A MEMS plate capacitance modeling method, system, device and medium

Abstract

The application discloses a MEMS flat plate capacitance modeling method, system, device and medium, relates to the micro-electro-mechanical system design technical field, and changes the paradigm that the traditional PINN relies on static, fixed data set for training, through active evaluation, screening the "difficult sample" with the maximum current prediction error of the model, the difficult sample is mainly located at the corner point and the edge of the polar plate with the maximum electric field gradient, and the dynamic feedback of the difficult sample is fed back to the loss function, the neural network is forced to learn in the area with the most complex physical mechanism, the model is forced to jump out of the "mediocre solution" caused by the pure physical constraint, and the characteristics of the corner point and the edge of the polar plate and other regions with the severe electric field change are accurately captured, so that the potential, the electric field distribution, the edge field and the capacitance of the parameterized flat plate capacitance are quickly solved.

Description

Technical Field

[0001] This invention relates to the field of microelectromechanical systems (MEMS) design technology, and in particular to a method, system, device, and dielectric for modeling MEMS planar capacitors. Background Technology

[0002] Microelectromechanical systems (MEMS) sensors have made significant progress and are widely used in consumer electronics, automotive industry and defense due to their advantages such as small size, light weight and low power consumption. Among various MEMS sensors, capacitive sensors dominate due to their high sensitivity, low temperature drift and low power consumption. MEMS capacitive sensors usually use capacitors to convert external physical quantities such as acceleration and pressure into measurable electrical signals, and parallel plate capacitors, as the most basic and common structure, are the core cornerstone of capacitive sensor design.

[0003] Currently, the main methods for modeling and calculating MEMS parallel plate capacitors include analytical calculation techniques, such as mathematical analytical models based on conformal transformation. These methods are usually based on idealized assumptions such as an infinitely large dielectric domain or zero plate thickness. However, these assumptions are seriously inconsistent with the finite physical boundaries that actually exist in MEMS devices, causing the model to systematically lose key physical characteristics such as edge effects when processing actual devices, and its accuracy and applicability are fundamentally limited.

[0004] In addition, finite element method (FEM) is also a commonly used modeling method. This method obtains physical quantities such as electric potential, electric field, and capacitance by dividing the computational domain into a fine mesh and solving the corresponding partial differential equations numerically. However, finite element simulation cannot directly provide a continuous and readily available functional mapping relationship from geometric parameters to capacitance values, thus failing to meet the needs of rapid iterative design or real-time prediction. Furthermore, the results of finite element simulation are affected by the mesh generation accuracy and mesh quality, and may fluctuate under different mesh settings.

[0005] To address the problems existing in analytical computation techniques and finite element numerical simulation techniques, researchers have proposed a modeling technique based on Physical Information Neural Networks (PINN). This method embeds physical laws as constraints into the neural network loss function. However, PINN models typically employ a "static" training paradigm, relying on a single, fixed training dataset or fixed physical constraint sampling points for global loss minimization optimization. Under this "static" training paradigm, the model "prioritizes" and "overfits" dominant smooth regions during training. When the model reaches a low loss in most areas of the field (such as the uniform electric field region between plates), local errors in complex physical regions such as plate corners and edges where the electric field gradient exhibits singularities are submerged in the global average loss, making them difficult to effectively identify and optimize. For MEMS capacitor design, edge effects are precisely the key physical mechanism determining device performance (such as sensitivity and linearity). Therefore, the traditional PINN model's global average optimal training method is insufficient to train a MEMS capacitor model that can specifically fit and optimize singular regions to achieve high-precision prediction of potential, electric field, and capacitance. Summary of the Invention

[0006] This invention provides a MEMS planar capacitor modeling method, system, device, and medium, which can solve the problems existing in the prior art.

[0007] This invention provides a method for modeling the capacitance of a MEMS planar magnetic field, comprising the following steps: A simulation dataset of a two-dimensional electrostatic field physical model of a MEMS parallel plate capacitor under multiple sets of geometric parameters is obtained. The simulation dataset includes the spatial coordinates, geometric parameters, simulation potential values, simulation electric field components, and simulation capacitance values ​​of the physical model sample points. A physical information neural network model is constructed, with spatial coordinates and geometric parameters as inputs and the predicted potential value at the corresponding location as output. The partial derivative of the predicted potential value with respect to the spatial coordinates is obtained through automatic differentiation. Define a hybrid physics data loss function, which includes a physics loss for constraining the predicted potential value to satisfy the physical laws of electrostatics and a supervisory data loss for constraining the deviation between the predicted potential value and the simulation true value. Under the current calibration set, the physical information neural network model is trained using a hybrid physical data loss function. The trained model is then used to predict all sample points in the simulation dataset, and the absolute error between the predicted potential value and the simulated potential value for each sample point is calculated. Based on the absolute error, a predetermined number of sample points with the largest errors are selected from the simulation dataset as hard samples. The spatial coordinates, geometric parameters, simulated potential value, simulated electric field components, and simulated capacitance value corresponding to the hard samples are added to the calibration set to update the calibration set. In each training round, the supervised data loss is calculated by sampling data points from the updated calibration set, enabling the model to learn the physical field distribution of the region corresponding to the hard samples. The physical information neural network model is iteratively trained until a predetermined number of iterations is reached, resulting in the trained physical information neural network model. Here, the physical loss is used to constrain the model to satisfy physical laws in each training round, and the supervised data loss is used to constrain the model to fit the simulation data of the hard samples. The trained physical information neural network model is used as a surrogate model for MEMS planar capacitors, and the potential, electric field and capacitance are predicted based on the input geometric parameters and spatial coordinates.

[0008] Preferably, the construction of the simulation dataset includes: A two-dimensional electrostatic field physical model of MEMS parallel plate capacitor is constructed. Based on the two-dimensional electrostatic field physical model, the geometric parameters affecting the performance of MEMS parallel plate capacitor are determined. The geometric parameters include plate length, plate width and the distance between the upper and lower plates. The range of variation of plate length, plate width and the distance between the upper and lower plates is set. Multiple sets of geometric configuration cases are generated using a full factorial experimental design to cover the preset geometric parameter space. For each set of geometric configuration cases, the finite element method is used to mesh the two-dimensional electrostatic field physical model and solve for the steady-state electrostatic field, obtaining the spatial coordinates, potential values, and electric field components of each sample point in the computational domain, and obtaining the capacitance value corresponding to the geometric configuration case. The spatial coordinates, geometric parameter labels, simulated potential values, simulated electric field components, and simulated capacitance values ​​corresponding to each geometric configuration case are summarized to form the simulation dataset.

[0009] Preferably, the construction of the physical information neural network model includes: A parameterized physical information neural network model is constructed. The parameterized physical information neural network model adopts a fully connected neural network structure. Its input layer is used to receive a feature vector composed of spatial coordinates and geometric parameters. The feature vector includes normalized horizontal coordinates, vertical coordinates, electrode length, electrode width, and the distance between the upper and lower electrodes. The parameterized physical information neural network model includes a preset number of hidden layers. Each hidden layer adopts a fully connected layer structure and uses the hyperbolic tangent function as the activation function to establish a nonlinear mapping relationship between spatial coordinates, geometric parameters and electrostatic field potential distribution. The output layer of the parameterized physical information neural network model is a single neuron structure, which is used to output the predicted electric potential value at the corresponding position under the current spatial coordinates and geometric parameters, and obtain its partial derivative with respect to the spatial coordinates through automatic differentiation based on the predicted electric potential value.

[0010] Preferably, the construction of the hybrid physical data loss function includes: A hybrid physical data loss function, consisting of a weighted average of physical loss and supervised data loss, is constructed and used as the optimization objective of the parameterized physical information neural network model. The physical losses include partial differential equation residual loss, Dirichlet boundary loss, Neumann boundary loss, and overshoot regularization loss; The monitoring data loss includes potential monitoring loss and electric field component monitoring loss.

[0011] Preferably, the training of the physical information neural network model includes: The calibration set is initialized to an empty set, and the physical information neural network model is iteratively trained according to a preset number of outer layer iterations. In each outer layer iteration, a preset number of training cycles are executed. In each training cycle, physical collocation points for calculating the residuals of the control equations and boundary points for boundary condition constraints are randomly generated to calculate the physical loss. When the calibration set is not empty, a preset number of data points are randomly sampled from the current calibration set. The supervision data loss is calculated based on the simulation potential and electric field components corresponding to the data points. The physical loss and the supervision data loss are combined to form a hybrid physical data loss function to update the parameters of the physical information neural network model. After each round of outer layer iteration training is completed, the electric potential of all sample points in the simulation database is predicted using the currently trained physical information neural network model. The absolute deviation between the predicted electric potential and the true value of the simulated electric potential at each sample point is calculated. Based on the absolute deviation, all sample points in the simulation database are sorted in descending order, and the top Np sample points with the largest absolute deviation are selected as the difficult samples mined in this round of outer layer iteration, where Np is the preset number of newly added difficult samples in each round of outer layer iteration. The spatial coordinates, structural parameters, true values ​​of simulated electric potential, and true values ​​of simulated electric field components corresponding to the difficult samples are added to the current calibration set. The updated calibration set is deduplicated to form the calibration set used in the next round of outer layer iteration. The training, prediction, difficult sample mining, and calibration set update process are repeated until the number of outer layer iterations reaches the preset number of iterations, and the trained physical information neural network model is obtained.

[0012] Preferably, the prediction of potential, electric field, and capacitance includes: The trained parameterized physical information neural network model is used as a proxy model for MEMS parallel plate capacitors to obtain the geometric parameters of the MEMS parallel plate capacitors to be predicted, and spatial sampling points are generated in a preset two-dimensional computational domain. The coordinates of the spatial sampling points and the geometric parameters are input into the surrogate model to obtain the predicted potential values ​​at each spatial sampling point, thereby obtaining the potential distribution under the geometric parameter conditions; Based on the gradient relationship between electric potential and electric field, the partial derivative of the predicted electric potential value with respect to spatial coordinates is calculated by automatic differentiation, and the electric field component at the corresponding spatial sampling point is obtained from the partial derivative, thereby obtaining the electric field distribution; Based on the principle of electrostatic field energy, the electric field energy density in the computational domain is calculated based on the electric field distribution, and the electrostatic energy is obtained by integrating the electric field energy density. Based on the relationship between the electrostatic energy and the voltage difference between the upper and lower plates, the capacitance value of the MEMS parallel plate capacitor under the given geometric parameters is calculated.

[0013] Preferably, the partial differential equation residual loss is used to constrain the predicted potential values ​​at the physical collocation points sampled in the air domain of the two-dimensional electrostatic field physical model and satisfy the Laplace equation. The Dirichlet boundary loss is used to constrain the predicted potential values ​​at the boundaries of the upper and lower plates to approximate the corresponding set voltage values. The Neumann boundary loss is used to constrain the electric potential normal gradient at the outer boundary of the two-dimensional computational domain to satisfy the zero flux boundary condition. The overshoot regularization loss is used to penalize situations where the predicted potential value exceeds the set voltage range of the upper and lower plates. The potential monitoring loss is used to constrain the deviation between the predicted potential value and the corresponding true value of the simulated potential at the data points sampled from the calibration set; The electric field component monitoring loss is used to constrain the deviation between the electric field component obtained by automatic differentiation of the electric potential prediction value and the true value of the simulated electric field component of the corresponding data point in the calibration set.

[0014] This invention also provides a MEMS planar capacitance modeling system, comprising: The data module uses a simulation dataset of a two-dimensional electrostatic field physical model of a MEMS parallel plate capacitor under multiple sets of geometric parameters. The simulation dataset includes the spatial coordinates, geometric parameters, simulation potential values, simulation electric field components, and simulation capacitance values ​​of the physical model sample points. The module is used to build a physical information neural network model, which takes spatial coordinates and geometric parameters as inputs, outputs the predicted potential value at the corresponding location, and obtains the partial derivative of the predicted potential value with respect to the spatial coordinates through automatic differentiation. The loss function module is used to define the hybrid physical data loss function, which includes physical loss for constraining the predicted potential value to satisfy the physical laws of electrostatics and supervisory data loss for constraining the deviation between the predicted potential value and the simulation true value. The training module trains the physical information neural network model using a hybrid physical data loss function on the current calibration set. It then uses the trained model to predict all sample points in the simulation dataset, calculating the absolute error between the predicted and simulated potential values ​​for each sample point. Based on the absolute error, it selects a predetermined number of sample points with the largest errors from the simulation dataset as hard samples. The module adds the spatial coordinates, geometric parameters, simulated potential values, simulated electric field components, and simulated capacitance values ​​corresponding to these hard samples to the calibration set to update it. In each training round, it samples data points from the updated calibration set to calculate the supervised data loss, enabling the model to learn the physical field distribution of the region corresponding to the hard samples. Finally, it iteratively trains the physical information neural network model until a predetermined number of iterations is reached, resulting in the trained physical information neural network model. The physical loss is used to constrain the model to satisfy physical laws in each training round, while the supervised data loss is used to constrain the model to fit the simulation data of the hard samples. The prediction module is used to use the trained physical information neural network model as a surrogate model for MEMS parallel plate capacitors, and to predict the potential, electric field and capacitance based on the input geometric parameters and spatial coordinates.

[0015] This invention also provides an electronic device, including a memory and a processor; The memory is used to store computer programs; When the processor executes the computer program stored in the memory, it implements the steps of the MEMS planar capacitor modeling method described above.

[0016] This invention also provides a computer-readable storage medium for storing a computer program, which, when executed by a processor, implements the steps of a MEMS planar capacitor modeling method as described above.

[0017] This invention provides a MEMS planar capacitor modeling method, system, device, and dielectric, which have the following advantages compared with the prior art: This invention introduces a dynamic closed-loop mechanism for training, evaluation, hard sample mining, and calibration set updating during the training process of a parametric physical information neural network model. After each round of outer layer iteration training, the current model is used to predict the potential of all sample points in the simulation dataset, calculate the potential prediction error at each sample point, and select the top Np sample points with the largest errors as hard samples. These hard samples are typically distributed at plate corners, plate edges, and regions with drastic edge field changes. By dynamically updating the coordinates, geometric parameters, simulated potential values, and simulated electric field components corresponding to the hard samples to the calibration set, and sampling and calculating data loss from the calibration set in subsequent training cycles, the model can continuously strengthen its learning of high-error regions during iterative training. In this way, this invention transforms the traditional PINN training method, which relies on fixed training data or fixed sampling strategies, into an adaptive training method based on global error evaluation. This allows the model to gradually shift its training focus to regions with small spatial proportions but drastic electric field changes and high prediction difficulty, thereby improving the potential and electric field prediction accuracy of the MEMS planar capacitor surrogate model in complex edge field regions and further enhancing the capacitance prediction accuracy based on electric field energy calculation. Attached Figure Description

[0018] Figure 1 This is a schematic diagram of a two-dimensional model of a MEMS planar capacitor provided in an embodiment of the present invention; Figure 2 A schematic diagram of a physical information neural network architecture for constructing a planar capacitor model provided in an embodiment of the present invention; Figure 3 Training set samples provided for embodiments of the present invention ( l =63.3μm, d =4.0μm, w A schematic diagram comparing the potential and electric field distribution at 2.0 μm. Figure 4 This is a schematic diagram comparing the PINN predicted capacitance with the FEM true value under all 64 training configurations provided in the embodiments of the present invention. Figure 5 The validation set samples provided for embodiments of the present invention ( l =82.5μm, d =6.8μm, w A schematic diagram comparing the potential and electric field distribution at (λ = 4.6 μm); Figure 6 This is a schematic diagram illustrating the verification of the parameterized prediction results of the capacitor using the PINN model provided in this embodiment of the invention. Detailed Implementation

[0019] To make the above-mentioned objects, features, and advantages of the present invention more apparent and understandable, specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings. Many specific details are set forth in the following description to provide a thorough understanding of the present invention. However, the present invention can be practiced in many other ways different from those described herein, and those skilled in the art can make similar modifications without departing from the spirit of the present invention. Therefore, the present invention is not limited to the specific embodiments disclosed below.

[0020] In practical MEMS capacitor design, the capacitor plates are surrounded by a dielectric material. Due to the limited size of the plates, edge effects of the electric field are inevitably generated. Whether trying to suppress edge effects to reduce interference or using edge effects as the core working mechanism, accurately calculating the total capacitance value after taking edge effects into account is the key to MEMS device design optimization.

[0021] Currently, the main technical approaches for modeling and calculating MEMS planar capacitors are as follows: (1) Analytical calculation techniques: Theoretical analytical models based on mathematical methods such as conformal transformation. These methods attempt to provide analytical solutions for capacitance under specific assumptions through mathematical derivation. However, existing analytical models are usually based on idealized assumptions such as an infinitely large dielectric domain or zero plate thickness. These assumptions do not match the finite physical boundaries that MEMS devices actually have, resulting in the models failing to truly reflect the physical characteristics of actual devices, and severely limiting their accuracy and applicability.

[0022] (2) Finite Element Method (FEM): This is the industry standard method. It obtains numerical solutions for electric potential and electric field by dividing the computational domain into fine meshes and solving partial differential equations using numerical methods. However, finite element simulation cannot provide a continuous, readily available functional mapping relationship from geometric parameters to capacitance values. It relies on specialized software and its computational efficiency is lower than that of neural network models.

[0023] (3) Physical information neural network modeling technology: In recent years, the method of combining deep learning with physical laws has been gradually applied to electrostatic field modeling; Basic PINN method: The Laplace equation and other control equations are directly added as constraints to the loss function of the neural network, and the derivative is calculated using automatic differentiation technology. Theoretically, it can be solved without relying on external label data; Data-enhanced PINN method: In order to accelerate convergence or improve the accuracy under complex boundaries, some studies have begun to try to use the discrete data obtained from finite element simulation as a supervision signal to constrain the training process of the neural network together with the physical equations.

[0024] However, current neural network research faces gaps and technical bottlenecks in the field of parallel-plate capacitor modeling: ① Research gaps: Although physical information neural networks have been applied in fluid mechanics, heat conduction, and some general electrostatic field problems, there are currently no reports in the literature on their specific application to establishing parameterized models of MEMS parallel-plate capacitors; existing technologies lack dedicated neural network modeling methods for this fundamental component, resulting in a lack of efficient intelligent simulation tools when designing such devices. ② Existing general models lack accuracy and adaptability: Existing general electrostatic field neural network models mostly rely on static simulation data or pure physical equation constraints; the former lacks a feedback mechanism and cannot perceive and correct areas with large prediction errors (such as plate corners and edges with extremely large electric field gradients); the latter is prone to getting trapped in local minima when solving for the potential distribution in the air domain, making it difficult for the model to capture complex edge effects and failing to meet the requirements of high-precision modeling.

[0025] Based on this, the present invention provides a capacitance parameterization prediction method based on iterative adaptive hard example mining; this method utilizes a physical information neural network, combined with constraints from electrostatic field physical equations and finite element simulation data, to achieve rapid and high-precision prediction of the potential distribution, electric field distribution, and capacitance value of MEMS planar capacitors under different geometric dimensions; specifically including: Step 1: Construct a two-dimensional electrostatic physical model of a parameterized parallel-plate capacitor; such as... Figure 1 As shown, a two-dimensional computational model of a parallel-plate capacitor is established. This model includes an air domain and upper and lower plates located at the center of the air domain. Geometric parameters are defined: the geometric input parameters of the model are defined, including the plate lengths. l Electrode width w and the spacing between the two plates d The size of the air domain is set as follows: l x × l y Boundary condition setting: Apply a constant voltage to the upper plate. V top A constant voltage is applied to the lower plate. V bottom The plate boundary follows the Dirichlet boundary condition; an insulating condition is applied to the outermost boundary of the air domain, i.e., the normal derivative of the potential is zero. ∂V / ∂n =0), belonging to the Neumann boundary condition; governing equation: in the air domain, electric potential V ( x , y The distribution of ) satisfies the Laplace equation ( x , y (in spatial coordinates) .

[0026] Step 2: Construct a simulation dataset for training support; to prevent the neural network from getting trapped in local minima (i.e., generating mediocre solutions) when solving for the air domain potential, and to provide a benchmark for subsequent "hard example mining," a simulation dataset needs to be generated in advance; select geometric parameters. l , d , w Range of variation (e.g.) l ∈[40,110]μm, d ∈[2,8]μm, w (∈[2,6]μm), multiple sets of geometric configuration cases are generated using full factorial design or uniform sampling methods; electrostatic field simulation is performed on each set of geometric configurations using finite element analysis software to obtain the coordinates of grid points in the computational domain. x , y ) and the corresponding true value of electric potential V GT True values ​​of electric field components E x,GT and E y,GT The above data are compiled to form a simulation dataset, which will be used for supervised learning and hard example mining in subsequent steps.

[0027] Step 3: Construct a parameterized physical information neural network model; such as Figure 2 As shown in the dashed box on the left, a fully connected neural network is constructed as a function approximator; input layer design: the network input layer receives a five-dimensional feature vector ( x n , y n , l n , d n , w n To improve training convergence speed, the input physical quantities are normalized: spatial coordinates. x , y Divide by the feature length respectively to map to the interval [-1, 1]; geometric parameters l , d , w Dimensionlessness is achieved by dividing by the feature length; Hidden layer structure: multiple fully connected hidden layers are used; The hyperbolic tangent (Tanh) activation function is chosen, utilizing its smooth and infinitely differentiable properties to support subsequent automatic differentiation calculation of the second derivative. Output layer design: The output layer contains one neuron, outputting the predicted potential value. .

[0028] Step 4: Define the hybrid physical-data loss function; such as Figure 2As shown in the middle section, the first derivative (∂²⁻¹) is obtained by differentiating the network output using automatic differentiation techniques. x 、∂ y ) and second derivative (∂2) x ∂2 y Construct a hybrid loss function that incorporates both physical and data constraints. L total : .

[0029] Specifically, it includes the following seven components: (1) Physical loss: PDE residual loss ( L pde This is used to constrain the network output to satisfy the Laplace equation in the air domain, and its calculation expression is: .

[0030] Dirichlet boundary loss ( L dir The voltage value used to constrain the network to output a set value at the electrode position is calculated as follows: .

[0031] Neumann boundary loss ( L neu The normal derivative of the network at the external boundary is constrained to be zero, and its calculation expression is: .

[0032] Overshoot regularization loss ( L over This is used to penalize predicted values ​​that exceed the physical range of voltage, forcing the solution to conform to the principle of physical extrema. Its calculation expression is: .

[0033] (2) Data loss: Potential data loss ( L v This is used to minimize the mean square error between the predicted potential and the true potential at the corrected sample points. .

[0034] Electric field data loss ( L Ex , L Ey This is based on the relationship between electric field and electric potential. E =-▽ V Minimize the error between the predicted electric field and the actual electric field data in the calibration set: .

[0035] in: N pde , N dir , N neu and N c These represent the number of sampling points corresponding to each loss function. S c Representative of the revision set, Γ top Indicates the boundary of the upper plate, Γ bottom Indicates the lower electrode boundary, Γ outer Γ represents the boundary of the air domain. plates =Γ top ∪Γ bottom .

[0036] Step 5: Execute the iterative adaptive hard example mining training strategy; such as... Figure 2 As shown in the flowchart on the right, a dynamic closed-loop training method is adopted, which improves the model accuracy by continuously updating the "calibration set"; the specific loop process is as follows: (1) Initial stage: The calibration set is initialized to be empty. At this time, the model is mainly driven by physical loss; (2) Model training: Under the current calibration set, the optimizer is used to train the neural network for several epochs to minimize the total loss. L total (3) Difficult sample evaluation and mining: After training is paused, the current model is used to perform full-field prediction on the simulation dataset in step 2; calculate the absolute value of the potential prediction error of all data points; according to the error distribution, select the top Nw data points with the largest errors. These points are the difficult samples; (4) Calibration set update: extract the coordinates, geometric parameters and corresponding real field values ​​(potential and electric field) of these difficult sample points; add these data to the existing calibration set; (5) perform deduplication to form a new, more targeted calibration set; (6) feed the updated calibration set back to the "model training" stage to start the next round of optimization; through multiple iterations, the calibration set will automatically focus on complex physical regions with significant edge effects, forcing the neural network to "focus on learning" in these regions.

[0037] Step 6: Parameterized prediction of capacitance values; after training, for any given new set of geometric parameters... l , d , w The data is then input into a trained neural network to predict the global potential distribution and the derived electric field distribution. The total energy is then calculated based on the electrostatic energy principle. U :

[0038] .

[0039] in: Ω is the vacuum permittivity. air For the air domain; using the formula C =2 U / ( V top - V bottom ) 2 The capacitance value under this geometric dimension was calculated. C Through the above steps, the present invention can establish a mathematical model that covers a continuous geometric parameter space, achieving fast capacitance prediction while ensuring high accuracy.

[0040] Specific experiment: (1) Setting physical model and geometric parameters.

[0041] The parameterization range of the model's geometric variables is set as follows: plate length l ∈[40.0, 110.0]μm, electrode spacing d ∈[2.0, 8.0]μm, plate width w ∈[2.0, 6.0]μm; the size of the external air domain is fixed at […]. l x =120μm, l y =22.5μm, the out-of-plane thickness of the entire model is set to 60.0μm; the boundary voltage condition is set as follows: upper plate voltage V top =1V, lower plate voltage V bottom =0V.

[0042] (2) Training dataset construction parameters.

[0043] To construct an unbiased training set that covers the entire parameter space, each continuous parameter range is discretized into four equally spaced levels using a linear interval. The specific set of discrete values ​​is as follows: l The wavelength is [40.0, 63.3, 86.7, 110.0] μm. d The micrometer size is [2.0, 4.0, 6.0, 8.0] μm. w The set is [2.0, 3.3, 4.7, 6.0] μm; the Cartesian product of these three sets generates the final 64 geometric configuration training cases; for each of these 64 configurations, FEM simulation is used to generate a model containing the potential. V electric field components E x and E y and capacitors C A high-fidelity benchmark dataset was used for subsequent model training and evaluation.

[0044] (3) Neural network structure and training hyperparameters.

[0045] The neural network architecture consists of one input layer, eight fully connected hidden layers (each with 384 neurons), and one single-neuron output layer. The entire adaptive training process is configured with a total of 150 outer loop iterations. Each outer loop iteration contains 8000 training epochs, from which 2000 hard examples are extracted. The Adam optimizer is used to optimize the model parameters, with a learning rate set to 1.9 × 10⁻⁶. -5 The specific weighting coefficients for each term in the hybrid physical-data loss function are optimally set as follows: λ pde =1.0、 λ dir =36.16、 λ neu =1.93、 λ over =154.01、 λ V =162.70、 λ x =6.17 and λ y =17.51.

[0046] (4) Model performance evaluation.

[0047] Figure 3 Representative training set samples are shown. l =63.3μm, d =4.0μm, w Detailed comparisons of various physical fields (e.g., 2.0 μm) confirm that the trained model has extremely high prediction fidelity in the air domain; for example... Figure 3 As shown in (a)-3(c), the potential distribution predicted by PINN is in high agreement with the true finite element value, accurately reproducing the potential drop from 1V to 0V between the plates and the outward bending of the equipotential lines in the external edge field. Its maximum absolute error is only 4.28 × 10⁻⁶. -3 Furthermore, the error is mainly limited to the corner points of the plates, with extremely low error in most areas of the air domain; for the electric field components derived from the potential, the model also achieves high-precision prediction: for the components E x (like Figure 3 (d)-3(f) shows that the model accurately captures the local concentration of electric field at the ends of the plates and the sign reversal characteristics across the edges, with a maximum absolute error of 8.65×10. 4 V / m, limited to the area around the corner of the electrode; for E y Components (e.g.) Figure 3As shown in (g)-3(i), the model not only accurately reproduces the high-intensity, uniform dominant negative electric field between the plates, but also accurately captures the outward bending phenomenon of the electric field lines at the edge of the plates, with the absolute error still controlled within 1.17×10. 5 V / m, and concentrated in high gradient regions such as corners.

[0048] This invention evaluates the overall performance of the model in capacitance prediction across all 64 training configurations, such as... Figure 4 (Comparison of predicted and true capacitance values ​​in the training set) As shown in the figure, the capacitance values ​​predicted by PINN are in high agreement with the true values ​​in the finite element method, and all data points are closely distributed in the training set. y = x Near the diagonal (a magnified view further confirms this trend); specific quantitative indicators show: the coefficient of determination of the prediction results ( R 2 The accuracy is as high as 0.99986, the mean absolute percentage error is as low as 0.31%, and the root mean square error is only 0.084fF. The above results prove that the model of the present invention has extremely high fitting accuracy to the training data, laying a reliable foundation for verifying its generalization performance.

[0049] To evaluate the model's ability to generalize (interpolate) on unseen data, this invention selects a set of validation set samples not included in the training set. l =82.5μm, d =6.8μm, w Independent tests were conducted on samples with a diameter of 4.6 μm. The comparison of the potential and electric field distribution of the validation set samples is shown in the figure below. Figure 5 As shown; regarding the potential distribution ( Figure 5 (a)-5(c)): The model accurately predicted the smooth potential drop between the plates and the outward bending characteristic of the external equipotential lines. Its maximum absolute error is only 4.87 × 10⁻⁶. -3 V, and stability is limited to the corner points of the plates; for E x Quantity ( Figure 5 (d)-5(f)): The model successfully captured the distribution pattern of the edge field under the interpolation parameters (including the field concentration and sign reversal at the electrode ends); the maximum absolute error is 4.42 × 10⁻⁵. 4 V / m is also limited to the corner area; for E y Quantity ( Figure 5 (g)-5(i): The model accurately reproduces the outward bending effect of the electric field lines on the outer edge of the electrode. Its peak absolute error is 4.55 × 10⁻⁵. 4 V / m is mainly concentrated in the corner gradient region. Further continuous parameter sweep tests (such as...) Figure 6As shown in the figure, the model-predicted capacitance value is in high agreement with the finite element true value. The results accurately reflect the capacitance as a function of the plate spacing. d The physical laws of change, targeting d The maximum relative error of the scan is less than 0.4%, which confirms that the model of the present invention has high-fidelity and high-precision parameterized prediction capabilities in the continuous unseen parameter space.

[0050] This invention utilizes a pre-trained neural network model to achieve "meshless" computation, enabling real-time and rapid prediction of electric potential, electric field, and capacitance after training. Through an iterative adaptive difficult example mining strategy, this invention dynamically identifies "difficult" samples with large errors and feeds them back to the loss function, significantly improving the model's prediction accuracy in physically complex regions (such as regions with significant edge effects). This invention establishes a parameterized model (which can accept different lengths, widths, and spacings) that can not only predict field distribution but also obtain electric field distribution based on the predicted electric potential and calculate capacitance values ​​through electrostatic field energy integration.

[0051] This invention constructs a model based on the finite physical boundaries of actual capacitors, abandoning the idealized assumptions of "infinite dielectric domain or zero plate thickness" in traditional analytical methods. It can realistically reflect the physical characteristics of MEMS parallel plate capacitors, significantly expanding the application range of high precision. This invention transforms discrete numerical solutions into continuous explicit function mappings of neural networks, achieving "meshless" computation. After model training, the computation time complexity for arbitrary geometric parameters is reduced, significantly reducing numerical solution overhead and improving multi-parameter scanning efficiency. This invention introduces an "iterative adaptive difficult example mining strategy" and a "hybrid loss function supervised by electric field gradient." This mechanism forces the model to escape the "mediocre solutions" easily caused by pure physical constraints, accurately capturing the characteristics of regions with drastic electric field changes, such as plate corners and edges.

[0052] This invention utilizes highly optimized underlying hyperparameters (including the number of nodes in the fully connected architecture and specific weight configurations in the hybrid loss function) through extensive experiments. These key parameter configurations optimize the network structure and loss weights, improving training stability and prediction accuracy. This invention can serve as an auxiliary module for the development of MEMS capacitive sensors, reducing the time for single parameter verification from minutes to milliseconds. This significantly reduces the cost for enterprises to rely on high-end computing workstation hardware and expensive commercial simulation software, and also shortens the trial-and-error iteration cycle of MEMS device design and optimization by a factor of two, demonstrating extremely high industrial transformation value.

[0053] This invention directly constructs a parameterized model based on the finite physical boundaries of actual capacitors, without making any idealized truncation of the dielectric domain and plate thickness, thus fully preserving the real physical characteristics of edge effects. The "iterative adaptive hard example mining" strategy of this invention endows the model with a "closed-loop feedback training" capability: after each round of training, the model can "evaluate" its shortcomings, "locate" the region with the largest error, "actively request" more targeted supervision signals, and "focus on overcoming" these weak points in the next round of training. This adaptive, closed-loop learning method simulates the cognitive strategy of human experts in solving complex problems—"repeatedly checking and focusing on key breakthroughs"—allowing the model's predictive ability to continuously evolve with the number of iterations.

[0054] The "iterative adaptive hard example mining" strategy of this invention realizes real-time redistribution of the equivalent weights of the loss function by dynamically updating the calibration set: In the first iteration, the calibration set is empty, the model is driven by physical loss, and initially learns the overall trend; In the second iteration, the first batch of hard samples (concentrated at corners) are mined, and the calibration set begins to include data from singular regions; In the Nth iteration, the sample density of singular regions in the calibration set is much higher than that of smooth regions, because samples in these regions are repeatedly mined and added; As the iteration progresses, the sampling distribution of the data loss can be continuously changed, enabling the model to continuously strengthen its fitting ability to complex edge regions.

[0055] The embodiments described above are merely illustrative of several implementations of the present invention, and while the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the invention patent. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these all fall within the protection scope of the present invention. Therefore, the protection scope of this invention patent should be determined by the appended claims.

Claims

1. A method for modeling MEMS planar capacitors, characterized in that, Includes the following steps: A simulation dataset of a two-dimensional electrostatic field physical model of a MEMS parallel plate capacitor under multiple sets of geometric parameters is obtained. The simulation dataset includes the spatial coordinates, geometric parameters, simulation potential values, simulation electric field components, and simulation capacitance values ​​of the physical model sample points. A physical information neural network model is constructed, with spatial coordinates and geometric parameters as inputs and the predicted potential value at the corresponding location as output. The partial derivative of the predicted potential value with respect to the spatial coordinates is obtained through automatic differentiation. Define a hybrid physics data loss function, which includes a physics loss for constraining the predicted potential value to satisfy the physical laws of electrostatics and a supervisory data loss for constraining the deviation between the predicted potential value and the simulation true value. Under the current calibration set, the physical information neural network model is trained using the hybrid physical data loss function. The trained model is then used to predict all sample points in the simulation dataset, and the absolute error between the predicted potential value and the simulated potential value of each sample point is calculated. Based on the absolute error, a preset number of sample points with the largest error are selected from the simulation dataset as hard samples. The spatial coordinates, geometric parameters, simulated potential values, simulated electric field components, and simulated capacitance values ​​corresponding to the hard samples are added to the calibration set to update the calibration set. In each round of training, the supervised data loss is calculated by sampling data points from the updated calibration set, so that the model learns the physical field distribution of the region corresponding to the hard samples. The physical information neural network model is iteratively trained until a preset number of iterations is reached to obtain the trained physical information neural network model. Among them, the physical loss is used to constrain the model to satisfy physical laws in each round of training, and the supervised data loss is used to constrain the model to fit simulation data of difficult samples. The trained physical information neural network model is used as a surrogate model for MEMS planar capacitors, and the potential, electric field and capacitance are predicted based on the input geometric parameters and spatial coordinates.

2. The MEMS planar capacitor modeling method according to claim 1, characterized in that, The construction of the simulation dataset includes: A two-dimensional electrostatic field physical model of MEMS parallel plate capacitor is constructed. Based on the two-dimensional electrostatic field physical model, the geometric parameters affecting the performance of MEMS parallel plate capacitor are determined. The geometric parameters include plate length, plate width and the distance between the upper and lower plates. The range of variation of plate length, plate width and the distance between the upper and lower plates is set. Multiple sets of geometric configuration cases are generated using a full factorial experimental design to cover the preset geometric parameter space. For each set of geometric configuration cases, the finite element method is used to mesh the two-dimensional electrostatic field physical model and solve for the steady-state electrostatic field, obtaining the spatial coordinates, potential values, and electric field components of each sample point in the computational domain, and obtaining the capacitance value corresponding to the geometric configuration case. The spatial coordinates, geometric parameter labels, simulated potential values, simulated electric field components, and simulated capacitance values ​​corresponding to each geometric configuration case are summarized to form the simulation dataset.

3. The MEMS planar capacitor modeling method according to claim 2, characterized in that, The construction of the physical information neural network model includes: A parameterized physical information neural network model is constructed. The parameterized physical information neural network model adopts a fully connected neural network structure. Its input layer is used to receive a feature vector composed of spatial coordinates and geometric parameters. The feature vector includes normalized horizontal coordinates, vertical coordinates, electrode length, electrode width, and the distance between the upper and lower electrodes. The parameterized physical information neural network model includes a preset number of hidden layers. Each hidden layer adopts a fully connected layer structure and uses the hyperbolic tangent function as the activation function to establish a nonlinear mapping relationship between spatial coordinates, geometric parameters and electrostatic potential distribution. The output layer of the parameterized physical information neural network model is a single neuron structure, which is used to output the predicted electric potential value at the corresponding position under the current spatial coordinates and geometric parameters, and obtain its partial derivative with respect to the spatial coordinates through automatic differentiation based on the predicted electric potential value.

4. The MEMS planar capacitor modeling method according to claim 1, characterized in that, The construction of the hybrid physical data loss function includes: A hybrid physical data loss function, consisting of a weighted average of physical loss and supervised data loss, is constructed and used as the optimization objective of the parameterized physical information neural network model. The physical losses include partial differential equation residual loss, Dirichlet boundary loss, Neumann boundary loss, and overshoot regularization loss; The monitoring data loss includes potential monitoring loss and electric field component monitoring loss.

5. The MEMS planar capacitor modeling method according to claim 1, characterized in that, The training of the physical information neural network model includes: The calibration set is initialized to an empty set, and the physical information neural network model is iteratively trained according to a preset number of outer layer iterations. In each outer layer iteration, a preset number of training cycles are executed. In each training cycle, physical collocation points for calculating the residuals of the control equations and boundary points for boundary condition constraints are randomly generated to calculate the physical loss. When the calibration set is not empty, a preset number of data points are randomly sampled from the current calibration set. The supervision data loss is calculated based on the simulation potential and electric field components corresponding to the data points. The physical loss and the supervision data loss are combined to form a hybrid physical data loss function to update the parameters of the physical information neural network model. After each round of outer layer iteration training is completed, the electric potential of all sample points in the simulation database is predicted using the currently trained physical information neural network model. The absolute deviation between the predicted electric potential and the true value of the simulated electric potential at each sample point is calculated. Based on the absolute deviation, all sample points in the simulation database are sorted in descending order, and the top Np sample points with the largest absolute deviation are selected as the difficult samples mined in this round of outer layer iteration, where Np is the preset number of newly added difficult samples in each round of outer layer iteration. The spatial coordinates, structural parameters, true values ​​of simulated electric potential, and true values ​​of simulated electric field components corresponding to the difficult samples are added to the current calibration set. The updated calibration set is deduplicated to form the calibration set used in the next round of outer layer iteration. The training, prediction, difficult sample mining, and calibration set update process are repeated until the number of outer layer iterations reaches the preset number of iterations, and the trained physical information neural network model is obtained.

6. A MEMS planar capacitor modeling method according to claim 5, characterized in that, The prediction of potential, electric field, and capacitance includes: The trained parameterized physical information neural network model is used as a surrogate model for MEMS parallel plate capacitors to obtain the geometric parameters of the MEMS parallel plate capacitors to be predicted, and spatial sampling points are generated in a preset two-dimensional computational domain. The coordinates of the spatial sampling points and the geometric parameters are input into the surrogate model to obtain the predicted potential values ​​at each spatial sampling point, thereby obtaining the potential distribution under the geometric parameter conditions; Based on the gradient relationship between electric potential and electric field, the partial derivative of the predicted electric potential value with respect to spatial coordinates is calculated by automatic differentiation, and the electric field component at the corresponding spatial sampling point is obtained from the partial derivative, thereby obtaining the electric field distribution; Based on the principle of electrostatic field energy, the electric field energy density in the computational domain is calculated based on the electric field distribution, and the electrostatic energy is obtained by integrating the electric field energy density. Based on the relationship between the electrostatic energy and the voltage difference between the upper and lower plates, the capacitance value of the MEMS parallel plate capacitor under the given geometric parameters is calculated.

7. A MEMS planar capacitor modeling method according to claim 4, characterized in that, The partial differential equation residual loss is used to constrain the predicted potential values ​​at the physical collocation points sampled in the air domain of the two-dimensional electrostatic field physical model and satisfy the Laplace equation. The Dirichlet boundary loss is used to constrain the predicted potential values ​​at the boundaries of the upper and lower plates to approximate the corresponding set voltage values. The Neumann boundary loss is used to constrain the electric potential normal gradient at the outer boundary of the two-dimensional computational domain to satisfy the zero flux boundary condition. The overshoot regularization loss is used to penalize situations where the predicted potential value exceeds the set voltage range of the upper and lower plates. The potential monitoring loss is used to constrain the deviation between the predicted potential value and the corresponding true value of the simulated potential at the data points sampled from the calibration set; The electric field component monitoring loss is used to constrain the deviation between the electric field component obtained by automatic differentiation of the electric potential prediction value and the true value of the simulated electric field component of the corresponding data point in the calibration set.

8. A MEMS planar capacitor modeling system, characterized in that, include: The data module uses a simulation dataset of a two-dimensional electrostatic field physical model of a MEMS parallel plate capacitor under multiple sets of geometric parameters. The simulation dataset includes the spatial coordinates, geometric parameters, simulation potential values, simulation electric field components, and simulation capacitance values ​​of the physical model sample points. The module is used to build a physical information neural network model, which takes spatial coordinates and geometric parameters as inputs, outputs the predicted potential value at the corresponding location, and obtains the partial derivative of the predicted potential value with respect to the spatial coordinates through automatic differentiation. The loss function module is used to define the hybrid physical data loss function, which includes physical loss to constrain the predicted potential value to satisfy the physical laws of electrostatics and supervisory data loss to constrain the deviation between the predicted potential value and the simulation true value. The training module is used to train the physical information neural network model using the hybrid physical data loss function under the current calibration set, and to use the currently trained model to predict all sample points in the simulation dataset, and calculate the absolute error between the predicted potential value and the simulated potential value of each sample point. Based on the absolute error, a preset number of sample points with the largest error are selected from the simulation dataset as hard samples. The spatial coordinates, geometric parameters, simulated potential values, simulated electric field components, and simulated capacitance values ​​corresponding to the hard samples are added to the calibration set to update the calibration set. In each round of training, the supervised data loss is calculated by sampling data points from the updated calibration set, so that the model learns the physical field distribution of the region corresponding to the hard samples. The physical information neural network model is iteratively trained until a preset number of iterations is reached to obtain the trained physical information neural network model. Among them, the physical loss is used to constrain the model to satisfy physical laws in each round of training, and the supervised data loss is used to constrain the model to fit simulation data of difficult samples. The prediction module is used to use the trained physical information neural network model as a surrogate model for MEMS parallel plate capacitors, and to predict the potential, electric field and capacitance based on the input geometric parameters and spatial coordinates.

9. An electronic device, characterized in that, include: Memory and processor; The memory is used to store computer programs; When the processor executes the computer program stored in the memory, it implements the steps of the MEMS planar capacitor modeling method as described in any one of claims 1 to 7.

10. A computer-readable storage medium, characterized in that, Used to store a computer program, which, when executed by a processor, implements the steps of a MEMS planar capacitor modeling method as described in any one of claims 1 to 7.

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