A method for reverse realization of magnetization field of magnetic soft material based on manifold neural operator

Abstract

A method for inversely implementing the magnetization field of magnetic soft materials based on manifold neural operators is characterized by the following steps: S1, acquiring simulation data of the deformation field of the magnetic soft material under different magnetization distributions, and extracting the Laplace-Beltrammian operator characteristic function of the material surface geometry; S2, constructing a manifold neural operator model based on the characteristic function, and training the model using simulation data to obtain a magnetization deformation field prediction model; S3, using the magnetization deformation field prediction model as a surrogate model, coupling an optimization algorithm for global optimization, and inversely searching for the optimal magnetization distribution that produces the target deformation. This invention employs the above-mentioned inverse design method for magnetization fields based on manifold neural operators, and establishes a magnetization deformation field prediction model based on the Laplace-Beltrammian operator, significantly improving the efficiency of inverse design of magnetization distribution for magnetic soft materials.

Description

Technical Field

[0001] This invention relates to a robotics technology, particularly an intelligent design and machine learning technology, specifically a method for inversely realizing the magnetization field of magnetic soft materials based on manifold neural operators. Background Technology

[0002] Magnetic-driven soft robots, with their contactless manipulation, rapid response, and good biocompatibility, have shown broad application prospects in targeted drug delivery, minimally invasive surgery, and flexible electronics. As applications expand into complex environments, there is an urgent need to design functional structures with specific deformation modes to achieve advanced tasks such as precise grasping and multimodal motion. However, achieving reverse design from desired function to specific magnetization distribution faces significant challenges: to obtain rich deformation modes, materials are divided into a large number of independently programmed voxels, leading to an exponential increase in the design space. Traditional methods rely on magnetic-force coupled finite element simulations to iterate extensively in the high-dimensional design space, resulting in high accuracy but low design efficiency and difficulty in convergence. In recent years, researchers have attempted to introduce deep learning models to accelerate forward prediction; however, traditional neural networks such as convolutional neural networks are limited by regular grids and cannot be directly applied to complex surfaces described by unstructured grids. Emerging neural operator methods, such as Fourier neural operators, can learn function space mappings, but their core operations rely on regular Cartesian grids in Euclidean space, making them difficult to adapt to magnetic soft materials with complex geometries. To address the aforementioned problems, this invention proposes a method for inversely realizing the magnetization field of magnetic soft materials based on manifold neural operators. This method utilizes the Laplace-Beltrami operator eigenfunctions to construct an orthogonal basis that naturally fits the geometry, learns a continuous mapping from magnetization distribution to a three-dimensional deformation field in a low-dimensional spectral space, and achieves efficient inverse search through a coupled optimization algorithm, significantly improving the design efficiency of magnetic soft materials. Summary of the Invention

[0003] This invention addresses the problem that existing methods struggle to efficiently achieve magnetization configuration of magnetic soft materials based on target deformation modes. It proposes a method for inversely realizing the magnetization field of magnetic soft materials based on manifold neural operators. By constructing a neural operator based on the Laplace-Beltramian operator characteristic function, applicable to arbitrarily complex geometries, it achieves efficient prediction of deformation fields and inverse design of magnetization distribution under complex geometries.

[0004] A method for inversely realizing the magnetization field of magnetic soft materials based on manifold neural operators, characterized by comprising the following steps:

[0005] S1. Obtain simulation data of the deformation field of magnetic soft materials under different magnetization distributions, and extract the Laplace-Beltramian operator characteristic functions of the mesh nodes on the material surface;

[0006] S2. Construct a manifold neural operator model based on the Laplace-Beltramian operator characteristic function, and train the model using the deformation field simulation data to obtain a magnetization deformation field prediction model.

[0007] S3. Using the magnetization deformation field prediction model as a surrogate model, the coupled optimization algorithm performs global optimization and reverse searches to find the optimal magnetization distribution that generates the target deformation field.

[0008] Preferably, the Laplace-Beltrami operator characteristic functions described in S1 are obtained by solving the generalized eigenvalue problem on a triangular mesh, and the number of characteristic functions is... Taking 128, we construct the characteristic basis matrix. ,in This represents the number of nodes in the surface mesh.

[0009] Preferably, the manifold neural operator model described in S2 adopts... The architecture contains four Laplacian kernel integral modules, with a network width of 192, and uses... The optimizer is trained with an initial learning rate of 0.001, using a cosine annealing learning rate scheduling strategy.

[0010] Preferably, the Laplace kernel integral module described in S2 is implemented as follows: the encoding sub-step utilizes the pre-computed feature basis matrix. Input features of the current layer Projecting onto the spectral coefficient space yields the spectral coefficients. ,in The mass matrix; the spectral space transformation substep applies to the spectral coefficients. By applying a learnable linear transformation and a nonlinear activation function, the transformed spectral coefficients are obtained. The decoding sub-steps utilize the same set of feature basis matrices. Transformed spectral coefficients Reconstructing the manifold space yields feature updates. The residual connection sub-step adds the feature update to the pointwise linear transformation and bias, and then uses an activation function to form the output of this layer. ,in and For learnable parameters, This is the activation function.

[0011] Preferably, the training process described in S2 adopts a relative Error, as a loss function, takes the form: in For batch size, For the first The predicted deformation field for each sample For the first The true deformation field of each sample.

[0012] Preferably, the forward prediction process in S3 includes data preprocessing, Gaussian normalization, model inference, result inverse normalization, and visualization output. The Gaussian normalization formula is:

[0013]

[0014] in and For training set statistics, To prevent division by zero by small constants, the prediction results are... Save the file in the specified format.

[0015] Preferably, the simulated annealing algorithm in S3 includes the following steps: Initialization step: randomly generating... Initial magnetization encoding vector of individual elements Set the initial temperature and cooling rate The iterative optimization step is in the first... Step on the current solution Generate new solutions by applying random perturbations The rapid evaluation step will yield a new solution. Predict the deformation field by inputting a trained manifold neural operator model. Calculate the objective function value Metropolis acceptance criteria steps for calculating fitness change ,like Then accept the new solution; otherwise, proceed by probability.

[0016]

[0017] Accept the poor solution; the cooling process should follow the cooling plan to lower the temperature. The termination and output steps are repeated iteratively until the preset conditions are met, and the optimal magnetization distribution code is output. .

[0018] Preferably, the objective function described in S3 is defined according to the specific design objective, for the maximum Towards the displacement target, in the form of:

[0019] in For prediction Mean displacement The item is a constraint / penalty item. These are the weighting coefficients.

[0020] The beneficial effects of this invention are:

[0021] This invention constructs a neural operator based on the Laplace-Beltrami operator characteristic function, applicable to arbitrarily complex geometries, to achieve efficient prediction of deformation fields and rapid inverse design of magnetization distribution in magnetic soft materials, significantly improving design efficiency and providing an effective technical means for the development of smart materials. Attached Figure Description

[0022] Figure 1 This is a flowchart of a reverse design method for the magnetization field of magnetic soft materials based on manifold neural operators.

[0023] Figure 2 The diagram shows the architecture of the NORM manifold neural operator model. (a) is the model flowchart, and (b) is the internal flowchart of the Laplace layer.

[0024] Figure 3 The following are comparison charts of random sample prediction results for the embodiment: (a) is a comparison chart of random sample prediction and actual value of the package structure, including prediction deformation chart, actual deformation chart, prediction-actual deformation error and absolute error chart; (b) is a comparison chart of random sample prediction and actual value of the crawling structure, including prediction deformation chart, actual deformation chart, prediction-actual deformation error and absolute error chart. Detailed Implementation

[0025] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0026] like Figure 1-3 As shown,

[0027] This embodiment uses two typical magnetic soft structures—a wrapping structure and a crawling structure—as examples to apply the method of this invention for deformation field prediction and reverse design. Figure 1 As shown, the specific implementation process includes four core stages: data preparation and feature function calculation, manifold neural operator model construction and training, forward prediction application, and reverse design and implementation.

[0028] S1. Data Preparation and Calculation of the Laplace-Beltrami Operator Characteristic Functions. This step begins with... Export the calculation data of the magnetic soft material from the simulation software and save it as... File format. The data file contains key arrays: Represents the coordinates of surface mesh nodes, with the shape of... Indicates the connection relationship of triangular facets; and They represent direction and The magnetization component in the direction, each sample is a an array, in which The number of voxels; , , They represent , , The displacement field in the direction is defined on the surface mesh nodes. The surface mesh of the wrapping structure contains 1674 nodes, and the surface mesh of the crawling structure contains 1154 nodes. 100 sets of data samples are randomly generated for each, and they are divided into training set and test set in a 3:1 ratio.

[0029] Next, the characteristic functions of the Laplace-Beltrami operator are calculated. This process uses... of Library implementation, starting from Load grid data from the file and extract it. and The array is used to create a triangular mesh object and solve the eigenvalue problem. The characteristic equation of the solution is: ,in Here is the stiffness matrix. For the quality matrix, For the first 1 eigenvector For the first The first 128 eigenvalues ​​are used to calculate the eigenvectors corresponding to them, forming the eigenfunction matrix. These characteristic functions, serving as basis functions for the manifold neural operator, project the high-dimensional displacement field onto a low-dimensional characteristic space. The calculation results are saved as... The file contains and Fields such as...

[0030] S2, Construction and training of manifold neural operator models. For example... Figure 2 As shown, the model is based on The architecture is built, and the core parameters are configured as follows: (Using the first 128 feature functions) (Network bandwidth) (Number of network layers). Since the number of grid nodes varies among different samples in the actual data, data preprocessing is performed: find the maximum number of nodes among all samples. For samples with insufficient nodes, zero-padding is performed along the node dimension; for samples with excessive nodes, truncation is performed. The processed input data has the following shape: ,in This refers to the batch size, where 2 represents... and Two channels.

[0031] Data normalization was performed using Gaussian normalization, calculating the mean and standard deviation for both the input and output data of the training set. The normalization formula is as follows: ,in and It is a statistic calculated from the training data. It is a small constant to prevent division by zero. For the two channels of magnetized input, create separate... The instances are normalized.

[0032] The model loads the pre-computed Laplacian eigenfunction matrix and takes the first... Columns as a basis matrix And calculate its pseudo-inverse matrix. During the forward propagation of the manifold neural operator, the Laplace kernel integrator module performs an encoding-spectral space transformation-decoding operation. The mathematical expression for each Laplace kernel integrator module is: the encoding step yields... ; spectral space transformation steps Applying a learnable linear transformation and nonlinear activation yields The decoding steps yielded... The residual join step yields... ,in For the first Layer input features, For the first Layer output features, Let be the eigenfunction matrix of the Laplace-Beltrami operator. For the quality matrix, and For learnable parameters, This is the activation function. Multiple such modules are stacked through residual connections to form a complete manifold neural operator.

[0033] The training process adopts The optimizer has an initial learning rate of 0.001 and a weight decay rate of... The learning rate is scheduled using a cosine annealing strategy, with training lasting 500 epochs. The loss function is a relative loss function. Error, in the form of ,in For batch size, For the first The predicted deformation field for each sample For the first The true deformation field of each sample.

[0034] After training, the model achieved excellent prediction performance on the test set. For wrapping structures, the mean relative L2 error on the test set was 0.018679, and the coefficient of determination was [missing information]. The average relative L2 error reached 0.9962; for crawling structures, the average relative L2 error was 0.000049. It reached 0.9999.

[0035] S3. Forward Prediction Application and Reverse Design Implementation. For a new magnetization distribution input, the prediction process first loads the saved model checkpoints, restoring the model parameters and normalizer statistics. After the input data undergoes the same preprocessing and normalization steps, the normalized displacement field prediction is obtained through model forward propagation, and then the displacement value of physical magnitude is obtained through inverse normalization. The prediction results of a random sample from each structure are shown below. Figure 3 As shown, the model takes only about 14.7 milliseconds for a single prediction, compared to an average finite element simulation time of about 1500 seconds, representing a computational efficiency improvement of more than five orders of magnitude.

[0036] Based on a trained forward model, a simulated annealing algorithm is used to search for the optimal magnetization distribution in a large design space. First, the mapping relationship between magnetization direction and magnetic moment is defined: 1→(40, 0), 2→(-40, 0), 3→(0, 40), 4→(0, -40), corresponding to the right, left, front, and back directions, respectively. For the wrapping structure, the objective function maximizes displacement while introducing a central region constraint; for the crawling structure, it directly maximizes... Displacement. The objective function is in the form of: .

[0037] The specific process of the simulated annealing algorithm is as follows: Initialization step: Randomly generate the initial magnetization encoding vector for all voxels. Set the initial temperature Cooling rate The iterative optimization step is in the first... A new solution is generated by randomly selecting a voxel and changing its magnetization direction. The rapid evaluation step will yield a new solution. The trained manifold neural operator model is input to predict the deformation field, and the objective function value is calculated; the Metropolis acceptance criterion step is used to calculate the fitness change. ,like Then accept the new solution; otherwise, proceed by probability. Accept the poor solution; cooling steps are as follows Lower the temperature; terminate the judgment step and repeat the iteration until the maximum number of iterations is reached, then output the optimal magnetization distribution code. For the wrapping structure, after 1500 iterations, the fitness score reaches 2642.23 points, and the predicted average value corresponding to the optimal magnetization distribution is... The displacement was 2.294 mm, and the deformation pattern showed a gradual upward movement from the center outwards. For the crawling structure, convergence occurred after 860 iterations, with the optimal magnetization distribution corresponding to the largest predicted value. The displacement is 11.545 mm.

[0038] Therefore, this invention employs a manifold neural operator method for predicting and inversely designing deformation fields of magnetic soft materials using the steps described above. By constructing a neural operator based on the Laplace-Beltramian operator characteristic function, applicable to arbitrarily complex geometric shapes, it achieves efficient prediction of deformation fields and rapid inverse design of magnetization distributions for magnetic soft materials, significantly improving design efficiency and providing an effective technical means for the development of smart materials.

[0039] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

[0040] The parts not covered in this invention are the same as or can be implemented using existing technologies.

Claims

1. A method for inversely realizing the magnetization field of magnetic soft materials based on manifold neural operators, characterized in that, Includes the following steps: S1. Obtain simulation data of the deformation field of magnetic soft materials under different magnetization distributions, and extract the Laplace-Beltramian operator characteristic functions of the mesh nodes on the material surface; S2. Construct a manifold neural operator model based on the Laplace-Beltramian operator characteristic function, and train the model using the deformation field simulation data to obtain a magnetization deformation field prediction model. S3. Using the magnetization deformation field prediction model as a surrogate model, the coupled optimization algorithm performs global optimization and reverse searches to find the optimal magnetization distribution that generates the target deformation field.

2. The method according to claim 1, characterized in that, The Laplace-Beltrami operator characteristic function in step S1 is obtained by solving the eigenvalue problem defined on the material surface mesh, and is used to represent the physical field defined on the surface mesh in the spectral space.

3. The method according to claim 2, characterized in that, The Laplace-Beltrammian operator eigenfunctions in step S1 are obtained by solving the generalized eigenvalue problem on a triangular mesh, and the number of eigenfunctions is... Taking 128, we construct the characteristic basis matrix. ,in This represents the number of nodes in the surface mesh.

4. The method according to claim 1, characterized in that, The manifold neural operator model in step S2 includes at least one Laplace kernel integral module, which is used to: project the input features from the manifold space to the spectral space using the Laplace-Beltrami operator feature function; The projected features are parametrically transformed in the spectral space; Using the Laplace-Beltrami operator feature function, the transformed features are reconstructed from the spectral space back to the manifold space to update the input features.

5. The method according to claim 4, characterized in that, The specific implementation of the Laplace kernel integral module in step S2 is as follows: the encoding sub-step utilizes the pre-computed feature basis matrix. Input features of the current layer Projecting onto the spectral coefficient space yields the spectral coefficients. ,in This is the quality matrix; The spectral space transformation sub-step affects the spectral coefficients. By applying a learnable linear transformation and a nonlinear activation function, the transformed spectral coefficients are obtained. The decoding sub-steps utilize the same set of feature basis matrices. Transformed spectral coefficients Reconstructing the manifold space yields feature updates. ; The residual connection sub-step adds the feature update to the pointwise linear transformation and bias, and then forms the layer output through an activation function. ,in and For learnable parameters, The activation function is used; the training process employs a relative activation function. Error, as a loss function, takes the form: in For batch size, For the first The predicted deformation field for each sample For the first The true deformation field of each sample.

6. The method according to claim 1, characterized in that, The training process in step S2, which trains the model, aims to minimize the difference between the deformation field predicted by the model and the real deformation field obtained from the simulation, and updates the model parameters.

7. The method according to claim 1, characterized in that, The forward prediction process in step S3 includes normalizing and denormalizing the input magnetization distribution and the output predicted deformation field; the forward prediction process includes data preprocessing, Gaussian normalization, model inference, result denormalization, and visualization output. The Gaussian normalization formula is as follows: in and For training set statistics, To prevent division by zero by small constants, the prediction results are... Save the file in the specified format.

8. The method according to claim 1, characterized in that, The optimization algorithm in step S3 is a global optimization algorithm used to search for the optimal magnetization distribution in the design space.

9. The method according to claim 8, characterized in that, In step S3, the global optimization process defines an objective function based on a preset design goal and quickly evaluates the objective function values ​​corresponding to different magnetization distributions based on the surrogate model to guide the search direction.

10. The method according to claim 8, characterized in that, The optimization algorithm described in S3 is the simulated annealing algorithm, which includes the following steps: Initialization step: randomly generates... Initial magnetization encoding vector of individual elements Set the initial temperature and cooling rate ; The iterative optimization step is at the 1st Step on the current solution Generate new solutions by applying random perturbations ; The rapid evaluation step will provide a new solution. Predict the deformation field by inputting a trained manifold neural operator model. Calculate the objective function value ; Metropolis acceptance criteria steps to calculate fitness change ,like Then accept the new solution; otherwise, use probability: Accept the poor solution; the cooling process should follow the cooling plan to lower the temperature. The termination and output steps are repeated iteratively until the preset conditions are met, and the optimal magnetization distribution code is output. ; The objective function is defined according to the specific design objective, for the maximum Towards the displacement target, in the form of: in For prediction Mean displacement The item is a constraint / penalty item. These are the weighting coefficients.

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