Deep learning for non-parameterized MEMS structural design

Machine learning (ML) has achieved broad success across fields by automatically learning relationships between inputs and outputs. Prior work in MEMS has applied ML to signal analysis and to structural design, typically in parameterized settings where a topology is chosen first and then geometric parameters are optimized. Here, the authors introduce a data-driven nonparameterized design approach that constructs structures voxel-by-voxel without constraining topology, vastly enlarging the design space but historically posing computational challenges for traditional ML. Deep learning (DL) can learn complex hidden patterns from large datasets via multilayer nonlinear representations, making it suitable for nonparameterized MEMS design. As a demonstration, the study targets disk-shaped MEMS resonators, focusing on two key properties: resonant modes/frequencies and the quality factor (Q), particularly the anchor-loss-limited Q (Q_anchor). Achieving a desired mode/frequency with high Q often requires extensive trial-and-error and costly simulations. The authors aim to replace much of this process with a trained DL model that predicts target properties directly from pixelated geometry, enabling rapid, experience-free screening of candidates. They represent geometries as binary images and label them with finite element analysis (FEA) results for frequency and Q_anchor. After training on tens of thousands of samples, the DL model predicts these properties accurately and orders of magnitude faster than FEA, facilitating data-driven MEMS structural design beyond resonators.

The introduction surveys applications of ML across disciplines and prior uses in MEMS. Earlier MEMS design studies typically employed parameterized approaches, selecting a base topology and optimizing a limited set of geometric parameters via ML. These methods, while effective, rely on substantial domain knowledge and constrain topology. In contrast, nonparameterized design—constructing structures voxel-by-voxel without topological constraints—has been explored but was historically computationally expensive for traditional ML due to the massive design space and data volume. The present work leverages DL to overcome these challenges by learning complex mappings from nonparameterized, pixelated inputs to physical properties, thereby enabling topology-unconstrained exploration.

System architecture: The pipeline has two phases. In training, a structure generator creates binary images (0 = void, 1 = solid) of resonator geometries; FEA provides ground-truth labels for resonant frequency (f) and anchor-loss-limited quality factor (Q_anchor). A DL model is trained over many epochs on these labeled samples. In testing, new generated structures are fed to the trained DL calculator to predict f and Q_anchor without FEA.

Geometry representation and generation: Disk-shaped resonators are represented as 100 × 100 binary images of the structural layer. Each voxel measures 0.44 µm × 0.44 µm in-plane and 0.5 µm in depth. Material is polysilicon with density ρ = 2.3 × 10³ kg/m³, Young’s modulus E = 150 GPa, and Poisson’s ratio ν = 0.29. Fixed parameters include the inner and outer ring diameters, the central anchor stem diameter, and the thicknesses of the disk layer and anchor stem. A mobile “agent,” defined as a 2 × 2 × 4 solid element region, performs Brownian-like random steps along cardinal directions within one quadrant, starting near the anchor stem and ending near the inner annulus. The total covered area is programmable. Folding the trajectory along two symmetry axes yields a symmetric connection between anchor and outer annulus, producing nonparameterized, pixelated topologies. Porosity is defined as the ratio of void pixels to total pixels in the design region.

Deep learning model: Several CNN-based architectures were considered (ResNet, DenseNet, EfficientNet) using PyTorch; a customized ResNet was ultimately selected. The network comprises input images, convolutional layers producing 2D feature maps, max pooling, residual blocks to mitigate degradation with depth, flattening to 1D, fully connected layers, and outputs. The model can output one or multiple targets (frequency and/or Q_anchor). Convolutions capture local pixel neighborhood effects; residual connections facilitate training deeper models.

FEA labeling and modal identification: Two FEA analyses are performed per structure. (1) Natural frequency analysis provides undamped eigenfrequencies, mode shapes, and an effective mass tensor m_eff with components m_ij^α quantifying participation in six motions: translations (XT, YT, ZT) and rotations (XR, YR, ZR). Modes are automatically identified by ranking these components; for the out-of-plane flexural mode, Z-translation participation m_ZZ dominates. The flexural natural frequency ω_flex is extracted. (2) Complex frequency analysis estimates Q_anchor near the flexural mode by searching within 0.98ω_flex to 1.02ω_flex to account for damped/undamped differences. The resonator anchor bottom attaches to a hemispherical substrate (radius 0.5 mm, ~22.7× resonator size); an outer layer of infinite elements absorbs outgoing elastic waves to prevent reflection. The complex eigenfrequency ω yields Q_anchor = Real[ω] / (2·Imag[ω]). A representative case gives damped frequency 910,731 Hz and Q_anchor = 5.78 × 10⁵, consistent with steady-state frequency response.

Dataset: 29,984 unique patterns were generated with balanced porosity levels from 0.20 to 0.90 in steps of 0.05 (approximately equal counts per level). Structure generation averages ~1.2 s per instance. The dataset exhibits a dominant trend: higher frequency correlates with increased anchor loss (lower Q_anchor), reflecting typical MEMS behavior. The f·Q_anchor product is consistently on the order of 10¹¹, with an average of (2.2 ± 1.0) × 10¹¹. Visualization includes t-SNE projections of hidden-layer vectors and outputs (frequency and Q_anchor) for interpretation of learned representations.

Performance use: After training on tens of thousands of samples, forward inference by the DL calculator predicts target properties much faster than FEA, enabling rapid screening and design iteration. The model can be configured for single-task (frequency or Q_anchor) or multi-task (both) prediction to streamline design workflows.

• DL-based surrogate modeling predicts flexural mode frequency and anchor-loss-limited quality factor directly from nonparameterized, pixelated geometries with high accuracy.
• Reported speedups over conventional FEA: approximately 4.6 × 10³ times faster for flexural mode frequency and 2.6 × 10⁴ times faster for Q_anchor prediction.
• Reported accuracies: frequency prediction at 98.8 ± 1.6%; Q_anchor prediction at 96.8 ± 3.1%.
• When simultaneously predicting both frequency and Q_anchor, up to ~96% of total computation time in the design process can be saved.
• Dataset statistics: 29,984 unique resonator patterns with balanced porosity between 0.20 and 0.90; empirical trend shows increasing frequency associated with decreasing Q_anchor; average f·Q_anchor product (2.2 ± 1.0) × 10¹¹.
• Example labeled case via FEA: damped natural frequency 910,731 Hz and Q_anchor = 5.78 × 10⁵ for a representative geometry.
• The automated mode identification via effective mass component rankings reliably isolates the out-of-plane flexural mode for labeling and training.

The study addresses the challenge of exploring vast, topology-unconstrained MEMS design spaces by replacing expensive FEA evaluations with an accurate, fast DL surrogate. By encoding geometries as binary images and training on FEA-labeled data, the model captures complex structure–property relationships without hand-crafted parameters or predefined topologies. The significant speedups and high accuracies enable rapid screening of thousands of candidates, making it feasible to optimize for target frequencies and high Q_anchor in early-stage design. Observed dataset trends, such as the tradeoff between higher frequency and lower Q_anchor, are learned by the model and reflected in predictions, supporting informed design decisions. The DL approach reduces reliance on domain heuristics and manual mode inspection, streamlining workflows and potentially improving discovery of unconventional, high-performing topologies.

This work demonstrates an experience-free, nonparameterized DL framework for MEMS structural design that predicts modal frequency and anchor-loss-limited quality factor from pixelated geometries. Using a customized ResNet trained on ~30k FEA-labeled samples, the approach achieves high accuracy and orders-of-magnitude speedups relative to FEA, saving up to ~96% of computation time when jointly predicting both targets. Automated mode identification via effective mass components further accelerates data preparation. The method enables rapid screening of large design spaces and is expected to generalize beyond disk resonators to other MEMS device categories and target properties. Future directions include expanding material and geometric variability, incorporating additional loss mechanisms (e.g., viscous and material damping), multi-objective optimization, and active design loops that couple the DL surrogate with generative or search algorithms.

• Scope is restricted to disk-shaped resonators with fixed global parameters (inner/outer ring diameters, anchor stem diameter, and layer thicknesses), limiting generalizability beyond these settings without retraining.
• Material is fixed to polysilicon with specific properties; different materials would require new datasets and training.
• Quality factor predictions focus on anchor loss; other damping mechanisms (viscous, material) are not included in Q labeling.
• Geometries are constrained to symmetric patterns generated via a stochastic agent and symmetry folding, which may exclude some viable asymmetric designs.
• Reported performance and trends are based on the curated dataset; extrapolation to significantly different design regimes may reduce accuracy without additional training data.