Obstacle certificate generation method and system based on deep learning and symbol regression

Abstract

The invention provides an obstacle certificate generation method and system based on deep learning and symbolic regression, and relates to the technical field of data processing, and the method comprises the steps: obtaining an initial region, an unsafe region and a state domain in a continuous power system; respectively carrying out sampling operation on the initial region, the unsafe region and the state domain to generate a training set; in combination with a loss function and an Adam optimizer, the training set is input to a neural network for training, candidate obstacle certificates in a neural network form are output, and the loss function is established based on initial region loss, unsafe region loss and state domain loss; carrying out symbol regression on the candidate obstacle certificate through a distiller to obtain an analytic candidate obstacle certificate; and performing correctness verification on the form of the analytic candidate obstacle certificate through an SMT solver, if the form of the analytic candidate obstacle certificate is correct, outputting the candidate obstacle certificate as a target obstacle certificate, otherwise, generating a counter-example set, forming a new training set, and training the neural network again.

Description

Technical Field

[0001] The present invention relates to the field of data processing technology, and in particular to a method and system for generating an obstacle certificate based on deep learning and symbolic regression. Background Art

[0002] Deep learning is a method within machine learning that typically uses neural networks to simulate the workings of the human brain, learning features from large amounts of data and performing tasks such as prediction and classification. Symbolic regression is a technique for automatically searching for mathematical formulas. Its goal is to derive mathematical models that express the relationships between variables from data without requiring a predefined formula structure. An obstacle certificate is a mathematical function used to formally verify the safety of continuous dynamical systems. Its purpose is to prove that the system will not enter unsafe regions. The obstacle certificate generation method based on deep learning and symbolic regression refers to an obstacle certificate generation algorithm that combines deep learning and symbolic regression techniques.

[0003] Barrier certificate generation is an effective technology for verifying the security of dynamic models of cyber-physical systems. Cyber-physical systems are an indispensable component of hybrid systems and play a fundamental role in CPS modeling. By generating barrier certificates, the security boundaries of the system can be formally defined and verified, thereby improving the reliability and security of the system and avoiding potential security risks.

[0004] However, for polynomial dynamics models, while the sum-of-squares (SOS) technique can improve efficiency, it relies on predefined polynomial templates and lacks flexibility and scalability. Neural network-based methods can approximate arbitrary functions and have greater flexibility, but their "black box" nature makes them less interpretable and may not be suitable for safety-critical applications. Furthermore, existing counterexample-guided inductive synthesis (CEGIS) methods suffer from scalability issues during verification, impacting their effectiveness and practicality. In summary, existing barrier certificate generation methods lack flexibility and scalability, and the generated barrier certificates are poorly interpretable, making them unsuitable for safety-critical applications and impacting their effectiveness and practicality. Summary of the Invention

[0005] In view of the above-mentioned deficiencies in the prior art, the purpose of an embodiment of the present invention is to provide a barrier certificate generation method based on deep learning and symbolic regression, which can solve the technical problems of the existing barrier certificate generation method, such as lack of flexibility and scalability, poor interpretability of the generated barrier certificates, unsuitability for safety-critical applications, and affecting effectiveness and practicality.

[0006] A first aspect of an embodiment of the present invention provides a method for generating an obstacle certificate based on deep learning and symbolic regression, comprising:

[0007] S1: Obtain the initial region, unsafe region and state domain in the continuous dynamic system;

[0008] S2: Perform sampling operations on the initial area, unsafe area and state domain respectively to generate a training set;

[0009] S3: Combine the loss function and the Adam optimizer to input the training set into the neural network for training, and output the candidate obstacle certificate in the form of a neural network. The loss function is established based on the initial area loss, the unsafe area loss, and the state domain loss.

[0010] S4: Through the distiller, perform symbolic regression on the candidate obstacle certificate to obtain the analytical candidate obstacle certificate;

[0011] S5: Use the SMT solver to verify the correctness of the form of the analytical candidate obstacle certificate. If correct, the candidate obstacle certificate is output as the target obstacle certificate. Otherwise, a counterexample set is generated to form a new training set, and then return to step S3.

[0012] A second aspect of an embodiment of the present invention provides a barrier certificate generation system based on deep learning and symbolic regression, comprising: a processor and a memory;

[0013] The memory stores programs or instructions that can be run on the processor. When the programs or instructions are executed by the processor, the steps of the obstacle certificate generation method based on deep learning and symbolic regression in the first aspect are implemented.

[0014] According to a third aspect of an embodiment of the present invention, a readable storage medium is proposed, on which a program or instruction is stored. When the program or instruction is executed by a processor, the steps of the obstacle certificate generation method based on deep learning and symbolic regression as in the first aspect are implemented.

[0015] The beneficial effects brought about by the technical solution provided by the embodiment of the present invention include at least:

[0016] In this embodiment of the present invention, a dataset of the initial region, unsafe region, and state domain is generated through sampling, providing the necessary input for neural network training. Next, the training set is fed into the neural network for training, combining a loss function and an Adam optimizer. The training outputs candidate obstacle certificates in the form of a neural network. Symbolic regression is then performed on the candidate obstacle certificates using a distiller to obtain analytic candidate obstacle certificates, improving the interpretability and verification efficiency of the certificates. Finally, the correctness of the form of the analytic candidate obstacle certificates is verified using an SMT solver. If the candidate fails, a counterexample is generated and fed back to the training set, promoting loop optimization. This not only improves the automation and accuracy of obstacle certificate generation, but also enhances the scalability and flexibility of the method, overcoming the limitations of traditional methods. This demonstrates the powerful advantages of combining deep learning and symbolic regression, particularly in applications of high-dimensional complex systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] The accompanying drawings are only for the purpose of illustrating specific embodiments and are not to be considered as limiting the present invention. Throughout the drawings, the same reference symbols represent the same components. Obviously, the drawings described below are only some embodiments of the present invention. It is clear that those skilled in the art can derive other drawings based on these drawings without inventive effort.

[0018] Figure 1 This is a flowchart of a method for generating an obstacle certificate based on deep learning and symbolic regression provided by an embodiment of the present invention;

[0019] Figure 2 This is a structural diagram of an obstacle certificate generation system based on deep learning and symbolic regression provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0020] In order to enable those skilled in the art to better understand the technical solutions in the embodiments of the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the embodiments described are part of the embodiments of the present invention, rather than all of the embodiments. It should be understood that these descriptions are merely exemplary and are not intended to limit the scope of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative work should fall within the scope of protection of the present invention.

[0021] The following describes in detail the obstacle certificate generation method based on deep learning and symbolic regression provided by the embodiment of the present invention through specific embodiments and their application scenarios in combination with the accompanying drawings.

[0022] Reference Manual Figure 1 , shows a flow chart of a method for generating an obstacle certificate based on deep learning and symbolic regression provided by an embodiment of the present invention.

[0023] An embodiment of the present invention provides a method for generating an obstacle certificate based on deep learning and symbolic regression, which may include the following steps:

[0024] S1: Obtain the initial region, unsafe region and state domain in the continuous dynamic system.

[0025] A continuous dynamical system is a system whose state changes continuously over time, and its dynamic behavior is typically described by differential equations. The initial region represents the set of all possible states a system can be in at the beginning, i.e., the region where the system's initial position or starting point is located. The unsafe region represents the set of states that the system must never enter. The state domain is the set of all possible states of the system, including all states outside the initial region and the unsafe region.

[0026] It's important to note that obtaining and clearly defining the initial region, state domain, and unsafe region provides clear boundaries and constraints for the subsequent generation of obstacle certificates, ensuring the clear objectives of the obstacle certificate—that is, the generated obstacle certificate must be able to separate the initial region from the unsafe region, thereby ensuring system safety. Furthermore, by precisely defining these regions, the system can be flexibly adjusted and expanded in different application scenarios. For example, in fields such as autonomous driving and drone control, the definitions of the initial region and unsafe region can be adjusted according to specific mission requirements.

[0027] In a possible implementation, the expression of the continuous dynamic system is specifically:

[0028]

[0029] x=(x1,x2,...,x n ) T ∈R n

[0030] f(x)=(f1(x),...,f n (x)) T

[0031] in, represents the derivative of the continuous dynamic system with respect to time, x represents the state vector, f(x) represents the vector field defined in the state space, x i represents the i-th variable, i=1,2,…,n, n represents the total number of variables, T represents transpose, R n represents the set of real numbers.

[0032] S2: Perform sampling operations on the initial area, unsafe area and state domain respectively to generate a training set.

[0033] Sampling refers to selecting a set of representative points from a given region or state space for training or validation purposes. In machine learning, a training set is a dataset used to train a model, consisting of inputs and corresponding outputs.

[0034] It's important to note that sampling across the initial region, unsafe region, and state domain generates a diverse training set. This provides comprehensive training data for the neural network, enabling the model to learn from various perspectives, including starting states, potential unsafe states, and system states. This diverse sampling ensures that the obstacle certificates generated by the neural network not only meet constraints in the initial region and unsafe region but also work effectively across the entire state domain, thus ensuring system security.

[0035] S3: Combine the loss function and Adam optimizer to input the training set into the neural network for training, and output the candidate obstacle certificate in the form of a neural network. The loss function is established based on the initial area loss, unsafe area loss, and state domain loss.

[0036] The loss function is used to measure the difference between the model's predictions and the actual values. In neural network training, the loss function is used to optimize model parameters so that the model's output is as close to the actual value as possible. The Adam optimizer is a commonly used optimization algorithm that combines the momentum method with the adaptive learning rate method.

[0037] A neural network is a machine learning model that mimics the structure of a biological nervous system, processing data and learning through multiple neuron hierarchies. A candidate disability certificate refers to a preliminary form of disability certificate, which is the result of a neural network prediction.

[0038] It's important to note that by combining the powerful fitting capabilities of neural networks with efficient optimization algorithms, we can quickly and accurately generate candidate obstacle certificates that meet the constraints of the initial region, unsafe region, and state domain. This training method not only allows neural networks to adapt to the characteristics of different systems but also provides a flexible and scalable obstacle certificate generation solution.

[0039] In one possible implementation, the loss function is calculated as follows:

[0040] L sum =L init +L unsafe +L domain

[0041]

[0042] Among them, L sum Represents the loss function value, L init Represents the loss function corresponding to the initial region condition, L unsafe Represents the loss function corresponding to the unsafe area condition, L domainRepresents the loss function corresponding to the state domain condition, x i represents the variables collected from the initial domain, X I represents the initial domain, ReLU represents the activation function, B(x i ) means in state x i The certificate value at ε I , ε U and ε D Both represent non-negative constants, X U Indicates an unsafe domain, B(x u ) means in state x u The certificate value at X D represents the state domain, B(x d ) means in state x d The certificate value at ε B Represents a positive integer, L f represents the time derivative.

[0043] It should be noted that by unifying the losses of the initial area, unsafe area and state domain into one loss function, it is ensured that the obstacle certificates generated by the neural network meet the safety requirements at multiple levels and in multiple areas, thereby improving the reliability, efficiency and adaptability of the model and providing a more robust solution for system safety verification.

[0044] S4: Through the distiller, perform symbolic regression on the candidate obstacle certificate to obtain the analytical candidate obstacle certificate.

[0045] Here, distiller refers to a process of converting complex models (such as candidate obstacle certificates generated by neural networks) into more concise and interpretable analytical expressions, and symbolic regression is a technique for automatically discovering mathematical expressions from data, usually by searching the mathematical expression space through evolutionary algorithms (such as genetic algorithms).

[0046] The analytical candidate barrier certificate is a mathematical expression extracted and simplified from the candidate barrier certificate generated by the neural network through symbolic regression. Compared with the candidate barrier certificate in the form of a neural network, the analytical certificate has higher interpretability and verifiability.

[0047] It should be noted that symbolic regression (using a distiller) is used to convert the complex candidate obstacle certificates generated by the neural network into concise analytical obstacle certificates. Symbolic regression can extract a clear and easy-to-understand mathematical expression from the complex neural network model, which greatly enhances the interpretability of the obstacle certificate and facilitates verification and further analysis.

[0048] In a possible implementation, S4 is specifically:

[0049] The candidate obstacle certificates are symbolically regressed by genetic algorithm to obtain analytical candidate obstacle certificates.

[0050] Among them, genetic algorithm is an optimization method that simulates the natural evolution process, and gradually optimizes the quality of the solution through operations such as selection, crossover and mutation.

[0051] It should be noted that a genetic algorithm performs symbolic regression on the candidate obstacle certificates generated by the neural network to obtain an analytical obstacle certificate. This combines the powerful expressive power of neural networks with the interpretability of symbolic regression, thus addressing multiple challenges in the safety verification of complex systems. Secondly, the genetic algorithm can search for optimal solutions in a wide range of mathematical expression spaces, thereby improving the expressive power and accuracy of the obstacle certificate.

[0052] In a possible implementation, S4 specifically includes:

[0053] S401: Determine the input data set based on the relationship between the candidate obstacle certificate predicted by the neural network and the system status:

[0054] D={(x,y)|(x i ,y i )},i∈[1,...,N]

[0055] Where D represents the data pair set, y i represents the output result of the i-th variable, N represents the total number of data pairs in the data set, and (x, y) represents the specific form of the data pair.

[0056] S402: Simplify the input data set into a simple data set.

[0057] The input dataset consists of a set of data points sampled from the relationship between the system state and the candidate obstacle certificates. A simple dataset is one that simplifies the input dataset in some way, retaining the most representative features, thereby reducing the complexity of the dataset and facilitating subsequent symbolic regression processing.

[0058] S403: Define a search space based on the simple data set, where the search space specifically searches for possible mathematical expressions.

[0059] The search space defines the space of all possible combinations of mathematical expressions.

[0060] S404: Randomly generate a preset number of initial populations in the search space, and calculate the fitness of each individual in the initial population.

[0061] Among them, those skilled in the art can set the size of the preset number of groups according to actual conditions, and the present invention does not limit this.

[0062] The initial population is a starting set in the genetic algorithm, consisting of several randomly generated mathematical expressions. Each expression is a candidate solution, which is then optimized through evolutionary operations.

[0063] S405: Based on fitness, perform evolution operations on each initial population.

[0064] Among them, evolutionary operation is the core step in genetic algorithm, including selection, crossover and mutation operations.

[0065] S406: Construct the Pareto front solution set based on the evolved initial population.

[0066] The Pareto front solution set refers to the solution set that cannot be further optimized on some criteria without compromising other criteria. The Pareto front solution set contains the optimal barrier certificate expression.

[0067] S407: Perform population migration operations based on the Pareto front solution set.

[0068] Among them, the population migration operation is to avoid falling into the local optimum, and the genetic algorithm allows different populations to exchange parts of the solution to improve diversity and the ability to search for the global optimal solution.

[0069] S408: Repeat steps S405-S407 until the maximum number of iterations is reached, and determine that the expression with the highest fitness is the analytical candidate obstacle certificate.

[0070] Fitness is a measure of the quality of an expression and is usually related to prediction error and complexity. The higher the fitness, the better the expression.

[0071] It is important to note that, first, simplifying the dataset can reduce unnecessary computation, preserve key features, and optimize the subsequent search process. Second, defining the search space allows symbolic regression to freely search for optimal solutions within the space of mathematical expressions, avoiding the limitations of manually designed templates. Through evolutionary operations, genetic algorithms can explore a wider range of expression combinations and gradually generate the most suitable barrier certificates. Within the Pareto front solution set, the algorithm can filter out the optimal solution and avoid the trap of local optimality. Finally, population migration ensures population diversity, making the evolutionary process more comprehensive and efficient. This approach combines the powerful expressive power of deep learning with the global search capabilities of genetic algorithms. It is extremely flexible, adaptable, and scalable, and can automatically and accurately generate barrier certificates that meet security requirements.

[0072] In a possible implementation, basic components used in the mathematical expression in S403 include variable input, binary operators, unary operators, and constants.

[0073] In a possible implementation, the evolution operation in S405 specifically includes a selection operation, a crossover operation, and a mutation operation.

[0074] The selection operation is used to determine which individuals will enter the next generation based on the fitness of each candidate solution (such as the accuracy and complexity of the obstacle certificate). Individuals with high fitness are more likely to be selected and pass their genes (part of the structure of the mathematical expression) to the next generation, thereby improving the quality of the solution.

[0075] The crossover operation swaps some genes between two "parent" individuals to create a new "offspring" individual. In symbolic regression, the crossover operation creates a new mathematical expression by combining parts of different expressions. This helps explore new expression structures and increases the diversity of the search space.

[0076] Mutation involves making small, random changes to an individual's expression, such as changing a variable, constant, or operator. The goal of mutation is to enhance the diversity of the population, prevent the algorithm from becoming trapped in a local optimum, and potentially discover new, better solutions.

[0077] In a possible implementation, the selection operation is specifically:

[0078] By calculating the fitness of each initial population, the initial population with high fitness is determined:

[0079] l SR (E)=-l pred (E)·exp(λ·C(E))

[0080]

[0081] C(E)=N var +N con +N oper

[0082] Among them, l SR (E) represents the fitness value of expression E, l pred (E) represents the prediction error of expression E, λ represents the weight coefficient of complexity in fitness, f() represents the neural network prediction, C(E) represents the complexity of expression E, N var Indicates the number of variables in expression E, N con Indicates the number of constants in the expression E, N oper Indicates the number of operators in expression E.

[0083] The crossover operation is specifically as follows: for a pair of expressions in the initial population, some sub-expressions in the expression are exchanged through the crossover operation to generate a new expression.

[0084] The mutation operation is specifically: randomly mutating part of the new expression to generate a new population.

[0085] It's important to note that the selection operation ensures that the best expressions are preserved and passed on to the next generation. The crossover operation, by combining the characteristics of outstanding individuals, explores new solution spaces and finds potentially better expressions. The mutation operation, while ensuring diversity, prevents the evolutionary process from becoming trapped in local optima. Through these operations, the genetic algorithm can globally search the space of mathematical expressions, continuously optimizing the expression of the barrier certificate, and ultimately generating an optimal certificate that meets safety requirements. These advantages make this method highly flexible and adaptable in the security verification of complex systems.

[0086] S5: Use the SMT solver to verify the correctness of the form of the analytical candidate obstacle certificate. If correct, the candidate obstacle certificate is output as the target obstacle certificate. Otherwise, a counterexample set is generated to form a new training set, and then return to step S3.

[0087] The SMT solver (Satisfiability Modulo Theories Solver) is a tool used to determine whether mathematical logic formulas are satisfiable. It can handle formulas with different theories (such as integers, real numbers, and Boolean logic) and is used to verify whether candidate solutions meet certain constraints. During the barrier certificate generation process, the SMT solver verifies whether the candidate barrier certificate meets the system's safety requirements.

[0088] Among them, correctness verification refers to using the SMT solver to check whether the generated analytical candidate barrier certificate meets all security constraints. If the analytical barrier certificate is correct, it means that the certificate can effectively separate the initial area and the unsafe area, ensuring the security of the system.

[0089] The counterexample set refers to the set of counterexamples generated by the SMT solver when the parsed candidate barrier certificate is found to not meet the security conditions during the verification process. These counterexamples are specific states that lead to verification failures, and they help identify where the barrier certificate does not meet the requirements.

[0090] It's important to note that the SMT solver accurately verifies that barrier certificates satisfy all necessary constraints, ensuring that the generated certificates effectively prevent the system from entering unsafe areas and safeguarding system security. If the parsed barrier certificate candidate fails to meet the requirements, a set of counterexamples is generated and fed back into the training process to further optimize the certificate. This verification and feedback mechanism continuously improves the quality and effectiveness of barrier certificates, avoiding potential errors and deficiencies in manually designed barrier certificates.

[0091] In a possible implementation, S5 specifically includes:

[0092] When the parsed candidate barrier certificate meets the following constraints, the candidate barrier certificate is determined to be the target barrier certificate:

[0093]

[0094] Where B(x) represents the barrier certificate, X I represents the initial domain, X U Indicates an unsafe domain, X D Represents the state domain, L f represents the time derivative, and t represents the time index.

[0095] It's important to note that formal constraint verification ensures that the generated barrier certificates meet the system's security requirements, preventing the system from entering unsafe zones. This process ensures the validity of the target barrier certificates and enhances the automation and reliability of the entire verification process. This approach enables efficient and automated security verification of complex system dynamics, without relying on human intervention or traditional manual design. Furthermore, this approach improves scalability, enabling its application to large-scale, complex systems, avoiding the challenges of traditional methods in high-dimensional systems.

[0096] The beneficial effects brought about by the technical solution provided by the embodiment of the present invention include at least:

[0097] In this embodiment of the present invention, a dataset of the initial region, unsafe region, and state domain is generated through sampling, providing the necessary input for neural network training. Next, the training set is fed into the neural network for training, combining a loss function and an Adam optimizer. The training outputs candidate obstacle certificates in the form of a neural network. Symbolic regression is then performed on the candidate obstacle certificates using a distiller to obtain analytic candidate obstacle certificates, improving the interpretability and verification efficiency of the certificates. Finally, the correctness of the form of the analytic candidate obstacle certificates is verified using an SMT solver. If the candidate fails, a counterexample is generated and fed back to the training set, promoting loop optimization. This not only improves the automation and accuracy of obstacle certificate generation, but also enhances the scalability and flexibility of the method, overcoming the limitations of traditional methods. This demonstrates the powerful advantages of combining deep learning and symbolic regression, particularly in applications of high-dimensional complex systems.

[0098] Reference Manual Figure 2 , shows a structural diagram of an obstacle certificate generation system based on deep learning and symbolic regression provided by an embodiment of the present invention.

[0099] The embodiment of the present invention provides a barrier certificate generation system 20 based on deep learning and symbolic regression, comprising: a processor 201 and a memory 202;

[0100] The memory 202 stores programs or instructions that can be run on the processor 201. When the programs or instructions are executed by the processor 201, the steps of the above-mentioned obstacle certificate generation method based on deep learning and symbolic regression are implemented, and the same technical effect can be achieved. To avoid repetition, the present invention will not be described in detail.

[0101] It should be understood that the processor 201 in the embodiment of the present invention may be a central processing unit (CPU), and the processor may also be other general-purpose processors, digital signal processors (DSP), application-specific integrated circuits (ASIC), field programmable gate arrays (FPGA) or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc. The general-purpose processor may be a microprocessor or any conventional processor, etc.

[0102] It should also be understood that the memory 202 in the embodiment of the present invention can be a volatile memory or a non-volatile memory, or can include both volatile and non-volatile memories. Among them, the non-volatile memory can be a read-only memory (ROM), a programmable read-only memory (PROM), an erasable programmable read-only memory (EPROM), an electrically erasable programmable read-only memory (EEPROM), or a flash memory. The volatile memory can be a random access memory (RAM), which is used as an external cache. By way of example and not limitation, many forms of random access memory (RAM) are available, such as static RAM (SRAM), dynamic random access memory (DRAM), synchronous DRAM (SDRAM), double data rate synchronous dynamic random access memory (DDR SDRAM), enhanced synchronous dynamic random access memory (ESDRAM), synchronous link DRAM (SLDRAM), and direct rambus RAM (DR RAM).

[0103] The above embodiments can be implemented in whole or in part through software, hardware (such as circuits), firmware, or any other combination. When implemented using software, the above embodiments can be implemented in whole or in part in the form of a computer program product. A computer program product includes one or more computer instructions or computer programs. When the computer instructions or computer program are loaded or executed on a computer, the process or function according to the embodiments of the present invention is generated in whole or in part. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device. Computer instructions can be stored in a computer-readable storage medium or transmitted from one computer-readable storage medium to another. For example, computer instructions can be transmitted from one website, computer, server, or data center to another website, computer, server, or data center via a wired method (such as infrared, wireless, microwave, etc.). The computer-readable storage medium can be any available medium that can be accessed by a computer or a data storage device such as a server or data center that contains a collection of one or more available media. The available media can be magnetic media (such as floppy disks, hard disks, tapes), optical media (such as DVDs), or semiconductor media. The semiconductor media can be a solid-state drive.

[0104] It should be understood that in various embodiments of the present invention, the size of the serial numbers of the above-mentioned processes does not mean the order of execution. The execution order of each process should be determined by its function and internal logic, and should not constitute any limitation on the implementation process of the embodiments of the present invention.

[0105] Those skilled in the art will appreciate that the units and algorithm steps of each example described in conjunction with the embodiments disclosed herein can be implemented in electronic hardware, or a combination of computer software and electronic hardware. Whether these functions are performed in hardware or software depends on the specific application and design constraints of the technical solution. Professionals and technicians can use different methods to implement the described functions for each specific application, but such implementation should not be considered beyond the scope of the present invention.

[0106] Those skilled in the art will clearly understand that, for the convenience and brevity of description, the specific working processes of the above-described equipment, devices and units can refer to the corresponding processes in the aforementioned method embodiments and will not be repeated here.

[0107] In the several embodiments provided by the present invention, it should be understood that the disclosed devices, apparatuses, and methods can be implemented in other ways. For example, the device embodiments described above are merely illustrative. For example, the division of units is merely a logical function division. In actual implementation, there may be other division methods, such as multiple units or components can be combined or integrated into another device, or some features can be ignored or not executed. In addition, the mutual coupling or direct coupling or communication connection shown or discussed can be through some interfaces, indirect coupling or communication connection of devices or units, and can be electrical, mechanical, or other forms.

[0108] Units described as separate components may or may not be physically separate, and components shown as units may or may not be physical units, that is, they may be located in one place or distributed across multiple network units. Some or all of these units may be selected to achieve the purpose of this embodiment according to actual needs.

[0109] In addition, each functional unit in each embodiment of the present invention may be integrated into one processing unit, or each unit may exist physically separately, or two or more units may be integrated into one unit.

[0110] If the function is implemented in the form of a software functional unit and sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present invention, or the part that contributes to the prior art, or the part of the technical solution, can be embodied in the form of a software product. The computer software product is stored in a storage medium and includes a number of instructions for enabling a computer device (which can be a personal computer, server, or network device, etc.) to execute all or part of the steps of the various embodiments of the present invention. The aforementioned storage medium includes: U disk, mobile hard disk, read-only memory (ROM), random access memory (RAM), magnetic disk or optical disk, and other media that can store program code.

[0111] An embodiment of the present invention provides a readable storage medium including: a program or instruction stored on the readable storage medium, which, when executed by a processor, implements the steps of the above-mentioned obstacle certificate generation method based on deep learning and symbolic regression, and can achieve the same technical effect. To avoid repetition, the present invention will not be described in detail.

[0112] Finally, it should be noted that the above embodiments are merely illustrative of the technical solutions of the embodiments of the present invention, and are not intended to limit them. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they may still modify the technical solutions described in the aforementioned embodiments, or replace some of the technical features therein with equivalents; and such modifications or replacements do not deviate from the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the embodiments of the present invention. Any changes or replacements that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be covered by the scope of protection of the present invention.

Claims

1. A method for generating obstacle certificates based on deep learning and symbolic regression, characterized in that: include: S1: Obtain the initial region, unsafe region and state domain in the continuous dynamic system; S2: performing sampling operations on the initial area, the unsafe area, and the state domain respectively to generate a training set; S3: Combining a loss function and an Adam optimizer, inputting the training set into a neural network for training, and outputting a candidate obstacle certificate in the form of a neural network, wherein the loss function is established based on an initial region loss, an unsafe region loss, and a state domain loss; S4: performing symbolic regression on the candidate obstacle certificate through a distiller to obtain an analytical candidate obstacle certificate; S5: Use the SMT solver to verify the correctness of the form of the analytical candidate obstacle certificate. If correct, output the candidate obstacle certificate as the target obstacle certificate. Otherwise, generate a counterexample set to form a new training set, and return to step S3.

2. The obstacle certificate generation method based on deep learning and symbolic regression according to claim 1 is characterized in that The expression of the continuous dynamic system is specifically: x=(x1,x2,...,x n ) T ∈R n ; f(x)=(f1(x),...,f n (x)) T ; in, represents the derivative of the continuous dynamic system with respect to time, x represents the state vector, f(x) represents the vector field defined in the state space, x i represents the i-th variable, i=1,2,…,n, n represents the total number of variables, T represents transposition, R n represents the set of real numbers.

3. The obstacle certificate generation method based on deep learning and symbolic regression according to claim 1 is characterized in that The calculation formula of the loss function is specifically: L sum =L init +L unsafe +L domain ; Among them, L sum Represents the loss function value, L init Represents the loss function corresponding to the initial region condition, L unsafe Represents the loss function corresponding to the unsafe area condition, L domain Represents the loss function corresponding to the state domain condition, x i represents the variables collected from the initial domain, X I represents the initial domain, ReLU represents the activation function, B(x i ) means in state x i The certificate value at ε I , ε U and ε D Both represent non-negative constants, X U Indicates an unsafe domain, B(x u ) means in state x u The certificate value at X D represents the state domain, B(x d ) means in state x d The certificate value at ε B Represents a positive integer, L f represents the time derivative.

4. The method for generating an obstacle certificate based on deep learning and symbolic regression according to claim 1, characterized in that: The S4 is specifically: The candidate obstacle certificate is subjected to symbolic regression by a genetic algorithm to obtain the analytical candidate obstacle certificate.

5. The method for generating an obstacle certificate based on deep learning and symbolic regression according to claim 4, characterized in that: The S4 specifically includes: S401: Determine an input data set based on the relationship between the candidate obstacle certificate predicted by the neural network and the system state: D={(x,y)|(x i ,y i )},i∈[1,...,N]; Where D represents the data pair set, y i represents the output result of the i-th variable, N represents the total number of data pairs in the data set, and (x, y) represents the specific form of the data pair; S402: Simplifying the input data set into a simple data set; S403: defining a search space according to the simple data set, wherein the search space specifically searches for possible mathematical expressions; S404: randomly generating a preset number of initial populations in the search space, and calculating the fitness of each individual in the initial population; S405: performing an evolution operation on each of the initial populations based on the fitness; S406: Construct the Pareto frontier solution set based on the evolved initial population; S407: performing a population migration operation according to the Pareto front solution set; S408: Repeat steps S405-S407 until the maximum number of iterations is reached, and determine that the expression with the highest fitness is the analytical candidate obstacle certificate.

6. The method for generating an obstacle certificate based on deep learning and symbolic regression according to claim 5, characterized in that: The basic components used in the mathematical expression in S403 include: variable input, binary operators, unary operators and constants.

7. The method for generating an obstacle certificate based on deep learning and symbolic regression according to claim 5, characterized in that: The evolution operation in S405 specifically includes a selection operation, a crossover operation, and a mutation operation.

8. The method for generating an obstacle certificate based on deep learning and symbolic regression according to claim 7, characterized in that: The selection operation is specifically as follows: By calculating the fitness of each of the initial populations, an initial population with high fitness is determined: it SR (E)=-l pred (E)·exp(λ·C(E)); C(E)=N var +N con +N oper ; Among them, l SR (E) represents the fitness value of expression E, l pred (E) represents the prediction error of expression E, λ represents the weight coefficient of complexity in fitness, f() represents the neural network prediction, C(E) represents the complexity of expression E, N var Indicates the number of variables in expression E, N con Indicates the number of constants in the expression E, N oper Indicates the number of operators in expression E; The crossover operation is specifically as follows: for any pair of expressions in the population, a part of the sub-expressions is exchanged through the crossover operation to generate a new expression; The mutation operation specifically includes: randomly mutating part of the new expression to generate a new population.

9. The method for generating an obstacle certificate based on deep learning and symbolic regression according to claim 1, characterized in that: The S5 is specifically: When the candidate barrier certificate of the analytical type satisfies the following constraints, the candidate barrier certificate is determined to be the target barrier certificate: Where B(x) represents the barrier certificate, X I represents the initial domain, X U Indicates an unsafe domain, X D Represents the state domain, L f represents the time derivative, and t represents the time index.

10. A barrier certificate generation system based on deep learning and symbolic regression, characterized in that, include: processor and memory; The memory stores programs or instructions that can be run on the processor, and when the programs or instructions are executed by the processor, the steps of the obstacle certificate generation method based on deep learning and symbolic regression as described in any one of claims 1 to 9 are implemented.

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