experience complex dynamics that may affect overall performance and prevent effective control of the gyroscope during operation. The goal of our deep learning work was to build a model that was able to reproduce the device’s nonlinear dynamics with respect to the input excitation frequency 𝜔 and amplitude 𝑉𝐴𝐶 .

To reproduce the non-linear behavior of the DRG, we created a set of deep learning-based reduced order models (DL-ROMs). We were able to create these highly accurate, parameterized models starting from simulation data supplied by *MEMS+*. We will now provide a short outline of our methodology.

Deep learning is being extensively used in many applications to extrapolate results from patterns hidden in data. A currently popular use of deep learning techniques is to create text, images, and pieces of art by starting from a description of the desired output (see Figure 2a). This type of deep learning approach can be found in applications such as ChatGPT [7] and Midjourney [8]. In these applications, the DL algorithm is trained to learn about an image that corresponds to a given description and the pattern underlying it. How does this work? For these types of applications, the DL algorithm exploits databases of text and pictures that have been associated with specific tags and descriptions, e.g., an image of a dove corresponds to the tag “dove” and a picture of a human face corresponds to the tag “face” etc. A key point of DL approaches is the capability to produce scenarios unseen or not included in the available dataset. Once the algorithm has been trained, it can generate new text and pictures by exploiting the rules learned during the training phase. For example, if you ask a deep learning system to draw a “mechanical dove”, it can produce an abstract but reasonable image of something that does not exist [9]. This impressive capability to learn rules that are hidden in data, and to infer new results, represents an appealing feature in not only in consumer applications but also in scientific computing and simulation.

We obtained input data for the deep learning system using the output of simulations run in CoventorMP (MEMS+). The deep learning approach in this study involved trying to learn the rules (and recognize patterns) that underlie the simulation data used to train our DRG reduced order model (Figure 2b). In effect, we used *MEMS+* simulation data to create a “black box” model of the DRG using deep learning. In our case, MEMS simulation data is governed by the physical laws built into the MEMS design software (i.e. – finite element modeling). Unfortunately, the computational burden to create a real-time parameterized simulation of a Disk Resonating Gyroscope using finite element modeling is very high. This is where machine learning can help.

Our deep learning method combined two elements: a convolutional autoencoder and a deep feedforward neural network (DFNN). Details on the functions and use of these constructs can be found in References [1-6].

We trained our DL-ROM using 149,900 data snapshots from *MEMS*+ simulations of the DRG. We gathered 100 data output values from our *MEMS+* model per period, using the following operating parameter ranges to generate the training data:

𝑓 (frequency) = [32.620; 32.6498] kHz, 𝑉𝐴𝐶 = [2; 3; 4; 5; 6; 7; 8; 9; 10] mV.

The results generated by the deep learning-based ROM are shown in Figure 3, which includes the response of the DRG at a new voltage value (𝑉𝐴𝐶 = 9.5mV) that was not included in the training data. In Figure 3, we have compared the results from the deep learning approach to those obtained using actual reference data. The frequency response of the drive motion 𝑑_{1} (represented by the radial displacement of the node), is indicated by the red circle in Figure 3. The frequency response curves show a softening response of the drive mode due to electrostatic nonlinearities. In addition, a plateau in the response curve develops above a certain threshold amplitude as a result of auto parametric resonance. As can be seen in the graph, the DL-ROM almost perfectly reproduces the dynamics and the behavior of the reference data (the actual device data). It is worth highlighting that the results being reported are limited to a single data point on the device. Nonetheless, the DL-ROM approach could potentially reproduce the behavior at different locations and under different operating parameter ranges for the gyroscope, and in general for any field that is provided during the training stage.

In this article, we demonstrated how deep learning algorithms, and in general data driven methods, can be effectively used to create parametrized MEMS models for design and online control purposes. We used the simulation capabilities of Coventor *MEMS+* to generate data used in a deep learning model, and then exploited a deep learning system to create a highly accurate, real-time, reduced order model. The DL-ROM approach provides a real-time simulation tool that can infer new and different results beyond the training data, and accurately reflect non-linearities of the MEMS device. In our example, we limited ourselves to two parameter changes, but in principle more parameters could have been included in our study. Due to the robust nature, physical significance, accuracy, and reliability of these deep learning-based methods, we can foresee their widespread future use in MEMS design.

Download the full whitepaper “Reduced Order Modeling of Nonlinear Vibrating Multiphysics Microstructures with Deep Learning-Based Approaches” to learn more.

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*Mechanical Systems and Signal Processing*171 (2022): 108864. - Fresca, S. et al. “Deep learning‐based reduced order models for the real‐time simulation of the nonlinear dynamics of microstructures.”
*International Journal for Numerical Methods in Engineering*20 (2022): 4749-4777. - Gobat, G. et al. “Virtual twins of nonlinear vibrating multiphysics microstructures: physics-based versus deep learning-based approaches.”
*arXiv preprint arXiv:2205.05928*(2022). - Fresca, S. et al. “A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs.”
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*PloS one*15.10 (2020): e0239416. - https://chat.openai.com/
- https://midjourney.com/
- https://en.wikipedia.org/wiki/Midjourney
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