Fractional order diffusion equation efficient solving method based on POD dimension reduction and RBF

Abstract

The invention discloses a fractional order diffusion equation efficient solving method based on POD dimension reduction and RBF, and the method comprises the steps: carrying out the discrete solving of a time-space fractional order diffusion equation through combining an L1 format and a Grunwald-Letnikov difference format, and obtaining snapshot data for training; secondly, extracting a main mode of the system by applying a POD technology, constructing a radial basis function interpolation neural network model, and reducing a solution dimension; and training a radial basis function neural network model, learning a nonlinear mapping relationship between a parameter space and a POD coefficient space, and realizing rapid prediction of the POD coefficient under a target parameter, thereby reconstructing a high-dimensional numerical solution field. According to the method, dimension reduction modeling and neural network prediction technologies are effectively fused, the calculation amount can be greatly reduced, the solving precision and efficiency are improved, and the method is suitable for multi-parameter and high-dimension fractional order diffusion problem modeling and simulation calculation and has good engineering application prospects.

Description

Technical Field

[0001] The present invention relates to an efficient solution method for a fractional-order diffusion equation based on POD dimension reduction and RBF, and belongs to the technical field of numerical analysis. Background Art

[0002] Fractional differential equations (FDEs) are widely used in fields such as control systems, (viscoelastic) materials science, biomedicine, and groundwater seepage, demonstrating unique advantages in describing the behavior of systems with non-classical dynamic characteristics. For example, in control systems, many physical processes exhibit fractional-order inertia, non-integer-order damping, or long-memory feedback characteristics, and are widely used in sub-fields such as robust control, PID controller design, and process control modeling. In groundwater seepage problems, fluid particles at the microscopic level often follow a continuous-time random walk (CTRW) model, exhibiting long-term retention or non-uniform migration characteristics. In the biomedical field, drug release processes or cell membrane current conduction often exhibit dynamic nonlinearity and strong memory. In materials science, the stress-strain response of viscoelastic materials involves a "memory" of historical load paths, exhibiting significant nonlocal response behavior.

[0003] The common point of the above phenomena is that the system behavior depends not only on the current state, but also on the long-term influence of historical behavior. Fractional differential equations are solved by introducing fractional derivative operators, such as Figure 1 As shown, it naturally possesses non-locality and memory effects, making it a powerful mathematical tool for describing such complex systems.

[0004] Although fractional-order models offer enhanced modeling capabilities, their numerical solution faces numerous challenges. Due to the nonlocal integral form of fractional-order derivatives, the entire historical information of the system from the initial moment to the current moment must be tracked during the time evolution process, significantly increasing computational complexity. Furthermore, to ensure solution accuracy, state variables must typically be retained over a longer time span, leading to a high-dimensional data storage burden. Furthermore, in multi-parameter modeling or system analysis scenarios, the complete model must be numerically solved again for each set of parameters, making traditional finite difference methods (FDM), finite element methods (FEM), and spectral methods inefficient and limited in applicability for high-dimensional or multi-parameter problems. Summary of the Invention

[0005] The technical problem to be solved by this invention is to overcome the shortcomings of the existing technology and provide an efficient solution method for fractional-order diffusion equations based on POD dimensionality reduction and RBF, so as to quickly solve fractional-order diffusion equations. This method uses POD to extract the main modes and RBF to perform interpolation, which can significantly reduce computational complexity and achieve efficient prediction of fractional-order partial differential equation solutions. This improves solution efficiency and adaptability, and is beneficial to the current research on numerical solution of fractional-order differential equations.

[0006] Preferably, the present invention provides an efficient solution method for the fractional diffusion equation based on POD dimensionality reduction and RBF, comprising:

[0007] Inputting the physical modeling parameters to be predicted into the trained radial basis function interpolation neural network model, the physical modeling parameters to be predicted include the fractional order α of the time dimension and the fractional order β of the space dimension; outputting a POD coefficient vector;

[0008] The POD coefficient vector is linearly combined with the main mode basis function obtained by singular value decomposition to predict the high-dimensional numerical approximate solution field;

[0009] Among them, the training of the radial basis function interpolation neural network model includes:

[0010] 1) Based on the target physical process and empirical formula, a time-space fractional diffusion equation is established, and the Caputo fractional derivative operator is discretized using the L1 format, and the Riesz fractional derivative operator is discretized using the Grünwald-Letnikov difference format;

[0011] 2) Select representative physical modeling parameters (α, β) and perform numerical simulations on the discretized time-space fractional diffusion equation at different time steps. Collect the high-dimensional numerical solutions to form a snapshot matrix, which is a two-dimensional matrix consisting of the numerical solution vectors at each time step stacked in columns.

[0012] 3) Perform singular value decomposition on the snapshot matrix to extract the first several energy-dominated main modal basis functions, and project the numerical solution vectors of each time step stored in the snapshot matrix into the low-dimensional subspace spanned by the main modal basis functions to obtain the corresponding POD coefficient vectors;

[0013] 4) normalizing the training set and constructing a radial basis function interpolation neural network model using the mapping relationship between the physical modeling parameters (α, β) in the training set and the POD coefficients, wherein the radial function interpolation kernel function in the radial basis function interpolation neural network model is selected from one of a thin plate spline kernel function, a linear kernel function, a cubic spline kernel function, a Gaussian kernel function, and a multi-quadratic kernel function;

[0014] 5) Linearly combine the POD coefficient vector obtained in step 4 with the main mode basis function retained in step 3 to predict the high-dimensional numerical approximate solution field;

[0015] 6) Compare the predicted high-dimensional numerical approximate solution field with the corresponding true high-fidelity numerical solution, and calculate the reconstruction error index. If the error between the predicted high-dimensional numerical approximate solution field and the corresponding true high-fidelity numerical solution is within the preset error threshold, the radial basis function interpolation neural network model is determined to be qualified.

[0016] Preferably, in step 1), the left-hand Caputo fractional derivative is discretized in the time dimension using the L1 format, and the Riesz fractional derivative is discretized in the space dimension using the Grünwald-Letnikov approximation to obtain a numerical expression in a fully discrete format.

[0017] Preferably, the discretization format of the temporal and spatial fractional derivatives is selected from one of the L1 format, the L2 format, the Grünwald-Letnikov format, the matrix construction method or other discretization methods with a preset accuracy threshold.

[0018] Preferably, the snapshot matrix is ​​a collection of spatial node solution vectors at multiple time steps, stacked in columns to form a two-dimensional matrix.

[0019] Preferably, in step 3), the snapshot matrix is ​​subjected to singular value decomposition, and the first several main modes are selected according to the energy accumulation criterion to form the dimensionality reduction basis function matrix. The energy accumulation criterion is:

[0020] Take η = 99% (1),

[0021] Where r is the number of selected main mode basis functions, σ i is the i-th singular value in the singular value decomposition of the snapshot matrix, and η is the set energy accumulation threshold.

[0022] Preferably, in step 4), when constructing the radial basis function neural network interpolation model, the kernel function is selected from one of the following functions: thin plate spline kernel function, linear kernel function, cubic spline kernel function, Gaussian kernel function, multi-quadratic kernel function or other known radial basis function kernels.

[0023] Preferably, in step 4), the training set p (i) =[α (i) ,β (i) ] to perform linear normalization:

[0024]

[0025] Where, is the normalized parameter vector, p (i) is the i-th training sample α (i) , β (i) The corresponding original parameter vector, pmin is a vector consisting of the minimum values ​​of each parameter dimension in the training set, p max is a vector consisting of the maximum values ​​of each parameter dimension in the training set, and N is the total number of training samples.

[0026] Preferably, the formula for the error indicator is:

[0027]

[0028] Where ε is the error value, U (x) To predict the high-dimensional numerical approximate solution field, U true It is a true high-fidelity numerical solution.

[0029] Preferably, the present invention provides an electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor implements the steps of any one of the methods described in the first aspect when executing the program.

[0030] Preferably, the present invention provides a computer-readable storage medium having a computer program stored thereon, which implements the steps of any one of the methods described in the first aspect when executed by a processor.

[0031] The beneficial effects achieved by the present invention are:

[0032] 1. The present invention uses the left-side Caputo fractional derivative and Riesz fractional derivative discrete formats to calculate the numerical solution, and improves the calculation accuracy based on the high-order interpolation method.

[0033] 2. The present invention adopts the POD method to perform data dimensionality reduction, extract the main modes of the data, and reduce the calculation dimension.

[0034] 3. The present invention uses RBF neural network for interpolation to quickly estimate the numerical solution under different parameters.

[0035] 4. This paper combines POD with RBF to rapidly predict the solution of the fractional diffusion equation under new parameter input conditions. This method can effectively reduce computational costs, maintaining high accuracy and efficient computational capabilities even for complex, high-dimensional problems.

[0036] 5. The method presented here is suitable for solving diffusion-type partial differential equations with fractional derivative structures. By performing modal compression and mapping learning on high-fidelity numerical solutions for multiple parameters, this method enables rapid reconstruction of solutions and high-precision prediction. Compared with traditional numerical methods, this method significantly reduces computational complexity while maintaining accuracy, making it particularly suitable for high-performance computing scenarios such as multi-parameter sweeps, model inversion, and real-time simulation. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] In order to more clearly illustrate the technical solution of the present application, the following is a brief introduction to the drawings required for use in the embodiments. Obviously, for ordinary technicians in this field, other drawings can be obtained based on these drawings without any creative work.

[0038] Figure 1 A schematic diagram comparing the numerical calculation characteristics of integer-order derivatives and fractional-order derivatives in the time direction;

[0039] Figure 2 is a flow chart of the present invention;

[0040] Figure 3 This is a visual comparison diagram of the high-dimensional numerical approximate solution field output by the model under different POD modal orders k=1~5 and the true high-fidelity numerical solution under the same parameters. DETAILED DESCRIPTION

[0041] See also Figure 1 The present application discloses an efficient solution method for fractional diffusion equations based on POD dimensionality reduction and RBF, which is characterized by comprising the following steps:

[0042] Inputting the physical modeling parameters to be predicted into the trained radial basis function interpolation neural network model, the physical modeling parameters to be predicted include the fractional order α of the time dimension and the fractional order β of the space dimension; outputting a POD coefficient vector;

[0043] The POD coefficient vector is linearly combined with the main mode basis function obtained by singular value decomposition to predict the high-dimensional numerical approximate solution field;

[0044] Among them, the training of the radial basis function interpolation neural network model includes:

[0045] 1) Based on the target physical process and empirical formula, a time-space fractional diffusion equation is established, where the Caputo fractional derivative is used in the time dimension and the Riesz fractional derivative is used in the space dimension. The Caputo fractional derivative operator is discretized using the L1 format, and the Riesz fractional derivative operator is discretized using the Grünwald-Letnikov difference format;

[0046] 2) Select representative physical modeling parameters (α, β) and perform numerical simulations on the discretized time-space fractional diffusion equation at different time steps. Collect the high-dimensional numerical solutions to form a snapshot matrix, which is a two-dimensional matrix consisting of the numerical solution vectors at each time step stacked in columns.

[0047] 3) Perform singular value decomposition (SVD) on the snapshot matrix to extract the first several energy-dominated main modal basis functions, and project the numerical solution vectors of each time step stored in the snapshot matrix into the low-dimensional subspace spanned by the main modal basis functions to obtain the corresponding POD coefficient vectors, thereby achieving a low-dimensional representation of the high-dimensional solution;

[0048] 4) normalizing the training set and utilizing the mapping relationship between the physical modeling parameters (α, β) in the training set and the POD coefficients to construct a radial basis function interpolation neural network model for predicting the POD coefficient vector corresponding to any physical modeling parameter input in the training set, wherein the radial function interpolation kernel function in the radial basis function interpolation neural network model is selected from one of a thin plate spline kernel function, a linear kernel function, a cubic spline kernel function, a Gaussian kernel function, and a multi-quadratic kernel function;

[0049] 5) Linearly combine the POD coefficient vector obtained in step 4 with the main mode basis function retained in step 3 to predict the high-dimensional numerical approximate solution field;

[0050] 6) Compare the predicted high-dimensional numerical approximate solution field with the corresponding true high-fidelity numerical solution, calculate the reconstruction error index, and evaluate the prediction accuracy. If the error between the predicted high-dimensional numerical approximate solution field and the corresponding true high-fidelity numerical solution is within the preset error threshold, the radial basis function interpolation neural network model is determined to be qualified;

[0051] 7) Based on the reconstruction error index, by gradually increasing the number of POD modes and performing error comparison verification, this method can obtain the approximate accuracy of high-fidelity solutions at lower modal dimensions and has good numerical reconstruction performance.

[0052] Furthermore, the discrete formats of the time and space fractional derivatives are selected from one of the L1 format, the L2 format, the Grünwald-Letnikov format, the matrix construction method or other discrete methods with a preset accuracy threshold.

[0053] Furthermore, the snapshot matrix is ​​a collection of spatial node solution vectors at multiple time steps, stacked in columns to form a two-dimensional matrix.

[0054] Furthermore, in step 3), the snapshot matrix is ​​subjected to singular value decomposition, and the first several main modes are selected to form a dimensionality reduction basis function matrix according to the energy accumulation criterion. The energy accumulation criterion is:

[0055] Take η = 99% (1),

[0056] Where r is the number of selected main mode basis functions, σ i is the i-th singular value in the singular value decomposition of the snapshot matrix, and η is the set energy accumulation threshold.

[0057] Furthermore, in step 4), when constructing the radial basis function neural network interpolation model, the kernel function is selected from one of the following functions: thin plate spline kernel function, linear kernel function, cubic spline kernel function, Gaussian kernel function, multi-quadratic kernel function or other known radial basis function kernels.

[0058] Furthermore, in step 4), the training set p (i) =[α (i) ,β (i) ] to perform linear normalization:

[0059]

[0060] Where, is the normalized parameter vector, p (i) is the i-th training sample α (i) , β (i) The corresponding original parameter vector, p min is a vector consisting of the minimum values ​​of each parameter dimension in the training set, p max is a vector consisting of the maximum values ​​of each parameter dimension in the training set, and N is the total number of training samples.

[0061] Furthermore, the formula for the error index is:

[0062]

[0063] Where ε is the error value, U (x) To predict the high-dimensional numerical approximate solution field, U true It is a true high-fidelity numerical solution.

[0064] Step 1: Construct the fractional-order diffusion equation and discretize it:

[0065] This step aims to construct a fractional diffusion equation with a physical background, and use the time fractional derivative defined by Caputo and the space fractional derivative defined by Riesz to perform numerical discretization on them using the difference method, respectively, to obtain a calculation format that can be used for numerical simulation and snapshot generation.

[0066] Construct the fractional diffusion equation:

[0067] Assume that the diffusion process in a one-dimensional space region can be described by the following fractional-order diffusion equation:

[0068]

[0069] Where: u(x,t) is the state variable function, which represents the state variable (such as concentration, temperature, etc.) at position x∈[a,b] and time t∈[0,t];

[0070] is the Caputo fractional derivative in the time dimension, order α∈(0,1);

[0071] is the Riesz fractional derivative in the spatial dimension, order β∈(0,2);

[0072] D is the diffusion coefficient, f(x,t) is the external source term;

[0073] The initial and boundary conditions depend on the specific setup of the problem.

[0074] Discretization of Caputo derivative in time dimension:

[0075] For the function u(x,t) on [0,t], find the α-order derivative, α∈(0,1), and the Caputo fractional derivative operator is:

[0076]

[0077] Divide the time interval [0, t] into N equal parts t Step, step length The discrete node is denoted as t n =nΔt,n=0,1,...,N t .

[0078] By L1 difference approximation method, we can get that at t=t n The difference at is approximately:

[0079]

[0080] in Indicates that the function u(x,t) is at the node (x j ,t n ), the weight coefficient is:

[0081] b k =(k+1) 1-α -k 1-α ,k=0,1,...,n-1 (7)

[0082] This format has first-order accuracy, versatility and good stability.

[0083] Discretization of Riesz derivatives in spatial dimensions:

[0084] For the function u(x,t) on [a,b], find the β-order derivative, β∈(0,2), and the Caputo fractional derivative operator is:

[0085] To approximate the diffusion behavior with spatial nonlocality, Riesz fractional derivatives are used to model the non-integer derivatives in the spatial dimension. Riesz fractional derivatives are essentially a symmetric combination of the left and right Riemann-Liouville fractional derivatives. For the β-order derivative of the function u(x, t) on [a, b], β∈(0,2), the Riesz fractional derivative operator is defined as follows:

[0086]

[0087] Among them, the left and right Riemann-Liouville fractional derivatives are defined as:

[0088]

[0089] Where n is the smallest integer not less than β.

[0090] Divide the space interval [a,b] into N equal parts x Step, step length The discrete node is denoted as x j =a+jh,j=0,1,...,N x .

[0091] In numerical implementation, the above integral Riesz derivative can be approximated using a difference format based on the Grünwald-Letnikov form, and its difference approximation is expressed as:

[0092]

[0093] in is the symmetric difference weight coefficient, which can be calculated based on the generalized binomial coefficient:

[0094]

[0095] M is a sufficiently large cutoff length to control the error. When close to the left endpoint, taking M = j is a reasonable cutoff.

[0096] 1. Get the fully discrete format:

[0097] Substitute the discrete format of time and space dimensions into the original equation, and at the node (x j ,t n ) to obtain the following fully discrete expression:

[0098]

[0099] Step 2: Dimensionality reduction based on POD:

[0100] This step aims to perform dimensionality reduction processing on the numerical solution data obtained in step 1, and use the POD method to extract the main eigenmodes of the system, thereby significantly reducing the calculation dimension and improving the subsequent interpolation and prediction efficiency.

[0101] 1. Construct the snapshot matrix:

[0102] set up The number of time steps is N t , the number of spatial nodes is N x +1, using the numerical format constructed in step 1 at each time step t n Get the spatial discrete solution vector:

[0103]

[0104] Select M representative time steps to construct the snapshot matrix

[0105]

[0106] Each column is a snapshot (i.e., the spatial distribution of the system state at a certain time).

[0107] Perform singular value decomposition (SVD) on the snapshot matrix:

[0108] Use SVD to decompose the snapshot matrix into:

[0109] U=VΣW T (16)

[0110] in:

[0111] is the left singular vector matrix, whose columns are POD modes (spatial modes);

[0112] Σ=diag(σ1,σ2,...,σ M ) is a singular value diagonal matrix;

[0113] W∈R M×M is the right singular vector matrix, corresponding to the time dimension coefficient.

[0114] Extract the main mode and construct the dimensionality reduction basis function:

[0115] According to the size of the singular value, retain the first r<<M modes and construct the dimensionality reduction basis function matrix:

[0116]

[0117] The i-th column φ i is the i-th main mode, which comes from the i-th left singular vector in the snapshot matrix SVD decomposition, and the corresponding singular value is the i-th largest singular value σi .

[0118] Mode selection can be controlled by cumulative energy threshold:

[0119] Take η = 99% (18)

[0120] 4.POD coefficient representation and dimensionality reduction expression:

[0121] For each snapshot vector, its projection in the dimensionality reduction basis function space is the POD coefficient vector:

[0122]

[0123] The coefficients of all snapshots are combined into the POD coefficient matrix:

[0124]

[0125] in:

[0126] The columns of A are parameter combinations (α (i) ,β (i) ) under the POD coefficient;

[0127] The rows of A are the amplitude changes of a certain POD mode under different parameters.

[0128] The snapshot matrix is ​​then approximated as:

[0129] U≈Φ r A(21)

[0130] Step 3: Use radial basis function (RBF) to interpolate and predict the POD coefficient:

[0131] This step aims to construct an efficient and generalizable prediction model based on the mapping relationship between training parameters and POD coefficients using the RBF interpolation method. This model can quickly predict the corresponding POD coefficients when any new parameters are given, thereby achieving a rapid estimation of the response of the fractional-order diffusion system.

[0132] 1. Parameter set settings and snapshot tags:

[0133] Assume that there are N groups of input parameters in the training set, and each group of parameters is:

[0134] p (i) =[α (i) ,β (i) ,...]∈R d ,i=1,2,...,N (22)

[0135] For each set of parameters, the POD coefficient vector is calculated in step 2:

[0136]

[0137] All sample combinations constitute the POD coefficient matrix:

[0138] A=[a (1) ,a (2) ,...,a (N) ]∈R r×N (twenty four)

[0139] 2. Parameter normalization processing:

[0140] In order to eliminate the dimensional differences between different parameter dimensions and improve interpolation accuracy, the input parameters need to be linearly normalized:

[0141]

[0142] where p min 、p max are the minimum and maximum values ​​of each dimension parameter in the training set respectively. For the new target parameter p (x) , which also needs to be normalized to

[0143] 3. RBF interpolation formula construction:

[0144] For each dimension of POD coefficient (r dimensions), an interpolation model is constructed. Let the radial basis kernel function be φ(r), then the predicted value of the kth POD coefficient at the new parameter point is:

[0145]

[0146] in:

[0147] w k,i is the interpolation weight of the kth coefficient;

[0148] ||·|| is the Euclidean distance;

[0149] φ(·) is the selected RBF kernel function.

[0150] Commonly used RBF kernel functions are as follows:

[0151] Thin-plate splines: φ(x) = ‖xx i ‖ 2 ln(‖xx i ‖);

[0152] Linear splines: φ(x) = ‖xx i ‖;

[0153] Cubic splines: φ(x) = ‖xx i ‖ 3 ;

[0154] Gaussian kernel:

[0155] Multiquadric:

[0156] Among them: c is an adjustable shape parameter that controls the response range of the kernel function to neighboring sample points.

[0157] The above interpolation formula is equivalent to the matrix form:

[0158] A=W·G (27)

[0159] in:

[0160] A∈R r×N is the POD coefficient matrix;

[0161] W∈R r×N is the POD coefficient matrix;

[0162] G∈R r×N is the kernel function matrix, and its (i,j)th item is

[0163] By solving the linear system:

[0164] W=A·G -1 (28)

[0165] It can be used to predict the POD coefficient under new parameters:

[0166] a (x) =W·φ (x) (29)

[0167] The target parameter The expression of the system response in the reduced-dimensional modal space.

[0168] in:

[0169]

[0170] Step 4: Recover the high-dimensional approximate solution:

[0171] In this step, the POD coefficient vector obtained by RBF interpolation in step 3 and the POD main mode matrix Φ extracted in step 2 are used. r , for a given new parameter p (x) =(α (x) ,β (x)) restores its corresponding high-dimensional physical solution approximate solution field.

[0172] 1. Calculate the low-dimensional representation of POD:

[0173] First, based on the POD dimensionality reduction formula, the original snapshot matrix can be approximately expressed as:

[0174] U≈Φ r ·A(31)

[0175] For the new input parameter p (x) , its corresponding POD coefficient vector α (x) ∈R r It has been obtained through RBF interpolation.

[0176] Therefore, we can use the linear combination form to project it back to the original space and recover the approximate high-dimensional solution field:

[0177] U (x) =Φ r ·a (x) (32)

[0178] 2. Error analysis:

[0179] To evaluate the accuracy of the approximate solution, we can compare it with a high-fidelity numerical solution (such as the finite difference method) and evaluate the accuracy of the POD approximation by calculating the reconstruction error ε:

[0180]

[0181] Among them: U true It is the true solution calculated by high-precision numerical methods.

[0182] U (x) It is an approximate solution of POD+RBF prediction.

[0183] If the error ε is small, it indicates that the prediction accuracy of the POD+RBF model is high.

[0184] 3. Order comparison analysis: Verify the approximation performance of the POD model:

[0185] To further verify the approximation ability of this method under different modes retention numbers, reconstruction solutions with increasing order are constructed and the errors are compared. The specific operations are as follows:

[0186] 1) Fixed a set of new parameter input p (x) =[α (x) ,β (x) ,...], retain the first k = 1, 2, 3,... POD main modes respectively;

[0187] 2) Use the corresponding POD coefficient a (x)Reconstruct the approximate solution by linear combination with the retained main modes:

[0188]

[0189] 3) Compare the results with those obtained using the high-fidelity solution (i.e., the direct numerical method) to observe changes in reconstruction error and image similarity. This visualization clearly demonstrates how the approximate solution gradually approaches the true solution as the modal order increases, demonstrating that only a small number of modes is required to achieve a highly accurate approximation and restore key features of high-dimensional physical fields.

[0190] To verify the numerical feasibility and modeling efficiency of the proposed method, a one-dimensional fractional diffusion equation was selected as a test model and solved using the proposed rapid prediction method based on a combination of POD dimensionality reduction and RBF interpolation. The model's performance in terms of prediction accuracy and computational time was evaluated by comparison with a high-fidelity numerical solution.

[0191] Assume that the spatial interval is x∈[0,1] and the time interval is t∈[0,T], and consider the following diffusion problem with a double fractional derivative term:

[0192]

[0193] in:

[0194] represents the Caputo-type time-dimensional fractional derivative, order α∈(0,1);

[0195] represents the symmetric Riesz space-dimensional fractional derivative, order β∈(0,2);

[0196] D = 1 is the diffusion coefficient;

[0197] f(x,t) is the external source function.

[0198] The initial conditions and boundary conditions are set as:

[0199] u(x,0)=0,u(0,t)=u(1,t)=0 (36)

[0200] The present invention provides an efficient solution method for fractional diffusion equation based on POD dimensionality reduction and RBF, such as Figure 2 As shown, the following steps are included:

[0201] S1: The Caputo derivative is discretized by constructing a time fractional derivative matrix based on generalized difference coefficients, and the spatial dimension non-local operator is discretized by the matrix method of symmetric Riesz space fractional derivatives. The specific implementation is as follows:

[0202] S1_1: The time dimension is defined by the Caputo fractional derivative on the left, and its mathematical expression is:

[0203]

[0204] In order to perform discrete processing, the time interval [0, T] is evenly divided into n small steps with a step length of Let the discrete time point be t k =kτ, the matrix method based on generalized difference coefficients can be used to discretely approximate the Caputo derivative:

[0205]

[0206] Among them, is the generalized difference weight coefficient, which is generated by the recursive formula:

[0207]

[0208] These coefficients are organized into matrices according to the lower triangular structure The discrete matrix representation of the fractional-order derivative in the time dimension can be obtained.

[0209] To form a space-time system, the Kronecker product form is further constructed:

[0210]

[0211] Among them I m is the unit matrix, corresponding to the number of nodes in the spatial dimension.

[0212] S1_2: The spatial dimension is defined using the symmetric Riesz fractional derivative, which is expressed as:

[0213]

[0214] In this embodiment, in order to simplify the constant coefficient processing, a numerically equivalent discrete representation is actually adopted, that is, a symmetric matrix is ​​constructed to approximate the average of the left and right derivatives. The space interval [a, b] is equally divided into m nodes with a node spacing of By constructing the Riemann-Liouville fractional derivative matrix And take the middle submatrix:

[0215]

[0216] Among them B M is the left derivative matrix after removing the first row.

[0217] Finally, the differential discrete matrix of the spatial dimension is obtained, and the Kronecker product expression is constructed:

[0218]

[0219] Among them I n is the time dimension identity matrix.

[0220] S1_3: Based on the above discrete results of time and space dimensions, the system matrix can be constructed:

[0221] A=T D -D.S. D (44)

[0222] Where D = 1 is the diffusion coefficient. Since the boundary condition is u(0, t) = u(1, t) = 0, the corresponding rows and columns are removed at the spatial nodes to obtain the matrix A reduced , solve the linear system:

[0223] A reduced ·U=F (45)

[0224] Solve the vector This is the space-time approximate solution after removing the boundary.

[0225] S2: To construct the feature space required for the dimensionality reduction model, we first need to collect representative high-fidelity numerical solution samples to form a snapshot matrix for the subsequent POD decomposition. In this step, we systematically scan different fractional derivative order parameters (α, β), solve the corresponding fractional diffusion equation, and extract the solution field to form a training sample set. The specific implementation is as follows:

[0226] S1_1: Uniformly sample the above parameter space to construct training set parameter pairs:

[0227] p (i) =(α (i) ,β (i) ),i=1,2,...,N (46)

[0228] In this embodiment, take: α∈{0.2, 0.4, 0.6, 0.8, 1.0}, β∈{0.2, 0.6, 1.0, 1.4, 1.8, 2.0}.

[0229] S1_2: For each parameter combination p (i) =(α (i) ,β (i) ), call the full discrete system solver function constructed in step 1 to solve the corresponding numerical solution U (i) , each solution is a two-dimensional matrix representing the joint distribution of space x and time t.

[0230] To facilitate subsequent processing, each solution matrix is ​​rearranged into a one-dimensional column vector in a column- or row-major manner:

[0231]

[0232] Right now:

[0233] u (i) =[u(x1,t1),u(x2,t1),...,u(x m ,t1),u(x1,t2),...,u(x m ,t n )] · (48)

[0234] S1_3: Stack all training solution vectors into a snapshot matrix

[0235] S=[u (1) ,u (2) ,...,u (N) ] (49)

[0236] The snapshot matrix will serve as the input data for the next step of POD modal decomposition to extract the main characteristic structure of the system.

[0237] S3: To reduce the computational and storage overhead of the high-dimensional solution and extract its main characteristic structure, the POD orthogonal mode decomposition method is used to reduce the dimensionality of the snapshot matrix. The specific implementation is as follows:

[0238] S3_1: To remove the influence of mean shift, first calculate the average vector of the snapshot matrix column by column:

[0239]

[0240] Reconstruct the mean-removed snapshot matrix:

[0241] S′=S-U0·1 · (51)

[0242] in is a column vector of all ones.

[0243] S3_2: Construct the covariance matrix of the snapshot matrix after removing the mean and perform eigenvalue decomposition:

[0244] R=S′·(S′) 晻 =Φ·Λ·Φ (52)

[0245] in:

[0246] Φ = [φ1, φ2, …] is the POD mode vector (characteristic function);

[0247] Λ=diag(λ1,λ2,…) is the eigenvalue matrix, representing the “energy” of each mode.

[0248] S3_3: Arrange the eigenvalues ​​in descending order λ1≥λ2≥…≥λ N , define the energy accumulation ratio:

[0249]

[0250] According to the set energy threshold, the number k of modes to be retained is determined.

[0251] Keep the first k main modes to form the POD basis function matrix:

[0252]

[0253] S3_4: Project the snapshot matrix to the main modal space to obtain the POD coefficient matrix:

[0254]

[0255] S4: To achieve rapid prediction of POD coefficients under arbitrary parameter input conditions, an interpolation model based on radial basis function (RBF) is constructed to learn the nonlinear mapping relationship between parameter space and POD coefficient space. The specific implementation is as follows:

[0256] S4_1: To avoid the influence of the dimension difference of each dimension parameter on the interpolation accuracy, all parameter vectors are first linearly normalized.

[0257] S4_2: In this embodiment, a linear kernel function φ(x)=‖xx is used i ‖Construct the radial basis function kernel matrix G.

[0258] S4_3: The interpolation coefficient matrix is ​​obtained by inversely solving the kernel matrix, and the prediction of the POD coefficient under the new parameters can be obtained at the same time.

[0259] S5: After completing the prediction of the POD coefficients under the parameter input, the approximate high-dimensional solution field under the parameter conditions can be reconstructed by linearly combining the predicted coefficients with the pre-extracted main modes. This step example details the solution reconstruction formula and the definition of the error index used to evaluate the approximation performance of the model. The specific implementation is as follows:

[0260] S5_1: The relative L2 norm error is used to measure the difference between the predicted solution and the high-fidelity solution:

[0261]

[0262] S5_2: Further analyze the approximation capability of the POD model for different modes, k. Repeat step S5 for multiple reconstructions with k = 1, 2, 3, 4, and 5, and calculate the reconstruction error at each order. Plot the trend to show that the error decreases as the number of modes increases.

[0263] The error index ε constructed in step 5 can be used to obtain the error distribution between the predicted solution and the high-fidelity solution under different modal orders. In this embodiment, the reconstruction error calculation is performed for the parameters (α = 0.55, β = 1.1) with modal numbers k = 1, 2, 3, 4, and 5. The results are as follows: Figure 3 :

[0264] When k = 3, the error has dropped to less than 2%;

[0265] When k ≥ 5, the reconstructed solution is almost completely consistent with the high-fidelity solution.

[0266] The results show that this method can achieve high-precision approximation of high-dimensional solutions while retaining only a small number of modes, and has good dimensionality reduction and compression capabilities.

[0267] Traditional methods for solving fractional-order partial differential equations (such as the difference method and the finite element method) require reassembling the system matrix and executing the solution process under each set of parameter inputs, which has high computational complexity. Especially when performing multi-parameter scanning or optimization tasks, the computational cost is extremely significant.

[0268] The innovation and superiority of the proposed method lies in the fact that the entire offline training process (POD modality extraction and interpolation model construction) is performed only once. During the prediction phase, only parameter normalization, construction of an RBF kernel vector, and a set of k×N matrix multiplications are required to obtain an approximate solution. This process does not involve numerical iteration of differential equations, significantly reducing the computational burden.

[0269] In summary, the present invention provides a method for solving diffusion-type partial differential equations with fractional derivative structures. This method utilizes modal compression and mapping learning on high-fidelity numerical solutions for multiple parameters, enabling rapid reconstruction of solutions and high-precision prediction. Compared with traditional numerical methods, this method significantly reduces computational complexity while maintaining accuracy, making it particularly suitable for high-performance computing scenarios such as multi-parameter sweeps, model inversion, and real-time simulation.

[0270] In an embodiment of the present application, the present invention provides an electronic device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor implements the steps of any of the above methods when executing the program.

[0271] In an embodiment of the present application, the present invention provides a computer-readable storage medium having a computer program stored thereon, which implements the steps of any of the above methods when executed by a processor.

[0272] The various embodiments in this specification are described in a progressive manner, and the same or similar parts between the various embodiments can be referred to each other. Each embodiment focuses on the differences from other embodiments.

[0273] Those skilled in the art will readily appreciate other embodiments of the present invention after considering the specification and practicing the invention as disclosed herein. This application is intended to cover any variations, uses, or adaptations of the present invention that follow the general principles of the invention and include common knowledge or customary techniques in the art not invented herein, and the description and examples are to be considered merely as exemplary.

[0274] The above specific implementation methods further illustrate the purpose, technical solutions and beneficial effects of this application in detail. It should be understood that the above are only specific implementation methods of this application and are not intended to limit the scope of protection of this application. Any modifications, equivalent replacements, improvements, etc. made on the basis of the technical solutions of this application should be included in the scope of protection of this application.

Claims

1. An efficient solution method for fractional-order diffusion equation based on POD dimensionality reduction and RBF, characterized by: include: Input the physical modeling parameters to be predicted into the trained radial basis function interpolation neural network model, where the physical modeling parameters to be predicted include the fractional order α of the time dimension and the fractional order β of the space dimension, and output the POD coefficient vector; The POD coefficient vector is linearly combined with the main mode basis function obtained by singular value decomposition to predict the high-dimensional numerical approximate solution field; Among them, the training of the radial basis function interpolation neural network model includes: 1) Based on the target physical process and empirical formula, a time-space fractional diffusion equation is established. The Caputo fractional derivative operator in the time-space fractional diffusion equation is discretized using the L1 scheme, and the Riesz fractional derivative operator in the time-space fractional diffusion equation is discretized using the Grünwald-Letnikov difference scheme. 2) Select representative physical modeling parameters (α, β) and perform numerical simulations on the discretized time-space fractional diffusion equation at different time steps. Collect the high-dimensional numerical solutions to form a snapshot matrix, which is a two-dimensional matrix consisting of the numerical solution vectors at each time step stacked in columns. 3) Perform singular value decomposition on the snapshot matrix to extract the first several energy-dominated main modal basis functions, and project the numerical solution vectors of each time step stored in the snapshot matrix into the low-dimensional subspace spanned by the main modal basis functions to obtain the corresponding POD coefficient vectors; 4) normalizing the training set and constructing a radial basis function interpolation neural network model using the mapping relationship between the physical modeling parameters (α, β) in the training set and the POD coefficients, wherein the radial function interpolation kernel function in the radial basis function interpolation neural network model is selected from one of a thin plate spline kernel function, a linear kernel function, a cubic spline kernel function, a Gaussian kernel function, and a multi-quadratic kernel function; 5) Linearly combine the POD coefficient vector obtained in step 4 with the main mode basis function retained in step 3 to predict the high-dimensional numerical approximate solution field; 6) Compare the predicted high-dimensional numerical approximate solution field with the corresponding true high-fidelity numerical solution, and calculate the reconstruction error index. If the error between the predicted high-dimensional numerical approximate solution field and the corresponding true high-fidelity numerical solution is within the preset error threshold, the radial basis function interpolation neural network model is determined to be qualified.

2. The efficient solution method for fractional-order diffusion equation based on POD dimension reduction and RBF according to claim 1 is characterized in that: In step 1), the L1 format is used to discretize the left-hand Caputo fractional derivative in the time dimension, and the Grünwald-Letnikov approximation is used to discretize the Riesz fractional derivative in the space dimension to obtain a numerical expression in a fully discrete format.

3. The efficient solution method for fractional-order diffusion equation based on POD dimension reduction and RBF according to claim 2 is characterized in that: The discretization format of the time and space fractional derivatives is selected from one of the L1 format, the L2 format, the Grünwald-Letnikov format, the matrix construction method or other discretization methods with a preset accuracy threshold.

4. The efficient solution method for fractional-order diffusion equation based on POD dimension reduction and RBF according to claim 1 is characterized in that: The snapshot matrix is ​​a collection of spatial node solution vectors at multiple time steps, stacked in columns to form a two-dimensional matrix.

5. The efficient solution method for fractional-order diffusion equation based on POD dimension reduction and RBF according to claim 4 is characterized in that: In step 3), the snapshot matrix is ​​subjected to singular value decomposition, and the first several main modes are selected to form the dimensionality reduction basis function matrix according to the energy accumulation criterion. The energy accumulation criterion is: Where r is the number of selected main mode basis functions, σ i is the i-th singular value in the singular value decomposition of the snapshot matrix, and η is the set energy accumulation threshold.

6. The efficient solution method for fractional-order diffusion equation based on POD dimension reduction and RBF according to claim 1 is characterized in that: In step 4), when constructing the radial basis function neural network interpolation model, the kernel function is selected from one of the following functions: thin plate spline kernel function, linear kernel function, cubic spline kernel function, Gaussian kernel function, multi-quadratic kernel function or other known radial basis function kernels.

7. The efficient solution method for fractional-order diffusion equation based on POD dimension reduction and RBF according to claim 1 is characterized in that: In step 4), for the training set p (i) =[α (i) ,β (i) ] to perform linear normalization: Where, is the normalized parameter vector, p (i) is the i-th training sample α (i) , β (i) The corresponding original parameter vector, p min is a vector consisting of the minimum values ​​of each parameter dimension in the training set, p max is a vector consisting of the maximum values ​​of each parameter dimension in the training set, and N is the total number of training samples.

8. The efficient solution method for fractional-order diffusion equation based on POD dimension reduction and RBF according to claim 1 is characterized in that: The formula for the error index is: Where ε is the error value, U (x) To predict the high-dimensional numerical approximate solution field, U true It is a true high-fidelity numerical solution.

9. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein: When the processor executes the program, the steps of the method according to any one of claims 1 to 8 are implemented.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the steps of the method according to any one of claims 1 to 8 are implemented.

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