An interpretable neural network and method for inverse modeling of a structural dynamics system

Abstract

The application provides an interpretable neural network and method for inverse modeling of a structural dynamics system, adopts a structural dynamics equation as an interpretable symbolic network type calculation architecture, embeds structural dynamics parameters into the constructed neural network, thereby designing and implementing a neural network model with physical parameters embedded and completely interpretable in a physical sense, on the basis of which, a training method of the interpretable physical embedded neural network is further designed, and a set of data-driven means for realizing full analytical inverse direct modeling of the structural dynamics system is provided. The application provides a novel and interpretable method for the field of structural dynamics modeling and the field of structural dynamics inverse problems, and promotes the development of theoretical understanding and practical application of structural dynamics.

Description

Technical Field

[0001] This invention relates to the fields of artificial intelligence and structural dynamics, and in particular to an interpretable neural network and method for inverse modeling of structural dynamic systems. Background Technology

[0002] Currently, research on explainable artificial intelligence (AI) has received significant attention. Explainable AI research represents a crucial direction for future breakthroughs in the field of AI; its application to solving challenging problems in aerospace and other fields forms the important background for this research direction. Inverse modeling of structural dynamics systems, as the foundation for inverse problems in structural dynamics research, is one of the key challenges that urgently requires such advanced technological support.

[0003] Modeling of structural dynamic systems is divided into two types: forward modeling and inverse modeling. Forward modeling is used to establish a theoretical model or numerical simulation model of the actual structure to solve the positive problem in the field of structural dynamics—accurately calculating the dynamic response of the actual structure under dynamic loads. Inverse modeling is mainly used for model identification and establishing an inverse model of the actual structure, thereby solving the inverse problem in the field of structural dynamics—the problem of identifying structural dynamic parameters and the problem of identifying structural dynamic loads.

[0004] Forward dynamic modeling techniques for structures are relatively mature, including theoretical modeling, finite element modeling, transfer matrix modeling, and machine learning techniques. However, inverse modeling methods and techniques for structural dynamics are quite limited, and each has its own shortcomings. Existing methods mainly fall into three categories: forward model inversion methods, statistical and estimation methods, and machine learning methods. Forward model inversion methods mainly include direct inversion of the frequency response function matrix and regularization methods, which are based on inverting an accurate forward structural dynamics model to obtain an inverse structural dynamics model. Statistical and estimation methods mainly include Kalman filtering and Bayesian estimation methods, which, based on a relatively accurate forward structural dynamics model, use statistical and probabilistic estimation methods to further estimate parameters, and are applicable to cases where some parameters are uncertain. Machine learning methods mainly rely on existing black-box artificial neural network models such as LSTM neural networks and convolutional neural networks, using data-driven methods to establish surrogate models for mapping inverse structural dynamic relationships. Forward model inversion methods and statistical and estimation methods can be collectively referred to as traditional mathematical methods; machine learning methods mainly rely on artificial neural network modeling methods.

[0005] In the field of aerospace structural engineering, applying artificial neural networks to solve the challenging problem of dynamic load identification undoubtedly has broader application prospects and deeper potential. However, in the process of implementing the inventive technical solutions in the embodiments of this application, the inventors of this application have discovered that the above-mentioned prior art has at least the following technical problems:

[0006] Traditional mathematical reasoning-based inverse modeling methods for structural dynamics systems are based on mathematical reasoning and are characterized by strong principles and clear physical meaning. However, because they require an accurate mathematical model of the loaded structure and involve unavoidable matrix inversion operations, they often encounter computational difficulties and ill-posed problems, resulting in inherent limitations that prevent further breakthroughs. In contrast, artificial neural network-based inverse modeling methods for structural dynamics systems do not require establishing theoretical expressions between vibration response and dynamic loads as in traditional methods. Instead, they utilize deep neural networks to build surrogate models for the inverse modeling problem of structural dynamics systems, fully leveraging the fitting ability of deep neural networks for strong correlations. This avoids many difficulties such as high-precision mathematical modeling and theoretical model error control. However, current modeling methods focus on the application and verification of artificial neural networks in the inverse modeling problem of structural dynamics systems, while existing technologies do not address the true integration of structural dynamics theory and machine learning methods at the underlying level, i.e., they do not provide artificial neural network models with a physically interpretable computational architecture. In other words, current deep neural network models used for modeling structural dynamics systems lack interpretability from a mechanical perspective in terms of their operational mechanisms. Therefore, the generalization ability of artificial neural network models in engineering practice cannot be fundamentally guaranteed. Consequently, key issues such as the robustness of artificial neural network models are difficult to resolve when applied to practical engineering. This directly results in existing reverse modeling methods for structural dynamics systems mostly remaining at the stage of proposing methodological ideas and initially verifying their feasibility, making it difficult to further explore the scientific mechanisms behind the problems and methods, and also hindering the true application of existing research results to engineering practice. Thus, the "interpretability of artificial neural network models" has become a critical barrier that is of utmost concern and urgently needs to be overcome by both academia and engineering. Summary of the Invention

[0007] To address the inherent defects of existing deep neural networks used in dynamic load identification, such as lack of physical interpretability, insufficient model robustness and generalization, and excessive reliance on data samples, this invention designs an interpretable neural network that can perform a fully analytical mapping of the inverse physical relationships of structural dynamics. It reveals the generalization mechanism of interpretable neural networks based on a physical knowledge framework and ultimately establishes a suitable method for inverse modeling of structural dynamic systems based on interpretable neural networks.

[0008] The technical solution of this invention is as follows:

[0009] An interpretable neural network for inverse modeling of structural dynamics systems, comprising an input layer, a hidden layer, and an output layer;

[0010] In the input layer, hidden layer and output layer, the number of neurons in each layer is taken as the number of degrees of freedom p of the dynamic response of the acquired structural dynamics system;

[0011] The input signal of the i-th neuron in the input layer is the dynamic response of the structural dynamics system [x1(t), ..., x]. p (t)] T For i = 1, 2, ..., p, the corresponding weight vector is: in For [φ] -1 In the vector, the i-th column vector, [φ] is the structural mode shape matrix;

[0012] The input signal of the i-th neuron in the output layer is the constructed intermediate variable. The corresponding weight vector is in For [φ T ] -1 The i-th column vector in the vector, and The self-power spectral density of yi f i (ω) and y i The cross-power spectral density of (ω), y i (ω) is the i-th modal dynamic response, f i (ω) represents the i-th modal dynamic load, and ω is the frequency;

[0013] The i-th modal dynamic response y of the input signal of the i-th neuron in the hidden layer i (ω), the output is the i-th modal dynamic load f i (ω), the corresponding weight vector is [m i c i k i ] T m i c i k i Let be the modal mass, modal damping, and modal stiffness of the i-th mode of the structural dynamics system, respectively.

[0014] The training method for the aforementioned interpretable neural network includes the following steps:

[0015] Step 1: Pre-training of the physical parameter neural network:

[0016] Step 1.1: Calculate the modal shape matrix [φ] of the structure based on the boundary conditions and external dimensions of the structure, and give the initial values ​​of the modal mass, modal damping and modal stiffness of the structure;

[0017] Step 1.2: Calculate [φ] based on the structural modal shape matrix [φ]. -1 The weight parameters are assigned to each neuron in the input and output layers of the neural network; and the weight parameters are assigned to each neuron in the hidden layer of the neural network based on the initial values ​​of modal quality, modal damping, and modal stiffness.

[0018] Step 2: Prepare training data:

[0019] Collect time-domain data samples of the dynamic response of the loaded structure X(t) and time-domain data samples of the dynamic load F(t);

[0020] By combining the structural modal shape matrix [φ] and using the acquired dynamic response time-domain data sample X(t) and dynamic load time-domain data sample F(t), the input signal of the hidden layer—the modal dynamic response time-domain signal y(t)—and the output signal of the hidden layer—the virtual modal dynamic load in the frequency domain—are obtained.

[0021] The input and output data of the hidden layer are used as training data samples;

[0022] Step 3: Train the neural network using the training data to obtain the modal mass, modal damping, and modal stiffness results of the structure;

[0023] Where the output of the i-th neuron in the hidden layer Specifically targeting its actual part and the virtual part Design loss function and

[0024]

[0025]

[0026] As can be seen, this invention first proposes a neural network model structure that is fully interpretable in a physical sense. By neuralizing the inverse computational relationships of a structural dynamics system with mechanical mechanisms, the overall computational architecture and neuron modules of the neural network are designed based on its computational flow. Physical parameters are embedded within the neurons, giving it clear modeling rules and full interpretability. This solves the bottlenecks of opaque computational flows and lack of physical meaning in network parameters in existing artificial neural networks. Furthermore, by designing a proprietary training method for the aforementioned interpretable neural network, it addresses a series of problems in traditional inverse modeling methods for structural dynamics systems, such as computational difficulties, matrix ill-conditioned problems, and ill-posed solutions. It also addresses the lack of interpretability in the operating mechanisms and model principles of existing artificial neural networks, leading to over-reliance on training samples and a lack of generalization and robustness. This invention achieves accurate and direct inverse modeling of structural dynamics systems based on a data- and knowledge-driven approach, exhibiting superior generalization ability, computational accuracy, and computational efficiency, and possessing the ability to solve inverse problems in structural dynamics.

[0027] Beneficial effects

[0028] The present invention has the following technical effects or advantages:

[0029] 1. By employing a neural network design to perform inverse calculations of the structural dynamics system, this invention effectively solves the problems of existing artificial neural networks used for modeling structural dynamics systems lacking interpretable computational processes based on mechanical mechanisms and having insufficient model generalization. This enables inverse modeling of structural dynamics systems that conforms to the laws of dynamics using an interpretable neural network architecture.

[0030] 2. By incorporating structural dynamic parameters into the neurons, the problem that the internal neuron parameters of existing artificial neural networks used for modeling structural dynamic systems do not have physical meaning is effectively solved. This enables the internal parameters of interpretable neural networks to perform analytical mapping of structural dynamic parameters.

[0031] 3. Because a training method specifically designed for interpretable neural networks is adopted, the problem of existing artificial neural network training processes applied to structural dynamics system modeling being overly dependent on sample datasets and unable to guarantee correct neural network training is effectively solved. This enables interpretable neural networks to achieve fast and accurate training based on small-scale sample datasets through a data-driven approach.

[0032] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0033] The above and / or additional aspects and advantages of the present invention will become apparent and readily understood from the description of the embodiments taken in conjunction with the following drawings, in which:

[0034] Figure 1 Schematic diagram of physical parameter neurons at the ends of the input and output layers; (a) Physical parameter neuron of the input layer, (b) Physical parameter neuron of the output layer;

[0035] Figure 2 A schematic diagram of the neuron's internal physical parameters;

[0036] Figure 3 A schematic diagram of a neural network for interpretable physical parameters;

[0037] Figure 4 This is a schematic diagram of the eight-degree-of-freedom discrete vibration system in Embodiment 1 of this application;

[0038] Figure 5 This is a schematic diagram of the convergence process of each neuron parameter in Embodiment 1 of this application; (a) Modal stiffness parameter k′ i Convergence process, (b) modal damping parameter c′ i Convergence process, (c) modal quality parameter m′ i Convergence process. Detailed Implementation

[0039] To address the core technical problems of existing artificial neural network models applied to structural dynamics system modeling, which lack physical interpretability and have difficulty guaranteeing generalization, this invention adopts structural dynamics equations as an interpretable symbolic network computational architecture. Structural dynamics parameters are embedded into the constructed neural network, thus designing and implementing a physically parameter-embedded neural network model with complete physical interpretability, called IPENN (Interpretable Physic-Embedded Neural Network). Based on this, a training method for IPENN is further designed, and a data-driven inverse direct modeling method for structural dynamics systems is proposed.

[0040] Step 1: Construct an interpretable physical parameter neural network

[0041] (1.1) Constructing the frequency domain inverse dynamic relationship of the structural dynamic system

[0042] Any linear time-invariant structural dynamic system can be equivalently modeled as a multi-degree-of-freedom discrete vibration system. For an n-degree-of-freedom discrete vibration system, according to modal analysis theory, the dynamic load F acting on it and its resulting dynamic response X can be expressed as the superposition of the n-order modal dynamic load f and the modal dynamic response y, respectively. Since the actual structure is continuous, the theoretical n often needs to approach infinity; while the frequency range of F(ω) is finite, the proportion of higher-order modal dynamic loads and modal dynamic responses is usually very small and can be ignored, so reasonable modal truncation can be performed.

[0043] For an n-degree-of-freedom discrete vibration system, assume it is subjected to q dynamic loads, i.e., F = [F1, ..., F2]. q The first p modes are truncated (q≤p≤n), and the dynamic response of the system's p degrees of freedom is collected, i.e., X=[x 1， ..., x p At this time, the relationship between X and y, and the relationship between F and f, can be expressed as Equation (1.a) and Equation (1.b) respectively; while in the frequency domain, the dynamic balance relationship between f(ω) and y(ω) is as shown in Equation (2).

[0044]

[0045]

[0046]

[0047] Where, φ i =[φ 1i … φ pi ] T φ is the mode shape vector composed of the modal coordinates of each dynamic response measurement point under the i-th mode; φ = [φ1 … φ n [] represents the modal shape matrix of the system; ω represents the frequency; m i c i k i Let represent the modal mass, modal damping, and modal stiffness under the i-th mode, respectively. From equations (1.a) and (1.b), equations (3.a) and (3.b) can be derived:

[0048]

[0049]

[0050] in, Indicates [φ] -1 The element in the i-th row and j-th column. Therefore, combining equations (2), (3.a), and (3.b), the inverse dynamic relationship of the system in the frequency domain can be obtained as shown in equation (4):

[0051]

[0052] (1.2) Extension of frequency domain inverse dynamics calculation relationship for multi-degree-of-freedom discrete vibration systems

[0053] From equation (2), it can be seen that in the i-th mode, f i (ω) and y i The corresponding dynamic relationship between (ω) is shown in equation (5.a):

[0054] f i (ω)=[-m i ω 2 +c i ωj+k i ]x i (ω) (5.a)

[0055] Furthermore, equation (5.a) can be generalized to equation (5.b):

[0056]

[0057] in, For y i The self-power spectral density, f i With y i The cross-power spectral density. Here, intermediate variables are constructed according to equations (6.a), (6.b), and (6.c), respectively. and

[0058]

[0059]

[0060]

[0061] but and It also satisfies the mathematical calculation relationship as shown in equation (7):

[0062]

[0063] at the same time, and The calculation relationship is shown in equation (8):

[0064]

[0065] For F i The self-power spectral density, To utilize calculate The operator. It is important to note that... and Here, it exists only as an intermediate variable to satisfy the operational relationship and does not represent any actual physical meaning; but and The construction of this is necessary to ensure the convex optimization of the network training process in subsequent interpretable physical parameter neural networks. Simultaneously, operators are defined. As in equation (9):

[0066]

[0067] Therefore, combining equations (1) to (9) above, we can re-examine the relationship from X(t) to S. FF The calculation process for (ω) is shown in equations (10.a) to (10.e):

[0068] ①y(t)=[φ] -1 X(t) (10.a)

[0069] ②

[0070] ③

[0071] ④

[0072] ⑤

[0073] Based on equations (10.a)-(10.e), we design interpretable neurons and artificial neural networks.

[0074] (1.3) Design of Physical Parameter Neurons and Interpretable Physical Parameter Neural Networks

[0075] According to equations (10.a) and (10.d), terminal physical parameter neurons are designed as the basic building blocks of the input and output layers of the physical parameter neural network; their computational structure is as follows: Figure 1 As shown. Similar to traditional artificial neurons, the terminal physical parameter neuron also receives external input signals [x1(t), ..., x...]. p (t)] T or The signal is then weighted and summed; however, the difference lies in the weight vector of the i-th terminal physical parameter neuron. or With [φ] -1 or [φ T ] -1 The i-th column vector in the equation directly corresponds to the i-th column vector; at the same time, its computational operations, inputs and outputs all have clear physical meanings.

[0076] According to equation (7), internal physical parameter neurons are designed as the basic building blocks of the hidden layers of the physical parameter neural network; their computational structure is as follows: Figure 2 As shown. For the i-th neuron with internal physical parameters, its weight vector is [m i c i k i ] T It directly corresponds to the modal mass parameters, damping parameters, and stiffness parameters of the i-th mode of the dynamic system; its input is the i-th modal dynamic response y. i (ω), the output is the i-th modal dynamic load f i (ω).

[0077] Based on the physical parameter neurons obtained from the above design, an interpretable physical parameter neural network is constructed as follows: Figure 3 As shown. The input signal of the neural network is the dynamic response time-domain signal X(t), and the output signal is the power spectral density S of the dynamic load. FF (ω). Wherein, the calculation module... Corresponding to equation (10.b), the calculation module This corresponds to equation (10.e).

[0078] Step 2: Establish an interpretable physical parameter neural network training method

[0079] (2.1) Physical parameter neural network pre-training

[0080] The pre-training method for physical parameter neural networks is based on prior knowledge to initialize the parameters of the neural network. In engineering practice, the boundary conditions and external dimensions of a structure are readily and accurately known; however, parameters such as the density, elastic modulus, and damping of the materials used in the structure are often difficult to measure directly. Reflected in the structural dynamics model, the modal shape matrix [φ] of the structure can be accurately calculated based on the boundary conditions and external dimensions; while the modal mass m... i Modal damping c i With modal stiffness k i The parameters are usually unknown parameters that are difficult to calibrate accurately.

[0081] Therefore, based on prior knowledge, the precise [φ] can be calculated, along with the initial m. i c i k i ; [φ] is obtained based on [φ]. -1 This involves assigning weights to the physical parameters of each neuron at the ends of the input and output layers of the neural network; and then assigning weights to m. i c i k iAs the parameters to be trained, and using m i c i k i The initial values ​​are assigned as weights to the neurons in the hidden layers of the neural network to represent the internal physical parameters. This completes the pre-training of the physical parameter neural network.

[0082] (2.2) Training Data Preparation

[0083] As can be seen from Section 2.1, the physical parameters to be trained in the physical parameter neural network are all concentrated in the hidden layer of the physical parameter neural network. Therefore, this invention directly uses the input and output data of the hidden layer as training data samples to train the hidden layer of the physical parameter neural network. Thus, it is necessary to prepare a training sample dataset according to the input and output signals of the hidden layer of the physical parameters.

[0084] First, dynamic response time-domain data samples X(t) and dynamic load time-domain data samples F(t) of the loaded structure are collected. Then, since the input signal of the hidden layer is the modal dynamic response time-domain signal y(t), the output data is the virtual modal dynamic load in the frequency domain. Therefore, by combining the structural modal matrix [φ] and the collected dynamic response time-domain data sample X(t) and dynamic load time-domain data sample F(t), we can obtain y(t) and

[0085] (2.3) Physical parameter neural network training algorithm

[0086] The output of the i-th neuron in the hidden layer Specifically targeting its actual part and the virtual part Design a loss function for training physical parameters. and As shown in equations (8.a) and (8.b);

[0087]

[0088]

[0089] in They are respectively The real and imaginary parts of the loss function are used. The Hessian criterion is applied to the loss function. and The convexity is tested as shown in equations (9.a) and (9.b) respectively. and They are respectively and Determinant of the Hessian matrix:

[0090]

[0091]

[0092] Obviously, and Both possess the characteristics of convex functions. This ensures that the training process of the physical parameter neural network is a convex optimization process, and the physical parameters inside the hidden layer neurons can be correctly optimized; this characteristic of IPENN also ensures that it can correctly identify modal parameters.

[0093] In the training process of a physical parameter neural network, firstly based on and The error value is calculated, and then the Adam-steepest gradient descent algorithm and backpropagation algorithm are used to train the physical parameter neural network.

[0094] In summary, the process of inverse modeling a structural dynamics system based on an interpretable physical parameter neural network for a specific structural dynamics system is as follows:

[0095] For a specific structural dynamic system, based on the above-mentioned method for constructing interpretable neural network models, a corresponding basic model of interpretable neural network is constructed.

[0096] Based on the above requirements, dynamic load and dynamic response information loaded on the structure are collected, and data processing is performed to prepare the training dataset for the interpretable neural network.

[0097] Subsequently, based on the aforementioned interpretable neural network training method, the interpretable neural network is first pre-trained based on prior physical knowledge; then, using the prepared training data and the training algorithm designed in this invention, the interpretable neural network is trained in a data-driven manner, thereby finally completing the inverse modeling of the structural dynamics system by the interpretable neural network.

[0098] To better understand the above technical solutions, the following will provide a detailed explanation of the technical solutions in conjunction with the accompanying drawings and specific embodiments.

[0099] This embodiment uses, as follows: Figure 4 The 8-DOF discrete vibration system shown is used to test the dynamic load recognition capability of an interpretable physical parameter neural network. The spring stiffness parameters of the 8-DOF discrete vibration system are K1 = ... = K8 = 600 N / m; the mass parameters of each lumped mass are M1 = ... = M8 = 5 kg; and the damping coefficient is C1 = ... = C8 = 1.225 N·s / m. Therefore, an interpretable physical parameter neural network is established, with 8 neurons in each of the input, hidden, and output layers.

[0100] Now assume that all the above parameters are unknown. Set the initial values ​​for the stiffness parameter as K′1=…=K′8=300N / m; the initial values ​​for the mass parameter as M′1=…=M′8=1kg; and the initial values ​​for the damping coefficient as C′1=…=C′8=0.301N·s / m. Based on the initial structural parameters, the modal mass m′ can be calculated. i Modal damping c′ i and modal stiffness k′ i The initial values, and the inverse of the mode shape matrix. According to the preceding text, Assign initial values ​​to the corresponding neuron parameters in the input and output layers of the interpretable physical parameter neural network; and assign the initial values ​​of the other structural parameters to the corresponding neuron parameters in the hidden layers.

[0101] Subsequently, a sinusoidal sweep dynamic load signal sample with a bandwidth of 0-50Hz, an amplitude of 200N, and a duration of 30s was applied to the end lumped mass, and dynamic response signal samples of each lumped mass were collected. The data was then processed according to the data processing method described above to form a training sample dataset.

[0102] In addition, a stationary random dynamic load signal sample with a bandwidth of 0-50Hz and a duration of 30s is applied to the end lumped mass, and dynamic response signal samples of each lumped mass are collected. The data is then processed according to the data processing method described above to form a random dynamic load test sample dataset.

[0103] A triangular wave impact load signal sample with a peak value of 1000N and a pulse width of 0.02 seconds is applied to the end lumped mass, and dynamic response signal samples of each lumped mass are collected. The data is then processed according to the data processing method described above to form the impact load test sample dataset.

[0104] The interpretable physical parameter neural network was trained using the training method described above. The initial learning rate was set to 0.1, and the training iterations were 300 steps. Training was completed using only an AMD Ryzen 9 5950X CPU (without GPU acceleration), and the total training time was only 2.85 seconds. The convergence process of each neuron's parameters during the entire training process is as follows: Figure 5 As shown.

[0105] After the interpretable physical parameter neural network has been trained, based on the neuron parameter k′ i c′ i and m′ i The convergence result can be obtained by calculating the natural frequency parameter ω′ using equation (10). i With modal damping ratio parameter ξ′ i Therefore, the identification results of the modal parameters can be further compared, as shown in Table 1 and Table 2 respectively.

[0106]

[0107]

[0108] Table 1 ω i Identification results

[0109]

[0110] Table 2ξ i Identification results

[0111]

[0112]

[0113] Note: The relative error of 0.00% in the table is due to the small amount of data error, with the rounding error after truncation reaching 0.00%; it is not an absolute error of 0.00%.

[0114] The comparison results in the table show that the identification results of the modal parameters are very accurate, with the maximum error of the natural frequency parameter not exceeding 0.05% and the maximum error of the modal damping ratio not exceeding 1%.

[0115] As can be seen, interpretable neural networks were used to achieve accurate and fully analytical inverse modeling of this 8-DOF dynamic system.

[0116] Interpretable Physical Embedded Neural Networks (IPENNs) embed physical parameters within neurons, possessing explicit modeling rules and full interpretability. By introducing physical meaning and interpretability, IPENN overcomes the limitations of existing artificial neural network-based dynamic load identification methods, providing a powerful tool for dynamic load identification and exhibiting superior generalization ability, computational accuracy, and computational efficiency. Furthermore, it successfully combines dynamic load identification with modal parameter identification, demonstrating broad applicability to solving inverse problems in structural dynamics. Therefore, this invention provides a novel and interpretable method for structural dynamics modeling and inverse problem solving, advancing the theoretical understanding and practical applications of structural dynamics.

[0117] Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make changes, modifications, substitutions and variations to the above embodiments within the scope of the present invention without departing from the principles and spirit of the present invention.

Claims

1. A training method for an interpretable neural network for inverse modeling of structural dynamic systems, characterized in that: The interpretable neural network includes an input layer, hidden layers, and an output layer; In the input layer, hidden layer and output layer, the number of neurons in each layer is taken as the structural mode truncation order p used in the structural dynamic response analysis; The input signal of the i-th neuron in the input layer is the dynamic response of the structural dynamics system. , The corresponding weight vector is ;in for The i-th column vector in the data, The structural modal shape matrix; The input signal of the i-th neuron in the output layer is the constructed intermediate variable. The corresponding weight vector is ;in for The i-th column vector in the vector, and , , , for The self-power spectral density, for and cross power spectral density, For the i-th modal dynamic response, For the i-th modal dynamic load, For frequency; The i-th modal dynamic response of the input signal of the i-th neuron in the hidden layer The output is the i-th modal dynamic load. The corresponding weight vector is , , Let be the modal mass, modal damping, and modal stiffness of the i-th mode of the structural dynamics system, respectively. The training method includes the following steps: Step 1: Pre-training of the physical parameter neural network: Step 1.1: Calculate the structural modal matrix based on the structural boundary conditions and external dimensions. The initial values ​​of the modal mass, modal damping, and modal stiffness of the structure are given. Step 1.2: Based on the structural modal shape matrix calculate The weight parameters are assigned to each neuron in the input and output layers of the neural network; and the weight parameters are assigned to each neuron in the hidden layer of the neural network based on the initial values ​​of modal quality, modal damping, and modal stiffness. Step 2: Prepare training data: Collect time-domain data samples of the dynamic response of the loaded structure. With dynamic load time domain data samples ; Combining structural modal matrix Using the collected dynamic response time domain data samples With dynamic load time domain data samples The input signal of the hidden layer is obtained: the modal dynamic response time domain signal. And the output signal of the hidden layer: virtual modal dynamic load in the frequency domain. ; The input and output data of the hidden layer are used as training data samples; Step 3: Train the neural network using the training data to obtain the modal mass, modal damping, and modal stiffness results of the structure; Where the output of the i-th neuron in the hidden layer , respectively targeting its actual part and the virtual part Design loss function and : 。 2. The training method according to claim 1, characterized in that: The input signal of the neural network is the dynamic response time domain signal. The output signal is the power spectral density of the dynamic load. ,from arrive The process is as follows: Operator Defined as: Operator Defined as: 。 3. The training method according to claim 1, characterized in that: In step 3, based on and The error value was calculated, and the Adam-steepest gradient descent algorithm and backpropagation algorithm were used to train the neural network.

4. The training method according to claim 1, characterized in that: Applying the Hessian criterion to the loss function and The convexity is tested. and They all exhibit the characteristics of convex functions.

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