A quantum-classical physical information KAN network solving method for fuzzy differential equations

Abstract

The application discloses a quantum-classical physical information KAN network solving method for fuzzy differential equations, and belongs to the technical field of physical information calculation, and the specific steps comprise the following steps: upper and lower endpoint control equations of the fuzzy partial differential equation are obtained by splitting based on the section set theory, and a joint input vector of a space-time coordinate and a membership level is constructed; a hybrid network architecture in series connection of a ChebyKAN preprocessing module, a parameterized quantum circuit core module and a ChebyKAN post-processing module is adopted, and upper and lower endpoint approximate solutions corresponding to the section set are output; a total loss function is constructed by fusing physical constraint loss and fuzzy structure constraint loss, and an approximate solving model is obtained through gradient descent optimization iterative training. The method can effectively improve the solving precision of the fuzzy partial differential equation, meanwhile, the legality of the fuzzy structure of the solution is guaranteed, and the method is suitable for multiple engineering scientific calculation scenes containing fuzzy parameters.

Description

Technical Field

[0001] This invention relates to the field of physical information computing technology, and in particular to a quantum-classical physical information KAN network solution method for fuzzy differential equations. Background Technology

[0002] Existing technologies are physical information machine learning systems for solving partial differential equations, including three main architectures: classical physical information neural networks based on multilayer perceptrons, physical information KANs based on Kolmogorov-Arnold networks, and quantum-classical hybrid physical information neural networks that integrate parameterized quantum circuits. For uncertain solutions with fuzzy parameters, existing technologies are generally based on… The solution scheme is constructed using cut set theory. By incorporating the physical equation residuals and initial boundary conditions into the loss function training network, the approximate solution in interval form at different membership levels is output.

[0003] However, the aforementioned existing technologies have inherent defects in their architectural characteristics and solution mechanisms. Specifically, these defects are as follows: Fuzzy physical information networks based on multilayer perceptrons are constrained by their own spectral bias characteristics, resulting in low fitting accuracy for high-gradient regions and motion fronts, and poor parameter utilization efficiency; pure classical physical information KANs are limited by the expression boundaries of classical feature spaces, and are insufficient in characterizing the strong coupling nonlinear relationships between spatiotemporal variables and membership levels; existing quantum-classical hybrid physical information networks are designed only for deterministic equations, and their single-output structure and single physical constraint system cannot adapt to the interval solution characteristics of fuzzy solutions; at the same time, the constraint mechanisms of existing fuzzy solution schemes themselves lack completeness, and during training, issues such as endpoint order reversal and disordered inter-layer nesting relationships easily occur. Structural instability issues such as core point overlap deviations make it difficult to reliably guarantee the mathematical validity and physical consistency of the solution results. Summary of the Invention

[0004] The purpose of this invention is to provide a quantum-classical physical information KAN network solution method for fuzzy differential equations, thereby solving the above-mentioned technical problems.

[0005] To achieve the above objectives, this invention provides a method for solving fuzzy differential equations using quantum-classical physical information KAN networks, comprising the following steps: S1. Based on the fuzzy partial differential equation to be solved and its initial and boundary value conditions, fuzzy parameters... Cut-set representation, through endpoint splitting and interval mapping of the control equation, yields the membership level. The lower and upper endpoint control equations are constructed simultaneously, including spatiotemporal coordinates and membership levels. The joint input vector; S2. Based on the joint input vector, feature mapping and function decoding are performed through a quantum-classical physical information KAN network composed of a ChebyKAN preprocessing module, a parameterized quantum circuit core module, and a ChebyKAN postprocessing module connected in series, to obtain the corresponding... The approximate solutions at the lower and upper endpoints of the cut set; S3. Based on the approximate solution of the lower endpoint, the approximate solution of the upper endpoint, the control equation of the lower endpoint, the control equation of the upper endpoint and the initial boundary conditions, the total loss function for network training is obtained by weighted combination of physical constraint loss and fuzzy structural constraint loss. S4. Based on the total loss function, all trainable parameters of the quantum-classical physical information KAN network are iteratively updated through a gradient descent optimizer. After the loss converges to a preset threshold, an approximate solution model for the fuzzy partial differential equation is obtained.

[0006] Preferably, S1 specifically includes: S11. Based on the triangular fuzzy number parameters in the fuzzy partial differential equation to be solved, through... The interval operation is used to obtain the corresponding membership level. The lower and upper bounds of the parameters are as follows; where the triangular fuzzy number is... The formula for calculating the cutoff set is: ; in, These are the triangular fuzzy type uncertain parameters in the fuzzy partial differential equation to be solved; The left support endpoint of the triangular fuzzy number; The kernel point of the triangular fuzzy number; The right support endpoint of the triangular fuzzy number; S12. Based on the lower bound and upper bound of the parameters and the monotonicity of the governing equations with respect to the parameters, the lower endpoint governing equations and the upper endpoint governing equations are obtained through endpoint extremum matching operations; among which the fuzzy solution... The cut set is represented in interval form, and the formula is: ; in, , They are respectively The lower and upper endpoints of the cut set; S13. Spatiotemporal coordinate variables and membership levels based on the original partial differential equation. By performing a dimension concatenation operation, a joint input vector of dimension d+1 is obtained, as shown in the formula: ; in, For joint input vectors; is the vector of independent variables of the original partial differential equation; It represents the subordinate level.

[0007] Preferably, the calculation process of the ChebyKAN preprocessing module in S2 specifically includes: S21. Based on the components of the joint input vector, nonlinear compression is performed using the hyperbolic tangent activation function to obtain normalized input components with values ​​ranging from [-1, 1]. The formula is as follows: ; in, For the normalized first One input component; The first of the original inputs One component; The hyperbolic tangent activation function is used to perform nonlinear compression operations. This represents the total number of input dimensions for the current ChebyKAN layer. S22. Based on the normalized input components, perform basis function expansion using the first-kind Chebyshev polynomial to obtain the values ​​of Chebyshev basis functions of each order; where the formula for the first-kind Chebyshev polynomial is: ; in, For the first Chebyshev polynomial of the first kind in order of order of input The function value at that location; Let be the order of the Chebyshev polynomial; S23. Based on the Chebyshev basis function values ​​of each order, weighted summation and cross-component aggregation are performed using trainable expansion coefficients to obtain a classical feature vector with dimensions matching the number of qubits, which is then input into the core module of the parameterized quantum circuit; the formula for calculating the output components of the ChebyKAN layer is: ; in, The first layer of ChebyKAN One output component; Let be the highest order of the Chebyshev polynomial expansion; For trainable Chebyshev expansion coefficients; This represents the total number of output dimensions for the current ChebyKAN layer.

[0008] Preferably, the calculation process of the parameterized quantum circuit core module in S2 specifically includes: S24. Based on the classical feature vectors output by the ChebyKAN preprocessing module, angle encoding is performed using the RX rotation gate to map the classical features to the quantum initial state, as shown in the formula: ; in, The initial quantum state after angle encoding. These are classic eigenvectors; For single-qubit gate sequential multiplication operators; This refers to the total number of qubits. To act on the first A rotation gate around the X-axis for 1 qubits, with a rotation angle equal to the classical eigencomponent. ; for All qubits are in a joint initial state where they are all in the ground state; S25. Based on the initial quantum state, the evolved final quantum state is obtained through an alternating parameterized quantum circuit containing multiple layers of single-qubit rotation gates and controlled entanglement gates. The calculation formula is as follows: ; in, The final quantum state after the parameterized quantum circuit has evolved; For the unitary evolution operator corresponding to the alternating parameterized quantum circuit; For the complete set of trainable parameters of a parameterized quantum circuit; This is the initial quantum state after RX rotating gate angle encoding; S26. Based on the quantum final state, the expected value of each qubit is measured using the Pauli-Z operator to obtain a quantum eigenvector with the same dimension as the number of qubits, which is then input into the ChebyKAN post-processing module. The formulas for calculating the expected value and quantum eigenvector of a single qubit are as follows: ; ; in, For the first Expected value of Pauli-Z operator measurement for 1 qubit; To act on the first Pauli-Z operator for 1 qubit; This is a quantum eigenvector.

[0009] Preferably, the calculation process of the ChebyKAN post-processing module in S2 specifically includes: S27. Based on quantum feature vectors, nonlinear mapping and feature dimensionality upscaling are performed through the first ChebyKAN layer to obtain the hidden layer feature vectors. S28. Based on the hidden layer feature vectors, feature decoding and dimensionality compression are performed through the second ChebyKAN layer to obtain... The approximate solutions at the lower and upper endpoints of the cut set.

[0010] Preferably, the process of constructing the physical constraint loss in S3 specifically includes: S31. Based on the approximate solutions at the lower and upper endpoints, the differential operators corresponding to the governing equations are calculated using automatic differentiation to obtain the lower and upper endpoint residuals at the placement points within the domain. The mean square error is then calculated to obtain the residual loss of the equations, as shown in the formula: ; in, This represents the residual loss of the equation. This represents the total number of residual configuration points within the domain. For the first Residuals of the lower endpoint control equations at each configuration point; For the first Residuals of the upper endpoint control equations at each configuration point; S32. Based on the approximate solutions of the lower and upper endpoints at the boundary configuration points and the given boundary conditions, the boundary condition loss is obtained by calculating the mean square error of the deviation. The formula is as follows: ; in, This represents the loss value due to boundary conditions. This represents the total number of boundary segments on the computational domain that require the application of boundary conditions. For the first Total number of configuration points on the segment boundary; This is an approximate solution for the lower endpoint; This is an approximate solution for the upper endpoint; For the first The first segment boundary The spatiotemporal coordinate vector of each configuration point; For the first The first on the boundary Membership level of each configuration point ; This is the complete set of trainable parameters for the QCPIKAN model; For the first The first segment boundary The target value of the lower endpoint boundary condition given at each configuration point; For the first The first segment boundary At each configuration point, the target value of the upper endpoint boundary condition is given; S33. For unsteady-state problems with a time dimension, based on the approximate solutions of the lower and upper endpoints at the initial time point and the given initial conditions, the initial condition loss is obtained by calculating the mean square error of the deviation. The formula is as follows: ; in, This is the initial conditional loss value; The total number of points configured at the initial time; For the first The spatial coordinate vectors of the initial configuration points; For the first Membership level of each initial configuration point ; For the first The target value of the initial condition of the lower endpoint given at each initial point; For the first The target value of the upper endpoint given at each initial point; S34. Based on the equation residual loss, boundary condition loss and initial condition loss, the physical constraint loss is obtained by weighted summation.

[0011] Preferably, the construction process of the fuzzy structural constraint loss in S3 specifically includes: S35. Membership levels based on the lower endpoint approximation solution and the upper endpoint approximation solution The partial derivatives are used to calculate the inter-layer nesting loss by penalizing the positive part of the inverse monotonicity deviation, which is then used to constrain the lower endpoint's tendency. Monotonic and undiminished, upper endpoint follows Monotonic non-increasing; nested loss between layers The calculation formula is: ; in, This represents the total number of nested constraint sampling points. It is a positive part function; For the approximate solution of the lower endpoint, the membership level The first-order partial derivative; For the approximate solution at the upper endpoint, the membership level The first-order partial derivative; For the first The spatiotemporal coordinate vector of nested sampling points; For the first Membership level corresponding to each nested sampling point ; S36. Based on the approximate solutions of the lower and upper endpoints under the same membership level, the endpoint orderliness loss is obtained by penalizing the deviation of the lower endpoint being greater than that of the upper endpoint. This loss is used to constrain the same... The validity of the solution interval; the loss due to the ordered nature of the endpoints. The formula is: ; in, The total number of sampling points is constrained by the orderliness requirement; For the first The spatiotemporal coordinate vector of an ordered sampling point; For the first Membership level corresponding to each ordered sampling point ; S37, Based on Membership Level The endpoint coincidence loss is obtained by calculating the mean square error of the deviation between the lower and upper endpoint approximate solutions at a given point. This loss is used to constrain the single-point kernel property of the triangular fuzzy number. The formula for the endpoint coincidence loss is: ; in, for Endpoint coincidence loss value; for The total number of constraint sampling points; For the first The spatiotemporal coordinate vector of each sampling point; S38. Based on the inter-layer nesting loss, endpoint ordering loss, and endpoint overlap loss, the fuzzy structure constraint loss is obtained through weighted summation; the total loss function... The weighted formula is: ; in, Weighting coefficients for controlling the residual loss of the equation; These are the weighting coefficients for the initial conditional loss; These are the weighting coefficients for the boundary condition loss; These are the weighting coefficients for the inter-layer nesting loss; The weighting coefficients for the endpoint orderliness loss; for Weighting coefficients for endpoint overlap loss.

[0012] Preferably, S4 specifically includes: S41. Based on the solution domain and membership level range of fuzzy partial differential equations, the domain-in-domain placement points, boundary placement points, and initial placement points are generated through the Latin hypercube sampling strategy to form a training sample set. S42. Based on the training sample set and the total loss function, iterative training is performed using the Adam optimizer, and the learning rate is dynamically adjusted in conjunction with the ReduceLROnPlateau learning rate scheduling strategy. S43. Based on the loss change curve during the training process, training is stopped when the total loss drops to a preset threshold or reaches the maximum number of iterations, resulting in a convergent fuzzy partial differential equation approximate solution model.

[0013] Preferably, the fuzzy partial differential equations in S43 include elliptic fuzzy Poisson equations, parabolic fuzzy heat conduction equations, parabolic fuzzy reaction-diffusion equations, and hyperbolic fuzzy convection equations; Specifically, for elliptic steady-state problems, the initial condition loss term in the total loss function is omitted. For the hyperbolic wave propagation problem, an additional wavefront position error is added as an auxiliary training constraint.

[0014] Preferably, after S4, there is also a precision verification step, specifically: Based on a set of independently sampled test points and a reference analytical solution or high-precision numerical solution, the average relative value is calculated. The accuracy evaluation results of the approximate solution model are obtained by combining the error and the wavefront position error; the wavefront position error is calculated by the coordinate deviation of the intersection point between the predicted profile and the set threshold.

[0015] Therefore, the present invention employs the aforementioned quantum-classical physical information KAN network solution method for fuzzy differential equations, which has the following beneficial effects: 1. By combining Chebyshev KAN with parameterized quantum circuits, a hybrid solution network for fuzzy partial differential equations is constructed. By leveraging the high-dimensional Hilbert space mapping capability of quantum entangled states and combining it with the adaptive fitting advantage of ChebyKAN's side functions, it can accurately capture the nonlinear coupling relationship between fuzzy parameters and spatiotemporal variables with a more streamlined parameter scale. Compared with the pure classical PIKAN architecture, it improves the approximation accuracy of fuzzy solutions and the training convergence efficiency.

[0016] 2. By using membership level as a continuous dimension and concatenating it with spatiotemporal coordinates to form a joint input vector, and simultaneously setting up dual output channels at the lower and upper endpoints, a one-time continuous solution for fuzzy solutions within the entire membership level range is achieved. The dual-output structure is naturally well-suited to this approach. The interval representation of the cut set ensures the structured expression of the understood fuzzy attributes in the network output layer, reducing the computational overhead and model redundancy of multi-level repetitive modeling.

[0017] 3. By constructing a multi-dimensional composite loss function of "physical conservation constraints + fuzzy structural constraints", in addition to the conventional equation residuals and initial boundary value conditions physical constraints, three new types of fuzzy mathematical constraints are added: endpoint orderliness, inter-layer nesting, and single-point kernel characteristics. This not only ensures that the solution results strictly follow the evolution law of the physical field, but also constrains the legality of the solution interval from the fuzzy mathematical definition level. This avoids structural anomalies such as interval nesting disorder, endpoint order reversal, and non-coincidence of kernel points that are prone to occur in traditional pure data-driven fuzzy solution schemes, thereby improving the physical consistency and mathematical validity of fuzzy solutions.

[0018] 4. A ChebyKAN layer based on the first type of Chebyshev polynomial is adopted as the pre- and post-processing module of the quantum circuit to complete the bidirectional mapping from classical input to quantum features and from quantum measurement values ​​to physical solutions. By utilizing the spectral approximation characteristics and orthogonality advantages of Chebyshev polynomials, it has a stronger ability to capture local high-frequency features compared with the fixed activation function of traditional MLP. It can better adapt to the solution requirements of complex physical scenarios such as fuzzy fronts and gradient abrupt changes, alleviate the numerical dispersion problem in strong gradient regions, and improve the local high-precision solution capability.

[0019] 5. A general solution paradigm covering elliptic, parabolic, and hyperbolic fuzzy partial differential equations has been formed. It can be adapted to different problem characteristics by flexibly adjusting the loss terms: the initial condition loss term can be omitted for steady-state elliptic problems, and the frontal position auxiliary constraint can be added for hyperbolic wave propagation problems. It can adapt to the problem-solving needs of uncertain partial differential equations in different engineering fields without reconstructing the main network architecture.

[0020] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0021] Figure 1 A schematic diagram of the quantum-classical physical information KAN (QCPIKAN) algorithm for solving fuzzy differential equations, provided by this invention. Figure 2 A comparison chart of training loss for Example 1 (fuzzy Poisson equation) in the embodiments provided by the present invention; Figure 3 Example 1 in the embodiments provided by the present invention The spatial distribution diagram of the solutions at the upper and lower endpoints, where... Figure 3 (a) in the figure is the QCPIKAN prediction result graph. Figure 3 (b) in the figure is the PIKAN prediction result graph; Figure 4 Example 1 in the embodiments provided by the present invention Spatiotemporal interval width and interlayer nesting violation distribution diagram, in which, Figure 4 (a) in the figure is the QCPIKAN solution interval width distribution diagram and the inter-layer nesting violation distribution diagram. Figure 4 (b) in the figure shows the distribution of the width of the PIKAN solution interval and the distribution of nested violations between layers; Figure 5 Example 1 in the embodiments provided by the present invention The absolute error plot of the upper and lower endpoint solutions, where, Figure 5 (a) in the figure is the absolute error distribution of the lower endpoint of QCPIKAN. Figure 5(b) in the figure is the absolute error distribution diagram of the upper endpoint of QCPIKAN. Figure 5 (c) in the figure is the absolute error distribution diagram of the lower endpoint of PIKAN. Figure 5 (d) in the figure is the distribution of the absolute error at the upper end of the PIKAN. Figure 6 This is a comparison diagram of the prediction results and reference solutions at different membership levels in Example 1 of the embodiments provided by the present invention, wherein... Figure 6 (a) The predicted endpoints of the center location in the middle location follow Change curve graph Figure 6 (b) in the figure is a comparison chart of the average relative error at different membership levels; Figure 7 The graph shows the sensitivity results of quantum circuit parameters in Example 1 of the embodiments provided by the present invention, wherein... Figure 7 (a) in the figure shows the effect of the number of qubits on the mean absolute error. Figure 7 (b) in the figure shows the effect of the number of quantum circuit layers on the mean absolute error; Figure 8 A comparison chart of training loss for the (fuzzy heat conduction equation) in Example 2 of the embodiments provided by the present invention; Figure 9 The time evolution diagram of the upper and lower endpoint solutions at different membership levels at spatial location 0.49 in Example 2 of the embodiments provided by the present invention, wherein, Figure 9 (a) in the figure is the QCPIKAN prediction result. Figure 9 (b) in the figure is the PIKAN prediction result graph; Figure 10 The spatial interval width and interlayer nesting violation distribution diagram at t=0.49 in Example 2 of the embodiments provided by the present invention, wherein, Figure 10 (a) in the figure is the QCPIKAN solution interval width distribution diagram and the inter-layer nesting violation distribution diagram. Figure 10 (b) in the figure shows the distribution of the width of the PIKAN solution interval and the distribution of nested violations between layers; Figure 11 Example 2 in the embodiments provided by the present invention The absolute error plot of the upper and lower endpoint solutions, where, Figure 11 (a) in the figure is the absolute error distribution of the lower endpoint of QCPIKAN. Figure 11 (b) in the figure is the absolute error distribution diagram of the upper endpoint of QCPIKAN. Figure 11 (c) in the figure is the absolute error distribution diagram of the lower endpoint of PIKAN. Figure 11 (d) in the figure is the distribution of the absolute error at the upper end of the PIKAN. Figure 12This is a comparison diagram of the prediction results and reference solution at different membership levels in Example 2 of the embodiments provided by the present invention, wherein... Figure 12 In the example, (a) represents the endpoint of a fixed spatiotemporal location. Change curve graph Figure 12 (b) in the figure is a comparison of the average relative L2 error at different membership levels; Figure 13 A comparison chart of training loss for the (fuzzy Fisher-KPP reaction-diffusion equation) in Example 3 of the embodiments provided by the present invention; Figure 14 The time evolution diagram of the upper and lower endpoint solutions at different membership levels at spatial location 0.49 in Example 3 of the embodiments provided by the present invention, wherein, Figure 14 (a) in the figure is the QCPIKAN prediction result graph. Figure 14 (b) in the figure is the PIKAN prediction result graph; Figure 15 The spatial interval width and inter-layer nesting violation distribution diagram at t=0.49 in Example 3 of the embodiments provided by the present invention, wherein, Figure 15 (a) in the figure is the QCPIKAN solution interval width distribution diagram and the inter-layer nesting violation distribution diagram. Figure 15 (b) in the figure is the distribution diagram of the width of the PIKAN solution interval and the distribution diagram of the nesting violation between layers; Figure 16 Example 3 in the embodiments provided by the present invention The absolute error plot of the upper and lower endpoint solutions, where, Figure 16 (a) in the figure is the absolute error distribution of the lower endpoint of QCPIKAN. Figure 16 (b) in the figure is the absolute error distribution diagram of the upper endpoint of QCPIKAN. Figure 16 (c) in the figure is the absolute error distribution diagram of the lower endpoint of PIKAN. Figure 16 (d) in the figure is the distribution of the absolute error at the upper end of the PIKAN. Figure 17 This is a comparison diagram of the prediction results and reference solutions at different membership levels in Example 3 of the embodiments provided by the present invention, wherein... Figure 17 In the example, (a) represents the endpoint at a fixed position. Change curve graph Figure 17 (b) in the figure is a comparison chart of the average relative error at different membership levels; Figure 18 A comparison chart of training loss for the (fuzzy convection equation) in Example 4 of the embodiments provided by the present invention; Figure 19 Example 4 in the embodiments provided by the present invention The spatiotemporal distribution diagram of the solutions at the upper and lower endpoints, where, Figure 19(a) in the figure is the QCPIKAN prediction result. Figure 19 (b) in the figure is the PIKAN prediction result graph; Figure 20 Example 4 in the embodiments provided by the present invention Spatiotemporal interval width and interlayer nesting violation distribution diagram, in which, Figure 20 (a) in the figure is the QCPIKAN solution interval width distribution diagram and the inter-layer nesting violation distribution diagram. Figure 20 (b) in the figure shows the distribution of the width of the PIKAN solution interval and the distribution of nested violations between layers; Figure 21 The wavefront profile prediction and average wavefront position error diagram in Example 4 of the embodiments provided by the present invention, wherein, Figure 21 (a) in the figure is a comparison diagram of different horizontal wavefront profiles at time t=0.24. Figure 21 (b) in the figure is a comparison of the average wavefront position error at different membership levels. Detailed Implementation

[0022] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the embodiments of the present invention will be further described in detail below with reference to the accompanying drawings and examples. It should be understood that the specific embodiments described herein are merely illustrative of the embodiments of the present invention and are not intended to limit the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of this application. Examples of the embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout.

[0023] It should be noted that the terms "comprising" and "having," and any variations thereof, are intended to cover non-exclusive inclusion, such as a process, method, system, product, or server that includes a series of steps or units, not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such process, method, product, or device.

[0024] The following is in conjunction with the appendix Figures 1-21 The embodiments of the present invention will be described in detail below.

[0025] Based on the above analysis, a specific embodiment is given. This embodiment provides a quantum-classical physical information KAN network solution method for fuzzy differential equations. The overall algorithm flow is as follows: Figure 1 As shown, the method specifically includes the following steps: S1. Fuzzy Equation Endpoint Decomposition and Joint Input Construction: Based on the fuzzy partial differential equation to be solved and its initial and boundary value conditions, fuzzy parameters... Cut-set representation, through endpoint splitting and interval mapping of the control equation, yields the membership level. The lower and upper endpoint control equations are constructed simultaneously, including spatiotemporal coordinates and membership levels. The joint input vector.

[0026] This embodiment is based on The fuzzy partial differential equations represented by truncated sets are solved. The general control form of the problem to be solved includes differential operators, initial condition operators and boundary condition operators. The fuzzy parameters, source terms and initial boundary value data in the equations are all represented in interval form.

[0027] S11. Based on the triangular fuzzy number parameters in the fuzzy partial differential equation to be solved, through... The interval operation is used to obtain the corresponding membership level. The lower bound and upper bound of the parameter; Among them, the triangular fuzzy number The formula for calculating the cutoff set is: ; in, These are the triangular fuzzy type uncertain parameters in the fuzzy partial differential equation to be solved; For the membership level, the value range is: , used to characterize the degree to which a parameter belongs to a fuzzy set; Triangular fuzzy number At the level of membership Below A cutoff set is in the form of a closed interval. The left support endpoint of the triangular fuzzy number, i.e., the membership level. The lower bound of the time parameter interval; The kernel point of the triangular fuzzy number, i.e., the membership level. The corresponding parameter value; The right support endpoint of the triangular fuzzy number, i.e., the membership level. The upper bound of the time parameter interval.

[0028] S12. Based on the lower bound of the parameters, the upper bound of the parameters, and the monotonicity of the control equations with respect to the parameters, the lower endpoint control equations and the upper endpoint control equations are obtained through endpoint extremum matching operations. Accordingly, fuzzy value solutions At the level of membership Below The cut set is represented as: ; in, For fuzzy value solutions At the level of membership Below Collection; for The function that controls the lower endpoint of a cutoff set; for The function at the upper endpoint of the cut set.

[0029] The fuzzy solution must satisfy a three-layer structure constraint: the same Order constraint that the lower endpoint is not greater than the upper endpoint, and the lower endpoint follows... Monotonic and does not decrease and the upper endpoint follows Monotonically non-increasing nested constraints between layers Single-point kernel constraint where the upper and lower endpoints coincide. The fuzzy parameters are then compared with the initial boundary value data. Substituting the cut set into the original governing equations, we can obtain the residual forms of the governing equations corresponding to the lower and upper endpoints, which serve as the basis for constructing subsequent physical constraints.

[0030] Among them, fuzzy solutions The cut set is represented in interval form, and the formula is: ; in, for The lower endpoint solution of the cut set. for The upper endpoint solution of the cut set; the lower endpoint solution with respect to... Monotony remains unchanged, the upper end is about It is monotonically non-increasing, forming a nested interval family that gradually shrinks as the membership level increases.

[0031] S13. Spatiotemporal coordinate variables and membership levels based on the original partial differential equation. By performing a dimension concatenation operation, a joint input vector of dimension d+1 is obtained, as shown in the formula: ; in, For joint input vectors; is the vector of independent variables of the original partial differential equation; It represents the subordinate level.

[0032] S2. QCPIKAN Network Forward Mapping Solution: Based on the joint input vector, feature mapping and function decoding are performed through a quantum-classical physical information KAN (QCPIKAN) network composed of a ChebyKAN preprocessing module, a parameterized quantum circuit core module, and a ChebyKAN postprocessing module connected in series, to obtain the corresponding... The approximate solutions at the lower and upper endpoints of the cut set.

[0033] The overall mapping relationship of the QCPIKAN network can be represented as: All trainable parameters consist of two parts: classical network coefficients and quantum circuit parameters. (1) ChebyKAN preprocessing module: S21. Based on the components of the joint input vector, nonlinear compression is performed using the hyperbolic tangent activation function to obtain normalized input components with values ​​ranging from [-1, 1]. The formula is as follows: ; in, For the normalized first One input component; The first of the original inputs One component; The hyperbolic tangent activation function is used to perform nonlinear compression operations. This represents the total number of input dimensions for the current ChebyKAN layer.

[0034] S22. Based on the normalized input components, the basis functions are expanded using the first-type Chebyshev polynomials to obtain the values ​​of Chebyshev basis functions of each order. The definition of a Chebyshev polynomial of the first kind is: ; in, For the first Chebyshev polynomial of the first kind in order of order of input The function value at that location; Let be the order of the Chebyshev polynomial.

[0035] S23. Based on the Chebyshev basis function values ​​of each order, weighted summation and cross-component aggregation are performed through trainable expansion coefficients to obtain a classical feature vector matching the dimension and number of qubits, which is then input into the core module of the parameterized quantum circuit. The formula for calculating the output components of the ChebyKAN layer is: ; In the formula, The first layer of ChebyKAN One output component; Let be the highest order of the Chebyshev polynomial expansion; For trainable Chebyshev expansion coefficients; This represents the total number of output dimensions for the current ChebyKAN layer.

[0036] In this embodiment, the preprocessing module includes two layers of ChebyKAN, which sequentially complete the mapping from the input dimension to the hidden dimension and from the hidden dimension to the qubit dimension.

[0037] The ChebyKAN layer mapping consisting of all trainable parameters is denoted as: ; in, This is the output feature vector of the ChebyKAN layer; (2) Core module of parameterized quantum circuit: For the parameterized mapping operator of the ChebyKAN layer; This is the complete set of trainable parameters for the ChebyKAN layer. This is the input feature vector of the ChebyKAN layer.

[0038] S24. Based on the classical feature vectors output by the ChebyKAN preprocessing module, angle encoding is performed using the RX rotation gate to map the classical features to the quantum initial state, as shown in the formula: ; in, The initial quantum state after angle encoding. These are classic eigenvectors; For multi-qubit tensor product chain multiplication operator, representing the chain multiplication of the first to the second qubits. Each qubit is sequentially subjected to its own RX rotation gate; This refers to the total number of qubits. This is a quantum rotation gate around the X-axis acting on the j-th qubit, with the rotation angle equal to the classical eigencomponent. ; for All qubits are in the joint initial state of the ground state.

[0039] S25. Based on the initial quantum state, the evolved final quantum state is obtained through an alternating parameterized quantum circuit containing multiple layers of single-qubit rotation gates and controlled entanglement gates. The calculation formula is as follows: ; in, The final quantum state after the parameterized quantum circuit has evolved; For the unitary evolution operator corresponding to the alternating parameterized quantum circuit; For the complete set of trainable parameters of a parameterized quantum circuit; This is the initial quantum state after RX rotation gate angle encoding.

[0040] In this embodiment, the alternating parameterized quantum circuit adopts a cross-entangled topology, which is composed of parameterized two-qubit modules of adjacent qubits interleaved. For the control qubits... With the target qubit The unitary transform of the parameterized two-qubit module is expressed as: ; in, For the action in the , No. Parameterized two-qubit unitary transform operator on two qubits; To act on the first A qubit orbit Axola revolving door; To act on the first A qubit orbit Axola revolving door; This is the symbol for the tensor product (direct product); It is a controlled NOT gate (CNOT gate); To act on the first The orbit of one quantum bit Axola revolving door; To act on the first The orbit of one quantum bit Axola revolving door; , , , These are all four trainable parameters of the double-bit module.

[0041] S26. Based on the quantum final state, by measuring the expectation value of each qubit using the Pauli-Z operator, a quantum feature vector with the same dimension as the number of qubits is obtained and input into the ChebyKAN post-processing module. The formulas for calculating the expected value of a single quantum bit measurement and the quantum eigenvector are as follows: ; ; in, For the first Expected value of Pauli-Z operator measurement for 1 qubit; To act on the first Pauli-Z operator for 1 qubit; This is a quantum eigenvector.

[0042] (3) ChebyKAN post-processing module: S27. Based on quantum feature vectors, nonlinear mapping and feature dimensionality upscaling are performed through the first ChebyKAN layer to obtain the hidden layer feature vectors. S28. Based on the hidden layer feature vectors, feature decoding and dimensionality compression are performed through the second ChebyKAN layer to obtain... The approximate solutions at the lower and upper endpoints of the cut set.

[0043] The overall mapping of the post-processing module is represented as follows: ; in, This is an approximate result of the fuzzy solution output by the post-processing module; This is the overall mapping function for the post-processing (output) ChebyKAN network; Mapping to the first ChebyKAN layer of the post-processing module; This is the mapping for the second ChebyKAN layer of the post-processing module; This is the input vector for the post-processing module; for Approximate solution at the lower endpoint of the cut set; for Approximate solution at the upper endpoint of the cut set; , These are the sets of trainable parameters for the first and second ChebyKAN layers, respectively.

[0044] All trainable parameters of the QCPIKAN network are composed of preprocessing parameters, quantum circuit parameters, and postprocessing parameters, collectively denoted as: ; in, This is the total set of all trainable parameters for the QCPIKAN network; This is the set of trainable parameters for the first layer of the preprocessed ChebyKAN network; This is the set of trainable parameters for the second layer of the preprocessed ChebyKAN network; This is the complete set of trainable parameters for a parameterized quantum circuit.

[0045] S3. Construction of the composite loss function: Based on the approximate solutions of the lower and upper endpoints, the control equations of the lower and upper endpoints, and the initial and boundary conditions, the total loss function used for network training is obtained by weighted combination of physical constraint loss and fuzzy structural constraint loss.

[0046] (1) Physical constraint loss: S31. Based on the approximate solutions at the lower and upper endpoints, the differential operators corresponding to the governing equations are calculated using automatic differentiation to obtain the lower and upper endpoint residuals at the placement points within the domain. The mean square error is then calculated to obtain the residual loss of the equations, as shown in the formula: ; in, This represents the residual loss of the equation. This represents the total number of residual configuration points within the domain. For the first Residuals of the lower endpoint control equations at each configuration point; For the first The upper endpoint control equation residuals at each configuration point.

[0047] S32. Based on the approximate solutions of the lower and upper endpoints at the boundary configuration points and the given boundary conditions, the boundary condition loss is obtained by calculating the mean square error of the deviation. The formula is as follows: ; in, This represents the loss value due to boundary conditions. This represents the total number of boundary segments on the computational domain that require the application of boundary conditions. For the first Total number of configuration points on the segment boundary; This is an approximate solution for the lower endpoint; This is an approximate solution for the upper endpoint; For the first The first segment boundary The spatiotemporal coordinate vector of each configuration point; For the first The first on the boundary Membership level of each configuration point ; This is the complete set of trainable parameters for the QCPIKAN model; For the first The first segment boundary The target value of the lower endpoint boundary condition given at each configuration point; For the first The first segment boundary At each configuration point, the target value of the upper endpoint boundary condition is given.

[0048] S33. For unsteady-state problems with a time dimension, based on the approximate solutions of the lower and upper endpoints at the initial time point and the given initial conditions, the initial condition loss is obtained by calculating the mean square error of the deviation. The formula is as follows: ; in, This is the initial conditional loss value; The total number of points configured at the initial time; For the first The spatial coordinate vectors of the initial configuration points; For the first Membership level of each initial configuration point ; For the first The target value of the initial condition of the lower endpoint given at each initial point; For the first The target value of the initial conditions of the upper endpoint given at the initial point.

[0049] S34. Based on the equation residual loss, boundary condition loss and initial condition loss, the physical constraint loss is obtained by weighted summation.

[0050] (2) Fuzzy structural constraint loss: S35. Membership levels based on the lower endpoint approximation solution and the upper endpoint approximation solution The partial derivatives are used to calculate the inter-layer nesting loss by penalizing the positive part of the inverse monotonicity deviation, which is then used to constrain the lower endpoint's tendency. Monotonic and undiminished, upper endpoint follows Monotonic non-increasing; nested loss between layers The calculation formula is: ; in, This represents the total number of nested constraint sampling points. It is a positive part function; For the approximate solution of the lower endpoint, the membership level The first-order partial derivative; For the approximate solution at the upper endpoint, the membership level The first-order partial derivative; For the first The spatiotemporal coordinate vector of nested sampling points; For the first Membership level corresponding to each nested sampling point .

[0051] S36. Based on the approximate solutions of the lower and upper endpoints under the same membership level, the endpoint orderliness loss is obtained by penalizing the deviation of the lower endpoint being greater than that of the upper endpoint. This loss is used to constrain the same... The validity of the solution interval; the loss due to the ordered nature of the endpoints. The formula is: ; in, The total number of sampling points is constrained by the orderliness requirement; For the first The spatiotemporal coordinate vector of an ordered sampling point; For the first Membership level corresponding to each ordered sampling point .

[0052] S37, Based on Membership Level The endpoint coincidence loss is obtained by calculating the mean square error of the deviation between the lower and upper endpoint approximate solutions at a given point. This loss is used to constrain the single-point kernel property of the triangular fuzzy number. The formula for the endpoint coincidence loss is: ; in, for Endpoint coincidence loss value; for The total number of constraint sampling points; For the first The spatiotemporal coordinate vector of each sampling point.

[0053] S38. Based on the inter-layer nesting loss, endpoint ordering loss, and endpoint overlap loss, the fuzzy structure constraint loss is obtained through weighted summation; the weighted formula for the total loss function is: ; in, Weighting coefficients for controlling the residual loss of the equation; These are the weighting coefficients for the initial conditional loss; These are the weighting coefficients for the boundary condition loss; These are the weighting coefficients for the inter-layer nesting loss; The weighting coefficients for the endpoint orderliness loss; for Weighting coefficients for endpoint coincidence loss; each weighting coefficient is a non-negative constant and its value can be adjusted according to the type of equation.

[0054] The optimization objective of network training is to find the optimal parameters to minimize the total loss. ; in, This represents the optimal set of trainable parameters. The operator is the minimum value of the independent variable; This is the total loss function.

[0055] S4. Network Training and Model Solving: Based on the total loss function, all trainable parameters of the quantum-classical physical information KAN network are iteratively updated by a gradient descent optimizer. After the loss converges to a preset threshold, an approximate solution model for the fuzzy partial differential equation is obtained.

[0056] Step S4 specifically includes: S41. Based on the solution domain and membership level range of fuzzy partial differential equations, the domain-in-domain placement points, boundary placement points, and initial placement points are generated through the Latin hypercube sampling strategy to form a training sample set. S42. Based on the training sample set and the total loss function, iterative training is performed using the Adam optimizer, and the learning rate is dynamically adjusted in conjunction with the ReduceLROnPlateau learning rate scheduling strategy. For elliptic steady-state problems, the initial condition loss term in the total loss function is omitted. , that is to say For hyperbolic wave propagation problems, wavefront position error can be added as an auxiliary training constraint.

[0057] S43. Based on the loss change curve during the training process, training is stopped when the total loss drops to a preset threshold or reaches the maximum number of iterations, resulting in a convergent fuzzy partial differential equation approximate solution model.

[0058] It also includes the QCPIKAN theoretical error advantage analysis: This implementation analyzes the reasons why QCPIKAN may achieve higher accuracy than PIKAN from three aspects: function approximation error, physical residual error, and fuzzy structure constraint error. Notably, parameterized quantum circuits cannot guarantee higher accuracy for arbitrary fuzzy differential equations without additional conditions. Therefore, the following conclusions are based on assumptions such as well-posed and stable equations, the use of the same physical information loss in both models, negligible quantum measurement errors, and the ability of the QCPIKAN feature space to effectively represent the non-separable coupling between input variables.

[0059] (1) Optimal approximation error and residual stability: Denote combined input as The exact upper and lower endpoint solutions of the fuzzy differential equation are , For the corresponding spatiotemporal-membership level domain, for any model , respectively and Let PIKAN and QCPIKAN be the functional representation spaces, and let their best approximation error under the norm be defined as: ; in, For the first The optimal approximation error of parameterized quantum circuits; For the infimum operator; For the first The set of all unitary transformations that can be expressed by parameterized quantum circuits; For the unitary transformation operator of the target to be approximated; This is a candidate unitary transform operator that can be realized by parameterized quantum circuits; It is the Hilbert-Schmidt norm; This serves as an identifier for the circuit topology category.

[0060] set up This includes the governing equations of the upper and lower endpoints, initial and boundary value conditions, endpoint order, inter-layer nesting, and... If the problem satisfies residual stability, then the joint operator consisting of endpoint coincidence conditions has a constant independent of the network parameters. Such that any approximate solution The formula is: ; in, It is the stability constant; For governing equation operators (residual operators); This is the norm of the output space.

[0061] If the weights of each loss term are positive and the sampling error can be controlled, and the total error of the model is decomposed into five parts: representation error, optimization error, sampling error, structural constraint error, and quantum noise error, then the upper bound of the total error of the model can be written as: ; in, For the first The total error of the model corresponding to parameterized quantum circuits; To express / approximate the error; To optimize error; This is the sampling error; For structural constraint errors; This is quantum noise error.

[0062] (2) The representational advantages of hybrid feature spaces: The potential advantage of QCPIKAN stems from its hybrid feature space. If we consider PIKAN as a model using only the classical ChebyKAN mapping, while QCPIKAN further introduces parameterized quantum feature maps on the same input, then the quantum measurement features can be written as: ; in, The feature vector output by the quantum module; for A qubit-zero quantum initial state; The total number of qubits in a parameterized quantum circuit; Encode the unitary operator for input; For the global unitary evolution operator of parameterized quantum circuits; For the first A Pauli measurement can be used for observable measurements; is the conjugate transpose of the operator.

[0063] Compared to classical representations that only perform one-to-one univariate mappings, entangled circuits can introduce mixed terms between variables into the measurement features. Therefore, QCPIKAN can provide additional interactive features at the same or similar classical parameter budgets when the fuzzy endpoint solutions simultaneously contain spatial-temporal coupling, spatiotemporal-membership coupling, or parameter-decoupling.

[0064] The exact solution is decomposed into separable and non-separable coupled parts that can be easily fitted by classical representation: ,in, This is the exact analytical solution to the partial differential equation to be solved; The solution is a separable partial solution; This is the solution for the non-separable coupled portion. If the approximation error of the quantum feature for the non-separable portion is significantly lower than that of the classical Cheby-Kan, i.e., it satisfies: ; in, For candidate approximation operators; For the operator expression space of the quantum-enhanced QCPIKAN model; This is the operator representation space for the pure classical Cheby-KAN model; This is an infimum operation; This represents the optimal approximation error of the quantum enhancement model for the non-separable coupled components; This represents the optimal approximation error of the pure classical Cheby-KAN model for non-separable coupled components.

[0065] QCPIKAN then exhibits a strict best approximation advantage in representing the structure of non-separable fuzzy solutions. This advantage is typically more pronounced in low to medium membership levels, propagation regions with large gradients, and when fuzzy parameters significantly influence the decay rate or wavefront position of the solution.

[0066] (3) Basic assumptions and representation space containment relationship: set up This represents the exact upper and lower endpoint solutions of the fuzzy differential equation. For the model The function representation space, where These correspond to PIKAN and QCPIKAN, respectively. This is an approximate solution obtained through training. The theoretical analysis in this paper adopts the following assumptions: A1: The endpoint problem has a well-defined and approximate solution. Residual stability of the formula; A2: Both models use the same physical operators. Cut set structure constraints, loss weights, sampling rules, and error norm; A3: The mixed feature space of QCPIKAN can at least reproduce the classical representation of PIKAN, or has a strict best approximation advantage under a fixed model size; A4: Sampling errors caused by finite collocation points, insufficient convergence errors in optimization, and soft fuzzy structure constraint errors can all be controlled by finite terms.

[0067] Specifically, the numerical experiments in this embodiment use analytical expectation values ​​in an ideal quantum simulation environment and calculate the network input and parameter gradients through automatic differentiation or analytical backpropagation, thus eliminating random measurement noise or stochastic gradient noise; simultaneously, the ideal quantum simulation does not generate hardware noise. For real quantum hardware, these error terms still need to be retained. Under assumption A3, if the representation space of PIKAN is embedded in the representation space of QCPIKAN, then: ; In the formula, This is the operator representation space for the pure classical PIKAN model; Let be the operator expression space of the QCPIKAN quantum enhancement model; The optimal approximation error of the QCPIKAN model; This represents the optimal approximation error of the pure classical PIKAN model. This equation theoretically proves that the lower bound of the representational power of QCPIKAN is no weaker than that of classical PIKAN.

[0068] To further obtain a strict advantage, the target solution needs to contain non-separable interaction components that are difficult to represent with a given network size, but which can be effectively approximated by the quantum eigenmaps of QCPIKAN, as shown in the formula: ; ; in, The remaining residual components of the solution; An orthogonal projection operator for quantum-enhanced QCPIKAN model expression space; For the orthogonal projection operator of the expression space of the pure classical PIKAN / ChebyKAN model; This represents the optimal approximation error of the quantum model for strongly coupled components. This represents the optimal approximation error of the classical model for strongly coupled components.

[0069] Let represent the separable part that is easily represented by the classical branch, represent the mixture of space, time, and membership level, represent the remaining approximation error, and represent the best projections on the two feature spaces, respectively. The above shows that the approximation error of quantum entanglement features for non-separable fuzzy structures is smaller than that of PIKAN; therefore, QCPIKAN has a positive representational gain in error decomposition. This condition is not automatically guaranteed by the existence of quantum circuits, but is determined by the specific circuit structure, input encoding, model size, and the interaction complexity of the target solution.

[0070] (4) Comparison of upper bounds of error and proof of the theorem: The composition of the upper bound of the total error is further refined, and the formula is as follows: ; ; in, To optimize error; This is the sampling error; For structural constraint errors; This represents the gradient estimation error; This is due to hardware noise error; For the first The upper bound of the final total error of the class model.

[0071] According to the triangle inequality, the error from the exact solution to the training solution can be divided into the optimal representation error and the training process error; further, from any approximate solution... The stability estimate of the formula and the control relationship of each loss term on the residual in the total loss function formula, as well as the ideal conditions in this embodiment, are discussed. Furthermore, PIKAN does not exhibit quantum hardware noise, therefore The upper bound estimate of the total error can be obtained as follows: ; in, For the first The actual approximate solution operator obtained by training the class model.

[0072] Assuming A1~A4 hold true, if the representational gain of QCPIKAN is greater than its increased computational error relative to PIKAN, the error formula is: ; in, To approximate the error gain.

[0073] For the ideal numerical experiment used in this embodiment, both the gradient error and hardware noise error in the above formula are zero. Therefore, the sufficient condition of the theorem is simplified to: ; The upper bound of the theoretical endpoint solution error of QCPIKAN is strictly less than that of PIKAN, that is: ; The proof is as follows: Based on the formula for the upper bound of the total error, the formula for the composition of the upper bound of the total error, and the formula for the estimation of the upper bound of the total error, the two models respectively satisfy the following formulas: ; ; Secondly, the equivalent formula for the error formula is: ; Due to stability constant Multiply both sides of the equivalent formula of the error formula by The formula is: ; You can get it immediately. The theorem holds.

[0074] For high-gradient fronts and discontinuous solution scenarios, ChebyKAN's spectral approximation properties can improve the convergence of high-frequency component errors from algebraic to exponential order. Combined with the sharpening effect of quantum entanglement on front features, QCPIKAN can further reduce gradient errors and front location errors. The Hausdorff distance of the front position is also controlled by the approximation error. Therefore, its advantages are more significant in fuzzy problems where convection is dominant and there are moving fronts.

[0075] Furthermore, step S4 is followed by an accuracy verification step: based on the independently sampled set of test points and the reference analytical solution or high-precision numerical solution, the average relative accuracy is calculated. The accuracy evaluation results of the approximate solution model are obtained by combining the error and the wavefront position error; the wavefront position error is calculated by the coordinate deviation of the intersection point between the predicted profile and the set threshold.

[0076] In this embodiment, the total number of training rounds is uniformly set to 20,000 rounds, the batch size is adjusted according to the scale of the test cases, and the initial learning rate is set to... The learning rate scheduling strategy has a decay factor of 0.9 and a patience value of 1000.

[0077] The numerical verification process is as follows: To systematically verify the performance of QCPIKAN in solving different types of fuzzy partial differential equations, this paper selects four typical examples—elliptic fuzzy Poisson equations, parabolic fuzzy heat conduction equations, nonlinear parabolic Fisher-KPP reaction-diffusion equations, and hyperbolic fuzzy convection equations—for comparative experiments, covering three mainstream partial differential equation forms. The experiments use the classic PIKAN as the benchmark model for comparison. Both models employ identical sampling rules, loss weights, and optimization hyperparameters, and are uniformly set with 20,000 training iterations and a batch size of 256. The initial learning rate is used, and the entire process is conducted in an ideal quantum simulation environment. The testing dimensions cover three categories of indicators: numerical accuracy of the solution, compliance of fuzzy structure, and accuracy of wavefront position. Accompanying charts include training loss curves, spatial / spatiotemporal distribution cloud maps, and error comparison curves, which intuitively present the differences in model performance.

[0078] Example 1: For elliptic fuzzy Poisson equations, with appendix Figure 2 The training loss curves show that QCPIKAN's loss remains lower than PIKAN's after approximately 8000 iterations, and its final loss after convergence is only about half that of PIKAN, demonstrating more thorough optimization of the composite loss objective. Figure 3 To be continued Figure 5 The solution space distribution and error contour plots show that both models can reproduce the central convex solution morphology under homogeneous boundaries. QCPIKAN's solution surface is smoother and the boundary transition is more natural; its accuracy advantage is particularly significant at medium to high membership levels. At the four levels, the average relative error of PIKAN was 1.4 times, 1.6 times, 1.9 times, and 2.0 times that of QCPIKAN, respectively; (See attached text) Figure 4 The nested violation distribution further shows that violations in both models are concentrated in the boundary and corner regions, with QCPIKAN exhibiting smaller and less severe violations. (Appendix) Figure 7 The parameter sensitivity experiment further verifies that increasing the number of qubits can stably reduce the solution error, but the circuit depth and accuracy are not monotonically correlated. The two-layer quantum circuit has the best overall performance under the current configuration.

[0079] Examples 2 and 3: For parabolic fuzzy heat conduction equations and nonlinear reaction-diffusion equations, appendices are provided. Figure 8 To be continued Figure 12 Appendix Figure 13 To be continued Figure 17 The complete results for the two types of examples are presented separately. For linear heat conduction problems, QCPIKAN's loss decreases more smoothly and exhibits smaller fluctuations in later stages, establishing an accuracy advantage after approximately 2000 rounds. The error difference is most pronounced at low membership levels. The average relative L2 error of the PIKAN is approximately 2.0 times that of the QCPIKAN; its characterization of the fuzzy characteristic that "the solution interval shrinks as the membership level increases" is more consistent with the reference solution. The endpoint overlap is also better. For the nonlinear Fisher-KPP problem, QCPIKAN's loss is consistently lower than PIKAN's from the early stages of training, with the final loss being only about 1 / 2.5 of the comparison model. At all membership levels, PIKAN's average relative error is 1.1 to 2.7 times that of QCPIKAN. Compared to PIKAN's tendency to reverse the endpoint order in the low-amplitude region, QCPIKAN's upper and lower endpoint curves are separated more smoothly, and its endpoint order is better.

[0080] Example 4: For hyperbolic fuzzy convection equations, appendix Figure 18 To be continued Figure 21 The results show that both models can reproduce the oblique wavefront pattern propagating from left to right, and the solution interval width is concentrated in the wavefront transition region, which conforms to the physical law that "velocity uncertainty is mainly manifested as wavefront position shift". The core advantage of QCPIKAN lies in its high accuracy in capturing the wavefront position: (See attached image) Figure 21Wavefront profiles and error statistics show that QCPIKAN's wavefront position error is lower than PIKAN's at all test membership levels. The overall average wavefront position error is only about 1 / 1.77 of PIKAN's, and the accuracy difference at intermediate membership levels can reach 3.24 times, making it more accurate in characterizing the uncertainty of the moving front.

[0081] Overall, the four types of examples jointly verified QCPIKAN's general ability to solve different types of fuzzy partial differential equations. The enhanced expression of the coupling between spatiotemporal variables and membership levels by quantum entanglement is the core reason why its accuracy is superior to that of classical PIKAN. However, both models have local fuzzy structure violations at the boundaries, in strong gradient regions, and near the wavefront. QCPIKAN did not show a consistent advantage in structural compliance, which also points the way for subsequent optimization of the constraint mechanism of the model.

[0082] In summary, existing classical physical information networks in fuzzy partial differential equation solving are insufficient in characterizing the strongly coupled nonlinear relationships between spatiotemporal variables and membership levels, and have low accuracy in approximating high gradients and wavefront regions. Furthermore, existing quantum-classical hybrid solution frameworks are not well-suited for the two-endpoint solution scenario of fuzzy α-cut sets, and the lack of fuzzy structural constraints easily leads to endpoint order reversal, inter-layer nesting violations, and other issues. To address mathematical legitimacy defects such as non-coincidence of core points, a QCPIKAN solution framework was constructed, consisting of a ChebyKAN preprocessing module, a parameterized quantum circuit core module, and a ChebyKAN postprocessing module connected in series. End-to-end training was achieved by incorporating a composite loss that integrates physical residuals, initial-boundary conditions, and three-layer fuzzy structure constraints. Verification was performed on four typical types of fuzzy partial differential equations (elliptic, parabolic, and hyperbolic). The method can stably output interval solutions that conform to fuzzy characteristics under an ideal quantum simulation environment. At most test membership levels, the average relative L2 error of classical PIKAN is 1.1 to 2.7 times that of QCPIKAN, and the average wavefront position error of PIKAN in the fuzzy convection example is 1.77 times that of QCPIKAN. This approach effectively ensures the rationality of the fuzzy structure while improving solution accuracy, providing high-precision meshless solution support for uncertainty quantification problems with fuzzy parameters in multiple fields.

[0083] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for solving fuzzy differential equations using quantum-classical physical information KAN networks, characterized in that, Includes the following steps: S1. Based on the fuzzy partial differential equation to be solved and its initial and boundary value conditions, fuzzy parameters... Cut-set representation, through endpoint splitting and interval mapping of the control equation, yields the membership level. The lower and upper endpoint control equations are constructed simultaneously, including spatiotemporal coordinates and membership levels. The joint input vector; S2. Based on the joint input vector, feature mapping and function decoding are performed through a quantum-classical physical information KAN network composed of a ChebyKAN preprocessing module, a parameterized quantum circuit core module, and a ChebyKAN postprocessing module connected in series, to obtain the corresponding... The approximate solutions at the lower and upper endpoints of the cut set; S3. Based on the approximate solution of the lower endpoint, the approximate solution of the upper endpoint, the control equation of the lower endpoint, the control equation of the upper endpoint and the initial boundary conditions, the total loss function for network training is obtained by weighted combination of physical constraint loss and fuzzy structural constraint loss. S4. Based on the total loss function, all trainable parameters of the quantum-classical physical information KAN network are iteratively updated through a gradient descent optimizer. After the loss converges to a preset threshold, an approximate solution model for the fuzzy partial differential equation is obtained.

2. The method for solving fuzzy differential equations using quantum-classical physical information KAN networks according to claim 1, characterized in that: S1 specifically includes: S11. Based on the triangular fuzzy number parameters in the fuzzy partial differential equation to be solved, through... The interval operation is used to obtain the corresponding membership level. The lower and upper bounds of the parameters are as follows; where the triangular fuzzy number is... The formula for calculating the cutoff set is: ； in, These are the triangular fuzzy type uncertain parameters in the fuzzy partial differential equation to be solved; The left support endpoint of the triangular fuzzy number; The kernel point of the triangular fuzzy number; The right support endpoint of the triangular fuzzy number; S12. Based on the lower bound and upper bound of the parameters and the monotonicity of the governing equations with respect to the parameters, the lower endpoint governing equations and the upper endpoint governing equations are obtained through endpoint extremum matching operations; among which the fuzzy solution... The cut set is represented in interval form, and the formula is: ； in, , They are respectively The lower and upper endpoints of the cut set; S13. Spatiotemporal coordinate variables and membership levels based on the original partial differential equation. By performing a dimension concatenation operation, a joint input vector of dimension d+1 is obtained, as shown in the formula: ； in, For joint input vectors; is the vector of independent variables of the original partial differential equation; This refers to the subordinate level.

3. The method for solving fuzzy differential equations using quantum-classical physical information KAN networks according to claim 2, characterized in that: The calculation process of the ChebyKAN preprocessing module in S2 specifically includes: S21. Based on the components of the joint input vector, nonlinear compression is performed using the hyperbolic tangent activation function to obtain normalized input components with values ​​ranging from [-1, 1]. The formula is as follows: ； in, For the normalized first One input component; The first of the original inputs One component; The hyperbolic tangent activation function is used to perform nonlinear compression operations. This represents the total number of input dimensions for the current ChebyKAN layer. S22. Based on the normalized input components, perform basis function expansion using the first-kind Chebyshev polynomial to obtain the values ​​of Chebyshev basis functions of each order; where the formula for the first-kind Chebyshev polynomial is: ； in, For the first Chebyshev polynomial of the first kind in order of order of input The function value at that location; Let be the order of the Chebyshev polynomial; S23. Based on the Chebyshev basis function values ​​of each order, weighted summation and cross-component aggregation are performed using trainable expansion coefficients to obtain a classical feature vector matching the dimension and number of qubits, which is then input into the core module of the parameterized quantum circuit; the output component calculation formula of the ChebyKAN layer is as follows: ； in, The first layer of ChebyKAN One output component; Let be the highest order of the Chebyshev polynomial expansion; For trainable Chebyshev expansion coefficients; This represents the total number of output dimensions for the current ChebyKAN layer.

4. The quantum-classical physical information KAN network solution method for fuzzy differential equations according to claim 3, characterized in that: The calculation process of the core module of the parameterized quantum circuit in S2 specifically includes: S24. Based on the classical feature vectors output by the ChebyKAN preprocessing module, angle encoding is performed using the RX rotation gate to map the classical features to the quantum initial state, as shown in the formula: ； in, The initial quantum state after angle encoding. These are classic eigenvectors; For single-qubit gate sequential multiplication operators; This refers to the total number of qubits. To act on the first A rotation gate around the X-axis for 1 qubits, with a rotation angle equal to the classical eigencomponent. ; for All qubits are in a joint initial state where they are all in the ground state; S25. Based on the initial quantum state, the evolved final quantum state is obtained through an alternating parameterized quantum circuit containing multiple layers of single-qubit rotation gates and controlled entanglement gates. The calculation formula is as follows: ； in, The final quantum state after the parameterized quantum circuit has evolved; For the unitary evolution operator corresponding to the alternating parameterized quantum circuit; For the complete set of trainable parameters of a parameterized quantum circuit; This is the initial quantum state after RX rotating gate angle encoding; S26. Based on the quantum final state, the expected value of each qubit is measured using the Pauli-Z operator to obtain a quantum eigenvector with the same dimension as the number of qubits, which is then input into the ChebyKAN post-processing module. The formulas for calculating the expected value and quantum eigenvector of a single qubit are as follows: ； ； in, For the first Expected value of Pauli-Z operator measurement for 1 qubit; To act on the first Pauli-Z operator for 1 qubit; This is a quantum eigenvector.

5. The method for solving fuzzy differential equations using quantum-classical physical information KAN networks according to claim 4, characterized in that: The calculation process of the ChebyKAN post-processing module in S2 specifically includes: S27. Based on quantum feature vectors, nonlinear mapping and feature dimensionality upscaling are performed through the first ChebyKAN layer to obtain the hidden layer feature vectors. S28. Based on the hidden layer feature vectors, feature decoding and dimensionality compression are performed through the second ChebyKAN layer to obtain... The approximate solutions at the lower and upper endpoints of the cut set.

6. The method for solving fuzzy differential equations using quantum-classical physical information KAN networks according to claim 5, characterized in that: The process of constructing the physical constraint loss in S3 specifically includes: S31. Based on the approximate solutions at the lower and upper endpoints, the differential operators corresponding to the governing equations are calculated using automatic differentiation to obtain the lower and upper endpoint residuals at the placement points within the domain. The mean square error is then calculated to obtain the residual loss of the equations, as shown in the formula: ； in, This represents the residual loss of the equation; This represents the total number of residual configuration points within the domain. For the first Residuals of the lower endpoint control equations at each configuration point; For the first Residuals of the upper endpoint control equations at each configuration point; S32. Based on the approximate solutions of the lower and upper endpoints at the boundary configuration points and the given boundary conditions, the boundary condition loss is obtained by calculating the mean square error of the deviation. The formula is as follows: ； in, This represents the loss value due to boundary conditions. This represents the total number of boundary segments on the computational domain that require the application of boundary conditions. For the first Total number of configuration points on the segment boundary; This is an approximate solution for the lower endpoint; This is an approximate solution for the upper endpoint; For the first The first segment boundary The spatiotemporal coordinate vector of each configuration point; For the first The first on the boundary Membership level of each configuration point ; This is the complete set of trainable parameters for the QCPIKAN model; For the first The first segment boundary The target value of the lower endpoint boundary condition given at each configuration point; For the first The first segment boundary At each configuration point, the target value of the upper endpoint boundary condition is given; S33. For unsteady-state problems with a time dimension, based on the approximate solutions of the lower and upper endpoints at the initial time point and the given initial conditions, the initial condition loss is obtained by calculating the mean square error of the deviation. The formula is as follows: ； in, This is the initial conditional loss value; The total number of points configured at the initial time; For the first The spatial coordinate vectors of the initial configuration points; For the first Membership level of each initial configuration point ; For the first The target value of the initial condition of the lower endpoint given at each initial point; For the first The target value of the upper endpoint given at each initial point; S34. Based on the equation residual loss, boundary condition loss and initial condition loss, the physical constraint loss is obtained by weighted summation.

7. The method for solving fuzzy differential equations using quantum-classical physical information KAN networks according to claim 6, characterized in that: The specific process of constructing the fuzzy structure constraint loss in S3 includes: S35. Membership levels based on the lower endpoint approximation solution and the upper endpoint approximation solution The partial derivatives are used to calculate the inter-layer nesting loss by penalizing the positive part of the inverse monotonicity deviation, which is then used to constrain the lower endpoint's tendency. Monotonic and undiminished, upper endpoint follows Monotonic non-increasing; nested loss between layers The calculation formula is: ； in, This represents the total number of nested constraint sampling points. It is a positive part function; For the approximate solution of the lower endpoint, the membership level The first-order partial derivative; For the approximate solution at the upper endpoint, the membership level The first-order partial derivative; For the first The spatiotemporal coordinate vector of nested sampling points; For the first Membership level corresponding to each nested sampling point ; S36. Based on the approximate solutions of the lower and upper endpoints under the same membership level, the endpoint orderliness loss is obtained by penalizing the deviation of the lower endpoint being greater than that of the upper endpoint. This loss is used to constrain the same... The validity of the solution interval; the loss due to the ordered nature of the endpoints. The formula is: ； in, The total number of sampling points is constrained by the orderliness requirement; For the first The spatiotemporal coordinate vector of an ordered sampling point; For the first Membership level corresponding to each ordered sampling point ; S37, Based on Membership Level The approximate solutions at the lower and upper endpoints are compared, and the mean square error of their deviations is calculated to obtain the endpoint coincidence loss, which is used to constrain the single-point kernel property of the triangular fuzzy number. The formula for the endpoint coincidence loss is: ； in, for Endpoint coincidence loss value; for The total number of constraint sampling points; For the first The spatiotemporal coordinate vector of each sampling point; S38. Based on the inter-layer nesting loss, endpoint ordering loss, and endpoint coincidence loss, the fuzzy structure constraint loss is obtained through weighted summation, and the total loss function is... The weighted formula is: ； in, Weighting coefficients for controlling the residual loss of the equation; These are the weighting coefficients for the initial conditional loss; These are the weighting coefficients for the boundary condition loss; These are the weighting coefficients for the inter-layer nesting loss; The weighting coefficients for the endpoint orderliness loss; for Weighting coefficients for endpoint overlap loss.

8. The method for solving fuzzy differential equations using quantum-classical physical information KAN networks according to claim 7, characterized in that: S4 specifically includes: S41. Based on the solution domain and membership level range of fuzzy partial differential equations, the domain placement points, boundary placement points and initial placement points are generated through the Latin hypercube sampling strategy to form a training sample set. S42. Based on the training sample set and the total loss function, iterative training is performed using the Adam optimizer, and the learning rate is dynamically adjusted in conjunction with the ReduceLROnPlateau learning rate scheduling strategy. S43. Based on the loss change curve during the training process, training is stopped when the total loss drops to a preset threshold or reaches the maximum number of iterations, resulting in a convergent fuzzy partial differential equation approximate solution model.

9. The quantum-classical physical information KAN network solution method for fuzzy differential equations according to claim 8, characterized in that: Fuzzy partial differential equations include elliptic fuzzy Poisson equations, parabolic fuzzy heat conduction equations, parabolic fuzzy reaction-diffusion equations, and hyperbolic fuzzy convection equations; Specifically, for elliptic steady-state problems, the initial condition loss term in the total loss function is omitted. For the hyperbolic wave propagation problem, an additional wavefront position error is added as an auxiliary training constraint.

10. The method for solving fuzzy differential equations using quantum-classical physical information KAN networks according to claim 9, characterized in that: Following S4, there is also a precision verification step, specifically: Based on a set of independently sampled test points and a reference analytical solution or high-precision numerical solution, the average relative value is calculated. The accuracy evaluation results of the approximate solution model are obtained by combining the error and the wavefront position error; the wavefront position error is calculated by the coordinate deviation of the intersection point between the predicted profile and the set threshold.

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