Reverse design method of local resonance topology metamaterial based on physical information network

Abstract

The invention discloses a physical information network-based local resonance topology metamaterial reverse design method, and relates to the field of design, optimization and application of elastic metamaterials, and the method comprises the steps: designing a unit cell structure based on a local resonance principle, and building a frequency dispersion relation data set of the unit cell structure; establishing a finite element equivalent model, and calculating the local resonance frequency of an oscillator in the unit cell structure; constructing a physical information neural network based on the unit cell structure and the finite element equivalent model, and constructing a topological metamaterial reverse design model by using the physical information neural network; and on the basis of sample data in the data set, constructing low-frequency broadband target frequency dispersion characteristics by performing linear transformation on the frequency dispersion relationship, and importing the target frequency dispersion characteristics into the reverse design model to realize topological metamaterial structure design. Through an innovative reverse design method, vibration control of a specific target frequency band in a low-frequency range can be realized, and the method is particularly suitable for low and medium frequency vibration suppression requirements of high-precision equipment such as a precision instrument working platform and a precision equipment vibration isolation base.

Description

Technical Field

[0001] The present invention relates to the fields of elastic metamaterial design, optimization and application, and in particular to a reverse design method of local resonance topological metamaterial based on physical information network. Background Art

[0002] Topological metamaterials, with their unique physical properties such as negative refraction, nonreciprocal transmission, and defect immunity, are emerging in fields such as acoustics, optics, and mechanical engineering, demonstrating promising applications. Efficient and accurate design methods for topological metamaterials are crucial for realizing their application.

[0003] Traditional topological metamaterial design methods typically rely on known physical models, using numerical simulations to fine-tune and optimize structural parameters locally. While this approach can meet basic design requirements to a certain extent, its heavy reliance on prior knowledge severely limits the optimization space for structural parameters, making it difficult to achieve efficient metamaterial performance.

[0004] Therefore, those skilled in the art are committed to developing a reverse design method for local resonance topological metamaterials based on physical information networks. Summary of the Invention

[0005] In view of the above-mentioned defects of the prior art, the technical problem to be solved by the present invention is that the traditional forward design method of topological metamaterials based on physical prior models is time-consuming and has high manpower and physical costs.

[0006] To achieve the above object, the present invention provides a method for inverse design of local resonance topological metamaterials based on a physical information network, the method comprising the following steps:

[0007] S101: Designing a unit cell structure based on the principle of local resonance, and establishing a dispersion relation data set of the unit cell structure;

[0008] S103: establishing a finite element equivalent model and calculating the local resonance frequency of the oscillator in the unit cell structure;

[0009] S105: Building a physical information neural network based on the unit cell structure and the finite element equivalent model, and constructing a topological metamaterial reverse design model using the physical information neural network;

[0010] S107: Based on the sample data in the data set, construct a low-frequency broadband target dispersion characteristic by performing a linear transformation on the dispersion relationship, and import the target dispersion characteristic into the inverse design model to realize the topological metamaterial structure design.

[0011] Furthermore, in the step S101, the dispersion relation data set is constructed based on the design parameters of the unit cell structure and the corresponding dispersion characteristics, and the dispersion relation data set is divided into a training set and a test set.

[0012] Furthermore, in step S103, in the finite element equivalent model, the outer frame structure is established using the solid element SOLID 185, the centers of the left and right vibrators are defined by the mass element MASS21, and the vibrator centers are rigidly coupled to the nodes on the surface of the inner hole of the vibrator. The support stiffness and natural frequency of the left and right vibrators are:

[0013]

[0014] Among them, k i is the oscillator support stiffness, f i is the natural frequency of the oscillator, F is the unit force, U i is the oscillator displacement, m i is the mass of the vibrator, i is the number of the left and right vibrators, i=l represents the left vibrator, and i=r represents the right vibrator.

[0015] Furthermore, in step S105, the physical information neural network includes a pre-training model, a reverse design model and a physical equivalent model, wherein:

[0016] The pre-trained model is trained using the dispersion relationship data set, and the trained pre-trained model predicts the dispersion curve of the unit cell structure according to given design parameters;

[0017] The inverse design model is configured as a multi-layer feedforward neural network architecture, which achieves accurate prediction of the design parameters by learning the mapping relationship between the input dispersion relationship and the output design parameters;

[0018] The physical equivalent model integrates the oscillator resonance frequency calculated by the finite element equivalent model into the inverse design framework, quickly solves the characteristic frequencies of the left and right oscillators at the wave vector K, and optimizes and verifies the first two order characteristic frequencies at the wave vector K of the predicted dispersion relationship as physical constraints.

[0019] Furthermore, the pre-training model includes an input layer, a hidden layer and an output layer, wherein:

[0020] The input layer is configured as 1 and is used to receive design variables;

[0021] The hidden layer is configured as 3, and the design variables are feature extracted and transformed through the neurons of the hidden layer. Each hidden layer adopts a nonlinear activation function SiLU, which is differentiable in the entire real number domain;

[0022] The output layer is configured as one, receives the data processed by the hidden layer, and outputs the predicted dispersion relationship.

[0023] Furthermore, the inverse design model includes an input layer, a hidden layer and an output layer, wherein:

[0024] The input layer flattens the received dispersion data from 24×5 to 120×1 dimensions;

[0025] The hidden layer is configured as 3, and the nonlinear activation function SiLU is used to process the input data;

[0026] The output layer contains 7 neurons, the number of which corresponds to the design parameters. The output layer uses the Sigmoid activation function for scaling processing to accurately limit the output data to the pre-set design variable value range.

[0027] The activation function scaling process is:

[0028] D i =W i ·Sigmoid(d i )+B i

[0029] Among them, D i is the i-th design parameter finally generated after scaling, d i The initial design parameters generated without scaling, W i is the weight of the activation process, B i is the bias for the activation process.

[0030] Furthermore, during the training process of the inverse design model, a loss function is constructed that integrates the three elements of design parameters, dispersion relationship, and physical information. The loss function is:

[0031]

[0032] Among them, the first one is the predicted design parameter The mean absolute error with the label result D; the second term is the predicted dispersion relationship The mean absolute error between the target dispersion relation R; the third term is the physical loss error term, D is the label result, is the predicted design parameter, R is the target dispersion relation, To predict the dispersion relation, f is the characteristic frequency at the wave vector K in the target dispersion relation, The local resonance frequency, w, is obtained by solving the finite element equivalent model based on the predicted design parameters. d ,w r ,w f is a hyperparameter.

[0033] Furthermore, the step S105 includes the following sub-steps:

[0034] S1051: Establishing the pre-training model: constructing the pre-training model using a multi-layer perceptron, inputting design parameters, and the pre-training model predicting the dispersion relation of a unit cell;

[0035] S1052: Establishing the reverse design model: integrating the finite element equivalent model and the pre-trained model to build a multi-layer neural network reverse design model for metamaterial unit cell structure design;

[0036] S1053: Model validity verification and error evaluation: Verify the validity of the reverse design model in the metamaterial reverse design, and evaluate the performance of the reverse design model in the metamaterial reverse design.

[0037] Furthermore, in the step S107, topological metamaterial design under a broadband design goal is included, and the topological metamaterial design under a broadband design goal includes the following sub-steps:

[0038] S10711: Randomly select a set of narrow bandgap dispersion relations from the sample set as the design basis;

[0039] S10712: Using the method of frequency band shifting or multiplication, the characteristic frequencies of the second band and above are expanded to construct a dispersion curve with broadband characteristics as the design goal;

[0040] S10713: Based on the proposed inverse design model and the constructed target dispersion curve, the unit cell design parameter prediction is performed to obtain a design parameter combination.

[0041] Furthermore, in the step S107, topological metamaterial design under a low-frequency design target is also included, and the topological metamaterial design under a low-frequency design target includes the following sub-steps:

[0042] S10721: Randomly select a set of dispersion relations from the sample set as the design basis;

[0043] S10722: Using the band multiplication method, the characteristic frequency of the first band is reduced, thereby constructing a dispersion curve with low-frequency bandgap characteristics as the design goal;

[0044] S10723: Based on the proposed inverse design model and the constructed target dispersion curve, perform unit cell design parameter prediction to obtain a design parameter combination.

[0045] In a preferred embodiment of the present invention, compared with the prior art, the present invention has the following beneficial effects:

[0046] 1. This invention is data-driven and predicts metamaterial design parameters through finite element simulation and physical information network. After the design model is trained, it can flexibly construct the target dispersion relationship and quickly and accurately predict the design parameters, which has positive significance for improving the efficiency of metamaterial design.

[0047] 2. This invention breaks through the limitations of traditional experience-based design methods and adopts a goal-oriented inverse design theory to achieve precise design of metamaterials with specific bandgap characteristics;

[0048] 3. The present invention uses a reverse design method to quickly respond to the needs of different application scenarios and achieve customized bandgap design in the range of 226 to 2210 Hz.

[0049] The concept, specific structure and technical effects of the present invention will be further described below in conjunction with the accompanying drawings to fully understand the purpose, characteristics and effects of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 This is a schematic diagram of the steps of a reverse design method for a local resonance topological metamaterial based on a physical information network according to a preferred embodiment of the present invention;

[0051] Figure 2 This is a flow chart of topological metamaterial design based on a physical information network model according to a preferred embodiment of the present invention;

[0052] Figure 3 Schematic diagram of a local resonance unit cell according to a preferred embodiment of the present invention;

[0053] Figure 4 The unit cell dispersion relation and the local motion mode under the wave vector K of a preferred embodiment of the present invention are shown;

[0054] Figure 5 It is a schematic diagram of a unit cell finite element equivalent model and the local motion of its oscillator in a preferred embodiment of the present invention;

[0055] Figure 6 1 is an overall schematic diagram of a physical information network model (PINN) of a preferred embodiment of the present invention;

[0056] Figure 7 This is a schematic diagram of a pre-training model and the changes in training / testing errors in a preferred embodiment of the present invention;

[0057] Figure 8 This is a comparison of the dispersion relations calculated by the pre-trained models of unit cells 1 and 2 and the finite element method in a preferred embodiment of the present invention;

[0058] Figure 9It is a schematic diagram of a reverse design model of a preferred embodiment of the present invention and the introduction of physical knowledge;

[0059] Figure 10 This is a preferred embodiment of the present invention, which incorporates physical knowledge into the inverse design model training / test loss change and prediction dispersion relationship;

[0060] Figure 11 The comparison between the inverse design model prediction results with and without the physical mechanism and the finite element results of a preferred embodiment of the present invention is as follows;

[0061] Figure 12 The broadband metamaterial design process (based on band addition) and the predicted dispersion relation of a preferred embodiment of the present invention are as follows;

[0062] Figure 13 The broadband metamaterial design process (based on frequency band multiplication) and the predicted dispersion relation of a preferred embodiment of the present invention are as follows;

[0063] Figure 14 This is a comparison between the prediction results of the model in this paper and the target results under the low-frequency target of a preferred embodiment of the present invention. DETAILED DESCRIPTION

[0064] The following describes several preferred embodiments of the present invention with reference to the accompanying drawings to make its technical content clearer and easier to understand. The present invention can be embodied in many different forms of embodiments, and the scope of protection of the present invention is not limited to the embodiments mentioned herein.

[0065] In the drawings, components with identical structures are denoted by the same reference numerals, and components with similar structures or functions are denoted by similar reference numerals. The size and thickness of each component shown in the drawings are arbitrary and are not limited by the present invention. For clarity, the thickness of components in some places in the drawings is appropriately exaggerated.

[0066] like Figure 1 、 Figure 2 As shown, in order to address the problems of traditional forward design methods of topological metamaterials based on physical models, which are time-consuming and have high manpower and physical costs, the present invention provides a reverse design method for topological metamaterials based on local resonance of physical information networks. Based on the principle of local resonance, a unit cell structure is designed, and the dispersion relationship is calculated using the finite element method to establish a model data set. The resonance frequency of the local oscillator of the unit cell structure is calculated using the finite element equivalent model and introduced into the metamaterial design model as a physical constraint. At the same time, considering the physical information of the local oscillator resonance frequency, a physical information neural network model is established for the reverse design of topological metamaterials. The present invention uses the proposed design model to verify the performance of the physical information network under the low-frequency and broadband design goals.

[0067] like Figure 1 As shown, an embodiment of the present invention provides a method for inverse design of a local resonance topological metamaterial based on a physical information network, the method comprising the following steps:

[0068] S101: Designing a unit cell structure based on the principle of local resonance, and establishing a dispersion relationship data set of the unit cell structure.

[0069] In this embodiment, a phononic crystal is designed based on the principle of local resonance, and its dispersion relationship data set is established for subsequent design model training and testing. In order to achieve the control of low-frequency elastic waveguides, a local resonance type unit cell is designed, such as Figure 3 shown.

[0070] In this embodiment, a dispersion relationship dataset is first constructed based on the design parameters of the unit cell structure and the corresponding dispersion characteristics, and the dispersion relationship dataset is divided into a training set and a test set, providing sufficient data support for the subsequent training and verification of the deep learning model.

[0071] Based on the unit cell design parameters and their corresponding dispersion characteristics, this example constructs a dataset containing multiple sets of design parameter and dispersion relationships. Each set of samples consists of seven design parameters and corresponding 24×5 dispersion curve data. To effectively preserve the key characteristics of the low-frequency energy band, the first five eigenfrequencies were selected as effective eigenfrequencies during the data preprocessing stage. This dataset is divided into training and test sets in a 4:1 ratio, providing sufficient data support for the subsequent training and validation of deep learning models.

[0072] S103: Establish a finite element equivalent model and calculate the local resonance frequency of the oscillator in the unit cell structure.

[0073] In this example, an efficient and accurate finite element equivalent model is established to calculate the resonant frequencies of the left and right local oscillators in the unit cell structure. These resonant frequencies provide key physical parameters for the design model and are used in the training and optimization of the deep learning model.

[0074] In the finite element equivalent model, the outer frame structure is established using the solid element SOLID 185, the centers of the left and right oscillators are defined by the mass element MASS 21, and the oscillator centers are rigidly coupled to the nodes on the surface of the inner hole of the oscillator.

[0075] The support stiffness and natural frequency of the left and right oscillators are:

[0076]

[0077] Among them, k i is the oscillator support stiffness, f i is the natural frequency of the oscillator, F is the unit force, U iis the oscillator displacement, m i is the mass of the vibrator, i is the number of the left and right vibrators, i=l represents the left vibrator, and i=r represents the right vibrator.

[0078] S105: Based on the unit cell structure and finite element equivalent model, a physical information neural network model is constructed, and based on the physical information neural network model, a topological metamaterial reverse design model is constructed.

[0079] In this embodiment, based on the designed unit cell structure and the established finite element equivalent model, a physical information network model (Physics-Informed Neural Networks, PINN) was built to realize the reverse design of local resonance metamaterials. Figure 6 As shown in Figure 2, the network model is mainly divided into three parts: pre-training model, reverse design model and physical equivalent model, among which,

[0080] 1) Pre-trained model

[0081] A pre-trained model trained on a dispersion relation dataset predicts the dispersion curve of a unit cell structure based on given design parameters. The pre-trained model consists of an input layer, a hidden layer, and an output layer. The input layer is configured as one, receiving the design variables; the hidden layers are configured as three, and the design variables are extracted and transformed through the neurons in the hidden layers. Each hidden layer uses a nonlinear activation function, SiLU, which is differentiable across the entire real domain. The output layer is configured as one, receiving the data processed by the hidden layers and outputting the predicted dispersion relation.

[0082] 2) Reverse design model

[0083] Configured as a multi-layer feedforward neural network architecture, it accurately predicts design parameters by learning the mapping between input dispersion relations and output design parameters. The inverse design model consists of an input layer, hidden layers, and an output layer. The input layer expands the received dispersion data from 24×5 to 120×1 dimensions. There are three hidden layers, and the nonlinear activation function SiLU is used to process the input data.

[0084] The output layer contains multiple neurons, the number of neurons corresponds to the design parameters, and the output layer uses the Sigmoid activation function for scaling processing to accurately limit the output data to the pre-set design variable value range.

[0085] The activation function scaling process is:

[0086] D i =W i ·Sigmoid(d i )+B i

[0087] Among them, D i is the i-th design parameter finally generated after scaling, d i The initial design parameters generated without scaling, W i is the weight of the activation process, B i is the bias for the activation process.

[0088] 3) Physical equivalent model

[0089] The oscillator resonance frequency calculated by the finite element equivalent model is integrated into the inverse design framework to quickly solve the eigenfrequencies of the left and right oscillators at the wave vector K, and the first two-order eigenfrequencies of the predicted energy band are optimized and verified as physical constraints.

[0090] In this embodiment, during the training process of the inverse design model, the constructed loss function integrates three aspects: design parameters, dispersion relationship, and physical information to ensure that the model can comprehensively and accurately reflect the relationship between the design goal and the actual output.

[0091] The loss function is specifically:

[0092]

[0093] Among them, the first one is the predicted design parameter The mean absolute error with the label result D, the second term is the predicted dispersion relationship The mean absolute error between the target dispersion relation R, the third term is the physical loss error term, and D is the label result. is the predicted design parameter, R is the target dispersion relation, is the predicted dispersion relation, f is the characteristic frequency in the target dispersion relation, To solve the local resonance frequency, w d ,w r ,w f is a hyperparameter.

[0094] In this embodiment, when constructing the physical information neural network model, the following sub-steps are specifically included:

[0095] S1051: Establish a pre-training model.

[0096] A pre-training model is constructed using a multi-layer perceptron, and the design parameters are input. The pre-training model predicts the dispersion relationship of the unit cell.

[0097] S1052: Building a reverse design model

[0098] By integrating the finite element equivalent model and the pre-trained model, a multi-layer neural network inverse design model is built for the design of metamaterial unit cell structures.

[0099] S1053: Model Validation and Error Assessment

[0100] Verify the effectiveness of the inverse design model in metamaterial reverse design, and evaluate the performance of the inverse design model in metamaterial reverse design.

[0101] S107: Based on the sample data in the dataset, a low-frequency broadband target dispersion characteristic is constructed by performing a linear transformation on the dispersion relationship, and the target dispersion characteristic is imported into the inverse design model to realize the topological metamaterial structure design.

[0102] To address the broadband, low-frequency design requirements of topological metamaterials, this example focuses on the application of a proposed physical information neural network (PINN) model. Based on sample data from a dataset, a linear transformation (including multiplication and addition) of the dispersion relation is performed to construct a low-frequency, broadband target dispersion characteristic. This characteristic is then used as an input parameter in an inverse design model, enabling the efficient design of topological metamaterial structures, specifically for both broadband and low-frequency design objectives.

[0103] 1) Topological metamaterial design for broadband design goals

[0104] Based on the inverse design method, the design of metamaterials with broadband characteristics is achieved by regulating the dispersion characteristics. In order to obtain a dispersion curve with broadband characteristics as the design target, numerical operations (including addition, subtraction, multiplication and division operations) are performed on frequency bands within different frequency ranges based on the dispersion relationship in the data set, and then the target dispersion curve with broadband characteristics is obtained.

[0105] When designing topological metamaterials with broadband design goals, the following sub-steps are included:

[0106] S10711: Randomly select a set of narrow bandgap dispersion relations from the sample set as the design basis;

[0107] S10712: Using the method of frequency band shifting or multiplication, the characteristic frequencies of the second band and above are expanded to construct a dispersion curve with broadband characteristics as the design goal;

[0108] S10713: Based on the proposed inverse design model and the constructed target dispersion curve, the unit cell design parameters are predicted to obtain the design parameter combination.

[0109] 2) Topological metamaterial design for low-frequency design goals

[0110] With the help of inverse design model, the design of topological metamaterials with low-frequency bandgap characteristics is realized.

[0111] When designing topological metamaterials for low-frequency design goals, the following sub-steps are included:

[0112] S10721: Randomly select a set of dispersion relations from the sample set as the design basis;

[0113] S10722: Using the band multiplication method, the characteristic frequency of the first band is reduced, thereby constructing a dispersion curve with low-frequency bandgap characteristics as the design goal;

[0114] S10723: Based on the proposed inverse design model and the constructed target dispersion curve, the unit cell design parameters are predicted to obtain the design parameter combination.

[0115] Judging from the above two topological metamaterial designs provided in this embodiment, the embodiment of the present invention flexibly constructs a target dispersion relationship with low-frequency and broadband characteristics through the control method of data concentration band shifting and multiplication, and then uses the proposed topological metamaterial inverse design model to achieve efficient reverse design of metamaterials.

[0116] Compared with the prior art, the inverse design method of the physical information network local resonance topological metamaterial provided by the embodiment of the present invention has the following specific features:

[0117] 1. To address the time-consuming and high-manpower and physical costs of traditional forward design methods for topological metamaterials based on physical models, this invention uses data-driven methods to predict metamaterial design parameters through finite element simulation and physical information networks. Based on the principle of local resonance, a dual-oscillator metamaterial structure is designed, and the oscillator resonant frequency is used as a physical constraint to establish a physical information network prediction model. Finally, the target dispersion relation is input, and the design parameters are output. After the design model is trained, the target dispersion relation can be flexibly constructed, and the design parameters can be quickly and accurately predicted, which has positive significance for improving the efficiency of metamaterial design.

[0118] 2. Addressing the low accuracy and efficiency of dispersion relation predictions for localized resonant metamaterials, the pre-trained sub-model designed in this invention can predict the dispersion relation of localized resonant metamaterial structures. Based on finite element simulation, a dataset of design parameters and dispersion relations is established. A pre-trained model is then built using a multi-layer perceptron model, enabling efficient prediction of dispersion relations. Given any design parameters that fit within the parameter space, the dispersion relation of a localized resonant unit cell can be efficiently predicted, with characteristic frequency prediction errors exceeding 3%.

[0119] 3. In view of the difficulty in designing topological metamaterials under the low-frequency broadband design goal, the present invention constructs a target dispersion relationship with low-frequency broadband characteristics through multiplication and addition operations of the dispersion relationship samples in the data set, and then imports the established physical information network design model to realize the design of topological metamaterials. The present invention adopts the principle of local resonance to design the metamaterial unit cell structure, which provides a basis for the construction of low-frequency band gap. At the same time, the dispersion relationship data set can flexibly construct a dispersion relationship with low-frequency broadband through addition and multiplication to serve the physical information network. Under the low-frequency broadband design goal, the proposed method can increase the band gap width of the initial sample by 6 times, and the lowest band gap frequency reaches 226Hz. At the same time, compared with the forward design method, it has higher prediction efficiency.

[0120] The present invention is described in detail below in conjunction with the preferred embodiments of the present invention.

[0121] The flowchart of this article is as follows Figure 1 、 Figure 2 As shown, the present invention provides a method for inverse design of topological metamaterials based on local resonance of a physical information network. First, a unit cell structure is designed based on the principle of local resonance, and the dispersion relationship is calculated using the finite element method to establish a model data set. The resonant frequency of the local oscillator of the unit cell structure is calculated using a finite element equivalent model and introduced as a physical constraint into the metamaterial design model. Considering the physical information of the local oscillator resonant frequency, a physical information neural network model is established for the design of topological metamaterials. Finally, the proposed design model is used to verify the performance of the physical information network under the low-frequency and broadband design goals.

[0122] The inverse design method of a local resonance topological metamaterial based on a physical information network provided by an embodiment of the present invention specifically includes the following steps:

[0123] Step 1: Design the unit cell structure based on the local resonance principle and establish the dispersion relationship data set

[0124] This section will design phononic crystals based on the principle of local resonance and establish a data set of its dispersion relationship for subsequent design model training and testing. In order to achieve the control of low-frequency elastic waveguides, a local resonance unit cell is designed, such as Figure 3 As shown. The lattice constant a and unit cell thickness h are 40 mm and 4 mm respectively. There are 7 main unit cell design variables (h b ,t b ,R f ,R l ,R r ,h l ,h r ), which is marked in the figure. The thickness of the oscillator h in the design variables i and radius R i (i=l or r, representing the left / right vibrator respectively) mainly determines the quality of the vibrator.b and t b Together they determine the stiffness of the oscillator support beam structure. f represents the geometric space in which the local oscillator structure resides. In the designed structure, the oscillator is made of lead metal, and the outer frame is made of epoxy resin. The value ranges of the unit cell design variables and material parameters are shown in Table 1.

[0125] Table 1 Unit cell design variables and their material parameters

[0126]

[0127] Next, we discuss the unit cell dispersion relation and its parameter influence. b =1.4mm,t b =0.8mm,R f =10mm,R l =R r =4.8mm,h l =h r =4.0mm, the unit cell A was constructed, and the band structure of the first irreducible Brillouin zone of the unit cell A was calculated using the finite element method, as shown in Figure 4 As shown in a. When the left and right oscillator parameters (h i and R i ) are consistent, there is a Dirac degeneracy point at the wave vector K in the band structure, which is mainly determined by the out-of-plane resonance frequency of the left and right oscillators. When the left and right oscillator parameters are adjusted (parameter R l and R r are modified to 3.8 mm and 5.8 mm respectively) to form a unit cell B. The Dirac degeneracy point in its dispersion relation will separate, resulting in a complete band gap. Figure 4 b. The band gap created by the separation of Dirac degenerate points is the basis for constructing topological boundary states. The solid points in the band structure are the out-of-plane motion modes of the oscillator, which are of particular interest.

[0128] In the process of constructing topological metamaterials, the eigenfrequency changes and motion patterns before and after the degeneration of the Dirac point at the wave vector K are important features that need to be paid attention to. In order to briefly explain the characteristics of the eigenfrequency at the wave vector K, the finite element method is used to calculate the motion modes of the unit cells A and B at the first two eigenfrequencies, as shown in the figure below. Figure 4 c. As can be seen from the figure, at the characteristic frequency K - and K + Under the finite element method, the eigenmodes of unit cells A and B appear as local resonances of the left and right oscillators, respectively, and are primarily out-of-plane motion modes. Accurately identifying these out-of-plane motion modes is crucial for subsequent research. It not only provides a theoretical basis for calculating the local resonance frequencies of the oscillators using the finite element equivalent model, but also lays the foundation for introducing physical constraints on the resonance frequencies into the design model.

[0129] Based on the unit cell design parameters and their corresponding dispersion characteristics, the present invention constructed a data set containing 22,300 sets of design parameter and dispersion relationships. Each set of samples consists of 7 design parameters (1×7 dimensions) and corresponding dispersion curve data (the first 5 order characteristic frequencies under 24 wave vectors, 24×5 dimensions). In order to effectively retain the key features of the low-frequency energy band, the first 5 order characteristic frequencies were selected as effective characteristic frequencies in the data preprocessing stage. The data set is divided into a training set (17840 groups) and a test set (4460 groups) in a ratio of 4:1, providing sufficient data support for the subsequent training and verification of deep learning models.

[0130] Step 2: Establish a finite element equivalent model to calculate the local resonance frequency of the oscillator

[0131] In this section, we will develop an efficient and accurate finite element equivalent model to calculate the resonant frequencies of the left and right local oscillators in the unit cell structure. These resonant frequencies will provide key physical parameters for the design model and are used in the training and optimization of the deep learning model.

[0132] In the finite element equivalent model, the out-of-plane support stiffness and mass of the oscillator in the unit cell are calculated, and then the out-of-plane resonance frequency of the local oscillator is solved. The finite element model of the unit cell structure and the outer frame is as follows Figure 5 In the finite element model, the outer frame structure is established using solid elements SOLID 185, and the center of the left and right oscillators O l and O r Defined by mass unit MASS21, and they are combined with the oscillator inner hole R l and R r When solving the support stiffness of the left and right oscillators, a fixed constraint is imposed on the four vertices of the diamond frame and the center of the oscillator O l and O r Apply a unit force F along the z direction and solve the concentrated mass unit O separately l and O r Displacement U along the z direction i (i=l or r). The mass of the left and right oscillators m i (i=l or r) can be calculated by the material density ρ and volume V i (i=l or r) is calculated. In summary, the support stiffness k of the left and right local oscillators is i With the natural frequency f i Expressed as:

[0133]

[0134] Where i=l or r represents the left and right oscillators respectively.

[0135] The design parameters of unit cell A and its dispersion relation are used to verify the finite element equivalent model of the unit cell. Unit force loads are applied to the left and right oscillator center mass units respectively, and the corresponding displacement fields are calculated, as shown in the following example: Figure 5 As shown in b. As can be seen from the figure, when the oscillator is subjected to out-of-plane load, the displacement field of the unit cell is Figure 4 The motion mode of the local oscillator in c at the wave vector K is the same. In addition, combined with the stiffness and mass parameters of the left and right oscillators, the natural frequencies of the left and right oscillators are calculated to be 802.6 and 803.1 Hz respectively. Figure 4 Compared to the Dirac point frequency of 800.5 Hz, the maximum error is approximately 0.32%. Therefore, based on the calculated results of the motion mode and resonance frequency of the local oscillator, it can be inferred that the proposed finite element mechanical equivalent model can effectively evaluate the out-of-plane motion eigenfrequency of the left and right local oscillators in the band structure at the wave vector K. This eigenfrequency will provide the basic physical constraints for the subsequent construction of the physical information neural network.

[0136] Step 3: Construct a topological metamaterial design model based on a physical information neural network

[0137] Based on the designed unit cell structure and the established finite element equivalent model, this section builds a physical information network model (Physics-Informed neural networks, PINN) to realize the reverse design of local resonance metamaterials. Figure 6 As shown in the figure, the network model is mainly divided into three parts: pre-training model, reverse design model and physical equivalent model. Next, the model construction ideas and the functions of each sub-model are explained.

[0138] In the pre-trained model, the dispersion curve of the phononic crystal can be predicted by inputting the design parameters. After training the model with a dispersion relation dataset, the model can quickly predict the dispersion curve of the unit cell based on the given parameters.

[0139] In the metamaterial inverse design model, the design parameters of the phononic crystal can be generated based on the target dispersion relationship. The model is trained based on a dataset and can accurately establish the nonlinear mapping relationship between the dispersion curve and the design parameters.

[0140] Furthermore, this section integrates the oscillator resonant frequencies calculated using a finite element equivalent model into the inverse design framework. This model quickly solves for the eigenfrequencies of the left and right oscillators at the wave vector K and uses them as physical constraints to optimize and verify the first two eigenfrequencies of the predicted energy band. This approach not only improves design accuracy but also enhances the model's physical interpretability. The following sections will detail the implementation of the pre-trained sub-model and the inverse design model.

[0141] Step 3.1 Pre-training model establishment

[0142] In this section, we use a multi-layer perceptron to build a pre-trained model. By inputting the design parameters, we can predict the dispersion relation of the unit cell. The architecture of the neural network model is as follows: Figure 7 As shown in a, it includes an input layer, three hidden layers, and an output layer. The input layer receives seven design variables, which are extracted and transformed through hidden layer neurons (200×300×200). Each hidden layer uses a nonlinear activation function SiLU (Sigmoid linear unit), which is differentiable in the entire real domain and can effectively alleviate the problem of vanishing gradients. After processing through the three hidden layers, the data information is finally passed to the output layer (120×1), which is used to output the predicted dispersion relation. The output layer (120×1) can be converted into a 24×5 dispersion relation, where 24 is the number of wave vectors in the first irreducible Brillouin zone and 5 is the number of eigenfrequencies calculated under a single wave vector.

[0143] For the pre-trained network model, the mean absolute error (MAE) was used to evaluate the performance of the model during training and testing. Figure 7 Figure 2b shows the MAE change during training, while the darker curve reflects the MAE change during testing. The figure clearly shows that both the test error and the training error show a steady downward trend with the increase in training epochs, and stabilize after 20 training epochs. This demonstrates that the model effectively learns the intrinsic characteristics of the data during training and demonstrates excellent generalization ability on the test set. Ultimately, both the test error and the training error stabilize at low levels, effectively verifying the reliability of the proposed model.

[0144] The prediction ability of the pre-trained model was evaluated. Based on the above pre-trained model, the dispersion curves of two unit cell structures with different design parameters were predicted and verified with the help of the finite element method. Here, unit cells 1 and 2 were used as the research objects. The design parameters of unit cell 1 are: h b =0.8mm, t b =0.5mm, R f =9mm, R l =2.8mm,h l =6.0mm, R r =3.8mm,h r =2.0mm, the design parameters of unit cell 2 are: h b =0.8mm, t b =0.5mm, R f =9mm, R l =2.8mm,h l =2.0mm, R r=4.8mm,h r =6.0mm. To simplify the expression, the design parameters of cells 1 and 2 are expressed using the arrays [0.8, 0.5, 9, 2.8, 6.0, 3.8, 2.0] and [0.8, 0.5, 9, 2.8, 2.0, 4.8, 6.0] respectively. Figure 8 It can be seen from a that the predicted dispersion results of unit cell 1 and the finite element results show good consistency in the low and medium frequency range (0-1000Hz). Figure 8 In Figure 2b, the predicted dispersion results for unit cell 2 also show a high degree of agreement with the finite element results in the low and mid-frequency bands. These two cases verify that the proposed pre-trained model can accurately predict the dispersion characteristics of locally resonant unit cells.

[0145] Step 3.2 Reverse design model establishment

[0146] This section will integrate the finite element equivalent model and the pre-trained model to build a multi-layer neural network inverse design model for the design of metamaterial unit cell structures. The inverse design model is a multi-layer feedforward neural network architecture, such as Figure 9 As shown in the figure, its core goal is to achieve accurate prediction of design parameters by learning the mapping relationship between the input dispersion relationship and the output design parameters. In the design model, the input layer receives 24×5 dispersion data, which is first flattened into a 120×1 array and input into the network structure. In the middle layer of the network model, three hidden layers are set up, with the number of neurons in each layer being 200, 300, and 200 respectively. These hidden layers process the input data with the help of the nonlinear activation function SiLU. Finally, the output layer contains 7 neurons, corresponding to 7 design parameters respectively. After these design parameters are derived from the network model, in order to ensure that the output data meets the value range of the actual design variables, the Sigmoid activation function is used here for scaling processing, that is, through the activation function scaling process (Activation Function Scaling), the output data is accurately limited to the pre-set design variable value range. The activation function scaling process can be expressed as:

[0147] D i =W i ·Sigmoid(d i )+B i ,i=1,2,...7, (2)

[0148] Where the subscript i represents the i-th design variable. i represents the i-th design parameter finally generated after the design model is scaled, and d i is the initial design parameter generated by the output layer without scaling, W i and B iare the weight and bias of the activation process respectively. According to the value range of the design variables in Table 1, the present invention W i and B i The arrays are W = [3.0, 1.5, 2.0, 5.0, 5.0, 4.0, 4.0] and B = [0.8, 0.5, 9.0, 2.8, 2.8, 2.0, 2.0]. The activation function scaling process helps to ensure that the predicted design parameters are within the design variable range, thereby ensuring the rationality of the design structure.

[0149] In the training process of the inverse design model, the construction of the loss function is crucial to the optimization of model training. The loss function proposed in this paper comprehensively considers the three factors of design parameters, dispersion relationship and physical information (oscillator resonance frequency) to ensure that the model can fully and accurately reflect the relationship between the design goal and the actual output. Specifically, the loss function of the inverse design model can be expressed as:

[0150]

[0151] The first term is the predicted design parameter The average absolute error with the label result D is used to measure the difference between the predicted design parameters and the design parameter labels. The second term is the predicted dispersion relation The mean absolute error between the target dispersion relation R and the predicted dispersion relation The error term is generated by generating design parameters from the inverse design model and importing them into the pre-trained model. This error term can evaluate the difference between the target dispersion relation and the dispersion relation generated by the predicted design parameters. The third term is the physical loss error term, and its calculation process mainly relies on the finite element equivalent model to quickly solve the local resonance frequencies of the left and right oscillators. The local resonance frequency The error is solved with the first two eigenfrequencies f at the wave vector K in the target dispersion relation to obtain the physical loss term. The physical loss term plays a vital role in model training optimization. It can effectively constrain the first two most important frequency bands of the dispersion relation in the low frequency range to make it close to the target dispersion relation. In the loss function, w d , w r and w f These are the hyperparameters of the three parts of the loss term. This paper uses a grid search strategy (GridSearch) to optimize these hyperparameters, aiming to find the parameter combination that can enable the model to achieve the best performance on the dataset.

[0152] Step 3.3 Model validity verification and error assessment

[0153] In order to verify the effectiveness of the proposed inverse design model in the inverse design of metamaterials, this section discusses the performance of the proposed design model. First, the loss changes of the design model in the training and testing phases are calculated, as shown in the figure below. Figure 10 As shown in a. As can be seen from the figure, the design model constructed by the present invention that incorporates physical knowledge has shown significant advantages in the initial stage of training, and its convergence speed is very fast (such as Figure 10 (a) The model's test and training errors exhibited a more stable and gentle fluctuation throughout the training process. The results demonstrate that the proposed inverse design model can quickly converge and maintain high prediction accuracy and stability even when presented with new test set data.

[0154] In order to verify the performance of the design model, a test set sample with a design parameter combination of [0.8, 0.5, 9.0, 2.8, 3.8, 6.0, 2.0] was selected, and its corresponding dispersion relationship was input into the inverse design model to calculate its output structural design parameters and the dispersion relationship of the predicted design parameters. The design parameter combination generated by the proposed model is [0.8, 0.69, 9.18, 3.39, 3.14, 2.52, 6.0]. After importing the design parameters into the pre-trained model, the predicted dispersion relationship is generated, as shown in Figure 2. Figure 10 At the same time, in order to intuitively show the difference between the predicted results and the input dispersion, the dispersion relationship of the model input is also plotted in Figure 10 b. The light-colored dispersion curve represents the input target dispersion relation, while the dark-colored dispersion curve represents the dispersion relation generated by importing the predicted design parameters into the pre-trained model. As can be seen from the figure, the predictions generated by the proposed model are highly consistent with the target dispersion relation at the first two eigenfrequencies, effectively verifying the design accuracy of the proposed design model.

[0155] To further verify the validity of the predicted design parameters, the predicted design parameters generated by the above model were used for finite element modeling, and the corresponding dispersion relations were calculated, such as Figure 11 As shown in Figure a. The light-colored curve in the figure represents the dispersion curve calculated using the finite element method, while the dark-colored curve represents the target dispersion curve. As can be seen from the figure, the dispersion relationship generated by the proposed design model is relatively close to the target result, showing a high degree of agreement, especially in the first two critical frequency bands. It should be noted that when achieving a specific dispersion relationship, the solution space of the design parameters exhibits a non-unique characteristic, that is, there are multiple parameter combinations that can achieve the same or similar design goals. This discovery not only reveals the multiplicity of solutions in the metamaterial inverse design process, but also expands the possible space of metamaterial design.

[0156] Next, the performance of the proposed model and the traditional model in the reverse design of topological metamaterials will be compared to verify the effectiveness of the proposed model. 2 ) is a statistical indicator for evaluating the goodness of fit of the regression model, which can be used to characterize the design performance of different models. It can be expressed as:

[0157]

[0158] Where y i is the true value. is the model predicted value. is the mean of the true values. 2 The value range of R is [0,1]. 2 The closer the value is to 1, the higher the agreement between the model's predicted value and the actual observed value; conversely, when R 2 When it approaches 0, it means that the model fails to effectively explain the variation of the data. In order to verify the superior performance of the proposed physical information neural network (PINN) model in metamaterial design, the R 2 Values, such as Figure 11 b. The three models include the PINN model proposed in this paper, the tandem neural network model (TNN), and the multi-layer perceptron (MLP) model. The TNN model does not consider the physical information constraints output by the finite element equivalent model during the design process, and its loss function only retains the first two terms in Equation (3). The MLP model is further simplified and does not consider the influence of the finite element equivalent model and the pre-trained model. Its loss function only retains the first term in Equation (3).

[0159] The results show that the R 2 The distribution of the values ​​is concentrated and stably maintained at a high level (0.7-0.8), which fully proves that the model has good stability and excellent fitting ability. 2 The distribution of values ​​is relatively discrete, and the performance on the test set drops significantly (0.1 to 0.8). This reflects the shortcomings of these two models in the prediction accuracy and generalization ability in the local resonant metamaterial design task. The comparative analysis further highlights the effectiveness and stability of the inverse design model proposed in this paper.

[0160] In addition, the mean absolute error (MAE), mean square error (MSE) and coefficient of determination (R 2) and other evaluation indicators to compare the performance of the three models on the test set dispersion relation, as shown in Table 2. The dispersion relation of the proposed model is smaller than the traditional TNN and MLP models in terms of MAE and MSE indicators. In addition, the determination coefficient R 2 It is also superior to traditional models such as TNN and MLP. In summary, the design model proposed in this invention combines physical principles with data-driven methods, which can more efficiently perform metamaterial reverse design.

[0161] Table 2 Evaluation indicators under different deep learning prediction models

[0162]

[0163] Step 4: Design and application of topological metamaterials under low-frequency and broadband design goals

[0164] This section focuses on the application of the proposed Physical Information Neural Network (PINN) model, targeting the broadband, low-frequency design requirements of topological metamaterials. Based on sample data from the dataset, the dispersion relation is linearly transformed (including multiplication and addition operations) to construct the low-frequency, broadband target dispersion characteristics. These are then used as input parameters in the inverse design model, enabling the efficient design of topological metamaterial structures.

[0165] Step 4.1 Topological metamaterial design under broadband design goal

[0166] This study is based on the inverse design method, and realizes the design of metamaterials with broadband characteristics by regulating the dispersion characteristics. In order to obtain a dispersion curve with broadband characteristics as the design target, numerical operations (including addition, subtraction, multiplication and division operations) are performed on frequency bands in different frequency ranges based on the dispersion relationship in the data set, and then the target dispersion curve with broadband characteristics is obtained. Specifically, in order to achieve the design goal of a Dirac frequency separation interval of 300 to 500 Hz at the wave vector K, a set of design parameters [0.8, 0.5, 10, 6.8, 3.8, 6.0, 6.0] are randomly selected from the sample set as the design basis, and its dispersion relationship is as follows: Figure 12 As shown in a. The Dirac frequency interval at wave vector K in the initial dispersion relation is 300.2~350.3Hz. Using the band addition method, the characteristic frequencies of the second band and above are increased by 150Hz, thereby constructing a dispersion curve with broadband characteristics as the design target. At this time, the first two frequency intervals at wave vector K in the target dispersion curve are extended to 300.2~500.3Hz, as shown in Figure 12 As shown in b. Based on the proposed inverse design model and the constructed target dispersion curve, the unit cell design parameter prediction is performed, and the final design parameter combination is [0.8, 0.84, 10.68, 5.11, 4.19, 2.09, 6.0]. In order to make the comparison more intuitive, the dispersion relationship of the predicted parameters is calculated, as shown in Figure 12 c. Comparative analysis shows that the dispersion curves obtained based on the predicted parameters are highly consistent with the design targets: at point K, the relative error between the predicted frequency interval (302.4-502.7 Hz) and the target value (300.2-500.3 Hz) is less than 1%. Furthermore, the predicted results for the first two critical frequency bands essentially coincide with the target curves, and the band gap width increases significantly by approximately four times compared to the initial value.

[0167] In the above case, the target dispersion relation was constructed by frequency band shifting within the high frequency range, and the wideband bandgap design goal was achieved with the proposed model. Next, the band multiplication method will be used to expand the bandgap range. The dispersion relation with design parameters of [0.8, 0.5, 9.0, 3.8, 4.8, 6.0, 4.0] is used as the initial design sample, and the second and subsequent frequency bands are expanded by 1.5 times to obtain the following: Figure 13 b shows the target dispersion relation. In this process, the frequency gap at wave vector K increases from the initial 386.6~468.0Hz to 386.6~702.1Hz. Figure 13 The dispersion relation in b is used as the design target and introduced into the established inverse design model to obtain the design parameter combination [0.8, 1.37, 10.38, 4.69, 3.47, 2.0, 6.0]. The dispersion relation generated by the design parameters is compared with the design target, as shown in Figure 13 c. As shown in the figure, the dispersion relationship generated by the predicted design parameters agrees well with the design target, especially in the low-frequency range. Although there are still slight deviations in the predicted characteristic frequency at wave vector K, the proposed inverse design method effectively achieves a significant widening of the bandgap. Through a control method that combines band shifting and multiplication, the target dispersion characteristics can be efficiently constructed based on the data set samples, and then the proposed inverse design model can be introduced to design a metamaterial structure with broadband characteristics.

[0168] Step 4.2 Topological metamaterial design under low-frequency design goals

[0169] This section will use the inverse design model to achieve the design of topological metamaterials with low-frequency bandgap characteristics. First, select samples with design parameters of [0.8, 1.7, 11.0, 2.8, 4.8, 2.0, 4.0] from the training set as the design basis. Its dispersion relationship is as follows: Figure 14 The frequency interval at wave vector K is 470.75~844.84Hz. By multiplying the first frequency band in the dispersion relation by 0.6, we can construct the following Figure 14The target dispersion relation in light color in b is input into the inverse design model, and the predicted design parameter combination is [0.8, 1.99, 10.96, 2.87, 3.78, 2.0, 6.0]. The predicted design parameters are imported into the pre-trained model, and the predicted dispersion relation is calculated, as shown in Figure 14 The dark dispersion curve in (b) shows that the predicted dispersion relation at wave vector K has a frequency range of 346.91 to 935.81 Hz. Comparing the target dispersion relation with the predicted dispersion relation shows that the predicted frequency range at wave vector K is 57.4% wider than the initial sample frequency range, while the lowest frequency is 26.3% lower. This case demonstrates the value of inverse design models in achieving low-frequency metamaterial design.

[0170] In summary, the present invention flexibly constructs a target dispersion relationship with low-frequency and broadband characteristics through the control method of data-centered frequency band shifting and multiplication, and then uses the proposed topological metamaterial inverse design model to achieve efficient reverse design of metamaterials.

[0171] The inverse design method for metamaterials based on a physical information network local resonance topology, provided in this embodiment of the invention, uses innovative reverse design methods to achieve vibration control within a specific target frequency band within a wide frequency range of 226 to 2210 Hz for mechanical vibration reduction / isolation components constructed using metamaterials. This structural component is primarily targeted at the high-end equipment manufacturing sector, and is particularly suitable for suppressing low- and medium-frequency vibrations in high-precision equipment such as precision instrument work platforms and vibration isolation bases for precision equipment.

[0172] Compared with the existing technology, the topological metamaterial inverse design method provided by the present invention has the following technical advantages and performance advantages:

[0173] 1. Innovative design method: This method breaks through the limitations of traditional experience-based design methods and adopts a goal-oriented inverse design theory to achieve precise design of metamaterials with specific bandgap characteristics.

[0174] 2. Engineering feasibility: The feasible range of physical parameters was considered at the beginning of model establishment to ensure that the design results are actually manufacturable;

[0175] 3. Metamaterial customization capability: Through reverse design methods, it can quickly respond to the needs of different application scenarios and achieve customized bandgap design in the range of 226 to 2210 Hz.

[0176] 4. Key performance indicators:

[0177] Minimum bandgap frequency: 226Hz

[0178] Customizable bandgap range: 226~2210Hz

[0179] Critical frequency band design accuracy: the error of characteristic frequency points is less than 10% (target dispersion relation and predicted dispersion relation).

[0180] The above describes in detail the preferred embodiments of the present invention. It should be understood that those skilled in the art can make numerous modifications and variations based on the concepts of the present invention without inventive effort. Therefore, any technical solutions that can be derived by those skilled in the art through logical analysis, reasoning, or limited experimentation based on the concepts of the present invention and the prior art should be within the scope of protection defined by the claims.

Claims

1. A method for reverse design of local resonance topological metamaterials based on physical information networks, characterized in that: The method comprises the following steps: S101: Designing a unit cell structure based on the principle of local resonance, and establishing a dispersion relation data set of the unit cell structure; S103: establishing a finite element equivalent model and calculating the local resonance frequency of the oscillator in the unit cell structure; S105: Building a physical information neural network based on the unit cell structure and the finite element equivalent model, and constructing a topological metamaterial reverse design model using the physical information neural network; S107: Based on the sample data in the data set, construct a low-frequency broadband target dispersion characteristic by performing a linear transformation on the dispersion relationship, and import the target dispersion characteristic into the inverse design model to realize the topological metamaterial structure design.

2. The method according to claim 1, wherein In the step S101, the dispersion relation data set is constructed based on the design parameters of the unit cell structure and the corresponding dispersion characteristics, and the dispersion relation data set is divided into a training set and a test set.

3. The method according to claim 2, wherein In step S103, in the finite element equivalent model, the outer frame structure is established using solid elements SOLID 185, the centers of the left and right vibrators are defined by mass elements MASS21, and the vibrator centers are rigidly coupled to the nodes on the inner hole surface of the vibrator. The support stiffness and natural frequency of the left and right vibrators are: Among them, k i is the oscillator support stiffness, f i is the natural frequency of the oscillator, F is the unit force, U i is the oscillator displacement, m i is the mass of the vibrator, i is the number of the left and right vibrators, i=l represents the left vibrator, and i=r represents the right vibrator.

4. The method according to claim 3, wherein In step S105, the physical information neural network includes a pre-training model, a reverse design model and a physical equivalent model, wherein: The pre-trained model is trained using the dispersion relationship data set, and the trained pre-trained model predicts the dispersion curve of the unit cell structure according to given design parameters; The inverse design model is configured as a multi-layer feedforward neural network architecture, which achieves accurate prediction of the design parameters by learning the mapping relationship between the input dispersion relationship and the output design parameters; The physical equivalent model integrates the oscillator resonance frequency calculated by the finite element equivalent model into the inverse design framework, quickly solves the characteristic frequencies of the left and right oscillators at the wave vector K, and optimizes and verifies the first two order characteristic frequencies at the wave vector K of the predicted dispersion relationship as physical constraints.

5. The method according to claim 4, wherein The pre-training model includes an input layer, a hidden layer and an output layer, wherein: The input layer is configured as 1 and is used to receive design variables; The hidden layer is configured as 3, and the design variables are feature extracted and transformed through the neurons of the hidden layer. Each hidden layer adopts a nonlinear activation function SiLU, which is differentiable in the entire real number domain; The output layer is configured as one, receives the data processed by the hidden layer, and outputs the predicted dispersion relationship.

6. The method according to claim 5, wherein The reverse design model includes an input layer, a hidden layer and an output layer, wherein: The input layer flattens the received dispersion data from 24×5 to 120×1 dimensions; The hidden layer is configured as 3, and the nonlinear activation function SiLU is used to process the input data; The output layer contains 7 neurons, the number of which corresponds to the design parameters. The output layer uses the Sigmoid activation function for scaling processing to accurately limit the output data to the pre-set design variable value range. The activation function scaling process is: D i =W i ·Sigmoid(d i )+B i Among them, D i is the i-th design parameter finally generated after scaling, d i The initial design parameters generated without scaling, W i is the weight of the activation process, B i is the bias for the activation process.

7. The method according to claim 6, wherein During the training process of the inverse design model, the loss function constructed integrates the three elements of design parameters, dispersion relationship and physical information. The loss function is: Among them, the first one is the predicted design parameter The mean absolute error with the label result D; the second term is the predicted dispersion relationship The mean absolute error between the target dispersion relation R; the third term is the physical loss error term, D is the label result, is the predicted design parameter, R is the target dispersion relation, To predict the dispersion relation, f is the characteristic frequency at the wave vector K in the target dispersion relation, The local resonance frequency, w, is obtained by solving the finite element equivalent model based on the predicted design parameters. d ,w r ,w f is a hyperparameter.

8. The method according to claim 7, wherein The step S105 includes the following sub-steps: S1051: Establishing the pre-training model: constructing the pre-training model using a multi-layer perceptron, inputting design parameters, and the pre-training model predicting the dispersion relation of a unit cell; S1052: Establishing the reverse design model: integrating the finite element equivalent model and the pre-trained model to build a multi-layer neural network reverse design model for metamaterial unit cell structure design; S1053: Model validity verification and error evaluation: Verify the validity of the reverse design model in the metamaterial reverse design, and evaluate the performance of the reverse design model in the metamaterial reverse design.

9. The method according to claim 8, wherein In the step S107, topological metamaterial design under a broadband design goal is included, and the topological metamaterial design under a broadband design goal includes the following sub-steps: S10711: Randomly select a set of narrow bandgap dispersion relations from the sample set as the design basis; S10712: Using the method of frequency band shifting or multiplication, the characteristic frequencies of the second band and above are expanded to construct a dispersion curve with broadband characteristics as the design goal; S10713: Based on the proposed inverse design model and the constructed target dispersion curve, the unit cell design parameter prediction is performed to obtain a design parameter combination.

10. The method according to claim 9, wherein In the step S107, the topological metamaterial design under the low-frequency design target is also included, and the topological metamaterial design under the low-frequency design target includes the following sub-steps: S10721: Randomly select a set of dispersion relations from the sample set as the design basis; S10722: Using the band multiplication method, the characteristic frequency of the first band is reduced, thereby constructing a dispersion curve with low-frequency bandgap characteristics as the design goal; S10723: Based on the proposed inverse design model and the constructed target dispersion curve, perform unit cell design parameter prediction to obtain a design parameter combination.

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