Limiter construction method and system based on WENO physical characteristics

Abstract

The invention provides a limiter construction method and system based on WENO physical characteristics, and the method comprises the steps: constructing a WENO limiter which is used for suppressing the non-physical oscillation of a neural network in a process of solving a partial differential equation; carrying out semi-discretization processing on the time domain, and realizing discretization of time steps by adopting an implicit Runge-Kutta method; a neural network structure fused with the WENO limiter is constructed, the neural network comprises a plurality of time step modules, and each module is composed of a feedforward neural network, an iterative approximation unit FNN + IA and the WENO limiter; training the neural network by minimizing the loss function containing the physical constraint to obtain a numerical solution of a partial differential equation; the method is deployed in an actual application occasion, and any one or more tasks of target detection, image classification, defect identification and target positioning are executed. According to the method, the limiter based on WENO physical characteristics is constructed and embedded into the neural network structure, and the non-physical oscillation phenomenon occurring in the hyperbolic partial differential equation solving process is remarkably inhibited.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of numerical solution of partial differential equations and scientific computing, in particular to a limiter construction method and system based on the physical characteristics of WENO. BACKGROUND

[0002] In the field of materials science, numerical simulation technology has become an important means to reveal the evolution of material microstructure, performance prediction and new material design. Through numerical simulation, researchers can efficiently and cost-effectively analyze the mechanical, thermal, electrical and other multi-physical field responses of materials under different working conditions in a virtual environment, providing theoretical guidance and technical support for the research and development and engineering application of new materials. However, with the diversification of material functions and the complexity of material structures, higher requirements are put forward for the accuracy, efficiency and description ability of complex physical processes of numerical simulation.

[0003] In recent years, as a new numerical method, Physics-Informed Neural Networks (PINN) has attracted widespread attention in the field of materials science because it can embed physical constraints in the loss function of neural networks in a soft form, realizing meshless solution of high-dimensional partial differential equations. PINN method shows strong flexibility and generalization ability in dealing with material internal interfaces, defects and discontinuous phenomena. However, when facing the existence of steep gradient changes and discontinuous regions (such as cracks, phase interfaces, etc.) in materials, PINN method often has difficulty in balancing high resolution and numerical stability, and is prone to produce non-physical oscillation or accuracy loss, limiting its further application in high-fidelity modeling of complex material behavior.

[0004] In traditional numerical methods, high-resolution non-oscillatory schemes (such as TVD, ENO, WENO, etc.) have been widely used to deal with discontinuous solutions in hyperbolic partial differential equations, which can effectively capture complex phenomena such as shock waves and contact discontinuities. In particular, the Weighted Essentially Non-Oscillatory (WENO) scheme, by introducing a nonlinear weight mechanism to switch between different sub-templates, effectively suppresses numerical oscillation while maintaining high accuracy. However, these methods usually rely on grid discretization, and face challenges in computational efficiency and adaptability when dealing with high-dimensional and complex geometric problems.

[0005] Although existing research has attempted to incorporate domain knowledge into neural networks to improve their physical consistency, such as through domain decomposition (such as cPINN), adaptive point sampling, or residual weighting strategies to alleviate oscillation problems, these methods often come at the cost of increased computational complexity or compromised generalization ability, and have not systematically integrated the numerical characteristics of traditional non-oscillatory schemes with deep learning frameworks.

[0006] Therefore, there is an urgent need to develop a new numerical method that can maintain the flexibility of neural network representation while possessing the stability and non-oscillatory characteristics of traditional high-resolution schemes, so as to further improve the ability to solve problems with discontinuities and large gradients, and meet the urgent needs of materials science and other fields for high-precision and high-efficiency numerical simulation. Summary of the Invention

[0007] To address the shortcomings of existing technologies, the purpose of this invention is to provide a method and system for constructing a limiter based on the physical characteristics of WENO.

[0008] According to the present invention, a limiter construction method based on WENO physical characteristics is provided, the method comprising the following steps: Step S1: Construct a WENO limiter to suppress non-physical oscillations that occur in the neural network during the solution of partial differential equations; Step S2: Perform semi-discretization on the time domain, and use the implicit Runge-Kutta method to discretize the time steps; Step S3: Construct a neural network structure that integrates the WENO limiter. The neural network includes multiple time step modules, each of which consists of a feedforward neural network, an iterative approximation unit FNN+IA, and a WENO limiter. Step S4: Train the neural network by minimizing the loss function that includes physical constraints to obtain the numerical solution of the partial differential equation; The method is deployed in practical applications to perform one or more tasks, including target detection, image classification, defect recognition, and target localization.

[0009] Preferably, the construction of the WENO limiter in step S1 includes the following steps: Step S1.1: Discretize the solution domain uniformly into multiple elements and define the element boundaries; Step S1.2: Select a calculation template containing five nodes and divide it into three sub-templates; Step S1.3: Calculate the flux value at the interface of the sub-template calculation unit and introduce nonlinear weights to suppress non-physical oscillations; Step S1.4: Calculate the corrected flux value at the center of the cell using a linear interpolation method.

[0010] Preferably, the time semi-discretization process in step S2 using the implicit Runge-Kutta method includes the following steps: Step S2.1: Discretize the time domain into M time intervals; Step S2.2: Within each time step, implement the iterative approximation process using the weight parameters defined in the Butcher table; Step S2.3: Calculate the partial derivative terms using automatic differentiation techniques.

[0011] Preferably, the loss function of the neural network in step S3 includes: loss terms for initial conditions and boundary conditions, physical constraint loss term of the FNN+IA module in each time step, and output correction loss term of the WENO limiter module. The total loss function is a weighted sum of all loss terms and is optimized using gradient descent.

[0012] Preferably, the method is applied to any of the following scenarios: Product defect detection: The input is an image of the front screen or back surface of a smart mobile terminal or tablet computer, and the output is whether the product contains defects, the location of the defects, or the type of defects. Sea surface vessel target recognition: The input is a sea surface image, and the output is whether the image contains a vessel, the vessel's location, or the vessel's type.

[0013] The present invention also provides a limiter construction system based on the physical characteristics of WENO, the system comprising the following modules: Module M1: Constructs a WENO limiter to suppress non-physical oscillations that occur in the neural network during the solution of partial differential equations; Module M2: Performs semi-discretization processing on the time domain, using an implicit Runge-Kutta system to discretize the time steps; Module M3: Constructs a neural network structure that integrates the WENO limiter. The neural network includes multiple time-step modules, each of which consists of a feedforward neural network, an iterative approximation unit FNN+IA, and a WENO limiter. Module M4: Trains the neural network by minimizing a loss function that includes physical constraints to obtain numerical solutions to partial differential equations; The system is deployed in real-world applications and performs one or more tasks, including target detection, image classification, defect recognition, and target localization.

[0014] Preferably, the WENO limiter in module M1 is constructed using the following modules: Module M1.1: Discretizes the solution domain uniformly into multiple elements and defines the element boundaries; Module M1.2: Select a computation template containing five nodes and divide it into three sub-templates; Module M1.3: Calculates the flux value at the interface of the sub-template calculation unit and introduces nonlinear weights to suppress non-physical oscillations; Module M1.4: Calculates the corrected flux value at the center of the cell using a linear interpolation system.

[0015] Preferably, the time semi-discretization processing in module M2 employs an implicit Runge-Kutta system, including the following modules: Module M2.1: Discretizes the time domain into M time intervals; Module M2.2: Within each time step, the iterative approximation process is implemented using the weight parameters defined in the Butcher table; Module M2.3: Calculates partial derivatives using automatic differentiation techniques.

[0016] Preferably, the loss function of the neural network in module M3 includes: loss terms for initial conditions and boundary conditions, physical constraint loss term of the FNN+IA module in each time step, and output correction loss term of the WENO limiter module; The total loss function is a weighted sum of all loss terms and is optimized using gradient descent.

[0017] Preferably, the system is applied to any of the following scenarios: Product defect detection: The input is an image of the front screen or back surface of a smart mobile terminal or tablet computer, and the output is whether the product contains defects, the location of the defects, or the type of defects. Sea surface vessel target recognition: The input is a sea surface image, and the output is whether the image contains a vessel, the vessel's location, or the vessel's type.

[0018] Compared with the prior art, the present invention has the following beneficial effects: 1. This invention significantly suppresses non-physical oscillations that occur during the solution of hyperbolic partial differential equations by constructing a limiter based on the physical properties of WENO and embedding it into a neural network structure. 2. This invention organically combines physical information neural networks with classic high-resolution non-oscillatory schemes to construct the NN-WENO method; it retains the flexibility of neural networks in handling high-dimensional and complex problems, and introduces the advantages of the WENO scheme in capturing discontinuities such as interfaces and cracks, thus achieving high-precision, non-oscillatory solutions to partial differential equations in the continuous time domain. 3. This invention combines a time semi-discretization strategy with the implicit Runge-Kutta method, which significantly reduces computational complexity while maintaining high accuracy. It exhibits good generalization ability in multidimensional problems such as the two-dimensional inviscid Burgers equation, and does not require complex domain decomposition or adaptive sampling strategies, making it easy to implement and optimize in existing deep learning frameworks. Attached Figure Description

[0019] Other features, objects, and advantages of the present invention will become more apparent from the following detailed description of non-limiting embodiments with reference to the accompanying drawings: Figure 1This is a calculation template diagram for a 5th-order WENO; Figure 2 Network structure diagram for integrating the WENO limiter; Figure 3 This is a comparison chart of continuous and discontinuous functions before and after processing by the WENO limiter; Figure 4 A comparison of the results of solving the two-dimensional inviscid Burgers equation using the DeLISA method and the NN-WENO method at different time points; Figure 5 A comparison of the results of the two-dimensional inviscid Burgers equation at different locations at t=0.3s; Figure 6 This is a flowchart illustrating the principle of the present invention. Detailed Implementation

[0020] The present invention will now be described in detail with reference to specific embodiments. These embodiments will help those skilled in the art to further understand the present invention, but do not limit the invention in any way. It should be noted that those skilled in the art can make several changes and improvements without departing from the concept of the present invention. These all fall within the protection scope of the present invention.

[0021] Example 1: Reference Figure 6 According to the present invention, a limiter construction method based on the physical characteristics of WENO is provided, the method comprising the following steps: Step S1: Construct a WENO limiter to suppress non-physical oscillations that occur in the neural network during the solution of partial differential equations; the construction of the WENO limiter includes the following steps: Step S1.1: Discretize the solution domain uniformly into multiple elements and define the element boundaries; Step S1.2: Select a calculation template containing five nodes and divide it into three sub-templates; Step S1.3: Calculate the flux value at the interface of the sub-template calculation unit and introduce nonlinear weights to suppress non-physical oscillations; Step S1.4: Calculate the corrected flux value at the center of the cell using a linear interpolation method.

[0022] Step S2: Perform semi-discretization on the time domain, using the implicit Runge-Kutta method to discretize the time steps; the semi-discretization of time using the implicit Runge-Kutta method includes the following steps: Step S2.1: Discretize the time domain into M time intervals; Step S2.2: Within each time step, implement the iterative approximation process using the weight parameters defined in the Butcher table; Step S2.3: Calculate the partial derivative terms using automatic differentiation techniques.

[0023] Step S3: Construct a neural network structure incorporating the WENO limiter. The neural network includes multiple time-step modules, each consisting of a feedforward neural network, an iterative approximation unit FNN+IA, and a WENO limiter. The loss function of the neural network includes: loss terms for initial and boundary conditions, physical constraint loss terms for the FNN+IA module in each time step, and output correction loss terms for the WENO limiter module. The total loss function is a weighted sum of all loss terms, optimized using gradient descent.

[0024] Step S4: Train the neural network by minimizing the loss function that includes physical constraints to obtain the numerical solution of the partial differential equation; The method is deployed in practical applications to perform one or more tasks, including target detection, image classification, defect recognition, and target localization.

[0025] The method can be applied to any of the following scenarios: Product defect detection: The input is an image of the front screen or back surface of a smart mobile terminal or tablet computer, and the output is whether the product contains defects, the location of the defects, or the type of defects. Sea surface vessel target recognition: The input is a sea surface image, and the output is whether the image contains a vessel, the vessel's location, or the vessel's type.

[0026] The present invention also provides a limiter construction system based on WENO physical characteristics. The limiter construction system based on WENO physical characteristics can be implemented by executing the process steps of the limiter construction method based on WENO physical characteristics. That is, those skilled in the art can understand the limiter construction method based on WENO physical characteristics as a preferred embodiment of the limiter construction system based on WENO physical characteristics.

[0027] Example 2: The present invention also provides a limiter construction system based on the physical characteristics of WENO, the system comprising the following modules: Module M1: Constructs the WENO limiter to suppress non-physical oscillations that occur in neural networks during the solution of partial differential equations; the construction of the WENO limiter includes the following modules: Module M1.1: Discretizes the solution domain uniformly into multiple elements and defines the element boundaries; Module M1.2: Select a computation template containing five nodes and divide it into three sub-templates; Module M1.3: Calculates the flux value at the interface of the sub-template calculation unit and introduces nonlinear weights to suppress non-physical oscillations; Module M1.4: Calculates the corrected flux value at the center of the cell using a linear interpolation system.

[0028] Module M2: Performs semi-discretization of the time domain, using an implicit Runge-Kutta system to discretize the time steps; the semi-discretization of time using an implicit Runge-Kutta system includes the following modules: Module M2.1: Discretizes the time domain into M time intervals; Module M2.2: Within each time step, the iterative approximation process is implemented using the weight parameters defined in the Butcher table; Module M2.3: Calculates partial derivatives using automatic differentiation techniques.

[0029] Module M3: Constructs a neural network structure incorporating the WENO limiter. The neural network includes multiple time-step modules, each consisting of a feedforward neural network, an iterative approximation unit FNN+IA, and a WENO limiter. The loss function of the neural network includes: loss terms for initial and boundary conditions, physical constraint loss terms for the FNN+IA module in each time step, and output correction loss terms for the WENO limiter module. The total loss function is a weighted sum of all loss terms and is optimized using gradient descent.

[0030] Module M4: Trains the neural network by minimizing a loss function that includes physical constraints to obtain numerical solutions to partial differential equations; The system is deployed in real-world applications and performs one or more tasks, including target detection, image classification, defect recognition, and target localization.

[0031] The system is applicable to any of the following scenarios: Product defect detection: The input is an image of the front screen or back surface of a smart mobile terminal or tablet computer, and the output is whether the product contains defects, the location of the defects, or the type of defects. Sea surface vessel target recognition: The input is a sea surface image, and the output is whether the image contains a vessel, the vessel's location, or the vessel's type.

[0032] Example 3: This invention considers a class of partial differential equations in a bounded region Ω:

[0033] in This represents the unknown solution to the equation. For differential operators, for right The first derivative. Initial and boundary conditions are given by the practical problem. The purpose of this invention is to solve the above partial differential equations in the continuous time domain using the proposed time semi-discretization method within a neural network framework, and this method depends only on the initial and boundary points.

[0034] This invention considers bounded domains The general form of hyperbolic partial differential equations within:

[0035] in, It is a solution to the equation. It is about The flux function. Represents the initial conditions. Represents boundary conditions. Representative solution Relative to time The first-order partial derivative, Representative flux function Relative to space The first-order partial derivatives. The goal of solving this equation is to find a solution that satisfies the given initial and boundary conditions. .

[0036] The constraint construction based on WENO physical properties is as follows: First, the entire solution domain is uniformly discretized, where:

[0037] Indicates the element length. The element boundary is defined as... Subsequently, the entire template consisting of five nodes was divided into three sub-templates. ,in , , Therefore, the computational template for the 5th-order WENO method is as follows: Figure 1 As shown.

[0038] For each of the above sub-templates, the unit interface Flux The value can be calculated using the WENO method. The calculation expression is:

[0039] Similarly, unit interface Flux The expression for calculating the value is:

[0040] To suppress non-physical oscillations, the core of the WENO method is the introduction of nonlinear weights in discontinuous regions. By adjusting the magnitude of these nonlinear weights, the computational contribution of sub-templates containing discontinuous regions is reduced. Therefore, the expression for calculating the flux correction value is: (1.1) in, This represents a non-linear weight. In the WENO method, the non-linear weight is defined as:

[0041] in, It is a positive sensitivity parameter designed to prevent division by zero in the weighting formula (its value is set here). ); The ideal weight is defined as follows: , , ; Represents the global smoothness factor; This represents the smoothing factor for each sub-template. The expression for calculating the smoothing factor for each sub-template is:

[0042] Using equation (1.1), we can obtain and The correction value at that point. Then, a linear interpolation method is used to calculate... The correction value at that location. The expression for calculating the correction value is:

[0043] Thus, we have successfully constructed a limiter based on the physical properties of WENO, which we will simply call the WENO limiter.

[0044] A Network Structure Integrating the WENO Limiter: To address non-physical oscillations that may arise during the solution of hyperbolic partial differential equations, this invention proposes a numerical method, abbreviated as NN-WENO, that integrates the physical properties of the WENO limiter with deep learning. This method combines the semi-discretization concept of the DeLISA method with the WENO limiter to handle non-physical oscillations. Its network structure is as follows: Figure 2 As shown, the lower left corner is the structure diagram of the FNN+IA module; the lower right corner is the structure diagram of the WENO limiter module.

[0045] This method discretizes the time domain into M time intervals. Therefore, the entire network consists of M modules, each composed of FNN+IA and WENO. FNN+IA represents a feedforward neural network employing an iterative approximation scheme, with the structure as follows: Figure 2As shown in the lower left corner. WENO represents the WENO limiter, and its structure is as follows: Figure 2 As shown in the bottom right corner. In the FNN+IA module, the network... Will and As input, and producing multiple outputs, that is At the same time, this method employs Implemented using the Runge-Kutta method. and The iterative process between these points. Therefore, the following equation can be obtained:

[0046] in, Represents the time step; , , These represent the weight parameters of the Butcher table in the implicit Runge-Kutta method. For the hyperbolic partial differential equations considered in this chapter, the differential operators... The partial derivatives in the differential operator can be calculated using the automatic differentiation module in the PyTorch framework. Therefore, the loss function expression between each time step is:

[0047] Accordingly, the loss function expression for the boundary conditions at each time step is:

[0048] Therefore, the overall loss function expression for the FNN+IA module is:

[0049] In the WENO module, Represents the original value (i.e., the input). This represents the value after correction by the WENO limiter (i.e., the output). Therefore, the loss function expression for the WENO limiter module is:

[0050] Meanwhile, the loss function expression for the initial boundary conditions of the entire network is:

[0051] Since the entire time domain consists of M modules, and each module is composed of FNN+IA and WENO, the overall loss function expression for the entire network model is:

[0052] in, and These are the weights of the loss function. After the training data is ready, gradient descent is used to optimize the neural network to minimize the loss function. The following algorithm describes the specific implementation steps of the method proposed in this invention.

[0053]

[0054] Verification of the WENO limiter: Consider the following one-dimensional continuous function and one-dimensional discontinuous function:

[0055] In this experiment, by adding a value of a certain amount to the precise function value... The random error is used to simulate non-physical oscillations. Therefore, the input to the WENO limiter is a function value with random error, and its output is a function value corrected by the WENO limiter module. Figure 3 The results of continuous and discontinuous functions before and after correction using the WENO limiter are shown. Comparing the results before and after correction reveals that the WENO limiter can effectively suppress non-physical oscillations in the function.

[0056] Table 1. Relative values ​​of the continuity function before and after correction by the WENO limiter for different numbers of error points. Error and Error Comparison

[0057] Table 2 shows the relative values ​​of discontinuity functions before and after correction by the WENO limiter for different numbers of error points. Error and Error Comparison

[0058] The table above shows the relative performance of two different types of functions before and after correction by the WENO limiter under different numbers of error points. Error and Error. Comparing the data in the table, it can be seen that the WENO limiter significantly improves the accuracy of both continuous and discontinuous functions, especially the accuracy of discontinuous functions. In conclusion, the WENO limiter module has a positive effect on suppressing non-physical oscillations.

[0059] Two-Dimensional Inviscid Burgers Equation: To verify the effectiveness of the NN-WENO method in multidimensional problems, this experiment considers the two-dimensional inviscid Burgers equation under Dirichlet boundary conditions. The specific expression of the equation is as follows:

[0060] in, This represents the solution to the equation. Represents the time coordinate. and Represents spatial coordinates. Consider the following initial and boundary conditions:

[0061]

[0062]

[0063] Accordingly, the exact solution to the Burgers equation with two-dimensional inviscid terms is:

[0064] In this experiment, the neural network consisted of six hidden layers, each containing 20 neurons, and used the Tanh function as the activation function. The network's training data consisted of an initial number of points. Number of internal points Number of WENO restricted points Number of boundary points and time interval Composition. During training, the time step is fixed at... Thus, the time domain is discretized into A time interval. The initial number of points is set to [number]. In each time step, the number of boundary points is fixed. The number of internal points and the number of WENO-restricted points remain consistent with the initial number of points. Therefore, throughout the training process, , The training process employs both the Adam optimizer and the L-BFGS optimizer to optimize the loss function.

[0065] Experimental results are as follows Figure 4 As shown, it illustrates the situation at three different times ( , and Experimental results are presented below for solving the Burgers equation with two-dimensional inviscid terms using the DeLISA and NN-WENO methods. The experimental results show that the NN-WENO method can effectively suppress the generation of non-physical oscillations in two-dimensional space.

[0066] To allow for a direct comparison of the results before and after the correction, Figure 5 Showing time At time, position and The solution results of the DeLISA method and the NN-WENO method are presented in Table 3. Furthermore, Table 3 lists the relative results of the DeLISA method and the NN-WENO method for solving the two-dimensional inviscid Burgers equation. Error and error.

[0067] Table 3. Relative values ​​of the two-dimensional inviscid Burgers equation before and after correction using the NN-WENO method. Error and Error Comparison

[0068]

[0069] This invention utilizes the physical properties of the WENO numerical method to construct a WENO limiter, suppressing non-physical oscillations that occur during the solution of hyperbolic partial differential equations. The method employs a semi-discretization concept, using the implicit Runge-Kutta method to discretize time. This approach combines neural networks with the traditional WENO numerical method, proposing a novel numerical method for solving hyperbolic partial differential equations.

[0070] Those skilled in the art can understand this embodiment as a more specific description of Embodiment 1 and Embodiment 2.

[0071] Those skilled in the art will understand that, besides implementing the system and its various devices, modules, and units provided by this invention in the form of purely computer-readable program code, the same functions can be achieved entirely through logical programming of the method steps, making the system and its various devices, modules, and units of this invention function in the form of logic gates, switches, application-specific integrated circuits, programmable logic controllers, and embedded microcontrollers. Therefore, the system and its various devices, modules, and units provided by this invention can be considered as a hardware component, and the devices, modules, and units included therein for implementing various functions can also be considered as structures within the hardware component; alternatively, the devices, modules, and units for implementing various functions can be considered as both software modules implementing the method and structures within the hardware component.

[0072] Specific embodiments of the present invention have been described above. It should be understood that the present invention is not limited to the specific embodiments described above, and those skilled in the art can make various changes or modifications within the scope of the claims, which do not affect the essence of the present invention. Unless otherwise specified, the embodiments and features described in this application can be arbitrarily combined with each other.

Claims

1. A method for constructing a limiter based on the physical characteristics of WENO, characterized in that, The method includes the following steps: Step S1: Construct a WENO limiter to suppress non-physical oscillations that occur in the neural network during the solution of partial differential equations; Step S2: Perform semi-discretization on the time domain, and use the implicit Runge-Kutta method to discretize the time steps; Step S3: Construct a neural network structure that integrates the WENO limiter. The neural network includes multiple time step modules, each of which consists of a feedforward neural network, an iterative approximation unit FNN+IA, and a WENO limiter. Step S4: Train the neural network by minimizing the loss function that includes physical constraints to obtain the numerical solution of the partial differential equation; The method is deployed in practical applications to perform one or more tasks, including target detection, image classification, defect recognition, and target localization.

2. The method for constructing a limiter based on the physical characteristics of WENO according to claim 1, characterized in that, The construction of the WENO limiter in step S1 includes the following steps: Step S1.1: Discretize the solution domain uniformly into multiple elements and define the element boundaries; Step S1.2: Select a calculation template containing five nodes and divide it into three sub-templates; Step S1.3: Calculate the flux value at the interface of the sub-template calculation unit and introduce nonlinear weights to suppress non-physical oscillations; Step S1.4: Calculate the corrected flux value at the center of the cell using a linear interpolation method.

3. The method for constructing a limiter based on the physical characteristics of WENO according to claim 1, characterized in that, The time semi-discretization process in step S2, using the implicit Runge-Kutta method, includes the following steps: Step S2.1: Discretize the time domain into M time intervals; Step S2.2: Within each time step, implement the iterative approximation process using the weight parameters defined in the Butcher table; Step S2.3: Calculate the partial derivative terms using automatic differentiation techniques.

4. The method for constructing a limiter based on the physical characteristics of WENO according to claim 1, characterized in that, The loss function of the neural network in step S3 includes: loss terms for initial conditions and boundary conditions, physical constraint loss term of the FNN+IA module in each time step, and output correction loss term of the WENO limiter module. The total loss function is a weighted sum of all loss terms and is optimized using gradient descent.

5. The method for constructing a limiter based on the physical characteristics of WENO according to claim 1, characterized in that, The method can be applied to any of the following scenarios: Product defect detection: The input is an image of the front screen or back surface of a smart mobile terminal or tablet computer, and the output is whether the product contains defects, the location of the defects, or the type of defects. Sea surface vessel target recognition: The input is a sea surface image, and the output is whether the image contains a vessel, the vessel's location, or the vessel's type.

6. A limiter construction system based on the physical characteristics of WENO, characterized in that, The system includes the following modules: Module M1: Constructs a WENO limiter to suppress non-physical oscillations that occur in the neural network during the solution of partial differential equations; Module M2: Performs semi-discretization processing on the time domain, using an implicit Runge-Kutta system to discretize the time steps; Module M3: Constructs a neural network structure that integrates the WENO limiter. The neural network includes multiple time-step modules, each of which consists of a feedforward neural network, an iterative approximation unit FNN+IA, and a WENO limiter. Module M4: Trains the neural network by minimizing a loss function that includes physical constraints to obtain numerical solutions to partial differential equations; The system is deployed in real-world applications and performs one or more tasks, including target detection, image classification, defect recognition, and target localization.

7. The limiter construction system based on WENO physical characteristics according to claim 6, characterized in that, The WENO limiter in module M1 is constructed using the following modules: Module M1.1: Discretizes the solution domain uniformly into multiple elements and defines the element boundaries; Module M1.2: Select a computation template containing five nodes and divide it into three sub-templates; Module M1.3: Calculates the flux value at the interface of the sub-template calculation unit and introduces nonlinear weights to suppress non-physical oscillations; Module M1.4: Calculates the corrected flux value at the center of the cell using a linear interpolation system.

8. The limiter construction system based on WENO physical characteristics according to claim 6, characterized in that, The time semi-discretization processing in module M2 employs an implicit Runge-Kutta system, which includes the following modules: Module M2.1: Discretizes the time domain into M time intervals; Module M2.2: Within each time step, the iterative approximation process is implemented using the weight parameters defined in the Butcher table; Module M2.3: Calculates partial derivatives using automatic differentiation techniques.

9. The limiter construction system based on WENO physical characteristics according to claim 6, characterized in that, The loss function of the neural network in module M3 includes: loss terms for initial conditions and boundary conditions, physical constraint loss term of the FNN+IA module in each time step, and output correction loss term of the WENO limiter module. The total loss function is a weighted sum of all loss terms and is optimized using gradient descent.

10. The limiter construction system based on WENO physical characteristics according to claim 6, characterized in that, The system is applicable to any of the following scenarios: Product defect detection: The input is an image of the front screen or back surface of a smart mobile terminal or tablet computer, and the output is whether the product contains defects, the location of the defects, or the type of defects. Sea surface vessel target recognition: The input is a sea surface image, and the output is whether the image contains a vessel, the vessel's location, or the vessel's type.

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