Deep learning-driven performance prediction and design of high-DoF MEMS resonators

Abstract

The design of microelectromechanical systems (MEMS) resonators has long been hindered by high computational costs and restricted structural degrees of freedom. This study introduces a structural design framework based on a residual network-enabled solver (ResNES) for rapid physical performance prediction and efficient optimization of MEMS resonators. The resonant frequencies of in-plane vibration modes are employed as the representative physical properties to demonstrate the complete design workflow. A stochastic topology generation strategy is proposed to efficiently create diverse resonator designs by batch-generating elastic beams with high degrees of freedom (DoF) within a defined design space. Systematic evaluations reveal that fully-trained ResNES achieves a computational speed improvement of nearly three orders of magnitude compared to the finite element method (FEM), enabling millisecond-scale physical performance prediction with mean errors below 3%. The discrepancy between ResNES predictions and experimental measurements remains below 5%, validating the framework’s capability to bridge analysis and fabrication seamlessly. Furthermore, ResNES is integrated with a self-built particle swarm optimization algorithm to minimize resonant frequencies and maximize frequency spacing. Comparative analyses demonstrate that this collaborative optimization scheme enables multi-objective optimization, identifying optimal designs from tens of millions of potential topologies within minutes. Compared to conventional FEM-based methods, even when accounting for the time costs of data preparation and ResNES training, the collaborative optimization scheme reduces the overall design cycle by at least 70% while further improving physical performance metrics by 25%. This work establishes a scalable paradigm for intelligent MEMS structural design, advancing the development of high-performance MEMS devices.

Introduction

Microelectromechanical systems (MEMS) resonators serve as core components in diverse applications, such as inertial navigation^1,2, biomedical sensing^3,4, acoustic/optical detection^5–7, chemical analysis⁸, and fundamental research⁹. Their physical properties directly govern critical performance metrics such as bandwidth¹⁰, sensitivity¹¹, and noise levels¹². Different application scenarios impose divergent requirements on resonator properties due to fundamental differences in physical environments, signal characteristics, energy forms, and system objectives^13,14. For example, low-frequency resonators are preferred for energy harvesting¹⁵, biosensing¹⁶, and seismic monitoring^17–19, whereas high-frequency resonators dominate in mobile communications^20,21 and quantum computing²². Additionally, resonators with certain frequency spacing mitigate mode aliasing, enabling high stability and multifunctional integration potential, making them suitable for MEMS gyroscopes²³, actuators²⁴, and filters²⁵. The efficient design and optimization of resonator physical properties for increasingly complex applications present significant research value. However, conventional resonator design relies heavily on extensive finite element simulations and empirical parameter adjustments. For complex structures, the dimensionality of the design space grows exponentially, resulting in time-consuming analysis processes that often fail to achieve global optima. Machine learning has recently emerged as a transformative tool, capable of extracting patterns from large datasets and establishing end-to-end mappings between design parameters and multiphysical responses²⁶. This approach significantly accelerates analysis and optimization processes across multiple disciplines^27–35, thereby offering novel approaches for MEMS design. As a promising solution, data-driven machine learning has been successfully applied to MEMS structural design³⁶, signal analysis^37–40, error compensation^41–43, and manufacturing processes⁴⁴, yielding numerous groundbreaking achievements. For instance, Chimeh et al.⁴⁵ employed deep neural networks to optimize energy harvester geometries, reducing resonant frequencies and increasing output voltage. Cheng et al.³⁶ employed neural networks to achieve inverse design of accelerometer structures. Guo et al.⁴⁶ employed deep learning to predict modal frequencies and quality factors of disk resonators. Following a similar approach, Chen et al.⁴⁷ developed a convolutional neural network-based surrogate model to replace finite element analysis for exploring novel structural topologies in disk resonator gyroscopes. Despite these efforts to integrate machine learning into MEMS design, most studies remain focused on geometric modifications within specific topologies or lack consideration of optimization efficiency, failing to address comprehensive architectural innovation and verification. Currently, MEMS design still faces three major fundamental bottlenecks: (1) limited exploration of topologies with superior physical performance due to traditional design paradigms; (2) low efficiency of existing design methods, severely constraining design iteration; (3) Lack of manufacturability.

To overcome these limitations, we propose a deep learning-enabled framework for rapid MEMS resonator physical performance evaluation and efficient structural design while maintaining manufacturability. In this work, the first two resonant frequencies corresponding to in-plane vibration modes⁴⁸ are selected as target physical properties to demonstrate the design workflow. First, a stochastic topology generation strategy is developed, which can generate tens of thousands of valid resonator structures within hours by implementing rational constraints (e.g., no crossing, no repetition). The overall freedom and complexity of the design space are adjustable to meet practical fabrication and application requirements. The datasets comprising binarized resonator topologies and finite element method (FEM) results are constructed for training and testing the customized residual neural network. Experimental results demonstrate that the residual network-enabled solver (ResNES) achieves nearly three orders of magnitude speed improvement compared to conventional FEM, enabling physical property evaluation of hundreds of resonators at the second timescale. Regarding accuracy, ResNES serves as a perfect substitute for FEM, with prediction errors rigorously controlled below 3%. Systematic simulations and experiments verify that ResNES enables rapid and accurate prediction of physical performance and successfully incorporates manufacturability constraints into the design process. Finally, ResNES is embedded into a self-built Adaptive Elite Learning Particle Swarm Optimization (AELPSO) algorithm^17,49–51 to optimize resonator topologies with the objectives of minimizing the first-order resonant frequency and maximizing the adjacent frequency spacing. Comparative analyses show that the collaborative optimization scheme can evaluate and screen tens of thousands of candidate designs within minutes, significantly compressing the design cycle while achieving superior physical performance. Our work establishes a new paradigm for intelligent MEMS design, extensible to high-dimensional, nonlinear, and multi-constrained systems. These advancements validate the transformative potential of deep learning in MEMS design and establish a comprehensive methodology that bridges performance prediction with physical realization. The proposed structural design framework provides a robust foundation for developing next-generation, high-performance MEMS devices.

Results

ResNES architecture

The construction workflow of the ResNES is illustrated in Fig. 1, encompassing resonator structure generation, dataset construction, and network training and testing. First, diverse elastic beam structures are generated using a random topology generation strategy and combined with a central proof mass to form complete MEMS resonators, as shown in Fig. 1a. Resonant frequencies corresponding to in-plane vibration modes are identified and extracted using the FEM to establish the geometry-performance mapping. Binary matrices encoding resonator topologies, paired with FEM results, constitute the training and testing dataset, as illustrated in Fig. 1b.

Fig. 1. ResNES construction flowchart.

a MEMS resonator structure generation, including random elastic beam creation and complete resonator assembly. b Dataset construction, involving structural topology binarization and resonator physical property acquisition. c The implemented residual neural network architecture, including data input, residual blocks, neural network processing, and physical property output

To overcome the challenge of growing data dimensionality caused by complex structures, we implement a customized residual neural network, as shown in Fig. 1c. The network incorporates residual blocks with skip connections, mitigating gradient vanishing during deep training while enhancing feature extraction⁵². For network improvement, Leaky ReLU activation functions replace standard ReLU to stabilize parameter updates via non-zero gradients in negative intervals. An adaptive learning rate scheduler dynamically adjusts step sizes to avoid local optimum oscillation, and an early stopping mechanism halts training upon overfitting risks to improve generalization. The trained ResNES achieves accurate physical property predictions for various resonators at significantly accelerated speeds, serving as a FEM alternative. As an efficient surrogate model, ResNES can be integrated into optimization frameworks or employed for large-scale parametric screening, enabling rapid resonator performance design.

Dataset construction

The quality and scale of the dataset fundamentally determine network performance, making rapid generation of valid structural topologies essential to ResNES development. As presented in Fig. 2a, the resonators used in this study feature four elastic beams supporting a central proof mass in a symmetric configuration, with one quarter-region serving as the primary design space. A Brownian-motion-inspired topology generation strategy⁴⁶ is employed to create diverse elastic beam structures. As illustrated in Fig. 2b, each three-dimensional elastic beam is generated by extruding a two-dimensional path, starting from the vertex of the proof mass and ending at fixed anchor points on the outer frame. These two-dimensional paths are generated within uniformly discretized square grids matching the allowable design space dimensions, as shown in Fig. 2c. All paths originate at coordinate (0, 0) and terminate at randomly selected points either the right or upper boundaries, yielding 2 × n + 1 possible endpoints for an n × n grid.

Fig. 2. Resonator generation and physical property acquisition.

a Resonator configuration. b Three-dimensional elastic beam. c Two-dimensional path representing elastic beams within an 8 × 8 grid. d-f Schematics of resonators generated from grids with varying densities and their corresponding FEM results, showing the first two in-plane vibration modes and resonant frequencies. Structural features are encoded as binary matrices. d 8 × 8 grid. e 16 × 16 grid. f 24 × 24 grid

Paths are generated along grid lines, and each path consists of multiple segments with integer multiples of unit grid length. After generating each segment, one of three potential directions is randomly selected for subsequent extension. To ensure path validity and independence, the generation direction and length of each new segment are dynamically constrained by two mechanisms: (1) cross-checking against existing segments to prevent intersections through directional adjustment, and (2) maintenance of a real-time updated path library records historical coordinates to replan segments when excessive overlap occurs. Grid density directly determines design freedom, with finer grids enabling richer topological possibilities. Figure 2d-f display representative paths and corresponding resonator schematics generated from 8 × 8, 16 × 16, and 24 × 24 grids within the same design space. Remarkably, an 8 × 8 grid can produce over 10⁷ distinct topologies even after invalid path removal. This topology generation strategy enables batch production of valid resonator structures while preserving a high-DoF. For network input compatibility, elastic beams are encoded as a binary matrix (0: void, 1: solid), as shown in Fig. 2d-f. The matrix dimension relates to grid density, with an n × n grid producing (n + 1)-order binary matrices. Considering fabrication feasibility, the proof mass is designed with 5 mm edge length, while the elastic beams feature a width of 50 μm. Both the elastic beams and proof mass maintain a uniform thickness of 400 μm. The complete design space for the elastic beams occupies a 4 mm × 4 mm square region.

FEM analysis is performed on all fabricated resonators to extract their target physical properties. Modal analysis provides resonant frequencies, participation factors, and effective modal masses. Participation factors and modal masses serve as essential tools for evaluating the influence of eigenvalues on state variables in dynamic systems, thereby quantifying the contribution of specific variables to each mode. In modal analysis, the vibrational motion of the resonator can be projected into six directions: translations along the X, Y, and Z axes, and rotations about the X, Y, and Z axes. By comparing the participation factors and modal masses corresponding to these six directions, vibration modes associated with different resonant frequencies can be efficiently identified (Supplementary Note 1). Using this approach, the first two resonant frequencies corresponding to in-plane vibration modes are rapidly extracted, as demonstrated in Fig. 2d-f. The binary matrix representing the geometric topology of each resonator is flattened into a row vector and paired with the corresponding FEM results to form the input dataset for network training. All data are normalized to ensure uniform parameter scaling, enhancing training stability and convergence speed.

Performance analysis and verification of the ResNES

Following the aforementioned approach, 12,000 resonators are constructed using 8 × 8, 16 × 16, and 24 × 24 grids, with each labeled by corresponding FEM results. The dataset is divided into 10,000 samples for training and 2000 samples for testing. ResNES performance is systematically evaluated through prediction accuracy and time cost. As shown in Fig. 3a-f, ResNES predictions and FEM results demonstrate excellent agreement across all grid densities, with data points uniformly distributed along the y = x line with linear regression errors generally below 10%. These results indicate that the trained ResNES can effectively replace FEM for accurate multi-physical property prediction. However, prediction accuracy exhibits a gradual decline with increasing grid density. Table 1 demonstrates that with a training set of 10,000 samples, ResNES achieves a mean accuracy exceeding 99% for 8 × 8 grids while decreasing to 96% for 24 × 24 grids. This observed trend is attributed to the increased structural complexity and higher feature dimensionality associated with finer grid densities, which leads to insufficient feature learning given the fixed dataset size. Further analyses confirm a strong correlation between dataset size and solver accuracy. Figure 3g, h reveals that accuracy converges when the training set exceeds 10,000 samples for all grid densities, while 8 × 8 grids require significantly fewer samples to achieve comparable accuracy levels. Specifically, as shown in Table 1, 8 × 8 grids attain 98% accuracy with only 2000 training samples, whereas 16 × 16 and 24 × 24 grids require 10,000 and 15,000 samples, respectively, to reach the same performance level. This trend demonstrates the data requirements for learning high-dimensional features.

Fig. 3. Evaluation of ResNES results.

a-f Linear regression analysis of ResNES predictions across different grid densities. a 8 × 8 grid, first resonant frequency. b 16 × 16 grid, first resonant frequency. c 24 × 24 grid, first resonant frequency. d 8 × 8 grid, second resonant frequency. e 16 × 16 grid, second resonant frequency. f 24 × 24 grid, second resonant frequency. g, h Relationships between training set size, grid density, and ResNES accuracy. g First resonant frequency. h Second resonant frequency. i-n Resonant frequency distributions for different grid densities. i 8 × 8 grid, first resonant frequency. j 16 × 16 grid, first resonant frequency. k 24 × 24 grid, first resonant frequency. l 8 × 8 grid, second resonant frequency. m 16 × 16 grid, second resonant frequency. n 24 × 24 grid, second resonant frequency

Table 1.

Average prediction accuracy of ResNES for different training set sizes

Grid density	Resonant frequency order	Training set size
		1000	2000	5000	10000	15000	20000	25000	30000
8 × 8	First-order	0.97989	0.98058	0.99335	0.99735	0.99606	0.99712	0.99911	0.99893
	Second-order	0.98093	0.98838	0.99949	0.99998	0.99996	0.99950	0.99898	0.99991
16 × 16	First-order	0.84172	0.90446	0.95841	0.97944	0.98591	0.98721	0.99032	0.99532
	Second-order	0.85013	0.89932	0.96142	0.98086	0.98889	0.99123	0.99583	0.99883
24 × 24	First-order	0.78425	0.86137	0.92397	0.96083	0.98079	0.98433	0.98721	0.99025
	Second-order	0.77102	0.86081	0.94792	0.96689	0.98231	0.98553	0.98823	0.99205

Figure 3i-n present the resonant frequency distributions obtained from both ResNES and FEM across different grid densities, with blue and red representing first-order and second-order resonant frequencies, respectively. The frequency distributions demonstrate excellent agreement between both solvers, as evidenced by the highly consistent histogram trends fitted with log-normal functions. However, distinct frequency distribution characteristics emerge at different grid densities. Lower grid density (8 × 8) generates structures with first resonant frequencies concentrated between 400–600 Hz and second resonant frequencies dispersed from 800–4500 Hz, as shown in Fig. 3i, l. In contrast, higher grid density (24 × 24) produces first resonant frequencies distributed from 250–500 Hz and second resonant frequencies predominantly below 1500 Hz, as illustrated in Fig. 3k, n. These observations indicate that finer grids facilitate the exploration of more complex structural topologies that tend to exhibit lower resonant frequencies. Additionally, the frequency spacing of samples generated with low grid density is relatively larger, a feature potentially useful for specific frequency-response requirements. These patterns reflect general trends between grid density and vibrational characteristics, providing preliminary guidance for initial grid selection based on design objectives. The actual physical properties of resonators ultimately depend on specific geometric details rather than grid density alone.

Multiple resonators are fabricated to validate the effectiveness of ResNES. Figure 4a illustrates the architecture of the resonator used for vibration testing, comprising a proof mass, elastic beams, and an outer frame, with key structural parameters matching the simulation models. Considering practical fabrication tolerances, resonators are generated from three grid densities (8 × 8, 10 × 10, 12 × 12). Each resonator is packaged by bonding the frame to a glass substrate and a glass cover to minimize environmental interference. The vibration responses of the resonators are measured using a laser Doppler vibrometer. The frame is equipped with an observation window to allow the laser to directly illuminate the proof mass, as shown in Fig. 4b. The resonators undergo periodic mechanical excitation with precisely controlled amplitude and period to ensure multiple complete, independent free vibration decays during each measurement, as shown in Fig. 4c. As presented in Fig. 4d-i, the six tested samples exhibit significant variation in vibrational characteristics, with first resonant frequencies ranging from 243 Hz to 1824 Hz. For all samples, the first two peaks of the frequency-domain show excellent agreement with both ResNES predictions and FEM results. As shown in Table 2, the deviation between the ResNES predictions and the experimental results, although higher than that compared with the FEM results, remains strictly within 5%. This discrepancy can be primarily attributed to fabrication variations in the width of the elastic beams and fluctuations in material properties (Supplementary Note 2). The vibration experiments demonstrate that the proposed deep learning-driven structural design framework enables rapid, high-precision prediction of resonator physical properties and maintains manufacturability within constrained design spaces.

Fig. 4. Experimental validation of ResNES.

a Schematic diagram of the fabricated resonator. b Experimental setup. c Example of measured vibration response. d-i Frequency-domain responses of the resonators and the results from ResNES and FEM: d, e resonators generated from 8 × 8 grids, (f, g) from 10 × 10 grids, (h, i) from 12 × 12 grids. Blue dots represent the peak points in the frequency domain, the red dashed lines indicate the ResNES predictions, and the yellow dashed lines represent the FEM results

Table 2.

Comparison of ResNES predictions with FEM and experimental results

Grid density	Sample number	Frequency order	ResNES predictions (Hz)	FEM Results (Hz)	Deviation Between ResNES and FEM	Experimental Results (Hz)	Deviation Between ResNES and Experiment
8 × 8	1	First-order	255.4	249.1	2.53%	243.1	5.06%
		Second-order	327.2	321.2	1.87%	314.9	3.91%
	2	First-order	432.8	424.3	2.00%	453.0	-4.46%
		Second-order	595.2	598.3	-0.52%	620.2	-4.03%
10 × 10	3	First-order	832.4	844.5	-1.43%	814.1	2.25%
		Second-order	1573.2	1550.9	1.44%	1645.2	-4.38%
	4	First-order	865.0	872.7	-0.88%	908.7	-4.81%
		Second-order	2163.8	2207.3	-1.97%	2260.8	-4.29%
12 × 12	5	First-order	942.5	957.3	-1.55%	963.8	-2.21%
		Second-order	1620.3	1650.9	-1.85%	1665.1	-2.69%
	6	First-order	1890.3	1857.1	1.79%	1823.5	3.66%
		Second-order	4589.4	4501.2	1.96%	4338.3	5.79%

The most significant advantage of ResNES lies in its exceptional prediction speed. As detailed in Table 3, the prediction times required by ResNES for processing samples with varying grid densities are significantly lower than those of FEM. For structures generated from 8 × 8 grids, ResNES requires only 0.00852 (± 0.0006) s per inference, achieving a 930-fold speed improvement over FEM’s 7.94 (±0.5) s. The speed advantage of ResNES becomes even more pronounced for complex topologies. For structures generated from 24 × 24 grids, ResNES processes individual samples in merely 0.01458 (±0.0009) s, resulting in nearly a 3000-fold acceleration compared to FEM’s 43.42 (±2.5) s. Additionally, the robustness of ResNES’s computational efficiency advantage has been comprehensively validated through a wider range of FEM benchmarks (Supplementary Note 3).

Table 3.

Comparison of prediction time across different solvers

Grid density	Average prediction time per 10000 samples (min)		Average prediction time per single sample (s)
	ResNES	FEM	ResNES	FEM
8 × 8	1.42 (±0.10)	1323.36 (±75)	0.00852 (±0.0006)	7.94 (±0.5)
16 × 16	1.69 (±0.11)	2835.32 (±200)	0.01014 (±0.0007)	17.01 (±2)
24 × 24	2.43 (±0.15)	7237.12 (±350)	0.01458 (±0.0009)	43.42 (±2.5)

Resonator structure design based on ResNES

To further validate the potential of ResNES in enhancing design efficiency, it is integrated into the AELPSO algorithm for optimizing the resonator structure. The design space is constructed using a 16 × 16 grid, a medium grid density, and a specific elastic beam generation strategy is proposed to better align with the optimization algorithm. Specifically, each beam begins at the bottom-left corner (0, 0) of the grid and terminates at the top-right corner (17, 17). The elastic beam is composed of 32 path segments, consisting of 16 upward segments (encoded as 0) and 16 rightward segments (encoded as 1). The entire path can be represented by a 32-dimensional binary vector. Although this strategy compresses the number of candidate topologies, it maintains structural diversity, with the potential designs reaching the order of 10⁹. In this study, the minimum first resonant frequency and maximum resonant frequency spacing are selected as the optimization objectives. The frequency spacing coefficient $\psi$ is defined to quantify the spacing between the first two resonant frequencies, and its expression is

$$\psi =\frac{f_2-f_1}{f_2+f_1}$$ 1

where f₁ and f₂ denote the first and second resonant frequencies, respectively. As $\psi$ approaches 1, the frequency spacing increases.

Four structural optimization schemes are proposed, which are formed by pairing two distinct solvers with two search strategies. Scheme 1 integrates AELPSO with ResNES, Scheme 2 combines AELPSO with FEM, Scheme 3 employs the ranking method with ResNES, and Scheme 4 utilizes the ranking method with FEM. The particle and iteration parameters in Scheme 1 and Scheme 2 remain consistent, with a maximum iteration count of 300 and a total particle population of 200. Scheme 3 and 4 select the optimal design by calculating and ranking the physical properties of 150,000 randomly generated samples. Figure 5a-d illustrates the complete iteration processes for Schemes 1 and 2, where both approaches achieve satisfactory convergence after 300 iterations, with the optimized topologies exhibiting substantial performance enhancements compared to initial designs. The structural topology evolution processes corresponding to different solvers exhibit remarkable consistency, demonstrating that ResNES can effectively replace FEM implementations in embedded optimization algorithms.

Fig. 5. Resonator structure optimization.

a, b Optimization process for minimizing first resonant frequency. a AELPSO with ResNES. b AELPSO with FEM. c, d Optimization process for maximizing resonant frequency spacing. c AELPSO with ResNES. d AELPSO with FEM. e, f Results and time cost for four structural optimization schemes. e Minimization of resonant frequency. f Maximization of resonant frequency spacing. g Optimization results from the collaborative optimization scheme compared with randomly generated structures

Quantitative results presented in Table 4 and Fig. 5e, f reveal the superior performance of AELPSO-based schemes (1 and 2) in achieving both optimization objectives with significantly enhanced efficiency. Specifically, structures optimized through AELPSO exhibited a 25% reduction in resonant frequency and an 11% improvement in frequency spacing coefficient compared to ranking-based methods (3 and 4). For the FEM-based Schemes (2 and 4), Scheme 2 employing AELPSO achieved superior results within 70 hours, representing a 90% reduction in time cost compared to the 600 hours required by Scheme 4, as demonstrated in Fig. 5e, f. In particular, Scheme 1 (AELPSO + ResNES) takes only 1 second per iteration, which is 600 times faster than Scheme 2 (AELPSO + FEM), with the optimization result deviations less than 2%, as shown in Table 4. In summary, Scheme 1, which combines the computational efficiency of ResNES with the high-performance screening capability of AELPSO, offers exceptional advantages. Its per-iteration speed reaches the second level, enabling the identification of optimal solutions from tens of millions of candidate designs within minutes. Even when accounting for ResNES training and data preprocessing overhead, the collaborative optimization scheme integrating ResNES and AELPSO reduces the overall design cycle by over 70% compared to conventional FEM-based methods (Supplementary Note 4).

Table 4.

Performance comparison of different optimization schemes

Optimization schemes	Optimization objectives	First resonant frequency (Hz)	Second resonant frequency (Hz)	Frequency spacing coefficient	Time cost (hours)
Scheme 1 (AELPSO with ResNES)	Minimum resonant frequency	407.41	540.23	0.140	0.1
Scheme 2 (AELPSO with FEM)		402.43	596.94	0.195	65.4
Scheme 3 (Ranking method with ResNES)		546.58	727.44	0.141	0.4
Scheme 4 (Ranking method with FEM)		514.62	819.68	0.228	622.3
Scheme 1 (AELPSO with ResNES)	Maximum frequency spacing	997.01	3947.32	0.597	0.1
Scheme 2 (AELPSO with FEM)		980.37	4112.59	0.615	70.9
Scheme 3 (Ranking method with ResNES)		974.48	3241.82	0.538	0.4
Scheme 4 (Ranking method with FEM)		1025.72	3236.33	0.519	661.5

Furthermore, we consolidate the first resonant frequency and frequency spacing coefficient into a single objective function, expressed as

$$\phi =\alpha (1-\lambda )+(1-\alpha )(1-\psi )=\alpha (1-e^{-0.001f_1})+(1-\alpha )\left(1-\frac{f_2-f_1}{f_2+f_1}\right)$$ 2

where parameter $\lambda$ is derived from the exponentially attenuated first resonant frequency with a normalized range of 0 to 1. A larger $\lambda$ corresponds to smaller resonant frequencies. Parameter $\alpha$ serves as a weighting coefficient that dynamically adjusts the relative importance of the two physical performance metrics within the objective function, with its value constrained between 0 and 1. As $\alpha$ approaches 1, the optimization process shifts its focus from maximizing frequency spacing to minimizing resonant frequency. To visually demonstrate the collaborative optimization scheme outcomes, 20,000 resonator sets are randomly generated using the aforementioned specific elastic beam generation strategy, with their physical properties obtained through FEM. As shown in Fig. 5g, for single-objective optimization, the structures selected by the collaborative optimization scheme exhibit significantly superior physical performance compared to all randomly generated samples. The collaborative optimization scheme successfully explores extreme designs along the Pareto frontier in multi-objective optimization. These results demonstrate that the proposed ResNES integrated with optimization algorithms substantially enhances optimization efficiency and uncovers designs with superior physical performance. While this study primarily focuses on in-plane vibration modes and their corresponding first two resonant frequencies, the established workflow can be readily extended to other operational modes, including out-of-plane vibrations^53–55, and nonlinear resonances⁵⁶, potentially contributing to performance optimization for a broader range of MEMS devices.

Conclusions

In this study, a deep learning-driven MEMS design framework is constructed, wherein the proposed ResNES enables rapid and accurate prediction of physical properties for resonators. The design space is discretized into density-adjustable grids, facilitating a high-DoF elastic beam generation strategy that ensures manufacturability while automatically creating millions of resonator topologies. The structural topologies are encoded as binary matrices and labeled with FEM results to construct datasets. Systematic evaluation demonstrates that ResNES achieves exceptional accuracy and computational efficiency, serving as a competent alternative to FEM. Specifically, ResNES maintains errors below 3% while operating three orders of magnitude faster than FEM, enabling millisecond-scale evaluation of individual structures. Experimental vibration measurements show excellent agreement with ResNES predictions (deviation <5%), validating its reliability in practical resonator design workflows. The integration of ResNES with a self-built AELPSO algorithm enables rapid performance optimization of MEMS resonators. Compared to conventional FEM-based methods, this collaborative optimization scheme can simultaneously address multiple optimization objectives while reducing the overall design cycle by at least 70% even when including data preparation and ResNES training, and improving physical performance metrics by 25%. The dataset in this work is constructed based on FEM within the mechanical domain, and experimental validation is performed using mechanical resonators without integrated driving or sensing functionality. Nevertheless, ResNES is designed with strong extensibility. For more complex application scenarios, ResNES can be effectively extended by developing multiphysics datasets that accurately represent real-world operating conditions. In summary, this research facilitates the transition from local parameter tuning to global topological innovation in MEMS structural design. Compared to existing studies, the proposed framework establishes seamless integration between performance prediction, structural optimization, and fabrication, providing reproducible design templates for MEMS devices (Supplementary Note 5).

Methods

Machine learning model

In this work, the ResNES is obtained by training customized residual neural networks for 200 epochs on an NVIDIA V100 GPU. A ReduceLROnPlateau learning rate scheduler is employed during training, which halves the learning rate if the validation loss does not improve over 10 consecutive epochs. Training data is reshuffled before each training iteration to enhance model generalization and mitigate overfitting risks. To improve training stability, the standard ReLU activation function is replaced with Leaky ReLU. Since the geometric topology of the resonators is represented in the form of binary matrices, negative activations may arise after linear transformations within the residual network. Leaky ReLU provides a non-zero gradient in the negative region, ensuring that the corresponding neurons continue to participate in parameter updates and effectively alleviating the “dying neuron” problem. This modification proves particularly beneficial for capturing the complex geometry-performance mapping in high-DoF MEMS resonator structures across multiple grids (Supplementary Note 6). Notably, the proposed structural design framework demonstrates strong algorithmic compatibility and can be extended to other advanced machine learning models. To further evaluate the performance of ResNES, comprehensive comparisons are conducted with solvers based on various machine learning models in terms of computational accuracy and efficiency^57,58 (Supplementary Note 7).

Dataset expansion

The dataset employed in this work for training the neural network comprises binarized geometric topologies and corresponding resonant frequencies computed using the FEM. For geometric binarization, the elastic beam is conceptualized as consisting of multiple path segments. To present the modeling concept concisely, the number of potential generation directions for new path segments is limited to three, and consecutive endpoints are connected with straight lines. In practice, the proposed structural design framework can be extended to incorporate non-orthogonal beams, curved beams, or annular features by increasing the number of potential directions, modifying the connection strategy between endpoints, or adapting the resonator configuration (Supplementary Note 8). Furthermore, this study uses resonant frequency as the target physical property-a representative FEM-computed metric-to demonstrate the complete workflow from design generation to performance prediction. Beyond resonant frequency, the proposed framework also supports ResNES constructed based on other physical performance metrics, such as displacement, stress, mode shapes, and mode interference (Supplementary Note 3). Moreover, the method exhibits strong extensibility, with the design space readily expandable to include additional physical parameters such as material properties and cross-sectional dimensions (Supplementary Note 8).

Numerical simulations

All FEM computations in this study are performed using COMSOL Multiphysics commercial software. Although vibration modes can be identified and corresponding resonant frequencies extracted from the modal analysis results of 3D FEM models, we employ a 2D model instead of a 3D model to simulate the mechanical behavior in this study. This approach is adopted considering that the proof mass and elastic beams share the same thickness, and that the research focus is on in-plane vibration modes, thereby significantly reducing computational costs. The meshing process utilized mapped methods, with the mesh size set to one-tenth of the characteristic dimension through convergence verification. Material parameters follow standard silicon-based MEMS fabrication specifications, with density set to 2329 kg/m³ and Young’s modulus to 170 GPa, approximated as isotropic for model simplification. Fixed constraints are applied to all four boundaries of the elastic beams. Regarding computational efficiency, three-dimensional simulations for structures generated from 24 × 24 grids required approximately 180 seconds, while the two-dimensional simulation was completed in 40 seconds, representing a 70% improvement. Comprehensive validation against three-dimensional FEM results confirmed the accuracy of the two-dimensional simulations (Supplementary Note 9). The entire computational workflow is programmatically controlled, with all analysis results directly utilized for dataset construction or physical performance evaluation.

Manufacturing and packaging process

The resonators were fabricated using bulk silicon micromachining processes on 4-inch silicon wafers (thickness: 400 ± 5 μm). The fabrication sequence comprises: (1) Spin-coating AZ 4620 photoresist (3000 rpm for 30 s) followed by lithography to form etch masks. (2) Inductively coupled plasma (ICP) etching of 390 μm silicon, then temporary bonding to quartz glass carriers using adhesive. (3) Release of remaining 10 μm structures with over-etching controlled below 5%. (4) Ultrasonic removal of residual photoresist followed by oxygen plasma surface cleaning. (5) Temporary fixation of diced structures with paraffin to prevent handling damage. (6) Gradual heating (70 °C for 5 min) in a thermostatic water bath for stress-free release before packaging. (7) Permanent bonding to glass substrates using low-stress epoxy (EPO-TEK 353ND) and encapsulation with glass lids, followed by 24-hour curing on a leveled hotplate. Additional fabrication details are provided in Supplementary Note 10.

Experimental setups and analysis

All experimental measurements were conducted in a laboratory-grade environmental chamber under standard temperature and pressure conditions to ensure acoustic isolation. The chamber maintains a velocity noise root mean square (RMS) value below 3.16 × 10⁻⁸ m/s across the operational frequency range. The experimental environment was maintained at a constant temperature of 18.2 °C, with fluctuations within ±0.03 °C. To assess the temperature sensitivity of ResNES, externally controlled variable‑temperature tests confirmed that the influence of temperature variation on the prediction accuracy of ResNES remains within an acceptable range—specifically, an 80 °C temperature span leads to a fluctuation in prediction accuracy of less than 1.8% (Supplementary Note 11). A non-contact vibration measurement system was established using a Polytec IVS-500 laser Doppler vibrometer (LDV) to capture the vibrational response of the resonator. The LDV sampling frequency was set to 10 kHz, with a measurement range of 0 to 100 mm/s. Packaged resonators were horizontally mounted on a vibration-isolated optical table with air suspension, while the laser head was precisely aligned using a three-axis rotational stage. Prior to each measurement, coplanar alignment between the LDV and target resonator was verified to optimize optical signal collection. The resonant frequency was extracted from the obtained frequency spectrum through auto-spectral analysis of multiple oscillation cycles.

Optimization algorithm

The optimization employs the self-built AELPSO algorithm¹⁷, which incorporates population diversity analysis and complexity assessment based on dynamical system theory to guide sub-strategy selection and population partitioning. The core innovation lies in its “CODA” multi-strategy coordination mechanism, where crossover (C), orthogonal (O), dynamic (D), and all-component (A) subgroups co-evolve to dynamically balance global exploration and local exploitation capabilities. Specifically, subgroups C and A employ binomial crossover and weighted averaging strategies to enhance global search performance, while subgroups O and D utilize orthogonal testing and adaptive learning to improve local convergence efficiency. The algorithm features an elite preservation mechanism that periodically eliminates low-fitness particles to prevent premature convergence and adaptively adjusts strategy execution probabilities to maintain population diversity. Comprehensive benchmarking demonstrates that AELPSO outperforms several mainstream optimization algorithms in both convergence speed and solution quality (Supplementary Note 12).

Author contributions

Q.L. was responsible for formal analysis, manuscript writing, review, editing, and funding acquisition. Z.J. conducted investigations, data analysis, methodology development, and manuscript writing. C.W. and M.L. performed experimental design and visualization. W.Y. and S.W. fabricated the mechanical resonators. J.S. developed and optimized algorithms. W.L. and M.L. implemented neural network training and testing. C.S. and D.T. executed experiments and analyzed experimental data. P.L. and Y.W. carried out formal analysis and validation. X.W. managed project administration, resource allocation, and supervision.

Data availability

All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41378-026-01198-0.