High-precision fusion interpolation method for discontinuous physical fields in multi-physics coupling

Abstract

This application discloses a high-precision fusion interpolation method for discontinuous physical fields in multiphysics coupling, relating to the field of computer technology. The method includes: constructing a sample dataset based on the spatial coordinates of spatial sampling points on the wing and the pressure coefficient; constructing a nearest neighbor index based on the spatial coordinates of each training sampling point in the training set, and determining enhanced feature vectors by combining local standard deviation features and trigonometric function position encoding; constructing a fully connected feedforward neural network, training the neural network using the enhanced feature vectors, and determining the predicted pressure coefficient based on the target neural network; constructing an RBF kernel matrix based on the target wing grid points to be interpolated to determine the residual correction amount; determining the predicted target pressure coefficient based on the target enhanced feature vector of the target wing grid points and using the target neural network, and determining the final predicted pressure coefficient based on the residual correction amount. In the pressure field coupling and transmission of shock waves on the wing, this method balances interpolation efficiency in smooth regions with feature preservation accuracy in discontinuous regions.

Description

Technical Field

[0001] This invention relates to the field of computer technology, and in particular to a high-precision fusion interpolation method for discontinuous physical fields in multi-physics coupling. Background Technology

[0002] Coupled surface data interpolation is one of the core problems in multidisciplinary coupled numerical simulations, and it is widely used in engineering scenarios such as aerodynamic-structural coupled simulation of airfoils. Its purpose is to interpolate the nodal numerical characteristics of the source mesh to the target mesh nodes. However, for discontinuous functions, existing interpolation methods still have large interpolation errors at discontinuities. Classical global spectrum methods or high-order polynomial approximations have extremely high accuracy when dealing with smooth fields, but when the target physical field (such as the pressure field on the airfoil surface) contains discontinuities such as shock waves or material interfaces, the nonlocality of global basis functions will cause serious problems: the jump information at the discontinuity will pollute the entire computational domain through the basis function, and high-frequency numerical oscillations will be generated on both sides of the discontinuity, which cannot be eliminated simply by increasing the order.

[0003] In recent years, data-driven methods based on multilayer nonlinear mappings have been widely used in physical field reconstruction. Theoretically, given sufficient parameters, they can approximate any field. However, the actual optimization process is significantly affected by the convergence differences between high and low frequencies: in the early stages of gradient descent, the network prioritizes fitting the low-frequency Fourier components of the objective function, while the convergence speed for high-frequency components, such as sharp pressure jumps at wing shock waves, is exponentially slower than that for low-frequency components. For discontinuous functions with rich high-frequency features, this means that standard neural networks alone cannot accurately approximate the discontinuity interface within a finite number of training steps, wasting computational resources and failing to achieve the required engineering accuracy.

[0004] Another common approach to handling data transfer problems is the Radial Basis Function (RBF) method based on spatial distance. RBF methods can be divided into two categories: globally supported and tightly supported. When using globally supported basis functions such as Gaussian kernels and Multiquadric, each basis function has a non-zero influence on the entire approximation domain. For smooth regions, this globality helps improve accuracy by utilizing information from distant regions; however, near discontinuities, because the basis functions cross both sides of the discontinuity, they force the smoothing of discontinuous information, thus causing oscillations similar to the Gibbs phenomenon and contaminating the surrounding smooth regions. When using tightly supported RBFs, such as the Wendland function, the choice of the support radius faces a dilemma: a radius that is too large will cross discontinuities and introduce contamination; a radius that is too small requires a large number of nodes in the smooth region to maintain accuracy and is prone to numerical instability. More seriously, when sampling is very dense near discontinuities, the abrupt transitions of the basis functions can lead to a highly ill-conditioned linear system, resulting in highly distorted solutions or even unsolvable solutions.

[0005] As can be seen from the above, how to balance the interpolation efficiency in smooth regions and the preservation of feature accuracy in discontinuous regions during the pressure field coupling and transmission of shock waves in an airfoil is a problem that urgently needs to be solved. Summary of the Invention

[0006] In view of this, the purpose of this invention is to provide a high-precision fusion interpolation method for discontinuous physical fields in multi-physics coupling, which can balance the interpolation efficiency in smooth regions and the preservation of characteristic accuracy in discontinuous regions during the pressure field coupling and transmission of shock waves on an airfoil. The specific solution is as follows: Firstly, this application provides a high-precision fusion interpolation method for discontinuous physical fields in multi-physics coupling, including: A sample dataset is constructed based on the spatial coordinates of each spatial sampling point on the wing and the corresponding pressure coefficient, and the sample dataset is divided to obtain the corresponding training set and test set. A nearest neighbor index is constructed based on the spatial coordinates of each training sampling point in the training set. The local standard deviation feature of the training sampling point is determined based on the nearest neighbor index. A trigonometric function position code is constructed based on the spatial coordinates of the training sampling point. The spatial coordinates, the local standard deviation feature, and the trigonometric function position code are concatenated to obtain an enhanced feature vector. A fully connected feedforward neural network for the target structure is constructed. The enhanced feature vector and the corresponding pressure coefficient are input into the fully connected feedforward neural network for training based on the Huber loss function. The trained neural network is then tested using the test set to obtain the target neural network. Based on the target neural network, the predicted pressure coefficient values ​​corresponding to all spatial sampling points are determined. The residual vector is determined using the predicted pressure coefficient values ​​and the actual pressure coefficient. Based on the target wing mesh points to be interpolated, target interpolation points that meet the target distance conditions are determined from the training sampling points. An RBF kernel matrix is ​​constructed using the Euclidean distance between the target interpolation points. A weight vector is determined based on the RBF kernel matrix and the residual vector. An RBF interpolation function is constructed based on the weight vector. The residual correction amount of the target wing mesh points is determined using the RBF interpolation function. Determine the target enhanced feature vector corresponding to the target wing grid point, determine the target pressure coefficient prediction value based on the target enhanced feature vector and the target neural network, and determine the final pressure coefficient prediction value of the target wing grid point based on the residual correction amount and the target pressure coefficient prediction value.

[0007] Optionally, the step of constructing a nearest neighbor index based on the spatial coordinates of each training sampling point in the training set, and determining the local standard deviation features of the training sampling points based on the nearest neighbor index, includes: Based on the training set, determine the number of neighboring points of the first target closest to each training sampling point in Euclidean distance, and record the index corresponding to each neighboring point. Construct a nearest neighbor index based on each training sampling point and its corresponding index.

[0008] Optionally, determining the local standard deviation features of the training sampling points based on the nearest neighbor index includes: The mean pressure coefficient of each training sampling point is determined based on the nearest neighbor index, and the local standard deviation feature of the training sampling point is determined using the mean pressure coefficient.

[0009] Optionally, constructing the trigonometric function position code based on the spatial coordinates of the training sampling points includes: Based on the spatial coordinates of the training sampling points, two sets of codes with different frequencies are constructed to obtain the trigonometric function position codes; the trigonometric function position codes include a target low-frequency code for capturing global gradual components and a target high-frequency code for characterizing local sharp transitions.

[0010] Optionally, the concatenation of the spatial coordinates, the local standard deviation features, and the trigonometric function position encoding to obtain the enhanced feature vector includes: The spatial coordinates of each training sampling point, the local standard deviation feature, and the trigonometric function position encoding are concatenated to obtain an enhanced feature vector; The enhanced feature vectors of all training sampling points in the training set are stacked to obtain the corresponding feature matrix.

[0011] Optionally, the fully connected feedforward neural network for constructing the target structure is trained by inputting the enhanced feature vector and the corresponding stress coefficient into the fully connected feedforward neural network based on the Huber loss function, and then testing the trained neural network using the test set to obtain the target neural network, including: A fully connected feedforward neural network with hidden layers is constructed based on a multilayer perceptron; the number of neurons in the hidden layers of the fully connected feedforward neural network decreases layer by layer; the activation function of the hidden layers is the Swish activation function. The feature matrix and the corresponding pressure coefficients in the training set are input into the fully connected feedforward neural network and trained using the Huber loss function to obtain the trained neural network. The trained neural network is tested using the test set to obtain the target neural network.

[0012] Optionally, the step of determining the predicted pressure coefficient values ​​corresponding to all spatial sampling points based on the target neural network, and determining the residual vector using the predicted pressure coefficient values ​​and the actual pressure coefficient, includes: Pressure prediction is performed on all spatial sampling points based on the target neural network to obtain the corresponding pressure coefficient prediction value; The residual vectors are obtained by subtracting the actual pressure coefficient from the predicted pressure coefficient element by element.

[0013] Optionally, the step of determining target interpolation points that satisfy the target distance condition from the training sampling points based on the target wing mesh points to be interpolated, and constructing an RBF kernel matrix using the Euclidean distance between the target interpolation points, includes: Based on the training sampling points, determine the number of target interpolation points that are closest to the second target in terms of the Euclidean distance of the target wing grid points to be interpolated; Construct the RBF kernel matrix based on the Euclidean distance between any two target interpolation points.

[0014] Optionally, the step of determining the weight vector based on the RBF kernel matrix and the residual vector, constructing an RBF interpolation function based on the weight vector, and using the RBF interpolation function to determine the residual correction amount of the target wing grid points includes: A system of linear interpolation equations is constructed using the RBF kernel matrix and the residual vector corresponding to the target interpolation point, and the system of linear interpolation equations is solved to obtain the weight vector. Based on the weight vector, an RBF interpolation function is constructed for the target wing grid points. The spatial coordinates of the target wing grid points are substituted into the RBF interpolation function to obtain the residual correction amount of the target wing grid points.

[0015] Optionally, determining the target enhanced feature vector corresponding to the target wing grid point, determining the target pressure coefficient prediction value based on the target enhanced feature vector and using the target neural network, and determining the final pressure coefficient prediction value of the target wing grid point based on the residual correction and the target pressure coefficient prediction value, includes: Based on the training set, the target enhancement feature vector corresponding to the target wing grid point is determined, and the target enhancement feature vector is input into the target neural network to obtain the corresponding target pressure coefficient prediction value; The residual correction and the predicted target pressure coefficient are linearly superimposed to obtain the final predicted pressure coefficient of the target wing grid points.

[0016] This application constructs a sample dataset based on the spatial coordinates of each spatial sampling point on the wing and the corresponding pressure coefficient, and divides the sample dataset to obtain corresponding training and test sets; constructs a nearest neighbor index based on the spatial coordinates of each training sampling point in the training set, determines the local standard deviation feature of the training sampling point based on the nearest neighbor index, constructs a trigonometric function position code based on the spatial coordinates of the training sampling point, and concatenates the spatial coordinates, the local standard deviation feature, and the trigonometric function position code to obtain an enhanced feature vector; constructs a fully connected feedforward neural network for the target structure, inputs the enhanced feature vector and the corresponding pressure coefficient into the fully connected feedforward neural network based on the Huber loss function for training, and uses the test set to test the trained neural network to obtain the target neural network, based on the target The neural network determines the predicted pressure coefficient values ​​corresponding to all spatial sampling points, and uses the predicted pressure coefficient values ​​and the actual pressure coefficients to determine the residual vector. Based on the target wing grid points to be interpolated, target interpolation points that meet the target distance conditions are determined from the training sampling points, and an RBF kernel matrix is ​​constructed using the Euclidean distance between the target interpolation points. A weight vector is determined based on the RBF kernel matrix and the residual vector, an RBF interpolation function is constructed based on the weight vector, and the residual correction amount of the target wing grid points is determined using the RBF interpolation function. The target enhancement feature vector corresponding to the target wing grid points is determined, and the target pressure coefficient predicted value is determined based on the target enhancement feature vector and the target neural network. The final predicted pressure coefficient value of the target wing grid points is determined based on the residual correction amount and the target pressure coefficient predicted value.

[0017] As can be seen from the above, this application constructs a unified nearest neighbor index based on the training set, and uses the nearest neighbor index to determine the local standard deviation features and trigonometric function position codes of each training sampling point. The local standard deviation can automatically locate the shock wave region on the wing surface. The spatial coordinates, local standard deviation features, and trigonometric function position codes are concatenated to obtain an enhanced feature vector. The feedforward neural network is trained based on the enhanced feature vector and the corresponding pressure coefficient, and the generalization ability is verified using the test set to ensure the reliability of the model's interpolation for unknown grid points. Training sampling points near the target wing grid point are selected to construct a local RBF interpolator, which avoids oscillations caused by the basis function crossing both sides of the shock wave from the root, and further determines the residual correction amount of the target grid point to accurately compensate for the sharp shock wave jumps that the neural network fails to capture. In this way, the neural network is responsible for capturing the global low-frequency smooth trend, and the local RBF is responsible for correcting the high-frequency discontinuous residuals. The pressure coefficient prediction values ​​output by the two are combined with the residual correction amount to obtain the final pressure coefficient prediction value. This method can maintain stable interpolation accuracy for target structure grids with different topologies. Attached Figure Description

[0018] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0019] Figure 1 This is a flowchart of a high-precision fusion interpolation method for discontinuous physical fields in multi-physics coupling disclosed in this application. Detailed Implementation

[0020] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0021] Currently, existing interpolation methods for discontinuous functions still suffer from significant interpolation errors at discontinuities. While classical global spectral methods or high-order polynomial approximations offer high accuracy in smooth fields, the nonlocality of global basis functions leads to serious problems when the target physical field contains discontinuities such as shock waves or material interfaces: jump information at discontinuities pollutes the entire computational domain through the basis functions, generating high-frequency numerical oscillations on both sides of the discontinuity that cannot be eliminated simply by increasing the order. To address this, this application provides a high-precision fusion interpolation method for discontinuous physical fields in multi-physics coupling. A neural network captures the global low-frequency smoothing trend, while a local RBF corrects high-frequency discontinuity residuals. The pressure coefficient predictions output by both methods are combined with the residual corrections to obtain the final pressure coefficient prediction. This method maintains stable interpolation accuracy for target structure meshes with different topologies.

[0022] See Figure 1 As shown, this embodiment of the invention discloses a high-precision fusion interpolation method for discontinuous physical fields in multi-physics coupling, comprising: Step S11: Construct a sample dataset based on the spatial coordinates of each spatial sampling point on the wing and the corresponding pressure coefficient, and divide the sample dataset to obtain the corresponding training set and test set.

[0023] In this embodiment, a sample dataset is constructed, which includes the three-dimensional spatial coordinates of each spatial sampling point and the corresponding pressure coefficient value; the pressure coefficient value is the three-dimensional wing surface pressure coefficient obtained by computational fluid dynamics simulation; the sample dataset is randomly divided according to a preset ratio to obtain a training set and a test set; the preset ratio can be set according to the actual situation.

[0024] Step S12: Construct a nearest neighbor index based on the spatial coordinates of each training sampling point in the training set; determine the local standard deviation feature of the training sampling point based on the nearest neighbor index; construct a trigonometric function position code based on the spatial coordinates of the training sampling point; and concatenate the spatial coordinates, the local standard deviation feature, and the trigonometric function position code to obtain an enhanced feature vector.

[0025] In this embodiment, after obtaining the training set, a spatial nearest neighbor index structure is constructed using the spatial coordinates of all training sampling points in the training set. For each training point, its K nearest neighbor points in the training set are retrieved, and the indices of these neighbor points are recorded. The corresponding formula is as follows: ; in, Training sampling points The corresponding set of K nearest neighbor indices; Let K be the index values ​​of the K nearest neighbor index set; Indicates to make smallest ; This indicates the number of elements in the candidate set S; The summation symbol; Let i be the coordinates of the i-th point in the training set; It is an L2 norm index. This index structure remains unchanged throughout the training and prediction phases to ensure strict consistency in feature computation.

[0026] Specifically, the step of constructing a nearest neighbor index based on the spatial coordinates of each training sampling point in the training set, and determining the local standard deviation features of the training sampling points based on the nearest neighbor index, includes: determining the first number of neighbor points with the closest Euclidean distance to each training sampling point based on the training set, recording the index corresponding to each neighbor point, and constructing a nearest neighbor index based on each training sampling point and its corresponding index.

[0027] Understandably, after obtaining the nearest neighbor index set, the formula based on the local standard deviation of each training sampling point on the nearest neighbor index set is as follows: ; in, Sampling points The corresponding local standard deviation; This is the function for calculating standard deviation; For the first The attribute values ​​corresponding to each sampling point; Sampling points The set of nearest neighbor indices; K is the number of nearest neighbor indices; The summation symbol; The mean value is the average of the attribute values ​​within the K nearest neighbors. Specifically, determining the local standard deviation feature of the training sampling points based on the nearest neighbor index includes: determining the mean pressure coefficient of the neighbor points corresponding to each training sampling point based on the nearest neighbor index, and using the mean pressure coefficient to determine the local standard deviation feature of the training sampling points.

[0028] In this embodiment, for each training sampling point A trigonometric function position code is constructed. The formula for the trigonometric function position code is as follows: ; in, For the first The trigonometric function position encoding corresponding to each sampling point; This is a fundamental frequency sine code, where T is the transpose; For fundamental frequency cosine coding; This is a frequency-doubled sine coding. In one specific embodiment, the trigonometric function position coding includes a target low-frequency coding for capturing the gradual change component. and And high-frequency coding for characterizing sharp transitions and Specifically, the construction of trigonometric function position codes based on the spatial coordinates of the training sampling points includes: constructing two sets of codes with different frequencies based on the spatial coordinates of the training sampling points to obtain trigonometric function position codes; the trigonometric function position codes include a target low-frequency code for capturing global gradual transition components and a target high-frequency code for characterizing local sharp transitions.

[0029] It is understandable that the spatial coordinates of each training sampling point, the local standard deviation feature, and the trigonometric function position encoding are concatenated to obtain the enhanced feature vector, and the corresponding formula is as follows: ; in, The enhanced feature vector; This is a fundamental frequency sine code, where T is the transpose; For fundamental frequency cosine coding; It is a frequency-doubled sine code; Sampling points The corresponding local standard deviation. Then, the enhanced feature vectors based on N training sampling points are used to form the feature matrix. Specifically, the step of concatenating the spatial coordinates, the local standard deviation features, and the trigonometric function position codes to obtain the enhanced feature vector includes: concatenating the spatial coordinates, the local standard deviation features, and the trigonometric function position codes of each training sampling point to obtain the enhanced feature vector; and stacking the enhanced feature vectors of all training sampling points in the training set to obtain the corresponding feature matrix.

[0030] Step S13: Construct a fully connected feedforward neural network for the target structure. Based on the Huber loss function, input the enhanced feature vector and the corresponding pressure coefficient into the fully connected feedforward neural network for training. Test the trained neural network using the test set to obtain the target neural network. Determine the predicted pressure coefficient values ​​corresponding to all spatial sampling points based on the target neural network. Determine the residual vector using the predicted pressure coefficient values ​​and the actual pressure coefficient.

[0031] In this embodiment, a fully connected feedforward neural network is constructed, including an input layer, hidden layers, an activation function, and an output layer. The input is a 4d+1 feature matrix. The hidden layers can be configured as a three-layer hidden network, such as a funnel-shaped structure with 128, 128, and 64 layers, to extract deep physical correlations through layer-by-layer dimensionality reduction. The activation function can be the Swish activation function, and the formula corresponding to the Swish activation function is as follows: ; in, This is the output value of the Swish activation function, i.e., the result of the neuron after nonlinear activation; This is the input to the Swish activation function, i.e., the linear output of the neurons in the neural network; The function is Sigmoid. The output layer uses a linear mapping with no activation.

[0032] Understandably, to address the systematically large residuals generated by samples near discontinuous interfaces, a function with gradient truncation properties is configured as the loss function of the neural network. The Huber loss function can be used as the training objective of the fully connected feedforward neural network, and the formula for the Huber loss function is as follows: ; in, The output value of the Huber loss function is the loss calculation result for a single training sample. The truncation threshold controls the segmentation shape of the labeled loss. The prediction residual is the difference between the actual value and the model's predicted value. For the true value, This is the predicted value. Its gradient is: ; in, This represents the gradient of the loss function with respect to the residuals. This is a truncation function; This is the truncation threshold; To predict residuals, the gradient contribution of large residuals near discontinuities is shifted from... Suppressed to , The magnitude of the function jump at the point of discontinuity. The Huber loss is equivalent to... Lost Moreau envelope: ; in, This is the output value of the Huber loss function; As an auxiliary variable; To predict residuals; This is the L1 loss term; This is the truncation threshold. Its optimal solution... For soft thresholding operators, The reasonable range is , This represents the amplitude of the function jump at the point of discontinuity.

[0033] In one specific implementation, first determine After the neural network initially captures the global trend, the upper quartile of the residuals is calculated as... In this embodiment, the Adam optimizer is selected for gradient optimization. The Adam optimizer maintains the first-moment estimate of the gradient. and second-order moment estimation The parameter update formula is: ; in, These are the trainable parameters of the fully connected feedforward neural network; The assignment update symbol; The learning rate; This is the first-order moment estimate after bias correction; This is the second-order moment estimate after bias correction; This is a smoothing term, typically used to prevent extremely small positive numbers with a denominator of 0. Adam can maintain an effective parameter update step size in the smoothing region of Huber loss (the region after large residual samples are truncated), thus accelerating convergence.

[0034] Understandably, with the feature matrix as input and the pressure coefficient label value as the objective, the configured Huber loss is used, and the first layer of the neural network is optimized by an optimizer. Medium-batch gradient descent training is performed. The neural network adopts an MLP (Multilayer Perceptron) structure, with the Swish function chosen as the activation function for the hidden layers and a linear activation function chosen for the output layer. The batch size can be set to 64 or 128 to ensure that information from both smooth and discontinuous regions can be captured in a single update.

[0035] Specifically, the construction of the fully connected feedforward neural network for the target structure involves training the enhanced feature vector and the corresponding stress coefficient using the Huber loss function, and then testing the trained neural network using the test set to obtain the target neural network. This process includes: constructing a fully connected feedforward neural network with hidden layers based on a multilayer perceptron; decreasing the number of neurons in the hidden layers of the fully connected feedforward neural network layer by layer; using the Swish activation function for the hidden layers; inputting the feature matrix and the corresponding stress coefficient from the training set into the fully connected feedforward neural network and training it using the Huber loss function to obtain the trained neural network; and then testing the trained neural network using the test set to obtain the target neural network.

[0036] In this embodiment, the trained target neural network is used to predict the sampling points, and the residual vector is determined based on the predicted pressure coefficient and the actual pressure coefficient. The corresponding formula is as follows: ; in, For the residual vector, the calculation results of N points are concatenated in index order to obtain the complete residual vector; The true pressure coefficient at the i-th sampling point; The predicted value of the pressure coefficient output by the target neural network. The enhanced feature vector is described above. The magnitude of the residual vector is typically significantly smaller than the true pressure coefficient, and the main non-zero parts of the residual vector are concentrated near the discontinuity interface, exhibiting a spatial distribution characteristic of local concentration and sharp magnitude.

[0037] Specifically, the step of determining the predicted pressure coefficient values ​​corresponding to all spatial sampling points based on the target neural network, and determining the residual vector using the predicted pressure coefficient values ​​and the actual pressure coefficient, includes: performing pressure prediction on all spatial sampling points based on the target neural network to obtain the corresponding predicted pressure coefficient values; and obtaining each residual vector by subtracting the actual pressure coefficient from the predicted pressure coefficient values ​​element by element.

[0038] Step S14: Based on the target wing mesh points to be interpolated, determine the target interpolation points that meet the target distance conditions from the training sampling points, construct an RBF kernel matrix using the Euclidean distance between the target interpolation points, determine the weight vector based on the RBF kernel matrix and the residual vector, construct an RBF interpolation function based on the weight vector, and determine the residual correction amount of the target wing mesh points using the RBF interpolation function.

[0039] In this embodiment, a thin plate spline core can be selected. As the kernel function in this embodiment, The Euclidean distance between the two points is... The radial basis function is used. A local support strategy is employed, selecting the m nearest target interpolation points for each interpolation point. When m is 50, the nearest neighbor index formula for the interpolation point is as follows: ; in, The interpolation point The corresponding set of nearest neighbor indices; Training sampling points Interpolation point The Euclidean distance between them; The independent variable that minimizes the objective function is, in this case, the distance from the point to be interpolated. The most recent 50 training sampling points; This is the summation symbol.

[0040] It is understandable that an m×m local RBF kernel function matrix can be constructed using the Euclidean distance between any two target interpolation points. ,in, The corresponding formula is as follows: ; in, The element in the a-th row and b-th column of the RBF kernel function matrix; It is a radial basis kernel function; , Let be the spatial coordinates of the two points to be interpolated; It is an L2 norm.

[0041] Specifically, the step of determining target interpolation points that satisfy the target distance condition from the training sampling points based on the target wing mesh points to be interpolated, and constructing an RBF kernel matrix using the Euclidean distance between the target interpolation points, includes: determining the second number of target interpolation points that are closest to the target wing mesh points to be interpolated in terms of Euclidean distance based on the training sampling points; and constructing an RBF kernel matrix based on the Euclidean distance between any two target interpolation points.

[0042] Furthermore, for the linear interpolation equation Solve the problem to obtain the weight vector. , The matrix is ​​a local RBF kernel function. For the residual vector, the corresponding formula is: ; Where m represents the m nearest training points to the point to be interpolated. An RBF interpolation function is constructed based on the weight vector, and the corresponding formula is as follows: ; in, This is the residual correction amount output by the RBF interpolation function; The spatial coordinates of the target wing grid points to be interpolated; The interpolation point The corresponding set of nearest neighbor indices; The summation symbol; Let be the weight vector corresponding to the j-th nearest neighbor; For target wing grid points With the j-th nearest neighbor The Euclidean distance (L2 norm) between them, the subscript 2 is omitted here, as in the previous text. The meanings are the same.

[0043] Specifically, the step of determining the weight vector based on the RBF kernel matrix and the residual vector, constructing an RBF interpolation function based on the weight vector, and using the RBF interpolation function to determine the residual correction amount of the target wing grid point includes: constructing a system of linear interpolation equations using the RBF kernel matrix and the residual vector corresponding to the target interpolation point, and solving the system of linear interpolation equations to obtain the weight vector; constructing an RBF interpolation function for the target wing grid point based on the weight vector, and substituting the spatial coordinates of the target wing grid point into the RBF interpolation function to obtain the residual correction amount of the target wing grid point.

[0044] Step S15: Determine the target enhanced feature vector corresponding to the target wing grid point, determine the target pressure coefficient prediction value based on the target enhanced feature vector and the target neural network, and determine the final pressure coefficient prediction value of the target wing grid point based on the residual correction amount and the target pressure coefficient prediction value.

[0045] In this embodiment, for any target wing grid point to be interpolated First, determine the corresponding target enhancement feature vector. The target enhanced feature vector is input into the target neural network to obtain the predicted value of the target pressure coefficient. The final predicted pressure coefficient of the target wing grid point is determined based on the predicted target pressure coefficient and the residual correction amount of the target wing grid point, and the corresponding formula is as follows: ; in, This is the predicted value of the final pressure coefficient; The predicted value of the target pressure coefficient; This is the residual correction amount.

[0046] Specifically, determining the target enhanced feature vector corresponding to the target wing grid point, determining the target pressure coefficient prediction value based on the target enhanced feature vector and using the target neural network, and determining the final pressure coefficient prediction value of the target wing grid point based on the residual correction and the target pressure coefficient prediction value, includes: determining the target enhanced feature vector corresponding to the target wing grid point based on the training set; inputting the target enhanced feature vector into the target neural network to obtain the corresponding target pressure coefficient prediction value; and linearly superimposing the residual correction and the target pressure coefficient prediction value to obtain the final pressure coefficient prediction value of the target wing grid point. It should be noted that this embodiment can be used in aerodynamic-structural coupling simulations in the aerospace field, where the shock wave discontinuity on the surface of a supersonic wing is a typical strongly discontinuous physical field.

[0047] As can be seen from the above, this application constructs a unified nearest neighbor index based on the training set, and uses the nearest neighbor index to determine the local standard deviation features and trigonometric function position codes of each training sampling point. The local standard deviation can automatically locate the shock wave region on the wing surface. The spatial coordinates, local standard deviation features, and trigonometric function position codes are concatenated to obtain an enhanced feature vector. The feedforward neural network is trained based on the enhanced feature vector and the corresponding pressure coefficient, and the generalization ability is verified using the test set to ensure the reliability of the model's interpolation for unknown grid points. Training sampling points near the target wing grid point are selected to construct a local RBF interpolator, which avoids oscillations caused by the basis function crossing both sides of the shock wave from the root, and further determines the residual correction amount of the target grid point to accurately compensate for the sharp shock wave jumps that the neural network fails to capture. In this way, the neural network is responsible for capturing the global low-frequency smooth trend, and the local RBF is responsible for correcting the high-frequency discontinuous residuals. The pressure coefficient prediction values ​​output by the two are combined with the residual correction amount to obtain the final pressure coefficient prediction value. This method can maintain stable interpolation accuracy for target structure grids with different topologies.

[0048] Finally, it should be noted that in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0049] The technical solutions provided in this application have been described in detail above. Specific examples have been used to illustrate the principles and implementation methods of this application. The descriptions of the above embodiments are only for the purpose of helping to understand the methods and core ideas of this application. At the same time, for those skilled in the art, there will be changes in the specific implementation methods and application scope based on the ideas of this application. Therefore, the content of this specification should not be construed as a limitation of this application.

Claims

1. A high-precision fusion interpolation method for discontinuous physical fields in multiphysics coupling, characterized in that, include: A sample dataset is constructed based on the spatial coordinates of each spatial sampling point on the wing and the corresponding pressure coefficient, and the sample dataset is divided to obtain the corresponding training set and test set. A nearest neighbor index is constructed based on the spatial coordinates of each training sampling point in the training set. The local standard deviation feature of the training sampling point is determined based on the nearest neighbor index. A trigonometric function position code is constructed based on the spatial coordinates of the training sampling point. The spatial coordinates, the local standard deviation feature, and the trigonometric function position code are concatenated to obtain an enhanced feature vector. A fully connected feedforward neural network for the target structure is constructed. The enhanced feature vector and the corresponding pressure coefficient are input into the fully connected feedforward neural network for training based on the Huber loss function. The trained neural network is then tested using the test set to obtain the target neural network. Based on the target neural network, the predicted pressure coefficient values ​​corresponding to all spatial sampling points are determined. The residual vector is determined using the predicted pressure coefficient values ​​and the actual pressure coefficient. Based on the target wing mesh points to be interpolated, target interpolation points that meet the target distance conditions are determined from the training sampling points. An RBF kernel matrix is ​​constructed using the Euclidean distance between the target interpolation points. A weight vector is determined based on the RBF kernel matrix and the residual vector. An RBF interpolation function is constructed based on the weight vector. The residual correction amount of the target wing mesh points is determined using the RBF interpolation function. Determine the target enhanced feature vector corresponding to the target wing grid point, determine the target pressure coefficient prediction value based on the target enhanced feature vector and the target neural network, and determine the final pressure coefficient prediction value of the target wing grid point based on the residual correction amount and the target pressure coefficient prediction value.

2. The high-precision fusion interpolation method for discontinuous physical fields in multiphysics coupling according to claim 1, characterized in that, The step of constructing a nearest neighbor index based on the spatial coordinates of each training sampling point in the training set, and determining the local standard deviation features of the training sampling points based on the nearest neighbor index, includes: Based on the training set, determine the number of neighboring points of the first target closest to each training sampling point in Euclidean distance, and record the index corresponding to each neighboring point. Construct a nearest neighbor index based on each training sampling point and its corresponding index.

3. The high-precision fusion interpolation method for discontinuous physical fields in multiphysics coupling according to claim 2, characterized in that, The step of determining the local standard deviation features of the training sampling points based on the nearest neighbor index includes: The mean pressure coefficient of each training sampling point is determined based on the nearest neighbor index, and the local standard deviation feature of the training sampling point is determined using the mean pressure coefficient.

4. The high-precision fusion interpolation method for discontinuous physical fields in multiphysics coupling according to claim 1, characterized in that, The construction of trigonometric function position encoding based on the spatial coordinates of the training sampling points includes: Based on the spatial coordinates of the training sampling points, two sets of codes with different frequencies are constructed to obtain the trigonometric function position codes; the trigonometric function position codes include a target low-frequency code for capturing global gradual components and a target high-frequency code for characterizing local sharp transitions.

5. The high-precision fusion interpolation method for discontinuous physical fields in multiphysics coupling according to claim 1, characterized in that, The process of concatenating the spatial coordinates, the local standard deviation features, and the trigonometric function position encoding to obtain the enhanced feature vector includes: The spatial coordinates of each training sampling point, the local standard deviation feature, and the trigonometric function position encoding are concatenated to obtain an enhanced feature vector; The enhanced feature vectors of all training sampling points in the training set are stacked to obtain the corresponding feature matrix.

6. The high-precision fusion interpolation method for discontinuous physical fields in multiphysics coupling according to claim 5, characterized in that, The fully connected feedforward neural network for constructing the target structure is trained by inputting the enhanced feature vector and the corresponding stress coefficient into the fully connected feedforward neural network based on the Huber loss function, and then testing the trained neural network using the test set to obtain the target neural network, including: A fully connected feedforward neural network with hidden layers is constructed based on a multilayer perceptron; the number of neurons in the hidden layers of the fully connected feedforward neural network decreases layer by layer; the activation function of the hidden layers is the Swish activation function. The feature matrix and the corresponding pressure coefficients in the training set are input into the fully connected feedforward neural network and trained using the Huber loss function to obtain the trained neural network. The trained neural network is tested using the test set to obtain the target neural network.

7. The high-precision fusion interpolation method for discontinuous physical fields in multiphysics coupling according to claim 1, characterized in that, The step of determining the predicted pressure coefficient values ​​corresponding to all spatial sampling points based on the target neural network, and determining the residual vector using the predicted pressure coefficient values ​​and the actual pressure coefficient, includes: Pressure prediction is performed on all spatial sampling points based on the target neural network to obtain the corresponding pressure coefficient prediction value; The residual vectors are obtained by subtracting the actual pressure coefficient from the predicted pressure coefficient element by element.

8. The high-precision fusion interpolation method for discontinuous physical fields in multiphysics coupling according to claim 1, characterized in that, The method for determining target interpolation points that satisfy the target distance condition from the training sampling points based on the target wing mesh points to be interpolated, and constructing an RBF kernel matrix using the Euclidean distance between the target interpolation points, includes: Based on the training sampling points, determine the number of target interpolation points that are closest to the second target in terms of the Euclidean distance of the target wing grid points to be interpolated; Construct the RBF kernel matrix based on the Euclidean distance between any two target interpolation points.

9. The high-precision fusion interpolation method for discontinuous physical fields in multiphysics coupling according to claim 1, characterized in that, The steps of determining a weight vector based on the RBF kernel matrix and the residual vector, constructing an RBF interpolation function based on the weight vector, and using the RBF interpolation function to determine the residual correction amount for the target wing grid points include: A system of linear interpolation equations is constructed using the RBF kernel matrix and the residual vector corresponding to the target interpolation point, and the system of linear interpolation equations is solved to obtain the weight vector. Based on the weight vector, an RBF interpolation function is constructed for the target wing grid points. The spatial coordinates of the target wing grid points are substituted into the RBF interpolation function to obtain the residual correction amount of the target wing grid points.

10. The high-precision fusion interpolation method for discontinuous physical fields in multiphysics coupling according to any one of claims 1 to 9, characterized in that, The steps of determining the target enhanced feature vector corresponding to the target wing grid point, determining the target pressure coefficient prediction value based on the target enhanced feature vector and using the target neural network, and determining the final pressure coefficient prediction value of the target wing grid point based on the residual correction and the target pressure coefficient prediction value include: Based on the training set, the target enhancement feature vector corresponding to the target wing grid point is determined, and the target enhancement feature vector is input into the target neural network to obtain the corresponding target pressure coefficient prediction value; The residual correction and the predicted target pressure coefficient are linearly superimposed to obtain the final predicted pressure coefficient of the target wing grid points.

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