Security time keeping method and system based on physical information neural network, medium and product

By dynamically adjusting the weights of the loss function in the physical information neural network, the problem of error accumulation in long-term prediction by traditional models is solved, thereby improving timeliness accuracy and security.

CN121167149APending Publication Date: 2025-12-19NO 30 INST OF CHINA ELECTRONIC TECH GRP CORP
View PDF 11 Cites 0 Cited by

Patent Information

Application Number
CN202511130846.1
Authority / Receiving Office
CN · China
Patent Type
Applications(China)
Current Assignee / Owner
Filing Date
2025-08-13
Publication Date
2025-12-19

AI Technical Summary

Technical Problem

Traditional physical information neural networks do not fully consider the cumulative effect of data errors in long-term predictions, leading to model prediction saturation and affecting timekeeping accuracy and security.

Method used

By employing a fusion scheme of dynamic balancing mechanism and data item decay coefficient, the loss function is optimized. By dynamically adjusting the weights of the physical loss function and the data loss function, the interference of data error accumulation on long-term prediction is reduced, thereby improving the long-term prediction accuracy and stability of the model.

Benefits of technology

It significantly improves the long-term prediction accuracy and stability of the model, and enhances the system's ability to identify illegal attacks and its timeliness accuracy.

✦ Generated by Eureka AI based on patent content.

Smart Images

  • Figure CN121167149A_ABST
    Figure CN121167149A_ABST
Patent Text Reader

Abstract

The invention relates to the technical field of clocks, and provides a security time keeping method and system based on a physical information neural network, a medium and a product, the method comprises the following steps: receiving data, the received data being clock error data at the current moment; inputting the received data into a clock error model trained based on a physical information neural network to obtain a prediction result, and judging whether the received data can be used for punctuality updating or not based on the prediction result; local clock adjustments are accomplished with received data that can be used for punctuality updates. According to the method, the long-term prediction precision and stability of the model can be remarkably improved, so that the illegal attack recognition capability and punctuality precision of the system are improved.
Need to check novelty before this filing date? Find Prior Art

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of clock, in particular to a secure time keeping method and system based on a physics informed neural network, a medium and a product. BACKGROUND

[0002] With the rapid development of digital economy, the power, transportation, finance and other fields of the national economy have put forward higher requirements for high-precision and high-security time keeping technology. The current secure time keeping technology mainly faces two-dimensional challenges: on the one hand, the time keeping process is vulnerable to signal interference, information tampering and illegal time delay attacks, resulting in a decrease in the reliability of the system time source, thereby threatening the safety of system operation; on the other hand, the local clock of the system accumulates non-linear errors due to factors such as crystal oscillator aging and temperature drift, and it is difficult to provide stable and effective time information when the external reference time source fails, which cannot maintain high-precision time synchronization performance, and thus affects the long-term stable operation of the system.

[0003] The physics informed neural network (PINN) is a new type of deep learning framework that combines physical prior knowledge and data-driven, with strong function approximation ability and good generalization performance. The secure time keeping technology based on the physics informed neural network can achieve high-precision clock difference prediction, which is beneficial to the identification of time delay attacks and the protection of system time precision, but the difficulty lies in that the model does not fully consider the data error accumulation effect of different time steps in long-term prediction, i.e. the data items in each loss function have equal proportional weight coefficients, which causes the model to suddenly lose the prediction tracking ability in long-term prediction, i.e. the clock difference prediction saturation phenomenon occurs, thereby reducing the time keeping precision and security of the system. SUMMARY

[0004] In view of the fact that the traditional physics informed neural network does not fully consider the data error accumulation effect of different time steps in long-term prediction, resulting in the prediction saturation phenomenon of losing prediction tracking ability in long-term prediction, which causes security risks and a decrease in time keeping precision of the time system, the present application provides a secure time keeping method and system based on a physics informed neural network, a medium and a product.

[0005] In a first aspect, the present application provides a secure time keeping method based on a physics informed neural network, comprising: receiving data, the receiving data being clock difference data at the current time; inputting the receiving data into a clock difference model trained based on a physics informed neural network to obtain a prediction result, and determining whether the receiving data can be used for time keeping update based on the prediction result; completing local clock adjustment using the receiving data that can be used for time keeping update.

[0006] In a preferred embodiment, the clock error model trained based on a physical information neural network is obtained using the following method: Constructing neural network structures; The neural network structure is trained using clock difference data from past moments; During training, the model parameters of the neural network structure are updated using gradient information obtained through backpropagation, thereby minimizing the loss function; After training, a clock error model based on a neural network trained on physical information is obtained.

[0007] In a preferred embodiment, the loss function includes a physical loss function and a data loss function.

[0008] In a preferred embodiment, dynamic weights are used to balance the constraints of the physical loss function and the data loss function on the model output at different time steps.

[0009] In a preferred embodiment, a weight decay function is introduced into the data loss function.

[0010] In a preferred embodiment, the neural network structure employs a feedforward neural network or a recurrent neural network.

[0011] Secondly, the present invention provides a security timekeeping system based on a physical information neural network, comprising: A data storage unit is used to receive data, wherein the received data is the clock difference data at the current moment; The security identification unit is used to input the received data into a clock error model trained on a physical information neural network to obtain a prediction result, and to determine whether the received data can be used for timekeeping updates based on the prediction result. The timekeeping update unit is used to adjust the local clock using the received data that can be used for timekeeping updates.

[0012] Thirdly, the present invention provides an electronic device, comprising: At least one processor; and a memory communicatively connected to said at least one processor; The memory stores instructions that can be executed by the at least one processor, and the at least one processor executes the instructions stored in the memory to perform the above-described method.

[0013] Fourthly, the present invention provides a computer-readable storage medium, characterized in that the computer-readable storage medium is used to store instructions that, when executed, cause the above-described method to be implemented.

[0014] Fifthly, the present invention provides a computer program product that, when invoked by a computer, causes the computer to execute the above-described method.

[0015] In summary, due to the adoption of the above technical solution, the beneficial effects of the present invention are: This invention addresses the prediction saturation phenomenon in traditional physical information neural network models, which leads to security risks and decreased timekeeping accuracy in time systems. It proposes a fusion scheme of dynamic balancing mechanism and data item decay coefficient to enhance the performance of the model's loss function. This scheme dynamically balances the penalty factors of data-driven and physical-driven approaches in long-term prediction, reduces the interference of data error accumulation on long-term prediction accuracy, and significantly improves the accuracy and stability of long-term prediction. As a result, it enhances the system's ability to identify illegal attacks and improves timekeeping accuracy. Attached Figure Description

[0016] Figure 1 A flowchart of a secure timekeeping method based on a physical information neural network provided in an embodiment of the present invention.

[0017] Figure 2 This describes the principle of PINNs model training in this embodiment of the invention.

[0018] Figure 3 This is a schematic diagram of a security timekeeping system based on a physical information neural network, provided as an embodiment of the present invention.

[0019] Figure 4 This is a schematic diagram of the structure of an electronic device provided in an embodiment of the present invention. Detailed Implementation

[0020] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations.

[0021] Therefore, the following detailed description of the embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the invention without inventive effort are within the scope of protection of the invention.

[0022] Example The prediction accuracy and stability of clock error models trained on physical information neural networks are the foundation for ensuring the safety and timekeeping of systems such as power, transportation, and finance. On the one hand, the real-time predictions of the model can serve as a key basis for identifying illegal attacks such as time delay attacks. On the other hand, in the event of loss or inaccuracy of time source data, the high-precision and stable real-time predictions of the model can serve as backup data to update the local time, thereby ensuring the safety and timekeeping accuracy of the system.

[0023] In view of this, such as Figure 1 As shown, this embodiment of the invention provides a secure timekeeping method based on a physical information neural network, including: S100, Receive data, wherein the received data is the clock difference data at the current moment; S200: Input the received data into a clock error model trained on a physical information neural network to obtain a prediction result, and determine whether the received data can be used for timekeeping updates based on the prediction result; The S300 uses received data that can be used for timekeeping updates to complete local clock adjustments.

[0024] In other words, the system uses the prediction results at the current moment to determine whether there are any security risks in the received data of the local system. If the received data deviates from the prediction results by more than the set security threshold, it can be considered that the received data has been subjected to a delay attack, causing it to not conform to the historical change pattern of clock difference data and cannot be used for local clock updates. In this case, the prediction results are used to ensure the system's timekeeping accuracy and stability. If the received data is close to the prediction results and does not exceed the set security threshold, it is considered that the slight deviation is caused by system noise and temperature drift, and the received data can be used for timekeeping updates to complete the local clock adjustment.

[0025] The key to effectively and reliably identifying delay attacks lies in the accuracy and stability of the clock bias model trained on a physically based neural network. This clock bias model is obtained using the following method: S201, Construct the neural network structure; S202, the neural network structure is trained using clock difference data from past times; that is, the clock difference model is trained using clock difference data from past times in order to achieve clock difference prediction in future times. S203, During training, the model parameters of the neural network structure are updated through the gradient information obtained by backpropagation, so as to minimize the loss function; S204. After training, a clock error model based on a physical information neural network (hereinafter referred to as the PINNs model) is obtained.

[0026] The training principle of the PINNs model is as follows: Figure 2As shown, the PINNs model mainly includes input terms, neural network structure, output terms, loss function, and optimizer. Input terms represent the clock bias data required for training the PINNs model, and may also include input parameters such as temperature and humidity. The neural network structure can be selected from different network types, such as feedforward neural networks (FNN) and recurrent neural networks (RNN), depending on the characteristics of the training data and the application scenario. The output terms represent the prediction results output by the PINNs model. The optimizer's task is to update the model parameters using gradient information obtained through backpropagation to minimize the loss function. Different optimizer types, such as gradient descent and momentum gradient descent, can be selected based on the model's convergence speed and performance. The loss function measures the deviation between the model's predictions and the actual data and physical laws. The core difference between the PINNs model and deep learning lies in the introduction of a physical loss function, which constrains the model to converge quickly and conform to changes in physical characteristics through physical laws. That is, the loss function includes both a physical loss function and a data loss function.

[0027] In the PINNs model, the traditional loss function has multiple key components with equal weight coefficients or fixed weight coefficients of different sizes, as shown in the following formula:

[0028] In the formula, It is the total loss function of the PINNs model; It is the data loss function of the PINNs model, which is driven by data, and represents the deviation between the model's predicted value and the actual training data in this iteration; yes The data weighting coefficient represents The proportion of the total loss function, i.e. the ability of data characteristics to constrain the model output; It is the physics loss function of the PINNs model, which is driven by physics and represents the deviation between the model's prediction in this iteration and the physical law. yes The physical weight coefficient represents The proportion in the total loss function represents the ability of physical laws to constrain the model output.

[0029] In short-term predictions (e.g., the next few seconds or minutes), the data loss function better reflects the characteristics of short-term data changes, while the physical laws inherent in the physical loss function cannot be fully reflected in a short time. In this case, data-driven approaches should take precedence. In long-term predictions (e.g., the data loss function may become unreliable due to noise, temperature drift, etc.), and the physical loss function should have greater weight. Therefore, this invention utilizes dynamic weights to balance the constraints of the physical loss function and the data loss function on the model output at different time steps, optimizing the gradient propagation of the PINNs model in long-term predictions. The total loss function is shown in the following formula:

[0030] In the formula, Indicates the time step of the forecast; It is related to the time step The relevant dynamic equilibrium factors, in When smaller, It can approach 1, at which point the model mainly relies on data-driven approaches. When it is large, It can approach 0, at which point the model mainly depends on physics.

[0031] The classic data loss function is represented by the mean squared error between the model's predicted output data and the corresponding real data in the training sample set (composed of clock difference data from past times). The physical loss function is represented by the mean squared error between the partial derivative of the model's predicted output data and the derivative of the clock difference physical equation. The calculation methods are as follows:

[0032]

[0033] In the formula, n This indicates the size of the sample set during batch training; It is the time step in the training sample set; It is the time step The corresponding model prediction output, It is the corresponding partial differential solution; The time step in the training sample set The corresponding actual value; It is the time step The solution to the corresponding physical partial differential equation It is its corresponding derivative.

[0034] This invention reduces the proportion of unreliable data in the loss function during long-term prediction by introducing a weight decay function for data items. This reduces the negative impact of noise on gradient calculation and improves the model training convergence speed. The modified data loss function is shown in the following formula:

[0035] In the formula, It's about time step. The weight decay function, Different weight decay functions can be considered comprehensively based on factors such as the size of the sample training set and the application scenario. The most suitable weight decay function can be determined through multiple training sessions, thereby reducing the interference of cumulative errors caused by noise and other factors on model training.

[0036] By incorporating dynamic equilibrium factors With data item attenuation coefficient A completely new loss function can be obtained, as shown in the following formula:

[0037] This invention optimizes the loss function in the PINNs model using the methods described above, dynamically balancing the penalty factors between the physical loss function and the data loss function. This optimizes gradient propagation at different time steps and reduces the interference of data error accumulation on long-term prediction accuracy. Simultaneously, a weight decay function for data items is introduced into the data loss function to reflect changes in data reliability at different time steps, reducing the negative impact of noise on gradient calculation during long-term prediction. Ultimately, a multi-coefficient fusion loss function is obtained, significantly improving the model's long-term prediction accuracy and stability, and enhancing the system's timeliness and security.

[0038] An example: Taking the second-order clock bias partial differential equation as the physical constraint of the PINNs model as an example, we completed the safe timekeeping based on the physical information neural network.

[0039] The PINNs model employs a feedforward neural network architecture to learn the data characteristics of the training sample set and uses it as an embedding carrier for physical constraints. An adaptive moment estimation optimizer (Adam) is used to dynamically adjust the gradient update step size of the loss function. In the loss function... A linearly decreasing function is chosen to balance the contributions of the data loss function and the physical loss function to gradient calculation at different time steps. An exponential decay function is chosen to reduce the interference of accumulated data error on gradient calculation, as shown in the following formula:

[0040] The security threshold is set to 10ns. If the clock difference data received by the system deviates from the prediction result by more than 10ns, the received data is considered to have been illegally attacked and cannot be used for local clock updates. In this case, the prediction result is used to ensure the system's timekeeping accuracy and stability. If the clock difference data received by the system deviates from the prediction result by less than 10ns, the received data is considered to be a normal deviation caused by factors such as system noise and temperature drift. The received data can be used for timekeeping updates to complete local clock adjustments.

[0041] Based on the same technological concept, such as Figure 3 As shown, the present invention also provides a security timekeeping system based on a physical information neural network, comprising: The data storage unit is used to receive data, wherein the received data is the clock difference data at the current moment; in addition, the data storage unit also stores the clock difference data of past moments used for training a neural network based on physical information. The security identification unit is used to input the received data into a clock bias model trained on a physical information neural network to obtain a prediction result, and to determine whether the received data can be used for timekeeping updates based on the prediction result; wherein, the clock bias model trained on the physical information neural network is trained using PINNs units. The timekeeping update unit is used to adjust the local clock using the received data that can be used for timekeeping updates.

[0042] The working principles of the above-mentioned functional units can be referred to the description in the foregoing method embodiments, and will not be repeated here.

[0043] Based on the same technical concept, embodiments of the present invention also provide an electronic device that can implement the secure timekeeping method based on a physical information neural network provided in the above embodiments of the present invention. In one embodiment, the electronic device can be a server, a terminal device, or other electronic devices. Figure 4 As shown, the electronic device may include: At least one processor and a memory connected to the at least one processor. In this embodiment of the invention, the specific connection medium between the processor and the memory is not limited. Figure 4 The example used is the connection between the processor and memory via a bus. The bus... Figure 4 The connections between other components are indicated by thick lines and are for illustrative purposes only, not as limiting information. Buses can be divided into address buses, data buses, control buses, etc., but for ease of representation, [the specific bus type is not shown here]. Figure 4 The processor is represented by a single thick line, but this does not imply that there is only one bus or one type of bus. Alternatively, a processor can also be called a controller; there are no restrictions on the name.

[0044] In this embodiment of the invention, the memory stores instructions that can be executed by at least one processor. By executing the instructions stored in the memory, at least one processor can execute the security timekeeping method based on physical information neural networks discussed above.

[0045] The processor is the control center of the device. It can connect to various parts of the control equipment through various interfaces and lines. By running or executing instructions stored in memory and calling data stored in memory, it can monitor the various functions and data processing of the device as a whole.

[0046] In an alternative design, the processor may include one or more processing units. The processor may integrate an application processor and a modem processor, wherein the application processor primarily handles the operating system, user interface, and applications, while the modem processor primarily handles wireless communication. It is understood that the modem processor may also not be integrated into the processor. In some embodiments, the processor and memory may be implemented on the same chip; in some embodiments, they may also be implemented separately on separate chips.

[0047] The processor can be a general-purpose processor, such as a CPU, digital signal processor, application-specific integrated circuit, field-programmable gate array or other programmable logic device, discrete gate or transistor logic device, or discrete hardware component, capable of implementing or executing the methods, steps, and logic block diagrams disclosed in the embodiments of this invention. The general-purpose processor can be a microprocessor or any conventional processor. The steps of the security timekeeping method based on a physical information neural network disclosed in the embodiments of this invention can be directly manifested as execution by a hardware processor, or execution by a combination of hardware and software modules within the processor.

[0048] Memory, as a non-volatile computer-readable storage medium, can be used to store non-volatile software programs, non-volatile computer-executable programs, and modules. Memory can include at least one type of storage medium, such as flash memory, hard disk, multimedia card, card-type memory, random access memory (RAM), static random access memory (SRAM), programmable read-only memory (PROM), read-only memory (ROM), electrically erasable programmable read-only memory (EEPROM), magnetic memory, magnetic disk, optical disk, etc. Memory is any other medium capable of carrying or storing desired program code in the form of instructions or data structures, and accessible by a computer, but is not limited thereto. In embodiments of the present invention, memory can also be a circuit or any other device capable of implementing storage functions, used to store program instructions and / or data.

[0049] By designing and programming the processor, the code corresponding to the security timekeeping method based on a physical information neural network described in the foregoing embodiments can be embedded into the chip, enabling the chip to execute the steps of the method described in the foregoing embodiments during operation. How to design and program a processor is a technique well-known to those skilled in the art and will not be elaborated upon here.

[0050] Based on the same inventive concept, embodiments of the present invention also provide a storage medium storing computer instructions that, when executed on a computer, cause the computer to perform a security timekeeping method based on a physical information neural network as described above.

[0051] In some alternative embodiments, the present invention also provides a aspect of a security timekeeping method based on a physical information neural network that can also be implemented as a program product comprising program code that, when the program product is run on a device, causes the control device to perform the steps in a security timekeeping method based on a physical information neural network according to various exemplary embodiments of the present invention as described above.

[0052] It should be noted that although several units or sub-units of the apparatus have been mentioned in the detailed description above, this division is merely exemplary and not mandatory. In fact, according to embodiments of the invention, the features and functions of two or more units described above can be embodied in one unit. Conversely, the features and functions of one unit described above can be further divided and embodied by multiple units. Furthermore, although the operation of the method of the invention is described in a specific order in the drawings, this does not require or imply that these operations must be performed in that specific order, or that all the operations shown must be performed to achieve the desired result. Additionally or alternatively, certain steps may be omitted, multiple steps may be combined into one step, and / or one step may be broken down into multiple steps.

[0053] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0054] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a server, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the process. Figure 1One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0055] Program code for performing the operations of this invention can be written using any combination of one or more programming languages, including object-oriented programming languages ​​such as Java and C++, as well as conventional procedural programming languages ​​such as C or similar languages. The program code can be executed entirely on the user's computing device, partially on the user's device, as a standalone software package, partially on the user's computing device and partially on a remote computing device, or entirely on a remote computing device or server.

[0056] In cases involving remote computing devices, the remote computing device can be connected to the user's computing device via any type of network, including a local area network (LAN) or a wide area network (WAN), or it can be connected to an external computing device (e.g., via the Internet using an Internet service provider).

[0057] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0058] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0059] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A secure timekeeping method based on a physical information neural network, characterized in that, include: Receive data, wherein the received data is the clock difference data at the current moment; The received data is input into a clock error model trained on a physical information neural network to obtain a prediction result. Based on the prediction result, it is determined whether the received data can be used for timekeeping updates. Local clock adjustment is accomplished using received data that can be used for timekeeping updates.

2. The secure timekeeping method based on a physical information neural network according to claim 1, characterized in that, The clock error model trained based on physical information neural network is obtained using the following method: Constructing neural network structures; The neural network structure is trained using clock difference data from past moments; During training, the model parameters of the neural network structure are updated using gradient information obtained through backpropagation, thereby minimizing the loss function; After training, a clock error model based on a neural network trained on physical information is obtained.

3. The secure timekeeping method based on a physical information neural network according to claim 2, characterized in that, The loss function includes a physical loss function and a data loss function.

4. The secure timekeeping method based on a physical information neural network according to claim 3, characterized in that, The ability of dynamic weights to balance the constraints of the physical loss function and the data loss function on the model output at different time steps is utilized.

5. The secure timekeeping method based on a physical information neural network according to claim 4, characterized in that, A weight decay function is introduced into the data loss function.

6. The secure timekeeping method based on a physical information neural network according to claim 2, characterized in that, The neural network structure adopts either a feedforward neural network or a recurrent neural network.

7. A secure timekeeping system based on a physical information neural network, characterized in that, include: A data storage unit is used to receive data, wherein the received data is the clock difference data at the current moment; The security identification unit is used to input the received data into a clock error model trained on a physical information neural network to obtain a prediction result, and to determine whether the received data can be used for timekeeping updates based on the prediction result. The timekeeping update unit is used to adjust the local clock using the received data that can be used for timekeeping updates.

8. An electronic device, characterized in that, include: At least one processor; and a memory communicatively connected to the at least one processor; The memory stores instructions executable by the at least one processor, which executes the instructions stored in the memory to perform the method as described in any one of claims 1-6.

9. A computer-readable storage medium, characterized in that, The computer-readable storage medium is used to store instructions that, when executed, cause the method as described in any one of claims 1-6 to be implemented.

10. A computer program product, characterized in that, When the computer program product is invoked by a computer, it causes the computer to perform the method as described in any one of claims 1-6.

Citation Information

Patent Citations

  • Clock error prediction method and device based on deep learning, medium and terminal

    CN111738407A

  • Loss function adaptive balancing method of neural network with embedded physical knowledge

    CN114118405A

  • Differential privacy availability measurement method for deep learning

    CN114118407A

  • Fluid mechanics equation solving method based on physical information neural network

    CN117786286A

  • Time service method and device, equipment and storage medium

    CN118466156A