Micro-cantilever geometric structure parameter inversion system, method and device

Abstract

The invention discloses a micro-cantilever geometric structure parameter inversion system, method and device, and belongs to the technical field of micro-electro-mechanical systems. According to the system, a dynamic model containing geometry, material and electromechanical coupling nonlinearity is constructed, a PINNs network fused with physical constraints is designed, a staged training optimization strategy is adopted, and efficient inversion of micro-cantilever geometric parameters is achieved. The core of the method is that a nonlinear kinetic equation is used as a constraint to be embedded into a neural network for training, experimental data and physical residual errors are combined to construct a composite loss function, and the problems that a traditional method neglects a nonlinear effect, depends on finite element simulation and is low in inversion precision are solved. The average relative error of inversion of the system is lower than 1%, which is obviously superior to that of a traditional method, and the method can be widely applied to MEMS device design, online detection and closed-loop control scenes.

Description

Technical Field

[0001] This invention relates to the field of microelectromechanical systems (MEMS) technology, and more specifically to a system, method, and apparatus for inverting the geometric parameters of microcantilever beams. Background Technology

[0002] Microcantilever beams, as core components in the MEMS field, are widely used in sensors, actuators, and biological detection platforms. Their dynamic characteristics are fundamental to microscale detection and control. In existing technologies, the dynamic analysis of microcantilever beams is mostly based on a linear spring-damped mass model (…). This model is only applicable to simple scenarios with small amplitudes and low excitation. However, in practical applications, micro-cantilever beams often exhibit significant nonlinear behavior: geometric nonlinearity: strain and displacement have a nonlinear relationship under large deformation; material nonlinearity: the elastic modulus of microscale materials deviates from the linear assumption under multi-field coupling; electromechanical coupling nonlinearity: the electric field intensity and displacement are strongly coupled under capacitive driving, leading to dynamic changes in equivalent stiffness. Traditional linear models cannot accurately characterize these characteristics, resulting in large prediction errors; at the same time, existing parameter inversion relies on finite element simulation (computationally intensive and time-consuming) or data fitting (poor convergence and weak physical consistency), which is difficult to meet the requirements of high-dimensional parameters and high-precision inversion. Summary of the Invention

[0003] The purpose of this invention is to provide a system, method, and apparatus for inverting the geometric parameters of microcantilever beams, so as to solve the problem that existing technologies cannot meet the requirements of high-dimensional parameters and high-precision inversion.

[0004] The technical solution of the present invention to solve the above-mentioned technical problems is as follows: In a first aspect, this application provides a system for inverting the geometric parameters of a microcantilever beam, comprising: The nonlinear dynamics model building module is used to establish a dynamic model of a microcantilever beam that includes geometric, material, and electromechanical coupling nonlinearities. The model includes multi-order nonlinear correction terms. Input includes: Excitation angular frequency ( The angular frequency of the externally applied excitation signal is one of the key input parameters of the model. External electric field force ( The magnitude of the electric force applied through the electrodes affects the dynamic response of the microcantilever beam. The output includes: Displacement response ( The displacement of the microcantilever beam under excitation is the direct dynamic response of the model to external excitation. Excitation phase angle ( The phase difference between the displacement response and the excitation reflects the phase characteristics of the microcantilever beam vibration. This model uses the input excitation angular frequency and the applied electric field force, combined with the equivalent mass of the microcantilever beam itself ( ), damping coefficient ( ), structural stiffness ( ), electromechanical coupling stiffness ( ) and other parameters, output displacement response and excitation phase angle, so as to depict the dynamic behavior of micro cantilever beam under nonlinear conditions including geometry, material and electromechanical coupling; The physical information neural network design module adopts a neural network structure, takes the frequency response data and excitation frequency of the micro cantilever beam as input, and the geometric parameters to be inverted as output, and the training process of the neural network is embedded with the constraints of the nonlinear dynamic model. The training and optimization module is used to train the physical information neural network in stages and optimize the network parameters through a composite loss function, wherein the composite loss function includes at least the equation residuals of the nonlinear dynamic model and the observation errors of the experimental data. The inference and evaluation module is used to take unknown frequency response data as input, output the inverted geometric parameters through a trained physical information neural network, and evaluate the accuracy and physical consistency of the inversion results.

[0005] In a further embodiment, the model expression established by the nonlinear dynamics model construction module includes:

[0006] in, For displacement response, For the excitation angular frequency, To excite the phase angle, For equivalent quality, The damping coefficient is... For the linear and nonlinear stiffness of the structure, For electromechanical coupling stiffness, By applying an external electric field, the nonlinear effects caused by geometry, materials, and electromechanical coupling can be accurately quantified, providing strict physical constraint equations for neural networks. This solves the problem that traditional linear models cannot describe large deformation and strongly coupled scenarios, and improves the accuracy of dynamic behavior characterization.

[0007] In a further embodiment, the physical information neural network design module adopts a multilayer perceptron structure. The input of the neural network includes statistical features of frequency and response. These statistical features include at least the maximum value, minimum value, center value, and average value. This is to reduce the input dimensionality through statistical feature extraction, retain key information that is strongly correlated with structural parameters, enhance the neural network's ability to capture physical laws, improve the stability of the nonlinear mapping between input and output parameters, and reduce redundant data interference.

[0008] In a further embodiment, the composite loss function includes:

[0009] in, Represents the total loss function. For the residuals of the nonlinear dynamic equations, For boundary condition residuals, The error between the predicted value and the experimental data, Adjustable loss weights are used to balance the constraints of physical laws with the fit of experimental data, avoid physical inconsistencies caused by purely data-driven approaches, reduce the risk of overfitting, and ensure that the inversion results conform to both the dynamic equations and the actual measurement data, thereby improving the reliability of the results.

[0010] A further embodiment of the proposed training and optimization module includes a phased training strategy comprising: The first stage optimizes the equation residuals and boundary condition residuals in the composite loss function using the first learning rate. The second stage optimizes the observation error in the composite loss function using a second learning rate that is less than the first learning rate. This approach allows the network to quickly learn the framework of physical laws before finely fitting the details of experimental data, avoiding convergence instability caused by an excessively large initial learning rate, improving training efficiency and network generalization ability, and ensuring that physical constraints take precedence over data fitting.

[0011] In a further embodiment, the training and optimization module also includes a physical parameter normalization unit, which is used to normalize physical parameters such as equivalent mass, stiffness, and damping to a preset range. After training, the parameters are mapped back to the physical magnitude through inverse normalization, so as to eliminate the gradient explosion problem caused by the difference in the magnitude of physical quantities, ensure the numerical stability of the loss function during the training process, and enable the inversion results to directly correspond to the actual engineering scale, thereby improving the physical interpretability of the results.

[0012] In a further embodiment, the geometric parameters to be inverted include the beam width and / or beam length of the microcantilever beam. This allows for a focus on the core geometric parameters of MEMS devices, specifically addressing the problem that key dimensions such as beam width and beam length are difficult to accurately identify using traditional methods, thereby improving the system's adaptability to actual engineering needs.

[0013] In a further embodiment, the evaluation metrics of the inference and evaluation module include at least the average relative error, error distribution, and physical consistency test. The physical consistency test involves substituting the inversion parameters into the nonlinear dynamic model to verify the residual equation, so as to verify the inversion results from multiple dimensions of accuracy, stability, and physical rationality, thereby avoiding the problem of "good numerical fit but physical incompatibility" and ensuring that the results can be directly used in MEMS device design, online testing, and other scenarios, thus improving engineering practicality.

[0014] Secondly, based on the same inventive concept, this application also provides a method for inverting the geometric parameters of a microcantilever beam, comprising the following steps: S1: Construct a dynamic model of a microcantilever beam that includes geometric, material, and electromechanical coupling nonlinearities, and the model includes multi-order nonlinear correction terms; S2: Design a physical information neural network, taking frequency response data and excitation frequency as inputs and the geometric parameters to be inverted as outputs, and construct a composite loss function that includes the residuals of the dynamic model equations and the experimental data errors; S3: Train the physical information neural network in stages, and optimize the network parameters through the composite loss function; S4: Input unknown frequency response data, output inversion parameters through the trained network, and evaluate its accuracy and physical consistency.

[0015] In a further proposed approach, step S3 involves phased training, which includes: prioritizing the optimization of equation residuals and boundary condition residuals with a larger learning rate, and then optimizing experimental data errors with a smaller learning rate. During the training process, physical parameters are normalized, integrating the advantages of phased training and normalization. This approach accelerates the learning of physical laws while ensuring numerical stability, resulting in an average relative error of less than 1%, which is significantly better than traditional methods (3%~5%) and meets the requirements for high-precision parameter identification.

[0016] Thirdly, based on the same inventive concept, this application also provides a microcantilever beam geometric structure parameter inversion device, including the aforementioned microcantilever beam geometric structure parameter inversion system, further comprising: a data acquisition unit, the data acquisition unit being connected to the physical information neural network design module of the inversion system, for acquiring the original frequency response data and excitation frequency signal of the microcantilever beam, and processing the original data into statistical features including maximum value, minimum value, center value and average value before inputting it into the physical information neural network; and an execution unit, the execution unit being connected to the reasoning and evaluation module of the inversion system, for receiving the inverted geometric parameters, and outputting control commands to the microcantilever beam production or testing equipment according to the geometric parameters.

[0017] This invention offers the following advantages: It proposes a system, method, and apparatus for inverting the geometric parameters of microcantilever beams, aiming to overcome the limitations of traditional spring-mass models and finite element methods in microscale modeling and parameter inversion. By deeply integrating the nonlinear dynamic model of the microcantilever beam with a PINNs neural network, this invention achieves: constraint modeling combined with theoretical physical equations; efficient inversion of structural parameters based on experimental data; and adaptability to modeling requirements involving multi-field coupling and complex boundaries. Compared with existing technologies, this invention possesses significant advantages such as high physical consistency, strong scalability, and applicability to high-dimensional parameter inversion. Attached Figure Description

[0018] Figure 1 This is a schematic diagram of a microcantilever beam structure and its equivalent spring-damping-mass system model.

[0019] Figure 2 This is a flowchart of the physical loss calculation steps of the present invention.

[0020] Figure 3 This is a comparison chart of the actual structural parameters and the model's predicted parameters.

[0021] Figure 4 This is the convergence curve of the physical loss function.

[0022] Figure 5 A visualization of the parameter inversion error distribution. Detailed Implementation

[0023] The principles and features of the present invention are described below with reference to the accompanying drawings. The examples given are only for explaining the present invention and are not intended to limit the scope of the present invention.

[0024] This invention proposes a system for inverting the geometric parameters of microcantilever beams, aiming to effectively overcome the shortcomings of traditional spring-mass simplified models in nonlinear dynamic modeling and to replace the high computational cost of finite element simulation methods in high-dimensional parameter identification. Based on microscale physical modeling, and combined with the high-dimensional fitting capability and physical consistency constraints of neural networks, this system achieves efficient, stable, and interpretable inversion identification of the geometric parameters (such as beam width and length) of microcantilever beams.

[0025] The overall system adopts a four-stage process of "modeling-encoding-training-inference", which includes the following steps: S1: Constructing the nonlinear dynamic model of the microcantilever beam: In the system proposed in this invention, the primary task to achieve high-precision inversion of the structural parameters of the micro-cantilever beam is to establish a dynamic model with good physical consistency, interpretability, and nonlinear response capability. The dynamic response analysis of traditional microscale structures can typically be equivalent to a classical spring-damped-mass system, whose dynamic behavior can be described by the following linear second-order differential equation:

[0026] in, Indicates equivalent mass. Indicates the damping coefficient. Indicates stiffness, For structural response displacement, The external excitation force is used. These systems are characterized by their simple form and ease of solution, and are widely used in resonant frequency analysis, amplitude-frequency characteristic prediction, and parameter approximation modeling in microelectromechanical systems (MEMS).

[0027] However, it is worth emphasizing that microcantilever beams, as a common MEMS component, often operate under large displacements, high excitation voltages, or complex boundary conditions in practical applications. In these conditions, their mechanical behavior deviates from the ideal linear assumption, exhibiting significant nonlinear dynamic characteristics. These nonlinearities mainly originate from the following three aspects: Geometric nonlinearity: When a beam undergoes large deformation, the strain is no longer linearly related to the displacement, and a nonlinear stiffness term needs to be introduced for correction. Material nonlinearity: Under conditions of multi-field coupling or sudden load change, the elastic modulus of microscale materials may exhibit nonlinear changes; Electromechanical coupling nonlinearity: Under capacitor drive, there is a strong coupling relationship between electric field strength and displacement, which causes the equivalent stiffness to change dynamically with displacement.

[0028] Therefore, using only a linear model will severely underestimate the nonlinearity of the system's response and affect the accuracy of structural parameter inversion.

[0029] To address the aforementioned problems, this invention constructs a nonlinear dynamic model containing multi-order nonlinear correction terms based on the classical linear model, used to accurately characterize the response behavior of a micro-cantilever beam under harmonic excitation. This model, based on Euler-Bernoulli beam theory and combined with actual measurements and electric field driving conditions, establishes the following set of nonlinear ordinary differential equations:

[0030] in, Indicates displacement response, For the excitation angular frequency, For the excitation phase angle; For equivalent quality, The (equivalent) damping coefficient; These are the linear and nonlinear stiffnesses of the structure, respectively. For electromechanical coupling stiffness; The term represents the applied electric force, and its specific expression is as follows: ; in, This represents the amplitude of the small-signal driving voltage, which can be 5 millivolts (the harmonic excitation expression can be written as: ); The transfer factor is used to equate voltage excitation to a driving force, which is specifically determined by the dielectric and electrode geometry, and is equivalent to the electromechanical coupling force at the selected operating point. For voltage The linearization coefficients, specifically expressed as follows: ; Under the conditions of parallel plate approximation and small amplitude linearization, we can take: ; in, Represents the vacuum permittivity; This represents the static bias voltage (if there is no bias, it can be understood as the equivalent operating point voltage or a constant obtained through calibration). and These represent the length and thickness of the electrode, respectively. This represents the electrode spacing; therefore, we can obtain: .

[0031] This model not only maintains compatibility with traditional models in terms of mathematical structure, but also explicitly introduces the ability to characterize large amplitude responses through third-order nonlinear terms, making it suitable for modeling and analyzing complex phenomena such as asymmetric frequency responses and multiple solution region discrimination. Figure 1 This invention provides a model for a microcantilever beam structure and its equivalent spring-damped-mass system. Using this nonlinear model, the invention not only significantly enhances modeling accuracy but also lays a reliable physical and mathematical foundation for subsequent training of neural networks based on physical constraints. It provides interpretable and highly stable theoretical support for structural parameter inversion, overcoming the problem of predicting failure under high-amplitude responses in existing methods. Figure 1 The left side shows the physical structure of the microcantilever beam, including the fixed end, mass block, electrode structure and applied electric field; the right side shows its dynamic equivalent simplified model, which introduces structural and coupling nonlinear terms into the traditional mass-damped-spring frame to establish a nonlinearly corrected set of response equations.

[0032] S2: Design Physical Information Neural Networks (PINNs): After constructing the nonlinear dynamic model of the microcantilever beam, this invention further designs and constructs a parameter identification framework based on Physics-Informed Neural Networks (PINNs) to achieve efficient and physically consistent structural parameter inversion. This method organically combines the expressive power of the neural network with the constraint mechanism of the physical master equation, overcoming not only the uninterpretability problem caused by the "black box modeling" of traditional neural networks, but also significantly improving the stability and generalization ability of the model under conditions of sparse data or noise interference.

[0033] Specifically, PINNs networks take measured response data and excitation frequency as inputs, and output the structural parameters to be inverted, including but not limited to the beam width of the microcantilever beam. With Liang Chang To meet the needs of statistical feature modeling extracted from the frequency response data of micro-cantilever beam structures, this invention employs a Multi-Layer Perceptron (MLP) as the basic structure of the parameter inversion model. This network can establish nonlinear mapping relationships in a low-dimensional statistical feature space, possessing strong function approximation capabilities and engineering applicability. In terms of structural design, the network consists of multiple fully connected layers stacked sequentially. The internal activation function uses Tanh to enhance nonlinear modeling capabilities, and the output layer introduces a Sigmoid activation function to normalize the predicted structural parameters to a physically reasonable range of [0,1]. This structure has a moderate number of parameters, high computational efficiency, and good training stability and convergence, making it particularly suitable for microscale parameter inversion tasks based on normalized statistical feature inputs.

[0034] To incorporate the aforementioned nonlinear dynamics model into the training process of the neural network, this invention constructs a composite loss function of the following form to drive the network to approximate real observation data while satisfying physical laws:

[0035] in: Represents the total loss function; The residual of the equation obtained by substituting the predicted structural parameters of the neural network output into the nonlinear dynamic equation. Boundary condition residuals are mainly used to constrain the physical consistency of the model at fixed ends (such as zero displacement or rotation). This is the observation error term, representing the error between the predicted value and the (true) experimental data (such as the frequency response); These are the weight parameters for the three loss terms, which can be flexibly adjusted according to the training stage and task requirements.

[0036] By constructing the loss using this physically embedded method, this invention significantly improves the model's interpretability and robustness. Compared to traditional purely data-driven inversion methods, this network structure does not rely on a large amount of labeled data or simulation samples; it only requires a small number of measurement points to achieve accurate inversion of structural parameters. Furthermore, this structure overcomes the computational bottleneck of finite element simulation methods in high-dimensional spaces, possessing stronger real-time performance and scalability, making it suitable for industrial scenarios such as online detection and closed-loop control.

[0037] S3: Network Training and Optimization To ensure the good convergence and numerical stability of the physical information neural network in the structural parameter inversion task, this invention combines the characteristics of microscale physical modeling and designs a staged training strategy tailored to the frequency response data and the physical loss term. During training, while maintaining physical consistency, the learning pace and dominant loss term of the network are dynamically adjusted, ultimately achieving high-precision and generalizable prediction performance.

[0038] First, regarding the selection of optimization methods, the system employs the gradient-based first-order optimizer Adam for network training. This optimizer features an adaptive learning rate update mechanism, enabling stable training even with multiple coexisting loss terms. To further improve training efficiency and robustness, this invention introduces a two-stage segmented learning rate scheduling strategy: Phase 1: Initial training phase, setting a relatively large learning rate (e.g., ... The focus is on optimizing the network for the master dynamic equation ( The ability to fit the boundary conditions to quickly establish a preliminary understanding of physical laws; Phase Two: Fine-tuning Phase. After the loss has initially converged, the learning rate is reduced to... Or smaller, gradually increasing the impact of experimental observation data ( The fitting ability of the terms is considered, and the gradient updates are balanced between different loss terms.

[0039] During training, the system employs a mini-batch training mechanism, mixing multiple frequency response samples, different boundary conditions and location points, and data calculated from the current prediction results in each batch. Residual points. This hybrid input strategy can effectively improve the model's generalization ability and reduce the risk of overfitting due to sample distribution bias.

[0040] Considering the potential order-of-magnitude difference between structural parameters and physical response variables, this invention introduces a physical parameter normalization and denormalization mechanism to avoid gradient explosion in the early stages of training. In the early stages of training, all structural parameters (such as...) are normalized and denormalized. , All predictions and error calculations are performed within the normalized space; simultaneously, physical terms in the nonlinear dynamic equations (such as mass, stiffness, and driving terms) are also normalized according to a unified scale factor. After training, the system maps the output results back to the original physical magnitude through an inverse normalization operation to achieve engineering interpretability of the results.

[0041] Furthermore, to further enhance the stability and generalization ability of training, this invention can employ regularization strategies (such as Dropout and weight decay) and dynamic loss weight adjustment mechanisms to automatically adjust the weights according to the training phase. The relative weights of residuals and data items. These strategies collectively ensure robust training and physical consistency of the network under finite sample conditions.

[0042] In summary, by integrating the above-mentioned multi-stage optimization strategy with the physical perception mechanism, this invention effectively overcomes the problems of slow convergence, easy getting trapped in local optima and physical distortion in traditional parameter inversion methods during the training process, and significantly improves the practicality and industrial deployment value of the network in structural parameter inversion.

[0043] S4: Evaluation of Model Inference and Inversion Results: After completing network training, this invention further predicts and evaluates the structural parameters of the microcantilever beam through an inference module. The inference process uses unseen frequency response data as input and directly outputs normalized predicted values ​​of structural parameters through a pre-trained physical information neural network model. Subsequently, the system performs an inverse normalization operation using a previously saved scaler to map the results back to the real physical scale, thereby obtaining interpretable and deployable inverted values ​​of structural parameters.

[0044] In the evaluation phase, to fully verify the predictive performance of the model, this invention uses the following three types of indicators for quantitative and visual analysis: Mean Squared Error Index: Calculates the mean squared error (MSE) and relative error (such as the ratio of absolute error to the true value) between the predicted value and the actual value, reflecting the overall prediction accuracy; Error distribution analysis: The prediction errors of multiple samples in the dimension of structural parameters are statistically analyzed and plotted to identify possible systematic biases or areas of uncertainty. Physical consistency check: Substitute the predicted structural parameters back into the nonlinear dynamic model to verify whether the obtained response still satisfies the physical consistency check. The master equation and boundary conditions are used to ensure that the predicted values ​​are physically reasonable.

[0045] Furthermore, this invention introduces a multi-graph linkage analysis tool at the visualization level to compare the actual parameters and predicted parameters using scatter plots (e.g., vs The analysis includes the convergence curve of the loss function (such as the decreasing trend of the physical residual term with training epochs) and the visualization of the spatial distribution of prediction errors. Through these analyses, the predictive performance and stability of the model at different structural scales can be intuitively demonstrated.

[0046] Test results show that, under multiple experimental conditions, the parameter inversion system proposed in this invention has an average relative error of less than 1% on the test set, which is significantly better than the 3%~5% error level of traditional finite element modeling and fitting methods. Furthermore, the model possesses good generalization ability, adapting to prediction tasks under different frequency ranges and measurement accuracies, and can be extended for online inversion, rapid parameter tuning, and feedback control in device manufacturing processes.

[0047] In summary, the reasoning and evaluation mechanism of this invention not only has high accuracy and interpretability, but also enables the engineering closed-loop application of the results, meeting the dual requirements of parameter modeling accuracy and real-time performance in micro-nano systems. 1. This invention deeply integrates microscale nonlinear dynamics models with the PINNs framework, enabling effective description of: Geometric nonlinearity; Material nonlinearity; Electromechanical coupling nonlinearity; thus, it is suitable for high-amplitude resonance and strongly nonlinear scenarios.

[0048] 2. High-precision parameter inversion based on physical constraints During the training process, the neural network is strictly constrained by physical equations, and combines dynamic residuals, boundary constraints and experimental data to avoid overfitting and physical inconsistencies, thereby improving the reliability of the inversion results.

[0049] 3. Meshless Modeling for Multiphysics To avoid the dependence of traditional finite element methods on mesh density, a meshless sampling strategy combined with PINNs is adopted to improve the modeling adaptability to complex geometries and multi-field couplings.

[0050] 4. Verification and processing mechanism for variable-length physical valid data In actual microscale measurements, frequency is affected by physical constraints. With resonant response There are physically unsolvable regions, especially Within the sampling angle range, measurement results at certain angles may be invalid or violate physical laws.

[0051] Therefore, in the data preprocessing stage, this invention designs a validity verification process for variable-length data to ensure that sampling and training are only performed within the physically solvable angle range.

[0052] 5. Normalization and denormalization mechanisms based on physical consistency To prevent gradient explosion caused by tiny errors in the physical loss term during the early stages of training, this invention proposes a set of... The core logic of the parameter normalization-denormalization process is as follows: (1) Calculation of quadratic parameters In the data preprocessing stage, based on the raw data, the results are derived and calculated. The equation is directly related to 6 sets of physical parameters:

[0053] Among them, the first stiffness difference Second stiffness difference ;Will and As network input, it is processed using the Min-Max normalization method.

[0054] (2) Parameter grouping processing strategy and , irrelevant quantities (such as) Treat it as a constant ; Directly related to structural parameters After normalization, save the scaler for later use.

[0055] (3) The processing flow for calculating physical losses is as follows: Figure 2 As shown; 6. Improved convergence and training stability By using the above-mentioned physical constraint normalization mechanism, loss explosion caused by differences in the magnitude of physical quantities is effectively avoided, ensuring good convergence and numerical stability of the loss function during training.

[0056] Example: To verify the effectiveness of this invention, a Physical Information Neural Network (PINNs) based on statistical feature input was used to invert the structural parameters of the microcantilever beam, and the following implementation scheme was constructed: Neural Network Structure and Parameter Setting To efficiently invert the geometric parameters of micro-cantilever beams (such as beam width) With Liang Chang This embodiment employs a multi-layer perceptron (MLP) neural network structure based on statistical feature input. This method maps frequency response data to a set of physically representative statistics, and establishes a nonlinear relationship between these statistics and structural parameters through a fully connected neural network.

[0057] Specifically, the original response data for each sample group is dimensional. This data includes 180 frequency sampling points and corresponding composite response values ​​(such as displacement, current, or phase response). During the data preprocessing stage, this invention extracts the following eight key statistical features for modeling: For frequency ( ) and resonant response ( Extract the following values ​​respectively: maximum value (max), minimum value (min), response at center frequency (center), and mean value (mean); This feature extraction strategy can effectively avoid the "physically unsolvable region" that causes the system to be unresponsive under boundary structural conditions (such as excessive stiffness), while retaining global statistical features closely related to structural changes, effectively reducing the input dimension and improving training efficiency.

[0058] The neural network structure is as follows: Input feature dimensions: 8 (each sample corresponds to the above 8 statistical features); Number of hidden layers: 4; Number of neurons per layer: 128; Activation function: Tanh, used to enhance nonlinear modeling capabilities; Output layer dimension: 2 (corresponding to structural parameters) ); Output layer activation function: Sigmoid (restricts the output to the range of 0 to 1); Dropout: Optional, disabled by default; Implementation framework: PyTorch, the forward propagation process is implemented using the nn.Sequential function, the structure is clear, and it is easy to deploy and port.

[0059] This model has advantages such as being lightweight, stable, and easy to deploy. It is particularly suitable for microscale parameter inversion tasks under low-dimensional input statistical features and can achieve high-precision prediction of structural geometric parameters while ensuring physical consistency.

[0060] Training strategies and optimization methods To ensure that the model can be trained stably and achieve good prediction performance under limited samples and strong physical constraints, this embodiment adopts a two-stage segmented training strategy based on the Adam optimizer, combined with data normalization processing and mini-batch mixed input mechanism to improve the model's convergence speed, numerical stability and physical consistency.

[0061] Before model training, all input statistical features and target structural parameters are standardized and converted into a normalized expression with zero mean and unit variance. During training, the system saves the scaling factor so that the model output can be denormalized back to the original physical quantities in the subsequent inference stage, ensuring that the inversion results have clear physical meaning.

[0062] The specific training process is as follows: Optimizer configuration: The Adam optimizer is used, and the initial learning rate is set to... , The parameter is set to the default value ( ); Phase 1 (Warm-up Phase): Train for 10,000 epochs using the initial learning rate. The main goal is to enable the model to quickly capture the basic mapping relationship between structural parameters and frequency response statistical features, and avoid getting trapped in early local optima. Phase Two (Fine-tuning Phase): Reduce the learning rate to Continue training for 10,000 epochs, fine-tuning the network weights to improve the ability to fit structural details, and further optimize the physical residuals and prediction errors; Loss function construction: The total loss function during training consists of three parts, namely the residual term of the master equation. Boundary condition residuals Fitting error with observed values The weighted combination of the three is:

[0063] Among them, each weight During training, the algorithm can be automatically adjusted according to the residual order of magnitude to ensure that the gradient updates of different physical quantities are on the same numerical scale.

[0064] In addition, this embodiment also employs the following training assistance mechanism: Mini-batch sampling: Each training round uses a mini-batch mechanism to mix frequency response samples, residual sampling points and boundary condition points as input, which improves generalization ability and reduces local overfitting. Gradient clipping mechanism (optional): Gradient clipping can be activated to limit the maximum norm of the model when the gradient update is abnormal, thus preventing explosive weight updates; Breakpoint saving and validation evaluation: The model state is automatically recorded after every 500 training cycles, and error evaluation is performed on the validation set to ensure that the entire model training process is monitorable and recoverable.

[0065] Through the above optimization strategies and regularization control, the structural parameter inversion model proposed in this invention can maintain good training stability, physical consistency and prediction accuracy in the context of multi-physics coupling, which is significantly better than the inversion performance of traditional optimization or fitting methods in high-dimensional space.

[0066] Inversion results and performance evaluation To verify the accuracy, stability, and physical consistency of the inversion system of this invention, this embodiment systematically evaluates the trained model under multiple experimental conditions. The evaluation process covers the prediction accuracy of the test set, the convergence analysis of the loss function, the visualization of the error distribution, and the verification of the physical rationality of the inversion results.

[0067] (1) Test set prediction performance When performing inference on the test set, the system first converts the frequency response samples into corresponding 8-dimensional statistical features, performs normalization, and then inputs them into the trained neural network model. The predicted structural parameters output by the model are inversely normalized and compared with the true values. Experimental results show that in most test samples, the relative error between the predicted and true values ​​is less than 1%, with the beam width... The average error is 0.73%, beam length The average error is 0.92%, which is significantly better than the error level of traditional finite element inversion or least squares optimization (usually between 3% and 5%).

[0068] (2) Convergence of loss function and training stability During training, the residual terms of the physical master equation With observation error term The loss values ​​all showed a stable decreasing trend, without exhibiting numerical instability phenomena such as gradient explosion or oscillating convergence. Especially after adopting normalization and dynamic loss weight adjustment mechanisms, The optimization process of the items and data items maintained good numerical coordination, and the overall loss function tended to stabilize between 8,000 and 12,000 rounds.

[0069] (3) Visual analysis of error distribution To more comprehensively evaluate the model's prediction accuracy and error distribution characteristics, this embodiment presents three key visualization images based on the test set results, showcasing the model's performance from three dimensions: result accuracy, training convergence, and error structure. Figure 3 Comparison chart of actual structural parameters and model predicted parameters This figure, presented as a scatter plot, illustrates the correspondence between the predicted and actual structural parameters output by the model on the test set. The horizontal axis represents the actual structural parameters, and the vertical axis represents the model's predicted values, including beam width. With Liang Chang Two dimensions. The results show that most sample points are highly concentrated near the diagonal, indicating that the model has small prediction errors in both structural dimensions, and the overall trend is highly consistent with the target value.

[0070] Figure 4 Convergence curve of physical loss during training This figure records the physical residual loss term (e.g., ...) during model training. The numerical changes of the equation residuals in each training iteration are shown. The results show that the physical loss decreases rapidly in the initial training phase and tends to stabilize between about 8,000 and 12,000 iterations without abnormal fluctuations or gradient explosion, indicating that the model has good numerical stability and convergence under multi-loss term optimization conditions.

[0071] Figure 5 Visualization of prediction error distribution To reveal the prediction bias under different sample structure combinations, this figure presents the relative error distribution of beam width and beam length in the form of an error histogram and a box plot. Statistical results show that over 95% of the prediction samples have a relative error of less than 2%, and the average prediction error is less than 1%. The box plot does not show any significant outliers, indicating that the model has strong overall prediction stability and does not exhibit runaway prediction problems under extreme structural configurations.

[0072] Figure 3 , Figure 4 , Figure 5 The combined results show that the model not only exhibits good convergence during training, but also demonstrates high consistency in inversion accuracy and error control on the test set, enabling it to adapt to structural prediction tasks within different frequency response ranges. This visualization analysis of the error structure provides data support for the credibility assessment before model deployment.

[0073] (4) Physical consistency and reversibility verification This embodiment also re-introduces the structural parameters predicted by the model into the nonlinear dynamic model of the microcantilever beam, calculates the resulting frequency response, and performs a re-matching analysis with the original observation data. The results show that the predicted values ​​output by the model can effectively... The residual terms of the equation approach 0, which verifies the physical feasibility of the predicted structural parameters and further proves that the model not only has the ability to fit data, but also has consistency with the underlying physical model.

[0074] In summary, experimental results demonstrate that the structural parameter inversion system based on the PINNs framework proposed in this invention can achieve high-precision prediction and maintain physical consistency under various structural configurations and frequency response conditions, and has significant industrial practical value that is superior to traditional methods.

[0075] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A system for inverting the geometric parameters of a microcantilever beam, characterized in that, include: The nonlinear dynamics model building module is used to establish a dynamic model of a microcantilever beam that includes geometric, material, and electromechanical coupling nonlinearities. The model includes multi-order nonlinear correction terms. Inputs include excitation angular frequency External electric field force ; The output includes displacement response. Excitation phase angle ; The physical information neural network design module adopts a neural network structure, takes the frequency response data and excitation frequency of the micro cantilever beam as input, and the geometric parameters to be inverted as output, and the training process of the neural network is embedded with the constraints of the nonlinear dynamic model. The training and optimization module is used to train the physical information neural network in stages and optimize the network parameters through a composite loss function, wherein the composite loss function includes at least the equation residuals of the nonlinear dynamic model and the observation errors of the experimental data. The inference and evaluation module is used to take unknown frequency response data as input, output the inverted geometric parameters through a trained physical information neural network, and evaluate the accuracy and physical consistency of the inversion results.

2. The microcantilever beam geometric parameter inversion system according to claim 1, characterized in that, The model expressions established by the nonlinear dynamics model building module include: in, For displacement response, For the excitation angular frequency, To excite the phase angle, For equivalent quality, The damping coefficient is... These are the linear and nonlinear stiffnesses of the structure, respectively. For electromechanical coupling stiffness, This is an external electric field force.

3. The microcantilever beam geometric parameter inversion system according to claim 1, characterized in that, The physical information neural network design module adopts a multilayer perceptron structure. The input of the neural network includes statistical features of frequency and response, and the statistical features include at least the maximum value, minimum value, center value and average value.

4. The microcantilever beam geometric parameter inversion system according to claim 1, characterized in that, The composite loss function includes: in, Represents the total loss function. For the residuals of the nonlinear dynamic equations, For boundary condition residuals, The error between the predicted value and the experimental data, These are adjustable loss weights.

5. The microcantilever beam geometric parameter inversion system according to claim 1, characterized in that, The phased training strategy of the training and optimization module includes: The first stage optimizes the equation residuals and boundary condition residuals in the composite loss function using the first learning rate. The second stage optimizes the observation error in the composite loss function using a second learning rate that is less than the first learning rate.

6. The microcantilever beam geometric parameter inversion system according to claim 5, characterized in that, The training and optimization module also includes a physical parameter normalization unit, which is used to normalize physical parameters such as equivalent mass, stiffness, and damping to a preset range, and then map them back to the physical magnitude through inverse normalization after training.

7. The microcantilever beam geometric parameter inversion system according to claim 1, characterized in that, The geometric parameters to be inverted include the beam width and / or beam length of the microcantilever beam.

8. The microcantilever beam geometric parameter inversion system according to claim 1, characterized in that, The evaluation metrics of the reasoning and evaluation module include at least the average relative error, error distribution, and physical consistency test. The physical consistency test involves substituting the inversion parameters into the nonlinear dynamic model to verify the equation residuals.

9. A method for inverting the geometric parameters of a micro-cantilever beam, characterized in that, include: Constructing a nonlinear dynamic model: Establishing a dynamic model of the microcantilever beam that covers geometric, material, and electromechanical coupling nonlinearities. The model includes multi-order nonlinear correction terms to accurately describe the mechanical behavior of the microcantilever beam in actual operation. Design a physical information neural network: Build a neural network with a multilayer perceptron structure, take the frequency response data and statistical features such as the maximum, minimum, center and average values ​​of the excitation frequency as input, take the geometric parameters such as the width and length of the micro cantilever beam to be inverted as output, and embed the physical constraints of the constructed nonlinear dynamic model during the network training process. Implement phased training: First, adopt The learning rate is optimized to improve the residuals of the dynamic equations and boundary conditions in the composite loss function, enabling the network to quickly learn physical laws; subsequently, the following is employed... The learning rate is optimized to improve the observation error and the fitting accuracy of the experimental data; physical parameters such as mass and stiffness are normalized before training, and the scale factor is saved for subsequent inverse normalization. Inference and evaluation: The unknown frequency response data is input into the trained network to obtain the normalized geometric parameter prediction results. After inverse normalization, the geometric parameters of the microcantilever beam at the real scale are obtained. The accuracy and reliability of the inversion results are evaluated based on the average relative error, error distribution and physical consistency test.

10. A device for inverting the geometric parameters of a microcantilever beam, characterized in that, The microcantilever beam geometric parameter inversion system according to any one of claims 1-8 further includes: a data acquisition unit connected to the physical information neural network design module of the inversion system, used to acquire the original frequency response data and excitation frequency signal of the microcantilever beam, and process the original data into statistical features including maximum value, minimum value, center value and average value before inputting it into the physical information neural network; and an execution unit connected to the reasoning and evaluation module of the inversion system, used to receive the inverted geometric parameters and output control commands to the microcantilever beam production or testing equipment according to the geometric parameters.

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