Three-dimensional geometric spectrum analysis method and system based on model feature space

Abstract

The invention discloses a three-dimensional geometric spectrum analysis method and system based on a model feature space, and the method comprises the steps: inputting coordinates in an effective definition domain of a three-dimensional geometric shape and a shape code corresponding to the three-dimensional geometric shape into a trained neural network, outputting a group of first-order gradient vectors capable of representing feature function values and coordinates of the whole shape space; training the neural network: for the randomly sampled shape codes, calculating the energy value of each feature function output by the neural network according to the first-order gradient vector of the coordinates, and performing ascending sorting to obtain a feature function sequence; performing orthogonal projection on the feature function sequence to obtain a target feature function; and constructing a corresponding loss function based on a preset partial differential equation type, substituting the target feature function into the corresponding loss function, and updating parameters of the neural network by minimizing the loss function to obtain a trained neural network. According to the method, continuous shape spatial spectrum analysis of the characteristic value crossing problem can be realized without depending on grids.

Description

Technical Field

[0001] This invention belongs to the interdisciplinary fields of computer graphics, computational physics simulation, and deep learning. Specifically, it relates to a three-dimensional geometric spectrum analysis method and system based on model feature space. Background Technology

[0002] In modern industrial manufacturing, computer animation, virtual reality, and scientific computing, physical simulations of phenomena, including mechanical and acoustic behaviors, are crucial. Partial differential equations (PDEs) are the mathematical tools used to describe these physical phenomena. Characteristic analysis of these differential operators, solving for their eigenvalues ​​and characteristic functions, is a common method for analyzing the inherent properties of physical systems. Characteristic functions not only reveal the resonance modes and natural vibration frequencies of objective objects but also encode geometric information, which can be used for shape description, relational calculations, and physical simulations.

[0003] Currently, industry typically models and analyzes products in continuously changing shape spaces. However, existing spectral analysis techniques face significant technical bottlenecks when dealing with such continuous shape spaces. Traditional spectral analysis methods rely primarily on discretized meshes, following a process of mesh generation, matrix construction, and eigenvalue decomposition. For each new sample in the shape space, a high-quality mesh needs to be regenerated and a massive sparse matrix constructed. This process is computationally expensive, and changes in mesh topological connectivity can cause jumps in the dimensions of the operator matrices, making eigenvalues ​​and eigenfunctions non-differentiable with respect to shape encoding, thus hindering gradient-based automatic shape optimization.

[0004] In recent years, although implicit neural representation techniques have been introduced to solve partial differential equations without meshes, they face the challenge of eigenvalue intersection when extended to continuous shape spaces. According to perturbation theory, when the geometric domain undergoes continuous deformation, the energy curves of different physical modes may intersect. Existing neural spectrum methods typically employ a fixed ordering strategy, that is, forcing a specific output channel of the neural network to always fit the first... The smallest eigenvalue, that is, the eigenvalue after sorting from smallest to largest. The forced fixed ordering of eigenvalues ​​near eigenvalue intersections leads to drastic modal shifts in the output function, causing a non-differentiable kink in the loss function with respect to shape encoding. This not only disrupts the smoothness of the optimization, making it difficult for the neural network to converge, but also generates discontinuous or even physically erroneous feature fields in shape space, rendering them unusable for subsequent differentiable design and real-time simulation tasks. Therefore, a continuous shape space spectral analysis method that can overcome mesh dependence and effectively handle eigenvalue intersection problems is urgently needed. Summary of the Invention

[0005] To address the shortcomings of existing technologies, the present invention aims to provide a three-dimensional geometric spectrum analysis method and system based on model feature space. This method and system address the technical problems of existing technologies, such as the need to independently discretize each shape to solve for features, high computational costs and non-differentiability, and modal confusion and optimization divergence caused by eigenvalue crossover during continuous shape changes.

[0006] This invention utilizes implicit neural fields to uniformly solve eigenvalues ​​and eigenfunctions in a continuously parameterized shape space, solving the optimization problem caused by modal crossover, and realizing a method for real-time dimensionality reduction physical simulation and differentiable inverse design.

[0007] The present invention adopts the following technical solution.

[0008] A first aspect of the present invention provides a three-dimensional geometric spectrum analysis method based on a model feature space, comprising:

[0009] The coordinates within the effective domain of the 3D geometry and the shape code corresponding to the 3D geometry are input into the trained neural network, which outputs a set of first-order gradient vectors that can represent the feature function values ​​and coordinates of the entire shape space; one shape code corresponds to one effective domain.

[0010] Training a neural network includes: for a randomly sampled shape code, calculating the energy value of each feature function output by the neural network based on the first-order gradient vector of the coordinates, sorting the features in ascending order according to the energy values ​​to obtain the feature function sequence of the current shape code; wherein, the energy value is used to characterize the frequency or variational energy of the feature function;

[0011] The target feature function is obtained by orthogonally projecting the feature function sequence.

[0012] Based on the preset partial differential equation type, a corresponding loss function is constructed. The target feature function is substituted into the corresponding loss function. The parameters of the neural network are updated by minimizing the loss function, and the trained neural network is obtained.

[0013] Optionally, orthogonal projection can be performed on the feature function sequence to obtain the target feature functions, including:

[0014] Take the k-th eigenfunction of the eigenfunction sequence, determine the (k-1)-th eigenfunction as the orthogonal projection basis, project the k-th eigenfunction onto the orthogonal complement space of the subspace spanned by the (k-1)-th eigenfunction, calculate the projection components of the eigenfunction on the projection basis, and obtain the orthogonalized target eigenfunction.

[0015] Optionally, orthogonally projecting the feature function sequence to obtain the target feature function further includes:

[0016] The target feature function of the first k-1 feature functions after projection is taken as a constant without gradient;

[0017] Take the inner product of each of the k-1 constants with the corresponding target feature function, multiply the product by the corresponding constant, and sum the k-1 products. Subtract the sum from the k-th feature function to obtain the target feature function of the k-th feature function.

[0018] Optionally, the preset partial differential equation types include the Laplace operator and the linear elasticity operator. The energy of each feature function output by the neural network is calculated based on the first-order gradient vector of the coordinates, including:

[0019] The Rayleigh quotient of the feature function value output by the neural network is used as the energy value;

[0020] When the partial differential equation type is preset to Laplace operator, the sum of squares of the Euclidean norm of the vector is obtained by multiplying the characteristic function value with the first-order gradient vector of the corresponding coordinate, and then the ratio is calculated with the sum of squares of the characteristic function value. The ratio is used as the Rayleigh quotient of the characteristic function value.

[0021] When the partial differential equation type is set to linear elastic operator, the elastic potential energy density is calculated based on the linear strain tensor of the characteristic function and the Lamé constant of the material. The ratio of the elastic potential energy density to the sum of squares of the characteristic function values ​​is used as the Rayleigh quotient of the characteristic function values.

[0022] Optionally, when the preset partial differential equation type is Laplace operator, the loss function of the Laplace operator is constructed according to the following formula:

[0023]

[0024] in, This represents the loss function in the Laplace operator spectral analysis. The parameters representing the neural network; Represents the mathematical expectation operator, for shape space Shape encoding that follows a specific distribution sampling Calculate the expected value. This represents the total number of target characteristic functions to be solved. Indicates the shape encoding of the current sample. The determined valid domain, It is a differential volume element. This represents the output of the neural network after orthogonalization. One target feature function, Represents relative to spatial coordinates The first-order gradient operator, Let represent the square of the Euclidean norm of a vector.

[0025] Optionally, when the preset partial differential equation type is a linear elastic operator, the loss function of the linear elastic operator is constructed according to the following formula:

[0026]

[0027] in, This represents the loss function in spectral analysis of linear elastic operators. Indicates the first The linear strain tensor corresponding to each characteristic function is defined as follows: , Represents the displacement gradient tensor transpose; This represents the sum of the squares of all elements in the matrix. Represents the square of the trace of a matrix. Shear modulus These are material property constants. Represents the mathematical expectation operator, for shape space Shape encoding that follows a specific distribution sampling Calculate the expected value. It is a differential volume element. Indicates the shape encoding of the current sample. The determined valid domain.

[0028] Optionally, the method further includes:

[0029] A multilayer perceptron mechanism is configured in the neural network. A position encoding layer is set at the input of the neural network to map spatial coordinates into feature vectors representing continuous space. These feature vectors are then input into the multilayer perceptron mechanism, and a configuration is set at the output. There are 10 channels. The first channel is initialized to a constant value whose output is independent of the input. The rest... Each channel is configured to output the feature function values ​​to be learned.

[0030] Optionally, the method further includes:

[0031] The boundary of a three-dimensional geometry is defined as the zero isosurface of a scalar function. The sign of the scalar function value is used to distinguish between the interior and exterior of the geometry, thereby determining the effective domain of the three-dimensional geometry.

[0032] Optionally, updating the parameters of the neural network by minimizing the loss function includes:

[0033] Use gradient descent or the Adam optimizer to update the parameters of the neural network through backpropagation until the loss function converges to the minimum or reaches the preset number of iterations.

[0034] A second aspect of the present invention provides a three-dimensional geometric spectrum analysis system based on a model feature space, implementing the above-described three-dimensional geometric spectrum analysis method based on a model feature space, the system comprising:

[0035] The spectrum analysis module is used to input the coordinates within the effective domain of the 3D geometry and the shape code corresponding to the 3D geometry into the trained neural network, and output a set of first-order gradient vectors that can characterize the feature function values ​​and coordinates of the entire shape space; one shape code corresponds to one effective domain.

[0036] The training module is used to train the neural network. The training module includes:

[0037] The energy value calculation module is used to calculate the energy value of each feature function output by the neural network based on the first-order gradient vector of the coordinates for randomly sampled shape codes, and sort them in ascending order according to the energy values ​​to obtain the feature function sequence of the current shape code; wherein, the energy value is used to characterize the frequency or variable energy of the feature function;

[0038] The orthogonal projection module is used to orthogonally project the feature function sequence to obtain the target feature function.

[0039] The update module is used to construct the corresponding loss function based on the preset partial differential equation type, substitute the target feature function into the corresponding loss function, update the parameters of the neural network by minimizing the loss function, and obtain the trained neural network.

[0040] A third aspect of the present invention provides an electronic device including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the computer program, when loaded onto the processor, implements the above-described three-dimensional geometric spectrum analysis method based on model feature space.

[0041] A fourth aspect of the present invention provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the above-described three-dimensional geometric spectrum analysis method based on model feature space.

[0042] Compared with the prior art, the beneficial effects of the present invention include at least the following:

[0043] This invention inputs the coordinates within the effective domain of a 3D geometric shape and the corresponding shape code into a trained neural network. It outputs a set of feature function values ​​and first-order gradient vectors representing the entire shape space, directly mapping feature functions from the shape code and spatial coordinates. By calculating the energy value of each feature function output by the neural network based on the first-order gradient vector of the coordinates, and sorting them in ascending order according to the energy values, the invention obtains the feature function sequence of the current shape code. It evaluates modal energy in real time and dynamically adjusts the orthogonalization order, rearranging the modal index in real time based on the Rayleigh quotient energy value. This ensures that the feature value curves can smoothly cross rather than repel each other at intersection points, achieving complete differentiability of the feature basis with respect to the shape code, eliminating gradient singularities, and improving inference speed by several orders of magnitude. This provides an efficient and robust solution for real-time dimensionality reduction physical simulation and differentiable inverse shape design. Attached Figure Description

[0044] To more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort. Wherein:

[0045] Figure 1 This is a schematic diagram of a three-dimensional geometric spectrum analysis method based on model feature space provided in an embodiment of this disclosure;

[0046] Figure 2 This is a schematic diagram illustrating the result of handling feature value crossover using a fixed sorting mechanism in existing technologies.

[0047] Figure 3 This is a schematic diagram illustrating the result of the dynamic sequential sorting mechanism provided in this embodiment of the present disclosure handling the feature value crossover phenomenon. Detailed Implementation

[0048] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of this invention. The embodiments described in this application are merely some embodiments of this invention, and not all embodiments. Based on the spirit of this invention, all other embodiments obtained by those skilled in the art without creative effort are within the protection scope of this invention.

[0049] Combination Figure 1As shown, Embodiment 1 of the present invention provides a three-dimensional geometric spectrum analysis method based on model feature space. It demonstrates the extraction of features from geometric shape encoding and coordinate inputs via an implicit neural network, including core processing modules such as energy estimation, dynamic sorting, and gradient filtering. Finally, it generates a target feature function and calculates variational energy loss for backpropagation. The method includes the following:

[0050] The coordinates within the effective domain of the 3D geometry and the shape code corresponding to the 3D geometry are input into the trained neural network, which outputs a set of first-order gradient vectors that can represent the feature function values ​​and coordinates of the entire shape space; one shape code corresponds to one effective domain.

[0051] Training a neural network includes: for a randomly sampled shape code, calculating the energy value of each feature function output by the neural network based on the first-order gradient vector of the coordinates, sorting the features in ascending order according to the energy values ​​to obtain the feature function sequence of the current shape code; wherein, the energy value is used to characterize the frequency or variational energy of the feature function;

[0052] The target feature function is obtained by orthogonally projecting the feature function sequence.

[0053] Based on the preset partial differential equation type, a corresponding loss function is constructed. The target feature function is substituted into the corresponding loss function. The parameters of the neural network are updated by minimizing the loss function, and the trained neural network is obtained.

[0054] Optionally, the preset partial differential equation types include the Laplace operator and the linear elasticity operator. The energy of each feature function output by the neural network is calculated based on the first-order gradient vector of the coordinates, including:

[0055] The Rayleigh quotient of the feature function value output by the neural network is used as the energy value;

[0056] When the partial differential equation type is preset to Laplace operator, the sum of squares of the Euclidean norm of the vector is obtained by multiplying the characteristic function value with the first-order gradient vector of the corresponding coordinate, and then the ratio is calculated with the sum of squares of the characteristic function value. The ratio is used as the Rayleigh quotient of the characteristic function value.

[0057] When the partial differential equation type is set to linear elastic operator, the elastic potential energy density is calculated based on the linear strain tensor of the characteristic function and the Lamé constant of the material. The ratio of the elastic potential energy density to the sum of squares of the characteristic function values ​​is used as the Rayleigh quotient of the characteristic function values.

[0058] This disclosure uses dynamic sequential sorting based on energy to handle eigenvalue crossovers. Compared to existing technologies that employ fixed sorting strategies, where a forced fixed sorting near eigenvalue crossovers causes drastic modal shifts in the output function and results in a non-differentiable kink in the loss function with respect to shape encoding, this not only disrupts the smoothness of optimization and makes it difficult for the neural network to converge, but also generates discontinuous or even physically erroneous feature fields in shape space, making them unusable for subsequent differentiable design and real-time simulation tasks. This disclosure rearranges the modal index in real time based on the Rayleigh quotient energy value, enabling the eigenvalue curves to smoothly cross rather than repel each other at crossover points, thereby allowing the neural network to converge.

[0059] Specifically, for the Laplacian operator feature calculation, Rayleigh quotient is used. Approximate as Dirichlet energy and Norm ratio:

[0060]

[0061] in, Represents shape encoding The corresponding number The Rayleigh quotient of a characteristic function, i.e., the energy value of the characteristic function. Represents shape encoding and spatial points The corresponding output number One characteristic function, Represents a set of points in space. express In spatial coordinates The first-order gradient operator, Let represent the square of the Euclidean norm of a vector.

[0062] It is understandable that the Rayleigh quotient is the energy for each feature function. The approximation means that because the neural network outputs an estimate of the feature function, the calculated Rayleigh quotient is only an approximate estimate of the true feature value. This invention calculates energy values ​​based on the Rayleigh quotient to support subsequent dynamic ranking. The Dirichlet energy is used to describe the gradient distribution of a function, and measures the smoothness of the function by summing the squares of the function's gradients.

[0063] For linear elastic operators, the numerator is replaced by elastic potential energy density. Linear elastic operators are a common term in computer graphics for differential operators describing the elastic deformation of solids, specifically used to describe the deformation and vibration of three-dimensional entities, including but not limited to rubber and metals. Elastic potential energy density measures the deformation energy per unit volume of an object, and its calculation depends on the linear strain tensor of the characteristic function and the Lamé constant of the material.

[0064] Based on the calculated energy value The size of the output of the neural network output layer is determined by the channel index. Perform real-time ascending sorting to generate a shape code that depends on the current shape. Arrangement index ,satisfy:

[0065]

[0066] in, This represents the minimum energy value of the feature function corresponding to the shape code g after sorting. This represents the energy value of the second feature function after sorting. Indicates the sorting order The energy values ​​of each feature function, where N represents the output dimension of the neural network output layer.

[0067] Orthogonally projecting the sequence of feature functions yields the target feature functions, including:

[0068] Take the k-th feature function of the feature function sequence, determine the (k-1)-th feature function as the orthogonal projection basis, project the k-th feature function onto the orthogonal complement space of the subspace spanned by the (k-1)-th feature function, calculate the projection components of the feature function on the projection basis, and obtain the orthogonalized target feature function; where k is a positive integer.

[0069] It should be noted that orthogonal complement space refers to the space between the two previous spaces. A vector space perpendicular to the subspace spanned by a dominant mode.

[0070] Based on a defined dominant order Using the Schmidt orthogonalization principle, for the sorted... Modality Project it onto the front In the orthogonal complement space of the subspace spanned by the dominant mode. In this way, it is possible to eliminate modes that are related to the previous ones. The overlapping components of each mode, after this processing, result in a new mode that maintains orthogonality with all previously orthogonalized modes.

[0071] Optionally, orthogonally projecting the feature function sequence to obtain the target feature function further includes:

[0072] The target feature function of the first k-1 feature functions after projection is taken as a constant without gradient;

[0073] Take the inner product of each of the k-1 constants with the corresponding target feature function, multiply the product by the corresponding constant, and sum the k-1 products. Subtract the sum from the k-th feature function to obtain the target feature function of the k-th feature function.

[0074] To eliminate the non-physical reaction force gradient generated by secondary modes on the dominant mode in order to satisfy orthogonality, this invention introduces a gradient filtering operator in the projection calculation. The specific calculation formula is as follows:

[0075]

[0076] in, This represents the projected but unnormalized first... One target feature function, and Let be the sorted index number, satisfying , The order after sorting by energy value One characteristic function, This represents the projected but unnormalized first... One target feature function, This indicates that the projection calculation has been completed and normalized. One target feature function; This represents the inner product operation within the valid domain. This represents a gradient filtering operation, used in differential calculations to treat the variables within parentheses as constants. During differential calculations, i.e., backpropagation, their partial derivatives are always 0, thus blocking their gradient contribution to the network parameters. The gradient filtering operator avoids non-physical gradient interference from secondary modes to the dominant mode.

[0077] It is understandable that the first k-1 orthogonalized characteristic functions Gradient filtering operator After processing, it will be treated as a fixed known quantity, and the calculation process will only be performed on the k-th characteristic function to be orthogonalized. The calculations are performed without changing the values ​​of the first k-1 eigenfunctions. These first k-1 eigenfunctions are the dominant modes, arranged in ascending energy order, and their optimization should only be driven by their own energy loss. If they are not treated as constants, when orthogonalizing the k-th secondary mode, the backpropagation gradient will be transmitted to the first k-1 dominant modes, generating non-physical reaction force gradients that interfere with the optimization of the dominant modes. By setting them as constants, the gradient calculation will not update the parameters of these dominant modes, ensuring the optimization stability of the dominant modes.

[0078] More specifically, when the partial differential equation type is preset to Laplace operator, the loss function of the Laplace operator is constructed as follows:

[0079]

[0080] in, Let represent the loss function of the Laplace operator spectral analysis, and let be the expected physical energy of all N target eigenfunctions over the entire shape space D, i.e., the Dirichlet energy expectation. The parameters representing the neural network include the trainable weights and bias parameters in the implicit neural field network; Represents the mathematical expectation operator, for shape space Shape encoding that follows a specific distribution sampling Calculate the expected value. This represents the total number of target characteristic functions to be solved. Indicates the shape encoding of the current sample. The determined valid domain, It is a differential volume element. This represents the output of the neural network after orthogonalization. One target feature function, Represents relative to spatial coordinates The first-order gradient operator, The square of the Euclidean norm of the vector is used to quantize the magnitude of the gradient in order to calculate the Dirichlet energy.

[0081] More specifically, when the partial differential equation type is preset to linear elastic operator, the loss function of the linear elastic operator is constructed according to the following formula:

[0082]

[0083] in, Let $\mathbf{ ... Indicates the first The linear strain tensor corresponding to each characteristic function is defined as follows: , Represents the displacement gradient tensor transpose; Let Frobenius norm be the square of the matrix, which is the sum of the squares of all its elements. Represents the square of the trace of a matrix. and Lamé constant represents the elastic characteristics of a material, used to describe the elastic properties of homogeneous isotropic materials. Shear modulus These are material property constants. Represents the mathematical expectation operator, for shape space Shape encoding that follows a specific distribution sampling Calculate the expected value. It is a differential volume element. Indicates the shape encoding of the current sample. The determined valid domain.

[0084] Optionally, the method further includes:

[0085] A multilayer perceptron mechanism is configured in the neural network. A position encoding layer is set at the input of the neural network to map spatial coordinates into feature vectors representing continuous space. These feature vectors are then input into the multilayer perceptron mechanism, and a configuration is set at the output. There are 10 channels. The first channel is initialized to a constant value whose output is independent of the input. The rest... Each channel is configured to output the feature function values ​​to be learned.

[0086] It should be noted that for the Laplacian operator or elastic operator under von Neumann boundary conditions, there exist fixed zero eigenvalue modes. To improve training efficiency and avoid numerical instability, in this invention, the first channel of the neural network output is preset to a constant value independent of shape and spatial coordinates, enabling the neural network to learn subsequent non-zero frequency modes.

[0087] Optionally, the method further includes:

[0088] The boundary of a three-dimensional geometry is defined as the zero isosurface of a scalar function. The sign of the scalar function value is used to distinguish between the interior and exterior of the geometry, thereby determining the effective domain of the three-dimensional geometry.

[0089] Construct a parameterized shape space and implicit neural field model, defining a shape encoding... Continuously parameterized shape space Shape encoding yes A dimensional vector, for any given Uniquely determines the valid domain In this invention, boundary Implicitly defined by the level set function or the sign distance field, i.e. Where SDF() represents the symbolic distance field, and x is within the valid domain when it is inside the symbolic distance field. .

[0090] Specifically, the level set function defines the boundary of the geometry as the zero isosurface of a scalar function, using the sign of the function value to distinguish between the interior and exterior of the geometry. The signed distance field is a special form of the level set function, where the absolute value of its function represents the shortest Euclidean distance from any point in space to the boundary of the geometry. This implicit definition allows the invention to handle geometric families with complex topologies or even topological changes without explicitly generating a mesh. The convenience of using level set functions or signed distance fields to describe geometric structures instead of traditional meshes eliminates mesh dependency and supports continuous shape deformation and complex topologies.

[0091] Build parameters are Multilayer perceptron network This serves as a universal approximator for the feature function. The network input includes spatial coordinates. and shape encoding The output is a set of feature field vectors. ,in, To predetermine the number of feature modes to be solved, i.e., the N feature functions output by the neural network, This represents the original feature function value of the first channel of the neural network output. This represents the original feature function value of the Nth channel of the neural network output.

[0092] To mitigate the spectral bias problem of neural networks and capture high-frequency physical vibration modes, the neural network employs positional encoding techniques or sinusoidal activation functions. Specifically, for the input coordinates... The mapping formula for the first layer, i.e., the hidden layer, of a neural network is:

[0093]

[0094] in, and These are trainable weights and biases. By leveraging the smoothness and periodicity of the sine function, the grid's ability to represent high-frequency signals is enhanced, allowing for more accurate capture of high-frequency physical vibration modes. Furthermore, it ensures accurate and smooth gradient information in calculating the physical energy functional, thereby improving the convergence speed of training and the accuracy of physical simulations.

[0095] It is understandable that positional coding technology maps low-dimensional spatial coordinates x to a high-dimensional feature space through a set of high-frequency functions, thus solving the inherent spectral bias problem of multilayer perceptrons. Positional coding technology is existing technology, and the embodiments disclosed herein do not improve upon it, so they will not be described in detail here.

[0096] Optionally, updating the parameters of the neural network by minimizing the loss function includes:

[0097] Use gradient descent or the Adam optimizer to update the parameters of the neural network through backpropagation until the loss function converges to the minimum or reaches the preset number of iterations.

[0098] Specifically, during training, random sampling and forward propagation independent of discretization are performed. In each training iteration, from the shape space... Randomly sample a batch of shape codes Where B represents the number of shape codes in a single random sampling, i.e., the batch size, for shape coding. Uniform sampling spatial points are generated within their corresponding bounding boxes, and then eliminated using the defined SDF number. External points, thus obtaining the set of sampling points located inside the geometry. Where S represents the total number of spatial points sampled for a single shape code, and i represents the index number in the set of sampling points, where the i-th spatial point and the m-th shape code are sampled. To the implicit neural field network , obtain characteristic field values Using the automatic differentiation method, the characteristic field with respect to spatial coordinates is calculated. first gradient That is, for each component of the feature field vector, that is, the value of each feature function at that point, the spatial gradient is calculated to obtain the gradient of k feature functions at that point.

[0099] Understandably, the feature field values ​​are a set of vectors, where each component represents the function value of a single feature function at the corresponding point in space. The dimension of each feature field vector is equal to the predetermined number of feature functions. A complete feature function is a function of spatial coordinate x, covering all points within the geometrically valid domain.

[0100] Calculating the loss function using an automatic differentiation framework Regarding network parameters gradient By introducing gradient filtering, the gradient flow is prevented from backpropagating through orthogonal bases, thus ensuring the stability of the optimization. Stochastic gradient descent or the Adam optimizer is used to update the network weights until the loss function converges, resulting in a well-trained neural field model.

[0101] Figure 2 This is a schematic diagram illustrating the result of handling feature value crossover using a fixed sorting mechanism in existing technologies. Figure 2 This demonstrates a fixed sorting strategy used in the prior art. As the shape encoding changes, the two eigenmode curves, including eigenmode A and eigenmode B, repel each other when approaching the intersection point where they have equal energy. This results in non-differentiable cusps on the eigenvalue curves, disrupting the continuity of the gradient. Figure 3 This is a schematic diagram illustrating the result of the dynamic sequential sorting mechanism provided in this embodiment of the present disclosure handling the feature value crossover phenomenon. Figure 3 This invention demonstrates the use of a dynamic sequential sorting strategy that allows the energy magnitude of modes to be exchanged, enabling two characteristic mode curves to smoothly cross each other, thereby maintaining the smoothness and differentiability of eigenvalues ​​and characteristic functions throughout the entire effective domain of parameters.

[0102] This disclosure, through dynamic sorting, allows physical modes to flow freely between different output channels of the network. When shape encoding... When a change causes two eigenvalue curves to intersect, the indices are allowed to be swapped, thus ensuring that each sorted eigenvalue curve... and the corresponding characteristic function It is smooth and continuous within shape space. In contrast, existing techniques employ a fixed sorting strategy, forcing the first channel to always output the minimum eigenvalue, which leads to non-physical repulsion and bounce phenomena near the intersection, forming non-differentiable cusps that severely hinder subsequent gradient optimization.

[0103] This invention proposes a joint variational optimization framework comprising: an implicit neural network for directly mapping feature functions from shape encoding and spatial coordinates; a dynamic sequential sorting mechanism for real-time evaluation of modal energy and dynamic adjustment of the orthogonalization order; and a unidirectional gradient filtering projection to ensure orthogonality while blocking non-physical gradient backpropagation. During training, the implicit neural network receives randomly sampled shape encodings, and the dynamic sequential sorting mechanism rearranges the modal indices in real-time based on Rayleigh quotient energy values, ensuring that feature curves smoothly cross intersections rather than repel each other. Gradient filtering establishes a unidirectional energy dependency chain, ensuring the optimization stability of the dominant mode. Through this meshless continuous spectrum analysis, the model trained by this invention can uniformly represent the physical modes of the entire shape space. Compared with existing technologies, this invention achieves complete differentiability of feature bases with respect to shape encoding, eliminates gradient singularities, and improves inference speed by several orders of magnitude, providing an efficient and robust solution for real-time dimensionality reduction physical simulation and differentiable inverse shape design.

[0104] In summary, the embodiments of this disclosure, by constructing an implicit neural field and combining dynamic sequential sorting and gradient filtering techniques, achieve unified spectral analysis of continuous shape spaces without the need for mesh discretization, and successfully solve the theoretical and computational problems caused by eigenvalue crossover, providing an efficient and reliable solution for real-time physical simulation and intelligent design.

[0105] Embodiment 2 of the present invention provides a three-dimensional geometric spectrum analysis system based on model feature space, which runs the three-dimensional geometric spectrum analysis method based on model feature space as described in Embodiment 1. The system includes:

[0106] The spectrum analysis module is used to input the coordinates within the effective domain of the 3D geometry and the shape code corresponding to the 3D geometry into the trained neural network, and output a set of first-order gradient vectors that can characterize the feature function values ​​and coordinates of the entire shape space; one shape code corresponds to one effective domain.

[0107] The training module is used to train the neural network. The training module includes:

[0108] The energy value calculation module is used to calculate the energy value of each feature function output by the neural network based on the first-order gradient vector of the coordinates for randomly sampled shape codes, and sort them in ascending order according to the energy values ​​to obtain the feature function sequence of the current shape code; wherein, the energy value is used to characterize the frequency or variable energy of the feature function;

[0109] The orthogonal projection module is used to orthogonally project the feature function sequence to obtain the target feature function.

[0110] The update module is used to construct the corresponding loss function based on the preset partial differential equation type, substitute the target feature function into the corresponding loss function, update the parameters of the neural network by minimizing the loss function, and obtain the trained neural network.

[0111] Regarding the system in the above embodiments, the specific manner in which each unit performs operations has been described in detail in the embodiments related to the method, and will not be elaborated here.

[0112] Embodiment 3 of the present invention provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the computer program is loaded onto the processor, it implements the three-dimensional geometric spectrum analysis method based on model feature space described in Embodiment 1.

[0113] Embodiment 4 of the present invention provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the three-dimensional geometric spectrum analysis method based on model feature space as described in Embodiment 1.

[0114] It should be understood that the sequence number of each step in the above embodiments does not imply the order of execution. The execution order of each process should be determined by its function and internal logic, and should not constitute any limitation on the implementation process of the embodiments of the present invention.

[0115] This disclosure can be a system, method, and / or computer program product. A computer program product may include a computer-readable storage medium having computer-readable program instructions loaded thereon for causing a processor to implement various aspects of this disclosure.

[0116] 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 it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the specific implementation of the present invention. Any modifications or equivalent substitutions that do not depart from the spirit and scope of the present invention should be covered within the protection scope of the claims of the present invention.

Claims

1. A three-dimensional geometric spectrum analysis method based on model feature space, characterized in that, include: The coordinates within the effective domain of the 3D geometry and the shape encoding corresponding to the 3D geometry are input into the trained neural network, which outputs a set of first-order gradient vectors that can represent the feature function values ​​and coordinates of the entire shape space. One shape code corresponds to one valid domain; Training a neural network includes: for a randomly sampled shape code, calculating the energy value of each feature function output by the neural network based on the first-order gradient vector of the coordinates, sorting the features in ascending order according to the energy values ​​to obtain the feature function sequence of the current shape code; wherein, the energy value is used to characterize the frequency or variational energy of the feature function; The target feature function is obtained by orthogonally projecting the feature function sequence. Based on the preset partial differential equation type, a corresponding loss function is constructed. The target feature function is substituted into the corresponding loss function. The parameters of the neural network are updated by minimizing the loss function, and the trained neural network is obtained.

2. The three-dimensional geometric spectrum analysis method based on model feature space according to claim 1, characterized in that: Orthogonally projecting the sequence of feature functions yields the target feature functions, including: Take the k-th eigenfunction of the eigenfunction sequence, determine the (k-1)-th eigenfunction as the orthogonal projection basis, project the k-th eigenfunction onto the orthogonal complement space of the subspace spanned by the (k-1)-th eigenfunction, calculate the projection components of the eigenfunction on the projection basis, and obtain the orthogonalized target eigenfunction.

3. The three-dimensional geometric spectrum analysis method based on model feature space according to claim 2, characterized in that: Orthogonally projecting the sequence of feature functions yields the target feature function, which also includes: The target feature function of the first k-1 feature functions after projection is taken as a constant without gradient; Take the inner product of each of the k-1 constants with the corresponding target feature function, multiply the product by the corresponding constant, and sum the k-1 products. Subtract the sum from the k-th feature function to obtain the target feature function of the k-th feature function.

4. The three-dimensional geometric spectrum analysis method based on model feature space according to claim 1, characterized in that: The preset partial differential equation types include the Laplace operator and the linear elasticity operator. The energy of each feature function output by the neural network is calculated based on the first-order gradient vector of the coordinates, including: The Rayleigh quotient of the feature function value output by the neural network is used as the energy value; When the partial differential equation type is preset to Laplace operator, the sum of squares of the Euclidean norm of the vector is obtained by multiplying the characteristic function value with the first-order gradient vector of the corresponding coordinate, and then the ratio is calculated with the sum of squares of the characteristic function value. The ratio is used as the Rayleigh quotient of the characteristic function value. When the partial differential equation type is set to linear elastic operator, the elastic potential energy density is calculated based on the linear strain tensor of the characteristic function and the Lamé constant of the material. The ratio of the elastic potential energy density to the sum of squares of the characteristic function values ​​is used as the Rayleigh quotient of the characteristic function values.

5. The three-dimensional geometric spectrum analysis method based on model feature space according to claim 4, characterized in that: When the partial differential equation type is preset to Laplace operator, the loss function of the Laplace operator is constructed as follows: in, This represents the loss function in the Laplace operator spectral analysis. The parameters representing the neural network; Represents the mathematical expectation operator, for shape space Shape encoding that follows a specific distribution sampling Calculate the expected value. This represents the total number of target characteristic functions to be solved. Indicates the shape encoding of the current sample. The determined valid domain, It is a differential volume element. This represents the output of the neural network after orthogonalization. One target feature function, Represents relative to spatial coordinates The first-order gradient operator, Let represent the square of the Euclidean norm of a vector.

6. The three-dimensional geometric spectrum analysis method based on model feature space according to claim 4, characterized in that: When the partial differential equation type is preset to linear elastic operator, the loss function of the linear elastic operator is constructed according to the following formula: in, This represents the loss function in spectral analysis of linear elastic operators. Indicates the first The linear strain tensor corresponding to each characteristic function is defined as follows: , Represents the displacement gradient tensor transpose; This represents the sum of the squares of all elements in the matrix. Represents the square of the trace of a matrix. Shear modulus These are material property constants. Represents the mathematical expectation operator, for shape space Shape encoding that follows a specific distribution sampling Calculate the expected value. It is a differential volume element. Indicates the shape encoding of the current sample. The determined valid domain.

7. The three-dimensional geometric spectrum analysis method based on model feature space according to claim 1, characterized in that: The method further includes: A multilayer perceptron mechanism is configured in the neural network. A position encoding layer is set at the input of the neural network to map spatial coordinates into feature vectors representing continuous space. These feature vectors are then input into the multilayer perceptron mechanism, and a configuration is set at the output. There are 10 channels. The first channel is initialized to a constant value whose output is independent of the input. The rest... Each channel is configured to output the feature function values ​​to be learned.

8. The three-dimensional geometric spectrum analysis method based on model feature space according to claim 1, characterized in that: The method further includes: The boundary of a three-dimensional geometry is defined as the zero isosurface of a scalar function. The sign of the scalar function value is used to distinguish between the interior and exterior of the geometry, thereby determining the effective domain of the three-dimensional geometry.

9. The three-dimensional geometric spectrum analysis method based on model feature space according to claim 1, characterized in that: Updating the parameters of a neural network by minimizing the loss function includes: Use gradient descent or the Adam optimizer to update the parameters of the neural network through backpropagation until the loss function converges to the minimum or reaches the preset number of iterations.

10. A three-dimensional geometric spectrum analysis system based on model feature space, implementing the three-dimensional geometric spectrum analysis method based on model feature space as described in any one of claims 1 to 9, characterized in that, The system includes: The spectrum analysis module is used to input the coordinates within the effective domain of the 3D geometry and the shape code corresponding to the 3D geometry into the trained neural network, and output a set of first-order gradient vectors that can characterize the feature function values ​​and coordinates of the entire shape space; one shape code corresponds to one effective domain. The training module is used to train the neural network. The training module includes: The energy value calculation module is used to calculate the energy value of each feature function output by the neural network based on the first-order gradient vector of the coordinates for randomly sampled shape codes, and sort them in ascending order according to the energy values ​​to obtain the feature function sequence of the current shape code; wherein, the energy value is used to characterize the frequency or variable energy of the feature function; The orthogonal projection module is used to orthogonally project the feature function sequence to obtain the target feature function. The update module is used to construct the corresponding loss function based on the preset partial differential equation type, substitute the target feature function into the corresponding loss function, update the parameters of the neural network by minimizing the loss function, and obtain the trained neural network.

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