Unstructured mesh optimization method based on radial basis function neural network

Abstract

The application discloses a non-structure grid optimization method based on a radial basis function neural network. The method comprises the following steps: generating an initial non-structure grid, iteratively processing by using an optimization type smoothing method, recording a free node, ring neighbor node coordinates and an optimal position to form a training sample set; performing data enhancement based on ring neighbor node cyclic symmetry; normalizing the coordinate data to a predetermined range; training a radial basis function neural network (RBFNN) model, taking the ring neighbor node coordinates as input and the optimal position coordinates as output; inputting the target free node and ring neighbor coordinates in a to-be-smoothed grid into the trained RBFNN model, predicting the optimal position and updating the grid. The application learns and optimizes the smoothing mapping relationship through data driving, ensures the accuracy close to the optimization method, greatly reduces the calculation cost, effectively avoids the generation of abnormal cells, and enhances the numerical calculation stability and the reliability of the inversion result.

Description

Technical Field

[0001] This invention relates to the field of computer-aided engineering, and more specifically to an unstructured mesh optimization method based on radial basis function neural networks (RBFNN). Background Technology

[0002] Unstructured mesh inversion technology is a method for inverting geophysical parameters based on irregular meshes such as tetrahedrons and triangles. Compared with traditional regular meshes, it can flexibly adapt to complex terrain, irregular observation layouts, and local densification requirements, significantly improving computational efficiency and imaging accuracy. In recent years, with the maturity of mesh generation libraries and efficient sparse matrix solvers, unstructured meshes have been widely used in three-dimensional gravity, magnetic, seismic, and electromagnetic inversions, becoming an important tool for high-resolution imaging under complex conditions.

[0003] However, unstructured meshes are prone to producing poor-quality elements in complex regions. These deformed elements can lead to ill-conditioned coefficient matrices and numerical instability in finite element calculations. To address mesh quality issues, traditional mesh smoothing techniques are widely used in the industry. Existing mesh smoothing methods are mainly divided into two categories: one is heuristic smoothing methods (such as Laplace smoothing), which are computationally efficient but have limited optimization capabilities and are prone to generating new low-quality elements, leading to numerical instability and affecting the reliability of inversion results; the other is optimization-based methods, which can obtain higher-quality meshes than heuristic methods, but are computationally expensive and difficult to meet the real-time efficiency requirements of large-scale inversions. Summary of the Invention

[0004] The technical problem to be solved by this invention is: how to provide an unstructured mesh optimization method that combines high quality and high efficiency, while ensuring near-optimization accuracy and significantly reducing computational overhead.

[0005] To address the aforementioned technical problems, this invention provides an unstructured mesh optimization method based on radial basis function neural networks (RBFNN), comprising the following steps:

[0006] Step 1, Data Preparation: Generate an initial unstructured mesh and iteratively process the initial unstructured mesh using an optimized smoothing method. Record the coordinate set of free nodes and their cyclic neighbor nodes in each iteration, as well as the optimal position coordinates of the free nodes after optimization, to form a training sample set.

[0007] Step 2, Data Augmentation: Based on the cyclic symmetry of the ring neighbor nodes, perform data augmentation on the training sample set to expand the number of samples;

[0008] Step 3: Normalization: Normalize the coordinate data in the enhanced training sample set to a predetermined range;

[0009] Step 4: Model Training: Train an RBFNN model using the normalized training sample set; the RBFNN model takes the coordinates of the ring neighbor nodes of the free node as input and the optimal position coordinates of the free node as output.

[0010] Step 5: Mesh smoothing prediction: Input the coordinates of the target free node and its ring neighbor nodes in the unstructured mesh to be smoothed into the trained RBFNN model to obtain the predicted optimal position of the target free node, and update the mesh node positions accordingly to complete the mesh smoothing.

[0011] Preferably, in the model training step, training the RBFNN model includes: determining the center points of the hidden layer radial basis functions of the RBFNN model using a clustering algorithm; determining the width parameters of each radial basis function based on the distribution of the center points; and after fixing the center points and width parameters, solving the output layer weights of the RBFNN model using the linear least squares method.

[0012] Preferably, the clustering algorithm is the K-means algorithm.

[0013] Preferably, the radial basis function is a Gaussian function, expressed as follows:

[0014]

[0015] Where c j With σ as the center point j The width parameter is determined by the following formula:

[0016]

[0017] Where c j and c l Let k be the center point, k be the number of center points, and α be the adjustment coefficient.

[0018] Preferably, the normalization process step employs a local normalization method, independently calculating the scaling factor for the coordinate set of the ring neighbors of each free node and performing normalization mapping; and after the grid smoothing prediction step, inverse normalization processing is performed to map the predicted coordinates back to the original coordinate system.

[0019] Preferably, the grid smoothing prediction step further includes a degradation mechanism: when the number of ring neighbor nodes of the target free node exceeds the range covered by the RBFNN model during training, the optimal position of the node is calculated using the Laplace smoothing method.

[0020] Preferably, the method is applied to the optimization of unstructured grids in geophysical inversion, wherein the geophysical inversion includes one of gravity inversion, magnetic inversion, seismic inversion, or electromagnetic inversion.

[0021] Compared with the prior art, the beneficial effects of the present invention are:

[0022] This invention proposes an unstructured mesh inversion smoothing method based on RBFNN, which significantly improves overall performance from multiple dimensions by addressing the key constraint of mesh quality on computational efficiency and accuracy in geophysical inversions such as gravity and magnetic inversions. Compared with traditional heuristic smoothing methods (such as Laplace smoothing), RBFNN optimizes the mesh through an analytical one-time solution, significantly reducing the number of iterations, avoiding the generation of new low-quality cells, and maintaining high mesh quality, thus solving the problem of limited optimization capabilities of traditional smoothing methods. RBFNN's lightweight architecture reduces construction and training costs, enabling real-time mesh optimization and meeting the real-time efficiency requirements of large-scale unstructured mesh inversions. Through high-quality mesh optimization, it effectively prevents the generation of malformed cells, avoiding ill-conditioned coefficient matrices and numerical instability problems in finite element calculations, improving the reliability of inversion results, and making the inversion algorithm more stable under complex terrain and observation deployment conditions. RBFNN's local response characteristics ensure strong adaptability in complex geological models, and the method is robust under extreme conditions, capable of handling irregular observation deployments and local densification requirements, improving the applicability of high-resolution imaging. The smoothed mesh reduces the numerical instability of finite element calculations, thereby accelerating the overall convergence of inversion iterations, improving the overall computing speed, and achieving higher resolution imaging. It is applicable to various inversion scenarios such as seismic and electromagnetic inversions. Attached Figure Description

[0023] Figure 1 This is a schematic diagram of the grid smoothing principle provided in an embodiment of the present invention;

[0024] Figure 2 This is a diagram of the neural network training structure provided in an embodiment of the present invention. Detailed Implementation

[0025] The present invention will be further described in detail below with reference to specific embodiments and accompanying drawings.

[0026] This invention employs a neural network-based mesh smoothing technique (NN-Smoothing) to optimize the quality of the computational mesh in real time during the inversion iteration process, thereby improving the efficiency, stability, and accuracy of the final result of the inversion calculation. Furthermore, it uses a radial basis function neural network (RBFNN) to improve computational efficiency and reduce the cost of constructing the neural network.

[0027] like Figure 1As shown, this invention first processes a large number of Delaunay training grids using the Laplace smoothing method, and collects the neighborhood coordinates of free nodes and their optimized positions as training data.

[0028] Subsequently, a radial basis function (RBF) neural network is constructed, using Gaussian functions as radial basis functions. A clustering algorithm is used to determine the network center points, and the output weights are solved using linear least squares, learning the mapping relationship from local geometric features to the optimal position. In the application phase, for each free node in the grid to be smoothed, the coordinates of its neighboring nodes are input into the trained RBF network. The network directly outputs the optimized position of that node, replacing the traditional iterative optimization process.

[0029] The lightweight grid smoothing method based on RBFNN adopted in this invention mainly includes four core steps: data preparation and augmentation, normalization processing, RBFNN model training, and grid smoothing prediction. Figure 2 As shown.

[0030] The first step is data preparation. A point set is randomly generated within the circular domain, and a Delaunay triangulation is constructed to obtain the initial mesh.

[0031] An optimized smoothing method based on the conjugate gradient method is used to iterate the mesh T times. In each iteration, the v of each free node is extracted. i The coordinate information of the node and its ring neighbor set is used to determine the optimal position of the free node obtained by iteratively solving the problem using the conjugate gradient method with the goal of minimizing the unit. i *. Each training sample pair can be represented as: ({v i1 ,v i2 ,...,v in}, v i *), where n is the number of ring neighbor nodes, the left side of the mapping is the feature vector composed of the ring neighbor coordinates, and the right side is the corresponding unique optimal position coordinates.

[0032] The second step is data augmentation. Since the ring is a cyclic structure, changing the starting point will not affect the output. For a ring with n neighbors, n equivalent inputs can be generated. Therefore, the number of augmented samples is m×n, where m is the initial number of training samples.

[0033] The third step is data normalization and denormalization. To eliminate the scale effect, the coordinates of the ring neighbors and free nodes are mapped to the [0,1]×[0,1] region. Specifically, the method is to calculate the coordinate boundary values ​​x of all ring neighbors of the current free node. min x max y min y max Take the scaling factor s for the coordinates (x, y) of the j-th ring neighbor node. j, y j Normalize;

[0034] The scaling factor s is calculated using the following formula:

[0035]

[0036] The formula for normalization is:

[0037]

[0038] and Let be the normalized coordinates of the j-th ring neighbor node; after prediction, the output will be mapped back to the original coordinate system through inverse normalization.

[0039] .

[0040] The fourth step is model training, which uses a radial basis function (RBF) network to approximate the optimal position function of free nodes. The input layer of this network is a vector x (with dimensions 2n, where n is the number of neighboring nodes) formed by concatenating the normalized coordinates of the ring neighbor nodes, represented as (x1, y1, x2, y2, ..., x n ,y n ).

[0041] The RBF network contains one hidden layer, the number of neurons in which is equal to the number of centroids k determined by the clustering algorithm, and the output of each neuron in the hidden layer is...

[0042]

[0043] Where x is the input vector, and c j The coordinate vector of the j-th cluster center is obtained after clustering the input vector of the training samples using the K-means clustering algorithm before training. The number of neurons in the hidden layer is equal to the number of center points k determined by the clustering algorithm.

[0044] σ j Let be the width parameter of the j-th neuron. Its set value is determined based on the center distribution, by multiplying the average distance from each center point to other center points by the adjustment coefficient α (α is a hyperparameter).

[0045]

[0046] The output layer is a linear combination, which is a weighted sum of the output values ​​of each neuron.

[0047]

[0048] Where w jLet be the weight coefficient of the output value of the j-th neuron. This is solved directly using the least squares method. The objective function is to minimize the mean squared error, and the output layer weights are solved directly using linear least squares. The objective function expression is as follows:

[0049]

[0050] in, The total number of training samples, For sample index, For the first The true optimal location of each sample For the neural network to the first The predicted output for each sample.

[0051] The fifth step is mesh smoothing prediction, which iterates through all free nodes sequentially. For each free node, its cyclic neighbors are extracted and normalized, then input into the corresponding RBF model to obtain the predicted optimal coordinates. Inverse normalization is then used to restore the original coordinate system, and the free node positions are updated. After one iteration, this process is repeated for the entire mesh until the maximum number of iterations is reached. The mesh quality metrics converge, or the mesh quality indicators converge. These metrics include minimum cell angle, average minimum cell angle, or aspect ratio quality measures. When the absolute value of the index change in two consecutive iterations is less than the preset threshold, it is determined to be converged.

[0052] In real-world meshes, the number of cyclic neighbors for a free node is typically 3 to 9. When the number of cyclic neighbors for a free node exceeds the range covered by the trained model (≥10), the algorithm degenerates into a Laplace smoothing method, which takes the arithmetic mean of the coordinates of all cyclic neighbors as the new position. The calculation formula is as follows:

[0053]

[0054] This invention leverages the analytical one-time solution capability of RBFNN to significantly improve mesh smoothing quality and reduce computational costs. It effectively avoids the generation of distorted cells, greatly enhancing the numerical stability of gravity and magnetic inversion. Its local response characteristics ensure the adaptability and robustness of mesh optimization in inversions of complex geological models, thereby accelerating the inversion convergence process with limited computational resources and comprehensively improving the accuracy and efficiency of gravity and magnetic exploration data interpretation.

[0055] Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make changes, modifications, substitutions and variations to the above embodiments within the scope of the present invention.

Claims

1. An unstructured mesh optimization method based on radial basis function neural networks, characterized in that, The method includes the following steps: Step 1, Data Preparation: Generate an initial unstructured mesh and iteratively process the initial unstructured mesh using an optimized smoothing method. Record the coordinate set of free nodes and their cyclic neighbor nodes in each iteration, as well as the optimal position coordinates of the free nodes after optimization, to form a training sample set. Step 2, Data Augmentation: Based on the cyclic symmetry of the ring neighbor nodes, perform data augmentation on the training sample set to expand the number of samples; Step 3: Normalization: Normalize the coordinate data in the enhanced training sample set to a predetermined range; Step 4: Model Training: Using the normalized training sample set, train the radial basis function neural network model; the radial basis function neural network model takes the coordinates of the ring neighbor nodes of the free node as input and the optimal position coordinates of the free node as output; Step 5: Mesh smoothing prediction: Input the coordinates of the target free node and its ring neighbor nodes in the unstructured mesh to be smoothed into the trained radial basis function neural network model to obtain the predicted optimal position of the target free node, and update the mesh node positions accordingly to complete the mesh smoothing.

2. The unstructured mesh optimization method based on radial basis function neural network according to claim 1, characterized in that, The model training step includes training the radial basis function neural network model by: determining the center points of the hidden layer radial basis functions of the radial basis function neural network model through a clustering algorithm; determining the width parameters of each radial basis function according to the distribution of the center points; and solving the output layer weights of the radial basis function neural network model by fixing the center points and the width parameters through linear least squares.

3. The unstructured mesh optimization method based on radial basis function neural network according to claim 2, characterized in that, The clustering algorithm is the K-means algorithm.

4. The unstructured mesh optimization method based on radial basis function neural network according to claim 2, characterized in that, The radial basis function is a Gaussian function, expressed as follows: Where c j With the center point, σ j The width parameter is determined by the following formula: Where c j and c l Let k be the center point, k be the number of center points, and α be the adjustment coefficient.

5. The unstructured mesh optimization method based on radial basis function neural network according to claim 1, characterized in that, The normalization process in step three employs a local normalization method, independently calculating the scaling factor for the coordinate set of the ring neighbors of each free node and performing normalization mapping; and after the grid smoothing prediction step, performing inverse normalization processing to map the predicted coordinates back to the original coordinate system.

6. The unstructured mesh optimization method based on radial basis function neural network according to claim 1, characterized in that, The grid smoothing prediction step also includes a degradation mechanism: when the number of ring neighbor nodes of the target free node exceeds the range covered by the radial basis function neural network model during training, the optimal position of the node is calculated using the Laplace smoothing method.

7. The unstructured mesh optimization method based on radial basis function neural network according to claim 1, characterized in that, The method is applied to the optimization of unstructured grids in geophysical inversion, which includes one of gravity inversion, magnetic inversion, seismic inversion, or electromagnetic inversion.

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