Framework for synthesizing Fourier KAN obstacle certificate

Abstract

The invention provides a framework for synthesizing a Fourier KAN obstacle certificate, and relates to the field of computer science and artificial intelligence. The method comprises two main stages: a learning stage and a verification stage. In the learning stage, a candidate obstacle certificate is generated through a Fourier Kolmogorov-Arnold network (Fourier KAN), and network parameters are optimized through gradient descent so as to minimize a loss function; in the verification stage, an SMT solver is used for verifying the validity of the candidate certificates, and the global validity of the obstacle certificates is ensured through Lipschitz continuity conditions. The framework disclosed by the invention can efficiently generate and verify the Fourier KAN obstacle certificate, has good stability and accuracy, and provides a new technical scheme for research and application in related fields.

Description

Technical Field

[0001] The present invention relates to the fields of computer science and artificial intelligence, and particularly to a framework for synthesizing Fourier KAN barrier certificates. Background Art

[0002] In previous work on constructing barrier certificates, the barrier certificates were usually represented in the form of feedforward neural networks (FNNs). This method requires designers to manually select and set activation functions. However, the efficiency differences of different types of activation functions in generating barrier certificates are significant, and these differences may lead to a significant increase in computational costs. Especially when dealing with complex systems, selecting a suitable activation function often requires a large number of trials and adjustments, which not only increases the development time and cost but also limits the universality of the method. Many existing methods rely on traditional activation functions such as ReLU, Sigmoid, etc. These functions may have performance bottlenecks in different application scenarios and are difficult to efficiently handle complex constraints, resulting in the generated barrier certificates being ineffective in large-scale or high-dimensional problems. In addition, existing technologies often fail to fully consider how to automatically select and optimize activation functions under different network structures and system requirements, thus reducing manual intervention and computational overhead.

[0003] With the continuous development and complexity of deep neural network models, how to automatically select the most suitable activation function has become a key issue in constructing efficient barrier certificates. In this context, how to overcome the high cost of activation function selection in traditional methods and improve the efficiency of barrier certificate generation has become a technical problem to be solved urgently. The present invention precisely addresses the high-cost problem of selecting activation functions in the prior art and proposes a new method based on the Fourier Kolmogorov - Arnold network (FourierKAN). This method can automatically optimize the activation function when generating barrier certificates, thereby reducing the burden of manual selection and adjustment and improving the generation efficiency and system adaptability. Summary of the Invention

[0004] To solve the technical problem of manually selecting activation functions in the prior art and the low efficiency in generating barrier certificates, the present invention provides a framework for synthesizing FourierKAN barrier certificates.

[0005] The technical solution provided by the present invention is as follows:

[0006] A framework for synthesizing FourierKAN barrier certificates provided by the present invention includes:

[0007] Learning stage: Generate a candidate barrier certificate B(x) through the Fourier Kolmogorov - Arnold network (FourierKAN). The network structure of the FourierKAN consists of activation functions parameterized by learnable Fourier coefficients, and the network parameters are optimized by gradient descent to minimize the loss function;

[0008] Verification stage: Use an SMT solver to verify the validity of the candidate barrier certificate. If there is a counterexample, iterate and optimize; at the same time, based on the Lipschitz continuity condition L ∈ ≤ -η, ensure the global validity of the barrier certificate, where L is the Lipschitz constant of the barrier certificate, ∈ is the sampling grid density, and η < 0;

[0009] The loss function L B is defined as:

[0010]

[0011] where D I 、D U and D D are the sampling data sets of the initial region, the unsafe region, and the domain region respectively, and τ I 、τ U and τ D are numerical stability offsets.

[0012] The beneficial effects brought by the technical solution provided by the present invention at least include:

[0013] (1) In the present invention, the method of automatically optimizing the activation function by using the Fourier Kolmogorov - Arnold network (FourierKAN) avoids the high computational cost brought by manually selecting and adjusting the activation function in the traditional method. This automatic activation function optimization method not only improves the efficiency of barrier certificate generation but also ensures that the system can adapt more quickly and accurately to different tasks, thus greatly reducing the time and cost of development and experiments;

[0014] (2) In the present invention, by combining gradient descent optimization and the Lipschitz continuity condition, it is possible to ensure the global validity of the generated barrier certificate. Through this method, the generated certificate is not only applicable to a specific region but also can cover a wider input space, ensuring the robustness and stability of the network. This effectively avoids the local failure problem that may occur in the traditional method for high - dimensional data or complex models, and improves the security and reliability of the network model;

[0015] (3) In the present invention, by introducing an activation function parameterized by Fourier coefficients, the adaptability of the neural network to complex data sets is significantly enhanced. Compared with traditional fixed activation functions, the activation function parameterized by Fourier coefficients can more flexibly adjust the response characteristics of the network, thereby enabling more efficient and accurate generation of barrier certificates in different types of systems. This innovation not only improves the computational efficiency but also expands the applicability and flexibility of this method in various deep learning applications. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] To more clearly illustrate the technical solutions in the embodiments of the present invention, the following briefly introduces the drawings required for the description of the embodiments. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on these drawings.

[0017] Figure 1 Flow chart of a framework for synthesizing Fourier KAN barrier certificates provided by an embodiment of the present invention;

[0018] Figure 2 FourierKAN diagram of a framework for synthesizing Fourier KAN barrier certificates provided by an embodiment of the present invention;

[0019] Figure 3 Algorithm diagram of a Lipschitz verification method for a framework for synthesizing Fourier KAN barrier certificates provided by an embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0020] The following describes the technical solutions in the present invention with reference to the drawings.

[0021] In the embodiments of the present invention, words such as "exemplarily" and "for example" are used to represent examples, illustrations or explanations. Any embodiment or design solution described as "example" in the present invention should not be construed as being more preferred or having more advantages than other embodiments or design solutions. Exactly, the use of the word "example" is intended to present concepts in a specific way. In addition, in the embodiments of the present invention, the meaning expressed by "and / or" can be both, or either one of the two.

[0022] In the embodiments of the present invention, "image" and "picture" can sometimes be used interchangeably. It should be noted that when the difference is not emphasized, their intended meanings are the same. "(of)", "corresponding" and "corresponding" can sometimes be used interchangeably. It should be noted that when the difference is not emphasized, their intended meanings are the same.

[0023] In the embodiments of the present invention, sometimes the subscript, such as W 1 may be miswritten as a non-subscript form, such as W1. When the difference is not emphasized, the meanings they express are the same.

[0024] To make the technical problems, technical solutions and advantages to be solved by the present invention clearer, the following will be described in detail with reference to the accompanying drawings and specific embodiments.

[0025] Refer to the attached Figure 1 which shows a schematic flowchart of a framework for synthesizing Fourier KAN barrier certificates provided by an embodiment of the present invention.

[0026] An embodiment of the present invention provides a framework for synthesizing Fourier KAN barrier certificates, and the processing flow may include the following steps:

[0027] Learning stage: Generate a candidate barrier certificate B(x) through the Fourier Kolmogorov - Arnold network FourierKAN. The network structure of the FourierKAN consists of activation functions parameterized by learnable Fourier coefficients, and the network parameters are optimized by gradient descent to minimize the loss function;

[0028] Verification stage: Use an SMT solver to verify the effectiveness of the candidate barrier certificate. If there is a counterexample, iterate and optimize; at the same time, ensure the global effectiveness of the barrier certificate based on the Lipschitz continuity condition L∈≤ - η, where L is the Lipschitz constant of the barrier certificate, ∈ is the sampling grid density, and η < 0;

[0029] The loss function L B is defined as:

[0030]

[0031] where D I , D U and D D are the sampling data sets of the initial region, the unsafe region, and the domain region respectively, and τ I , τ U and τ D are numerical stability offsets.

[0032] In a possible implementation manner, the activation function of the Fourier KAN is defined as:

[0033]

[0034] where a k and b kare trainable Fourier coefficients, g is the grid size, which controls the number of basis functions of the Fourier series.

[0035] In a possible implementation, the network structure of the Fourier KAN is defined by the shape array [n 0 , n 1 ,..., n L , where n i represents the number of nodes in the i-th layer, and the value of the nodes in the (l + 1)-th layer is calculated as:

[0036]

[0037] where φ l,i,j is the Fourier activation function connecting the i-th node in the l-th layer and the j-th node in the (l + 1)-th layer.

[0038] In a possible implementation, the SMT solver verification step includes:

[0039] S401. Construct the following negative conditions and verify their unsatisfiability:

[0040]

[0041] S402. Use the dReal solver for verification. If "unsat" is returned, confirm that the barrier certificate is valid; otherwise, add the counterexample points to the training dataset to optimize the network parameters.

[0042] In a possible implementation, the Lipschitz continuity condition is derived from the following inequality:

[0043] |B(x j ) - B(x k )| ≤ L||x j - x k ||

[0044] where L is the Lipschitz constant of the barrier certificate, ∈ is the maximum distance between sampling points, and L∈ ≤ -η is satisfied.

[0045] In a possible implementation, the construction method of the sampling datasets D I , D U and D D is as follows:

[0046] S601. Divide the state set X D into finite elements X 1 , X 2 ,..., X N according to the discretization parameter ∈;

[0047] S602. Select a sampling point \(x\) from each cell i , satisfying \(\left\|\mathbf{x}-\mathbf{x}^{*}\right\| \leq \epsilon\) for all \(\mathbf{x} \in \mathcal{X}\) i ; i

[0048] S603. Extract sample points from \(\mathcal{X}_{1}\), \(\mathcal{X}_{2}\), and \(\mathcal{X}_{3}\) respectively to form a data set. I , \(\mathcal{X}_{2}\) U and \(\mathcal{X}_{3}\) D

[0049] In a possible implementation, the Lie derivative of the barrier certificate is calculated as:

[0050]

[0051] where is the gradient of the barrier certificate and \(f(\mathbf{x})\) is the system vector field.

[0052] In a possible implementation, the output dimension of the FourierKAN is 1, the input dimension is consistent with the system state dimension \(n\), and the network depth is achieved by stacking multiple KAN layers.

[0053] In a possible implementation, the Lipschitz constant \(L\) is determined as follows:

[0054] S901. Constrain the magnitude of the Fourier coefficients such that \(L=\max \left(\left|a_{k}\right|+\left|b_{k}\right|\right) \cdot g\); k (\left|a_{k}\right|+\left|b_{k}\right|) \cdot g\); k |) \cdot g; k

[0055] S902. Combine the grid density \(\epsilon\) and the offset \(\eta\) to ensure that \(L \epsilon \leq-\eta\) holds.

[0056] In a possible implementation, the numerical stability offset satisfies \(\tau_{1}<0\), \(\tau_{2}>0\), \(\tau_{3}>0\), which are used to strengthen the constraint conditions of the initial region, the unsafe region, and the domain region respectively. I <0, \(\tau_{2}\) U >0, \(\tau_{3}\) D >0, respectively for strengthening the constraint conditions of the initial region, the unsafe region, and the domain region.

[0057] This embodiment considers a continuous dynamic system, whose dynamics can be represented by an ordinary differential equation.

[0058]

[0059] where \(\mathbf{x}=\left(x_{1}, x_{2}, \ldots, x_{n}\right)\) 1 , \(x_{2}\) 2 , \(\ldots\), \(x_{n}\) n ) T \(\in \mathbb{R}^{n}\) n is the system state, Denote the derivative of \(x\) with respect to the time variable \(t\), and \(f(x):\Omega\rightarrow\mathbb{R}\) n is a vector field defined on the open subset . Assume it is Lipschitz continuous, which ensures that for a given initial state \(x\) 0 , the solution of the ordinary differential equation exists and is unique. Specifically, there exists a unique time trajectory \(x(t,x\) 0 ), which represents the value of the system state at time \(t\), where \(t > 0\).

[0060] This embodiment focuses on the safety verification problem. In particular, given the initial state \(x(0)=x\) 0 \(\in X\) I , where \(X\) I represents the set of initial states, and given the unsafe set Assume \(X\) I and \(X\) U are both subsets of \(X\) D . For safety verification, consider whether there exists any trajectory starting from the initial region \(X\) I that can enter the unsafe region \(X\) U .

[0061] It should be noted that regarding safety, for a constrained continuous dynamic system, given the initial region and the unsafe region If the following conditions are met, the system is safe:

[0062]

[0063] It should be noted that the barrier certificate plays an important role in verifying the safety of the system. For a continuous dynamic system \(S\), given the sets \(X\) I , \(X\) U and \(X\) D , if there exists a continuous real-valued function \(B(x):X\) D \(\rightarrow\mathbb{R}\), satisfying:

[0064]

[0065] s.t. \(B(x)=0\)

[0066] Then the function \(B(x)\) is called a barrier certificate, and the safety of the system is guaranteed, where is the Lie derivative of the barrier function \(B\) with respect to the vector field \(f\), defined as follows:

[0067]

[0068] Consider a trajectory \(x(t,x\) 0 ), observe \(B(x(t,x\)0 ) evolution, the initial conditions ensure that B(x) ≤ 0, while the final condition requires that B(x(t, x 0 )) ≤ 0 along the trajectory x(t, x 0 ) decreases monotonically, so such a trajectory x(t, x 0 ) is prevented from entering the unsafe region X U , where B(x) > 0, thus ensuring the safety of the system.

[0069] It should be noted that the Kolmogorov - Arnold network (KAN) is a new network architecture inspired by the Kolmogorov - Arnold representation theorem.

[0070] Kolmogorov - Arnold representation theorem: For any continuous function f from [0, 1] n to the real numbers R, there exists a set of continuous functions (where i = 1, 2,..., 2n + 1 and j = 1, 2,..., n + 1) such that:

[0071]

[0072] According to the above formula, KAN can be expressed as a nested combination of two univariate functions. In matrix form, KAN is defined as:

[0073]

[0074] where, Φ in is a matrix composed of univariate functions, expressed as:

[0075]

[0076] And Φ out is a row vector composed of univariate functions:

[0077] Φ out =(φ 1 (·)... φ 2n+1 (·))

[0078] Therefore, the KAN here is just a combination of two KAN layers. In practical applications, a deeper KAN can be achieved by simply stacking more KAN layers.

[0079] It is known that the feed - forward neural network FNN can be expressed as a series of alternating affine transformations W and a fixed non - linear function σ:

[0080]

[0081] Different from the FNN, the KAN uses a scientific activation function at the edge of the network It is expressed as:

[0082]

[0083] where b(x) is the basis function:

[0084]

[0085] and spline(x) is the spline function:

[0086]

[0087] where c i is trainable, B i (x) is the B-spline basis function, w b and w s are trainable parameters.

[0088] Previous work on constructing barrier certificates usually represents them in the form of feedforward neural networks FNNs, which requires manual setting of activation functions. In addition, different activation functions vary greatly in efficiency when generating barrier certificates for different systems. Selecting a suitable activation function for FNNs can be very costly. Therefore, this embodiment attempts to replace FNNs with KANs with learnable activation functions because KANs have stronger scalability and expressive power.

[0089] Therefore, this embodiment considers using an efficient KAN - Fourier KAN to synthesize barrier certificates. Given a continuous dynamical system S, use Fourier KAN to generate a barrier certificate and verify whether it can ensure the safety of the system.

[0090] This embodiment provides a new framework for synthesizing Fourier KAN barrier certificates, which can ensure the safety of continuous dynamical systems. The complete process is divided into two stages: the learner and the verifier.

[0091] This embodiment provides a learner component to synthesize candidate barrier certificates. This component trains Fourier KAN on a sampled dataset and uses a specific loss function to learn the activation function of the network. This embodiment also provides a verifier component for formally verifying the validity of candidate Fourier KAN barrier certificates, which includes two verification methods, namely the Lipschitz method and the SMT solver.

[0092] Regarding the Fourier KAN network architecture in this embodiment, the essence of KAN is to approximate any function by superimposing multiple non-linear functions. Since B-spline functions are recursively generated, training KAN is more challenging than training a feed-forward neural network (FNN), making it difficult to achieve the goal of efficiently synthesizing barrier certificates. In addition, spline functions cannot be formalized with explicit expressions, which increases the difficulty of verification. Therefore, we replace the B-spline function in KAN with more simple and explicitly representable Fourier coefficients. The replaced activation function is expressed as follows:

[0093]

[0094] where d represents the number of dimensions of the features, and a ik and b ik are trainable Fourier coefficients. The hyperparameter g (referred to as the grid size, gridsize) is crucial for. Affecting the number of frequencies in the Fourier series expansion. It specifically determines the types of sine and cosine terms included in each input dimension. Fourier coefficients play a key role in computational efficiency while effectively addressing the training and verification challenges brought by spline functions.

[0095] The shape of Fouier KAN can be represented by an integer array: [n 0 , n 1 ,..., n L , where n i represents the number of nodes in the i-th layer. Let x l,i be the i-th neuron in the l-th layer. There are n 1 n 1+1 activation functions between the l-th layer and the l+1-th layer. Let be the activation function connecting x l,i and x l+1,j . Then the value of the l+1-th layer can be calculated by the following formula:

[0096]

[0097] In matrix representation, it is:

[0098]

[0099] Therefore, a Fourier KAN containing L layers can be constructed as follows:

[0100]

[0101] where Φ l is the activation function matrix corresponding to the l-th layer Fourier KAN.

[0102] As Figure 2As shown, where the left figure is a two - layer Fourier KAN with a shape of [2, 2, 1]. The right figure shows the activation function parameterized by the Fourier curve. Figure 2 A simple FourierKAN is shown, which has a two - dimensional input and a one - dimensional output, consists of two FourierKAN layers, and the learnable activation function is on the edge rather than at the node. Assuming gridsize = 1, taking the node x 1,1 as an example, its calculation formula is as follows:

[0103]

[0104] where a 1 、a 2 、b 1 、b 2 are Fourier coefficients.

[0105] By observing the network structure of the Fourier KAN, the learnable activation function parameterized by Fourier coefficients enables the FourierKAN to represent complex functions with fewer parameters. Since the FourierKAN uses sine and cosine functions as basis functions, which are naturally smooth, its training speed is faster than that of the original KAN. The hyperparameter gridsize controls the number of sine and cosine functions, thereby reducing the model complexity and improving the computational efficiency. A significant advantage of the FourierKAN is its scalability because it can automatically learn to adapt the activation function to different systems.

[0106] Therefore, it is reasonable to use the Fourier KAN as the network for synthesizing barrier certificates. For a continuous dynamical system S, the input dimension of the FourierKAN matches the dimension of the system, and the output dimension is set to 1.

[0107] It should be noted that regarding the training of the barrier Fourier Kolmogorov - Arnold network, based on the barrier certificate condition, this embodiment provides a loss function for the training dataset. During the training process, the Fourier coefficients in the FourierKAN are updated by minimizing the loss function values of the sampled data points.

[0108] Without loss of generality, consider a state set X. This embodiment divides X into a finite number of cells (x 1 , x 2 ,..., x N ,) by choosing the discretization parameter ∈. Then sampling points x i ∈X i are selected from each cell such that for all x ∈ X i , there is ||x i - x i||≤∈. Let D denote the set of all these sampling points. Using this method, the dataset D can be constructed I D U and D D , where each batch is sampled from X I X U and X D .

[0109] The loss function is defined as follows:

[0110]

[0111] where τ I τ U τ D are offsets introduced to enhance numerical stability during training, and the three terms in the formula correspond to three barrier certificate conditions respectively. To generate the FourierKAN barrier certificate candidate function B(x), the gradient descent method is used to minimize L B . When the loss value is reduced to 0, it means that the learned FourierKAN can be used as a candidate barrier certificate.

[0112] It should be noted that regarding the verification of the FourierKAN barrier certificate, this embodiment provides two methods for formally verifying the candidate barrier certificate represented by the Fourier KAN, because the candidate barrier certificate generated by the gradient descent method cannot provide formal security guarantees. The first method is based on the SMT solver to search for counterexamples that violate the barrier certificate properties. The second method is the Lipschitz method to ensure that the trained network provides formal correctness guarantees. The main algorithms are as Figure 3 shown. It should be noted that the candidate barrier certificate will first be verified by the SMT solver. If there is a counterexample within the maximum number of SMT iterations (for example, MaxIter = 10), the Lipschitz method will be used for verification.

[0113] Verification is performed based on the counterexample-guided framework of the SMT solver, which aims to find the states that violate the barrier conditions. For this purpose, this embodiment provides the negation of the following three conditions and verifies whether these conditions are unsatisfiable, as follows:

[0114] x ∈ X I ∧ B(x) > 0

[0115] x ∈ X U ∧ B(x) ≤ 0

[0116]

[0117] Select dReal as the SMT verifier, which can ensure the correctness of its unsatisfiable decisions. Therefore, when dReal returns "unsat" for the given above formula, it confirms that there is no solution within the specified precision, and the candidate barrier certificate B(x) is valid. Otherwise, it means that dReal has found a counterexample that violates the safety property of the barrier certificate.

[0118] During the verification process, dReal can only find one counterexample each time. To improve the verification efficiency, the points around the counterexample will be sampled and added to the dataset to help the learner further optimize the barrier certificate. Repeat this process of optimization and verification until no more counterexamples are found.

[0119] It should be noted that the structure of the Fourier KAN makes the verification based on the SMT solver easier.

[0120] The SMT-based verification method lacks termination guarantee and may have a low success rate in synthesizing valid barrier certificates. Therefore, in this embodiment, an effectiveness condition is additionally implemented during the training process of the Fourier KAN to ensure the correctness of the barrier certificate without post-verification.

[0121] Consider a system S with its initial region as X I , its unsafe region as X U , and its domain region as X. Let D denote the set of points sampled from X. Assume that B is a Lipschitz continuous barrier certificate learned from the sampled data D and applicable to the given system S. If

[0122] L ∈ + η ≤ 0

[0123] where L is the Lipschitz constant of B, η is a negative number, and ∈ is the grid density of the samples, then B is a valid barrier certificate.

[0124] The proof process is as follows:

[0125] First, consider the initial constraint. Assume that there exists a negative number η such that for all x i ∈ D I we have B(x i ) ≤ η. Since B is Lipschitz continuous, we can obtain:

[0126]

[0127] where x j , x k ∈ X. In addition, for all x ∈ X I , there exists an x i such that ||x - x i || ≤ ∈. Therefore, it can be deduced that:

[0128]

[0129] where x i represents the center of the hyper-rectangle. By adjusting the equation, we get:

[0130]

[0131] Since according to L ∈ ≤ -η, we can get:

[0132] B(x) ≤ B(x i ) - η

[0133] We can get B(x i ) ≤ η, and by combining these two inequalities, we get: B(x) ≤ 0 for all x ∈ X I . Similar reasoning can be used to prove the other two conditions.

[0134] The beneficial effects brought by the technical solutions provided by the embodiments of the present invention at least include:

[0135] (1) In the present invention, the method of automatically optimizing the activation function by using the Fourier Kolmogorov - Arnold network (Fourier KAN) avoids the high computational cost brought by manually selecting and adjusting the activation function in the traditional method. This automated activation function optimization method not only improves the efficiency of obstacle certificate generation but also ensures that the system can adapt more quickly and accurately to different tasks, thus greatly reducing the time and cost of development and experiments;

[0136] (2) In the present invention, by combining gradient descent optimization and Lipschitz continuity conditions, it can ensure the global effectiveness of the generated obstacle certificate. Through this method, the generated certificate is not only applicable to a specific area but also can cover a wider input space, ensuring the robustness and stability of the network. This effectively avoids the local failure problem that may occur in the traditional method for high - dimensional data or complex models, and improves the security and reliability of the network model;

[0137] (3) In the present invention, by introducing an activation function parameterized by Fourier coefficients, the adaptability of the neural network to complex data sets is significantly enhanced. Compared with the traditional fixed activation function, the activation function parameterized by Fourier coefficients can more flexibly adjust the response characteristics of the network, so as to achieve more efficient and accurate obstacle certificate generation in different types of systems. This innovation not only improves the computational efficiency but also expands the applicability and flexibility of this method in various deep - learning applications.

[0138] The above are only specific embodiments of the present invention, but the protection scope of the present invention is not limited thereto. Any person skilled in the art within the technical scope disclosed by the present invention can easily conceive of changes or substitutions, which should all be covered within the protection scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the claims.

[0139] The following points need to be explained:

[0140] (1) The attached drawings of the embodiments of the present invention only relate to the structures involved in the embodiments of the present invention, and other structures can refer to the general design.

[0141] (2) For clarity, in the attached drawings used to describe the embodiments of the present invention, the thickness of the layer or region is enlarged or reduced, that is, these drawings are not drawn according to the actual scale. It can be understood that when an element such as a layer, film, region or substrate is referred to as being "on" or "under" another element, the element can be "directly" on or under the other element or there can be an intermediate element.

[0142] (3) Without conflict, the embodiments of the present invention and the features in the embodiments can be combined with each other to obtain new embodiments.

[0143] The above are only specific embodiments of the present invention, but the protection scope of the present invention is not limited thereto. The protection scope of the present invention shall be subject to the protection scope of the claims.

Claims

1. A framework for synthesizing FourierKAN obstacle certificates, characterized in that include: Learning phase: Generate candidate obstacle certificates B(x) through Fourier Kolmogorov-Arnold network FourierKAN, where the network structure of FourierKAN consists of activation functions parameterized by learnable Fourier coefficients, and optimize network parameters by gradient descent to minimize the loss function; Verification phase: Use the SMT solver to verify the validity of the candidate barrier certificate. If there is a counterexample, it will be optimized iteratively. At the same time, the global validity of the barrier certificate is ensured based on the Lipschitz continuity condition L∈≤-η, where L is the Lipschitz constant of the barrier certificate, ∈ is the sampling grid density, and η<0. The loss function L B Defined as: Among them, D I , D U and D D are the sampling data sets of the initial area, the unsafe area and the domain area, respectively, I , τ U and τ D is the numerical stability offset.

2. A framework for synthesizing Fourier KAN barrier certificates according to claim 1, characterized in that: include: The activation function of the Fourier KAN is defined as: Among them, a k and b k are trainable Fourier coefficients, g is the grid size, and controls the number of basis functions of the Fourier series.

3. A framework for synthesizing FourierKAN obstacle certificates according to claim 2, characterized in that: include: The network structure of FourierKAN consists of an array of shape [n0,n1,...,n L ] definition, where n i represents the number of nodes in the i-th layer, and the value of the l+1-th layer node is calculated as: Among them, φ l,i,j is the Fourier activation function connecting the i-th node in the l-th layer and the j-th node in the l+1-th layer.

4. A framework for synthesizing Fourier KAN barrier certificates according to claim 1, characterized in that: include: The SMT solver verification steps include: S401. Construct the following negative condition and verify its unsatisfiability: S402, use the dReal solver for verification. If "unsat" is returned, the obstacle certificate is confirmed to be valid. Otherwise, the counterexample point is added to the training data set to optimize the network parameters.

5. A framework for synthesizing FourierKAN barrier certificates according to claim 1, characterized in that: include: The Lipschitz continuity condition is derived from the following inequality: |B(x j )-B(x k )|≤L||x j -x k || Where L is the Lipschitz constant of the obstacle certificate, ∈ is the maximum distance between sampling points, and L∈≤-η.

6. A framework for synthesizing FourierKAN obstacle certificates according to claim 1, characterized in that: include: The sample data set D I , D U and D D The construction method is as follows: S601, set state set X D Divide into finite elements X1, X2, ..., X according to the discretization parameter ∈ N ; S602, select a sampling point x from each unit i , satisfy||xx i ||≤∈ for all x∈X i Established; S603, respectively from X I , X U and X D Sample points are extracted from the dataset to form a data set.

7. A framework for synthesizing FourierKAN barrier certificates according to claim 1, characterized in that: include: Lie derivative of the barrier certificate Calculated as: in, is the gradient of the obstacle certificate, and f(x) is the system vector field.

8. A framework for synthesizing Fourier KAN barrier certificates according to claim 1, characterized in that: include: The output dimension of the Fourier KAN is 1, the input dimension is consistent with the system state dimension n, and the network depth is achieved by stacking multiple KAN layers.

9. A framework for synthesizing FourierKAN barrier certificates according to claim 2, characterized in that: include: The Lipschitz constant L is determined by: S901, constrain the amplitude of the Fourier coefficient so that L = max k (|a k |+|b k |)·g; S902. Combine the grid density ∈ and the offset η to ensure that L∈≤-η holds.

10. A framework for synthesizing FourierKAN barrier certificates according to claim 1, characterized in that: include: The numerical stability offset satisfies τ I <0,τ U >0,τ D >0, which are used to strengthen the constraints of the initial area, unsafe area and domain area respectively.

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