Machine learning method of dielectric elastomer film local modeling under equal biaxial prestretching

Abstract

The invention relates to a machine learning method of a dielectric elastomer film on-line modeling under equal biaxial prestretching. The method comprises the following steps of: 1, acquiring an experimental data set for generating input data of the physical information neural network in the step 2 and the Kolmogorov-Arnold network in the step 3; 2, building a physical information neural network; the prediction module is used for predicting a mapping relation between invariants I1 and I2 and a strain energy function, and is used for optimization training of the Kolmogorov-Arnold network in the step 3; step 3, constructing a Kolmogorov-Arnold network, and carrying out the construction of the network; an explicit expression used for predicting invariants I1 and I2 and a strain energy function, and generating a constitutive model capable of reflecting the nonlinear behavior of the dielectric elastomer film; and step 4, carrying out joint training of a physical information neural network and a Kolmogorov-Arnold network on the basis of an experimental data set of voltage and stretch ratio obtained by applying voltage after equal biaxial pre-stretching of the dielectric elastomer film obtained in the step 1, and finding out optimal network parameters. According to the invention, the mechanical-electric coupling characteristic of the dielectric elastomer material can be reflected.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of constitutive modeling of dielectric elastomer thin films in solid mechanics, and particularly relates to a machine learning method for constitutive modeling of dielectric elastomer thin films under equi-biaxial pre-stretching. BACKGROUND

[0002] Dielectric elastomers combine the electrical properties with the characteristics of elastic materials, usually with high capacitance performance and strong electric field response characteristics, while high elastic materials have good deformation recovery ability, large stress-strain range and high elastic modulus. Dielectric elastomers are a kind of functional materials that combine these two types of characteristics, and are widely used in flexible electronic devices, sensors, actuators, artificial muscles, energy storage devices and other fields. Such materials usually exhibit significant electrostriction or electro-mechanical properties under the action of an external electric field, i.e. the material can significantly change its shape under the action of an electric field, showing a high degree of elastic deformation ability. Researchers have found through experiments that pre-stretching can greatly improve the deformation ability of dielectric elastomer thin films. However, these dielectric elastomer thin films have a high failure rate when in operation, which greatly hinders their practical application. How pre-stretching affects the deformation ability of dielectric elastomer thin films is an important problem. Pre-stretching of dielectric elastomers can change the microstructure of the material, leading to enhanced polarization behavior, thereby improving the electrostriction and capacitance. There are some theoretical studies that have analyzed the electro-mechanical properties of dielectric elastomer thin films after pre-stretching, such as the Neo-Hookean model, the Mooney-Rivilin model, the Arruda-Boyce model, and the Gent model. However, there is no model that can perfectly fit the experiments. How to accurately predict and control the effects of equi-biaxial pre-stretching on the performance of dielectric elastomers, especially under complex load conditions, remains a challenge.

[0003] When constructing the constitutive model of dielectric elastomers, an appropriate constitutive expression needs to be selected first, which usually depends on the material's electro-mechanical coupling characteristics. After selecting the constitutive model, corresponding experimental tests such as uniaxial tensile test, equi-biaxial tensile test, etc. are needed to obtain the experimental data of stretch ratio and voltage. It is worth noting that the constitutive model of dielectric elastomer materials not only involves traditional mechanical response, but also needs to consider the dielectric properties of the material, therefore, engineering and technical personnel must have certain experience in electro-mechanical coupling constitutive model and experimental test experience in order to select the appropriate constitutive model. In practical work, engineering and technical personnel usually need to calibrate several or even more constitutive models to reflect the complex nonlinear response of the material under different loads and electric field as accurately as possible. Each model calibration needs to rely on the experimental data of the material and select the appropriate parameter calibration algorithm. For dielectric elastomer thin film materials, the existing constitutive models cannot accurately predict the regularity of experimental data.

[0004] In summary, how to select and construct a reasonable constitutive model to accurately describe the mechanical response of dielectric elastomer materials under mechanical coupling is still an important and challenging problem in research. SUMMARY

[0005] In order to overcome the defects existing in the prior art, the purpose of the present application is to provide a machine learning method for constitutive modeling of dielectric elastomer film under equi-biaxial pre-stretching, which uses machine learning method to automatically find out the constitutive model of dielectric elastomer material from experimental data set, so as to reflect the mechanical-electric coupling characteristics of dielectric elastomer material.

[0006] In order to achieve the above purpose, the technical scheme adopted by the present application is:

[0007] The machine learning method for constitutive modeling of dielectric elastomer film under equi-biaxial pre-stretching comprises the following steps:

[0008] Step 1: Obtain the experimental data set of dielectric elastomer film after equi-biaxial pre-stretching under applied electric field, which is used as input data for the physical information neural network in step 2 and the Kolmogorov-Arnold network in step 3;

[0009] Step 2: Physical information neural network construction; used to predict the mapping relationship of invariants I1, I2 and strain energy function W(I1, I2), used for optimization training of Kolmogorov-Arnold network in step 3;

[0010] Step 3: Kolmogorov-Arnold network construction; used to predict the explicit expression of invariants I1, I2 and strain energy function W(I1, I2), and generate a constitutive model that can reflect the nonlinear behavior of dielectric elastomer film;

[0011] Step 4: Based on the experimental data set of voltage and stretch ratio obtained by applying voltage to the dielectric elastomer film after equi-biaxial pre-stretching in step 1, the physical information neural network and the Kolmogorov-Arnold network are jointly trained to find the optimal network parameters to describe the nonlinear force-electric coupling characteristics of dielectric elastomer film under applied electric field.

[0012] The step 1 is specifically:

[0013] First, a dielectric elastomer film sandwiched between two flexible electrodes is subjected to relevant mechanical test, and an equi-biaxial pre-stretching test is carried out to obtain the elongation ratio of the pre-stretching state generated by the pre-stretching of external forces P1, P2 After continuing to apply electric field E, the elongation ratio after P1, P2 continue to act And the real voltage Experimental data set of the dielectric elastomer constructed;

[0014] Based on the theory of dielectric elastomer, the deformation tensor is

[0015]

[0016] Where λ1, λ2, λ3 are the elongation ratios of the three directions of the dielectric elastomer film respectively;

[0017] The right Cauchy-Green deformation tensor is the transpose of the deformation tensor and the product of the deformation tensor:

[0018]

[0019] Calculate the different stretch ratios obtained in step one The corresponding three invariants under the combined parameters are:

[0020]

[0021] Where I1, I2, I3 are the three invariants of the Cauchy-Green deformation tensor, Due to the incompressibility of the material, I3 = 1.

[0022] The step two is specifically:

[0023] Build a physical information neural network W net To predict different stretch ratios The corresponding I1, I2 and the mapping relationship of strain energy function W(I1, I2) under the combined parameters;

[0024] W pred (I1, I2) = W net (I1, I2; w, b) (4)

[0025] Where W pred is the predicted strain energy function based on two invariants I1, I2, W net represents a fully connected neural network, w represents the weight of the fully connected neural network, and b represents the bias of the fully connected neural network.

[0026] Combined with the automatic differentiation calculation ability of the neural network, the derivative of the strain energy function with respect to the invariant is calculated:

[0027]

[0028] Based on the theory of dielectric elastomer, the Cauchy stress tensor is

[0029]

[0030] where p is the hydrostatic pressure determined by the boundary conditions, I is the identity matrix;

[0031] The Cauchy stress in component form is

[0032]

[0033] To eliminate the hydrostatic pressure p, the first term is subtracted from the second term and the third term is subtracted from the second term:

[0034]

[0035] Under the action of the equal biaxial pre-stretching, the real stresses in the two in-plane directions are

[0036]

[0037] The real stresses σ p1 , σ p2 are the stresses generated by the external forces P1, P2, which are initially functions of the elongation ratio λ p1 , λ p2 , and continue to act after the electric field is applied, and σ p1 , σ p2 are again functions of the elongation ratio λ1, λ2.

[0038] The relationship between the real electric field and the real electric displacement is

[0039] E = D / ε, (10)

[0040] Under the action of the biaxial stretching force and the electric field, the in-plane equilibrium equation of the ideal dielectric elastomer thin film is

[0041]

[0042] The real stresses σ P1 , σ P2 are the in-plane real stresses generated by the external forces P1, P2; ε is the dielectric constant of the dielectric elastomer thin film, which is a constant independent of deformation; E is the real electric field under the action of the voltage, E = V / H, H = h / (λ1λ2); W pred is the free energy function, and this equilibrium equation reflects the situation in which the in-plane real stresses σ P1 , σ P2 generated by the external forces P1, P2 and the in-plane equivalent Maxwell stress εE 2 generated by the applied voltage V jointly act on the dielectric elastomer thin film to reach the equilibrium state.

[0043] The loss function Loss of the physical information neural network is defined as the real value of the voltage at the measurement point with the stretching ratio ​ The mean square error Loss of the predicted value V pred

[0044]

[0045] where V is the true value of the voltage at the measurement point, V pred is the predicted value of the voltage at the measurement point, N P is the number of sampling points at the nominal stress data point

[0046] The third step is specifically:

[0047] Build a Kolmogorov-Arnold network to predict the explicit expression of the strain energy function W(I1, I2);

[0048] Decompose the strain energy density function W(I1, I2) of the dielectric elastomer material into the Kolmogorov-Arnold expression form:

[0049]

[0050] where the function φ q,p has trainable parameters, φ q,p : and Φ q :

[0051] Φ = {φ q,p}, p = 1, 2, …, n in , q = 1, 2, …, n out , (14)

[0052] where n in is the number of nodes in the i-th layer of the computational graph, (l, i) represents the i-th neuron in the l-th layer, and x l,i represents the activation value of the (l, i)-neuron. Between layer l and layer l+1, there are n l n l+1 activation functions: the activation function connecting the (l, i)- and (l+1, j)- is represented as:

[0053]

[0054] Written in matrix form

[0055]

[0056] where Φ l ​​is the matrix of functions corresponding to the l-th layer of the Kolmogorov-Arnold network. A general Kolmogorov-Arnold network is a combination of L layers: given a dielectric elastomer material at different stretch ratios The corresponding invariants under the combination parameters as the input vector, here denoted by X, the output of the Kolmogorov-Arnold network is:

[0057]

[0058] Setting of the network activation function: includes a base function b(x) such that the activation function b(x) is the sum of the base function and the spline function.

[0059] φ(x) = w(b(x) + spline(x)) (18)

[0060] where

[0061] b(x) = silu(x) = x / (1 + e -x ) (19)

[0062] The spline(x) is parameterized as a linear combination of B-splines such that

[0063]

[0064] where c i is trainable, in principle, w is redundant as it can be absorbed into b(x) and spline(x), however, this w factor is still included to better control the overall magnitude of the activation function;

[0065] Initialization of the scaling: each activation function is initialized to have a spline(x) ~ 0, w is initialized according to the Xavier initialization;

[0066] Updating of the spline line grid: to address the issue that the spline is defined on a bounded region, but the activation values can evolve out of the fixed region during training, each grid is dynamically updated according to the input activation function.

[0067] The pruned Kolmogorov-Arnold network is more interpretable than the non-pruned one.

[0068] To make the Kolmogorov-Arnold network maximally interpretable, some simplification techniques are used;

[0069] Sparsification: the Kolmogorov-Arnold network uses LI regularization of the linear weights to support sparsity;

[0070] The LI norm of an activation function is defined as its average amplitude over N p inputs:

[0071]

[0072] Then, for a Kolmogorov-Arnold network layer Φ with n in inputs and n out outputs, the LI norm of Φ is defined as the sum of the LI norms of all activation functions, i.e.,

[0073]

[0074] Furthermore, the entropy of Φ is defined as

[0075]

[0076] The total training objective is the prediction loss l pred of all Kolmogorov-Arnold network layers plus LI and entropy regularization:

[0077]

[0078] where μ1, μ2 are relative amplitudes, typically set to μ1= μ2= 1, and λ controls the overall regularization amplitude.

[0079] Visualization: Set the transparency of an activation function φ l,i,j proportional to tanh(βA l,i,j ), where β = 3. Thus, functions with small amplitudes fade away so that we can focus on the important ones.

[0080] Pruning: After training with sparsity penalty, to prune the network to a smaller subnetwork. Sparsify the Kolmogorov-Arnold network on the node layer (not the edge layer). For each node (say the i-th neuron in the l-th layer), define its incoming and outgoing scores as

[0081]

[0082] If both the incoming and outgoing scores are larger than a threshold hyperparameter θ = 10 -2 , consider the node important. All unimportant neurons are pruned.

[0083] Symbolization: When it is suspected that some activation functions are actually symbolic in nature (e.g. cos or log), an interface is provided to set them to the specified symbolic form, fix_symbolic(l, i, j, f) can set the (l, i, j) activation to; however, it is not possible to simply set the activation function to the exact symbolic formula, as its input and output can have shifts and scalings; therefore, pre-activation x and post-activation y are obtained from the samples, and affine parameters (a, b, c, d) are fitted so that y ≈ cf(ax + b) + d. The fitting is done by an iterative grid search of a and b and linear regression.

[0084] The fourth step is specifically:

[0085] When the loss function Loss of the physical information neural network and the loss function of the Kolmogorov-Arnold network reach the condition at the same time, stop the network training;

[0086] Considering the difference between the training efficiency and convergence effect of the physical information network and the Kolmogorov-Arnold network, an iterative alternating training scheme is adopted, first reduce the loss function Loss of the physical information network, and then train the loss function Loss of the Kolmogorov-Arnold network, and realize the joint training of the network through the iterative algorithm;

[0087] In the back propagation phase, for the physical information network, the system calculates the gradient of the loss to the network parameters through automatic differentiation, and the parameter update adopts an adaptive optimization algorithm (such as the Adam optimizer), applies Xavier initialization to the linear weight, so that the loss function Loss is less than a certain threshold (such as 10 -4 ), stop the network training; for the Kolmogorov-Arnold network, the system calculates the gradient of the loss to the network parameters through automatic differentiation, including the linear combination weight and the output layer weight w l,i and the single variable basis function internal parameter (such as the spline control point coordinate, the polynomial coefficient), wherein the basis function gradient needs to consider its local support characteristic, for example, the node interval gradient of the spline function is only locally non-zero. The parameter update adopts an adaptive optimization algorithm (such as the Adam optimizer), applies Xavier initialization to the linear weight and superimposes an L2 regularization term to prevent overfitting, and applies a smoothness constraint to the basis function parameter to ensure that the strain energy function form is reasonable. In addition, if the validation set loss does not decrease for several consecutive rounds, the early stopping mechanism is triggered to terminate the training, and finally the network parameters are optimized through iteration to meet the loss minimization and the theoretical completeness of function representation at the same time, realizing high-precision approximation under data driving.

[0088] Through the above steps, the final output of the joint training of the physical information deep neural network and the Kolmogorov-Arnold network is an expression of the strain energy function W of the dielectric elastomer material obtained by the constitutive modeling method * (I1, I2), that is, a constitutive model reflecting the nonlinear behavior of the dielectric elastomer film, and the model parameters in the constitutive model strain energy function W * (I1, I2) are determined simultaneously by the Kolmogorov-Arnold network. Comparison can be made with the experimental data set to verify the obtained constitutive model of the dielectric elastomer material.

[0089] The beneficial effects of the present application are:

[0090] In summary, the method proposed in the present application realizes the acquisition of the experimental data set obtained by the mechanical test experiment, the joint training of the physical information deep neural network and the Kolmogorov-Arnold network, the obtaining of the explicit expression of the strain energy function in the constitutive model of the dielectric elastomer material and the model parameters in the constitutive model, and the reflection of the electromechanical coupling characteristics of the dielectric elastomer material.

[0091] The present application realizes an automatic modeling scheme of the machine learning method for the constitutive modeling of the dielectric elastomer film under equal biaxial pre-stretching based on the experimental data set. The constitutive model is completely trained by the neural network, and the strain energy function expression discovered by the model may be different from all the traditional constitutive models of the dielectric elastomer material. The more possibilities of the constitutive model enable it to automatically discover new constitutive models of the dielectric elastomer material under equal biaxial pre-stretching and have the ability to predict complex nonlinear behaviors that existing constitutive models cannot handle.

[0092] The machine learning method for the constitutive modeling of the dielectric elastomer film under equal biaxial pre-stretching provided by the present application does not require a large amount of experimental data, thereby significantly simplifying the establishment process of the material constitutive model and reducing the problem of unsuitable models caused by insufficient experience or insufficient testing. BRIEF DESCRIPTION OF DRAWINGS

[0093] Figure 1 Flowchart of the machine learning method for the constitutive modeling of the dielectric elastomer film under equal biaxial pre-stretching.

[0094] Figure 2 Voltage prediction diagram of a certain material under equal biaxial stretching of the dielectric elastomer based on the dielectric elastomer film test case of the present method. DETAILED DESCRIPTION

[0095] The present application will be further described in detail below with reference to the accompanying drawings.

[0096] The application is based on a constitutive model method of dielectric elastomer materials based on physical information neural networks and Kolmogorov-Arnold networks, and realizes an artificial intelligence neural network automatic modeling scheme of the constitutive model of dielectric elastomer materials based on an experimental data set.

[0097] The application will be described in detail in the following steps, and the specific operation process is as shown in the following Figure 1 :

[0098] Step one: experimental data set acquisition, used for generating input data of step two and step three; first, a dielectric elastomer film sandwiched between two flexible electrodes is subjected to relevant mechanical test experiments, equibiaxial tension test is carried out, and experimental data set is obtained, and the experimental data set of dielectric high-elastic polymer material composed of tensile ratio and voltage is obtained;

[0099] Taking equibiaxial tension test as an example, a flat test piece for testing is prepared, biaxial tension test is carried out, and the experimental data set of dielectric high-elastic polymer material composed of tensile ratio and voltage is obtained, as shown in the following Figure 1 . Specifically, a dielectric elastomer film is sandwiched between two flexible electrodes to carry out relevant equibiaxial tension test, in the reference state, the film is not affected by mechanical force and voltage, the thickness of the film is H, and the length is L1 and L2. When the dielectric high-elastic polymer film is in a pre-tension state, the film is subjected to external force P1 and P2, wherein P1=P2. The thickness becomes h1, and the two directions in the plane are respectively expanded by λ p1 times and λ p2 times, wherein λ p1 = λ p2 = λ p . Finally, the final equilibrium state of the dielectric high-elastic polymer film under the coupling action of electric field and force, on the basis of pre-tension in the plane, the horizontal direction maintains the external force P1 and P2, the thickness direction adds voltage V, the thickness of the film becomes h, and the two directions in the plane are respectively expanded by λ1 times and λ2 times, wherein λ1=λ2=λ. The experimental data of voltage under different tensile ratio combination parameters are finally obtained, so as to obtain the experimental data set of dielectric elastomer film composed of tensile ratio and voltage .

[0100] Based on the theory of dielectric elastomer, when three-dimensional tension occurs, the deformation tensor is

[0101]

[0102] where λ1, λ2, λ3 are the stretch ratios of the dielectric elastomer film in three directions, respectively.

[0103] The right Cauchy-Green deformation tensor is the transpose of the deformation tensor and the product of the deformation tensor

[0104]

[0105] The different stretch ratios obtained in the calculation step one The corresponding deformation invariants under the combined parameters.

[0106]

[0107] where I1, I2, I3 are the three invariants of the Cauchy-Green deformation tensor, Due to the incompressibility of the material, I3 = 1.

[0108] Step two: physical information neural network building; for different stretch ratios The corresponding I1, I2 and strain energy function W(I1, I2) mapping relationship under the combined parameters; build a physical information neural network W net to predict the strain energy function W(I1, I2).

[0109] First, build a fully connected neural network W net (w, b) to predict the strain energy function W(I1, I2), where the neural network input is the deformation invariants I1, I2 of the dielectric elastomer material, and the output is the strain energy function W pred (I1, I2), as shown in the attached Figure 1 .

[0110] W pred (I1, I2) = W net (I1, I2; w, b) (4)

[0111] where W pred is the predicted strain energy function based on two invariants I1, I2, W net represents a fully connected neural network, w represents the weight of the fully connected neural network, and b represents the bias of the fully connected neural network.

[0112] Combined with the automatic differentiation calculation ability of the neural network, the derivative of the strain energy function with respect to the invariant

[0113]

[0114] Based on the theory of dielectric elastomers, the Cauchy stress tensor is

[0115]

[0116] The Cauchy stress is expressed in the form of components:

[0117]

[0118] Where p is the hydrostatic pressure, which is determined by the boundary conditions, and I is the identity matrix. To eliminate the hydrostatic pressure p, the first and second terms of Equation (5) are subtracted from the third term to obtain

[0119]

[0120] Under equibiaxial pretension, the true stress in the two directions of the plane is

[0121]

[0122] True stress σ p1 ,σ p2 It is the stress generated by the pre-stretching of external forces P1 and P2, and the elongation ratio λ in the pre-stretching state is the initial elongation ratio p1 ,λ p2 function, P1 and P2 continue to act after the electric field is applied, σ p1 ,σ p2 It is also a function of the elongation ratios λ1 and λ2.

[0123] The relationship between true electric field and true electric displacement is:

[0124] E=D / ε, (10)

[0125] Under the combined action of biaxial tension and electric field, the in-plane equilibrium equation of an ideal dielectric elastomer film is:

[0126]

[0127] σ P1 ,σ P2 is the true in-plane stress generated by the pre-stretching action of the external forces P1 and P2; ε is the dielectric constant of the dielectric elastomer film, which is a constant that does not depend on deformation; E is the true electric field under the action of voltage, E = V pred / H, H=h / (λ1λ2); W is the free energy function. This equilibrium equation reflects the true in-plane stress σ generated by the pre-tensioning action of the external forces P1 and P2. P1 ,σ P2 and the in-plane equivalent Maxwell stress εE generated by the applied voltage V 2 The situation where the dielectric high elastomer film reaches equilibrium under the combined action.

[0128] The loss function of the physical information neural network is defined as the stretching ratio of the measurement point The true value of the voltage at and the predicted value V predMean Squared Error Loss

[0129]

[0130] where is the true value of the voltage at the measurement point, V pred is the predicted value of the voltage at the measurement point, N P is the number of sampling points at the nominal stress data point

[0131] Step three: Kolmogorov-Arnold network; for generating a constitutive model that can reflect the nonlinear behavior of dielectric elastomer thin films;

[0132] A Kolmogorov-Arnold network is built to predict the explicit expression of the strain energy function W(I1, I2); the strain energy density function W(I1, I2) of the dielectric elastomer material is decomposed into the Kolmogorov-Arnold representation form:

[0133]

[0134] where the function φ q,p has trainable parameters, φ q,p : and Φ q :

[0135] Φ = {φ q,p}, p = 1, 2, …, n in , q = 1, 2, …, n out , (14)

[0136] where n in is the number of nodes in the i-th layer of the computational graph. (l, i) denotes the i-th neuron in the l-th layer, with x l,i denoting the activation value of the (l, i)-neuron. Between layer l and layer l+1, there are n l n l+1 activation functions: the activation function connecting the (l, i)- and (l+1, j)-neurons is denoted as

[0137]

[0138] written in matrix form

[0139]

[0140] where Φ l ​is the matrix of functions corresponding to the l-th layer of the Kolmogorov-Arnold network. A general Kolmogorov-Arnold network is a combination of L layers: given a dielectric elastomer material at different stretch ratios The corresponding invariants under the combination parameters as input vector, here denoted by X, the output of the Kolmogorov-Arnold network is

[0141]

[0142] Setting of the network activation function: includes a base function b(x) such that the activation function b(x) is the sum of the base function and a spline function.

[0143] φ(x) = w(b(x) + spline(x)) (18)

[0144] where

[0145] b(x) = silu(x) = x / (1 + e -x ) (19)

[0146] The spline(x) is parameterized as a linear combination of B-splines such that

[0147]

[0148] where c i is trainable. In principle, w is redundant as it can be absorbed into b(x) and spline(x). However, this w factor is still included to better control the overall magnitude of the activation function.

[0149] Initialization of the scaling: each activation function is initialized to have a spline(x) ~ 0. w is initialized according to the Xavier initialization.

[0150] Updating of the spline grid: to address that the spline is defined on a bounded region, each grid is dynamically updated according to the input activation function.

[0151] The pruned Kolmogorov-Arnold network is more interpretable than the non-pruned one.

[0152] To maximize the interpretability of the Kolmogorov-Arnold network, some simplification techniques are used;

[0153] Sparsification: the Kolmogorov-Arnold network uses LI regularization of the linear weights to support sparsity

[0154] The LI norm of an activation function is defined as its average magnitude over N p inputs

[0155]

[0156] Then, for a Kolmogorov-Arnold network layer Φ with n in invariant inputs and n out strain energy function outputs, define the L1 norm of Φ as the sum of the L1 norms of all activation functions, i.e.,

[0157]

[0158] Also, define the entropy of Φ as

[0159]

[0160] The total training objective is the prediction loss l pred of all Kolmogorov-Arnold network layers plus L1 and entropy regularization:

[0161]

[0162] where μ1, μ2 are relative magnitudes, typically set to μ1 = μ2 = 1, and λ controls the overall regularization magnitude.

[0163] Visualization: Set the transparency of activation function φ l,i,j proportional to tanh(βA l,i,j ), where β = 3. Thus, functions with small magnitudes fade away so that we can focus on the important ones.

[0164] Pruning: After training with sparsity penalty, to prune the network into a smaller subnetwork. Sparsify the Kolmogorov-Arnold network on the node layer (not the edge layer). For each node (say the i-th neuron in the l-th layer), define its incoming and outgoing scores as

[0165]

[0166] If both the incoming and outgoing scores are larger than a threshold hyperparameter θ = 10 -2 , consider the node important. All unimportant neurons are pruned.

[0167] Symbolization: When it is suspected that some activation functions are actually symbolic in nature (e.g., cos or log), an interface is provided to set them to the specified symbolic form, fix_symbolic(l, i, j, f) can set the (l, i, j) activation to f. However, one cannot simply set the activation function to the exact symbolic formula, as its input and output can have shifts and scalings. Thus, pre- and post-activations x and y are obtained from samples, and affine parameters (a, b, c, d) are fitted so that y ≈ cf(ax + b) + d. The fitting is done by an iterative grid search of a and b, and linear regression.

[0168] Step four: neural network training, find the optimal network parameters;

[0169] Based on the experimental data set of the dielectric elastomer material obtained in step one, the network training is carried out, and the physical information neural network in step two and the Kolmogorov-Arnold network in step three are jointly trained;

[0170] When the loss function Loss of the physical information neural network and the loss function of the Kolmogorov-Arnold network reach the condition at the same time, stop the network training;

[0171] Considering the difference between the training efficiency and convergence effect of the physical information network and the Kolmogorov-Arnold network, an iterative alternating training scheme can be adopted, first reduce the loss function Loss of the physical information network, then train the loss function Loss of the Kolmogorov-Arnold network, and realize the joint training of the network through the iterative algorithm. In the back propagation phase, for the physical information network, the system calculates the gradient of the loss to the network parameters by automatic differentiation, and the parameter update adopts the adaptive optimization algorithm (such as Adam optimizer), and the Xavier initialization is applied to the linear weight, so that the loss function Loss is less than a certain threshold (such as 10 -4 ), stop the network training. For the Kolmogorov-Arnold network, the system calculates the gradient of the loss to the network parameters by automatic differentiation, including the linear combination weight and the output layer weight w l,iand the internal parameters of the univariate basis function (such as the coordinates of the spline control points, the coefficients of the polynomial), wherein the gradient of the basis function needs to consider its local support characteristics, for example, the interval gradient of the node of the spline function is only locally non-zero. The parameter update adopts an adaptive optimization algorithm (such as the Adam optimizer), and the Xavier initialization is applied to the linear weight and the L2 regularization term is superimposed to prevent overfitting, while the smoothness constraint is applied to the basis function parameters to ensure that the strain energy function form is reasonable. In addition, if the validation set loss does not decrease for consecutive rounds, the early stopping mechanism is triggered to terminate the training, and finally the network parameters are optimized iteratively to meet the loss minimization and the theoretical completeness of the function representation, realizing high-precision approximation under data-driven.

[0172] Through the above steps, the final output of the Kolmogorov-Arnold network is the strain energy function W of the dielectric elastomer constitutive model obtained by the constitutive modeling method * (I1,I2), that is, a dielectric elastomer constitutive model capable of reflecting the nonlinear behavior of the dielectric elastomer film is generated. And the strain energy function W * (I1,I2) of the constitutive model is determined by the Kolmogorov-Arnold network. The dielectric elastomer constitutive model obtained can be compared with the experimental data set to verify the dielectric elastomer constitutive model.

[0173] In summary, the method proposed in the present application realizes the joint training of the experimental data set obtained by the mechanical test experiment, the physical information neural network and the Kolmogorov-Arnold network, directly obtains the explicit expression of the strain energy function of the dielectric elastomer, and simultaneously determines the model parameters in the constitutive model.

[0174] Step one is used to generate the input data of the physical information neural network in step two and the Kolmogorov-Arnold network in step three; the physical information neural network in step two obtains the mapping relationship of I1, I2 and the strain energy function W(I1,I2), which is used for the optimization training of the Kolmogorov-Arnold network in step three; and the Kolmogorov-Arnold network in step three finally generates a constitutive model capable of reflecting the dielectric elastomer film under the condition of equal biaxial pre-stretching;

[0175] Embodiment:

[0176] The constitutive model scheme of the dielectric elastomer film of the present application is applied to the constitutive and deformation analysis of a certain material dielectric elastomer.

[0177] The equal biaxial stretching test is performed on a certain dielectric elastomer film, and the experimental data set of the dielectric elastomer composed of the stretching ratio and the voltage is obtained, as shown in the attached Figure 2The points shown in (a).

[0178] Via the dielectric elastomer constitutive model scheme of the application, combined with the joint training of physical information neural network and Kolmogorov-Arnold network, the explicit expression of the strain energy function in the constitutive model of the dielectric elastomer is directly obtained as

[0179]

[0180] And the model parameters in the constitutive model are determined as C 10 = 0.0008, C 20 = 3.2 x 10 -9 .

[0181] Based on the network model training, the constitutive model of the dielectric elastomer can be further calculated to obtain Cauchy stress and in-plane true stress, and finally obtain the voltage, and the results are shown in the line shown in the attached Figure 2 (a). Compared with the experimental data set, as shown in the attached Figure 2 , it can be seen that the constitutive model prediction can well predict the experimental data obtained by mechanical test, which verifies the obtained dielectric elastomer constitutive model. As shown in the attached Figure 2 (b), compared with the traditional New-Hookean and Gent model, it can be seen that the prediction of Neo-Hookean model deviates from the experimental points seriously, and cannot fit the deformation trend, and the prediction of Gent model is much better than that of Neo-Hookean model, but in the stage of 3.5<λ<5.5, it deviates from the experimental points obviously, while the prediction of the new free energy function can well fit the experimental points.

[0182] The method proposed in the application, combined with the joint training of physical information neural network and Kolmogorov-Arnold network, directly obtains the explicit expression of the strain energy function in the constitutive model of the dielectric elastomer, and simultaneously determines the model parameters in the constitutive model. In particular, the constitutive model is completely obtained by neural network training, and the constitutive model expression has more possibilities, so it has the ability to predict some complex nonlinear behaviors that existing constitutive models cannot handle.

Claims

1. A machine learning method for constitutive modeling of dielectric elastomer films under equibiaxial pre-tensioning, characterized in that: The following steps are included: Step 1: Obtain an experimental data set of biaxial pre-stretching of a dielectric elastomer film followed by an applied electric field, which is used to generate input data for the physical information neural network in step 2 and the Kolmogorov-Arnold network in step 3. Step 2: Construction of a physical information neural network; used to predict the mapping relationship between the invariants I1, I2 and the strain energy function W(I1, I2), which is used for the optimization training of the Kolmogorov-Arnold network in step 3; Step 3: Kolmogorov-Arnold network construction; used to predict the explicit expressions of the invariants I1, I2 and the strain energy function W(I1, I2), and generate a constitutive model that can reflect the nonlinear behavior of the dielectric elastomer film; Step 4: Based on the experimental data set of voltage and stretching ratio obtained after the dielectric elastomer film is subjected to equibiaxial pre-stretching and voltage is applied, the physical information neural network and the Kolmogorov-Arnold network are jointly trained to find the optimal network parameters.

2. The machine learning method for constitutive modeling of dielectric elastomer films under equibiaxial pre-tensioning according to claim 1, characterized in that: The step 1 is specifically as follows: First, a mechanical test was carried out on a dielectric elastomer film sandwiched between two flexible electrodes, and an equibiaxial pre-stretching test was performed to obtain the elongation ratio of the pre-stretched state caused by the external forces P1 and P2. After the electric field E is applied again, the elongation ratio of P1 and P2 after the action continues is and real voltage Experimental dataset of constructed dielectric elastomers; Based on the dielectric elastic body theory, when stretched in three directions, the deformation tensor is Where λ1, λ2, and λ3 are the elongation ratios of the dielectric elastic polymer film in three directions respectively; The right Cauchy-Green deformation tensor is the product of the transpose of the deformation tensor and the deformation tensor: Calculation step 1 for different stretch ratios The corresponding three invariants under the combined parameters are: Among them, I1, I2, I3 are the three invariants of the Cauchy Green deformation tensor, Due to the incompressibility of the material, I3=1.

3. The machine learning method for constitutive modeling of dielectric elastomer films under equibiaxial pre-tensioning according to claim 2, characterized in that: The step 2 is specifically as follows: Build a physical information neural network W net , to predict different stretch ratios The mapping relationship between I1, I2 and strain energy function W(I1, I2) under the combined parameters; W pred (I1,I2)=W net (I1,I2;w,b) (4) Among them, W pred is the predicted strain energy function based on the two invariants I1 and I2, W net represents a fully connected neural network, w represents the weight of the fully connected neural network, and b represents the bias of the fully connected neural network; Combining the automatic differentiation computing capability of the neural network, the derivative of the strain energy function with respect to the invariant is calculated: Based on the dielectric elastic body theory, the Cauchy stress tensor is Where p is the hydrostatic pressure, determined by the boundary conditions, and I is the identity matrix; Cauchy stresses should be expressed in component form as: To eliminate the hydrostatic pressure p, subtract the third term from the first and second terms respectively to obtain: Under equibiaxial pretension, the true stress in the two directions in the plane is: True stress σ p1 , σ p2 It is the stress generated by the pre-stretching of external forces P1 and P2, and the elongation ratio λ in the pre-stretching state is the initial elongation ratio p1 ,λ p2 function, P1 and P2 continue to act after the electric field is applied, σ p1 , σ p2 It is also a function of the elongation ratios λ1 and λ2.

4. The machine learning method for constitutive modeling of dielectric elastomer films under equibiaxial pre-tensioning according to claim 3, characterized in that: The relationship between true electric field and true electric displacement is: E=D / ε, (10) Under the combined action of biaxial tension and electric field, the in-plane equilibrium equation of an ideal dielectric elastomer film is: σ P1 ,σ P2 is the true in-plane stress generated by the pre-stretching action of external forces P1 and P2; ε is the dielectric constant of the dielectric elastomer film, which is a constant that does not depend on deformation; E is the real electric field under the action of voltage, E = V / H, H = h / (λ1λ2); W pred is the free energy function. The equilibrium equation reflects the true in-plane stress σ generated by the pre-tension of external forces P1 and P2. P1 ,σ P2 and the in-plane equivalent Maxwell stress εE generated by the applied voltage V 2 The dielectric elastomer film reaches equilibrium under the combined action; The loss function of the physical information neural network is defined as the stretching ratio of the measurement point The true value of the voltage at and the predicted value V pred The mean square error Loss: in is the true value of the voltage at the measurement point, V pred is the predicted value of the voltage at the measurement point, N P is the nominal stress data point The number of sampling points.

5. The machine learning method for constitutive modeling of dielectric elastomer films under equibiaxial pre-tensioning according to claim 4, characterized in that: The step three is specifically as follows: Build a Kolmogorov-Arnold network to predict the explicit expression of the strain energy function W(I1,I2); Decompose the strain energy density function W(I1,I2) of the dielectric elastomer material into the Kolmogorov-Arnold representation: where the function φ q,p With trainable parameters, and Φ={φ q,p },p=1,2,…,n in ,q=1,2…,n out , (14) where n in is the number of nodes in the i-th layer of the computational graph, (l, i) represents the i-th neuron in the l-th layer, and x l,i represents the activation value of (l, i)-neuron. Between layer l and layer l+1, there are n l n l+1 Activation function: The activation function connecting (l, i)- and (l+1, j)- is expressed as: Written in matrix form where Φ l is the function matrix corresponding to the lth layer of the Kolmogorov-Arnold network, given the dielectric elastomer material at different stretching ratios The corresponding invariant under the combined parameters is taken as the input vector, which is uniformly recorded as X here. The output of the Kolmogorov-Arnold network is: Setting of network activation function: including a basis function b(x), so that the activation function b(x) is the sum of the basis function and the spline function; φ(x)=w(b(x)+spline(x)) (18) in b(x)=shadow(x)=x / (1+e -x ) ( 19 spline(x) is parameterized as a linear combination of B-splines such that where c i It is trainable.

6. The machine learning method for constitutive modeling of dielectric elastomer films under equibiaxial pre-tensioning according to claim 5, characterized in that: Find explicit expressions for strain energy functions; Initialization scaling: Each activation function is initialized to have a spline spline(x)≈0, and w is initialized according to Xavier initialization; Spline grid update: Dynamically update each grid based on the input activation function; Sparsification: Kolmogorov-Arnold networks use L1 regularization of linear weights to support sparsity; The L1 norm of the activation function is defined as its p Average amplitude over inputs: Then, for n in input and n out The L1 norm of the Kolmogorov-Arnold network layer Φ with 1 output is defined as the sum of the L1 norms of all activation functions, that is: In addition, the entropy of Φ is defined as The total training target is the prediction loss l for all Kolmogorov-Arnold network layers pred Add L1 and entropy regularization: Where μ1 and μ2 are relative amplitudes, usually set to μ1 = μ2 = 1, and λ controls the overall regularization amplitude; Visualization: Setting the activation function φ l,i,j The transparency and tanh(βA l,i,j ) Functions with proportionally small amplitudes will gradually disappear; Pruning: After training with a sparsification penalty, the Kolmogorov-Arnold network is sparsified at the node level, where for each node, its incoming and outgoing scores are defined as If both incoming and outgoing scores are greater than a threshold hyperparameter θ = 10 -2 , then the node is considered important and all unimportant neurons are pruned; Symbolization: When it is suspected that some activation functions are actually symbolic, an interface is provided to set them to the specified symbolic form. fix_symbolic(l, i, j, f) sets the (l, i, j) activation to f; obtain the pre-activation x and post-activation y from the sample, and fit the affine parameters (a, b, c, d) so that y ≈ cf(ax+b)+d. The fitting is done by iterative grid search and linear regression of a and b.

7. The machine learning method for constitutive modeling of a dielectric elastomer film under equibiaxial pre-tensioning according to claim 6, wherein the fourth step comprises: When the loss function Loss of the physical information neural network and the loss function of the Kolmogorov-Arnold network meet the conditions at the same time, stop network training; Adopting an iterative alternating training scheme, the loss function of the physical information network is first reduced, and then the loss function of the Kolmogorov-Arnold network is trained, and the network joint training is achieved through an iterative algorithm; In the backpropagation phase, for physical information networks, the system calculates the gradient of the loss to the network parameters through automatic differentiation. The parameters are updated using an adaptive optimization algorithm. Xavier initialization is applied to the linear weights to make the loss function Loss less than a certain threshold, and the network training is stopped. For Kolmogorov-Arnold networks, the system calculates the gradient of the loss to the network parameters through automatic differentiation, including the linear combination weights and the output layer weights w l,i and the internal parameters of the univariate basis functions, where the gradients of the basis functions need to take into account their local support characteristics; Parameters are updated using an adaptive optimization algorithm. Xavier initialization and L2 regularization are applied to linear weights to prevent overfitting, while smoothness constraints are imposed on basis function parameters to ensure the reasonable shape of the strain energy function. In addition, if the validation set loss does not decrease over multiple consecutive rounds, an early stopping mechanism is triggered to terminate training. Ultimately, through iterative optimization, the network parameters are optimized to simultaneously meet the requirements of loss minimization and theoretical completeness of the function representation, achieving high-precision approximation driven by data. Through the above steps, the final output of the joint training of the physical information deep neural network and the Kolmogorov-Arnold network is the expression W of the strain energy function of the dielectric elastomer material obtained by this constitutive modeling method. * (I1, I2), which is the constitutive model reflecting the nonlinear behavior of the dielectric elastic film, and the strain energy function W of the constitutive model * The model parameters in (I1, I2) are determined simultaneously by the Kolmogorov-Arnold network.

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