Partial differential equation data processing method, system, storage medium, device and application
Through attention mechanism and sparse regression optimization in data sampling and loss functions, a differential term alternative library is constructed in combination with prior knowledge and basis function representation, which solves the problems of low efficiency and data dependence in learning partial differential equations in the existing technology. It realizes efficient and precise learning under small amounts of data.
Patent Information
- Application Number
- CN202110131994.0
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2021-01-30
- Publication Date
- 2024-12-13
- Estimated Expiration
- 2041-01-30
AI Technical Summary
The existing technology is inefficient and susceptible to subjective factors when learning partial differential equations implicit in the system, and it is difficult to find the mechanism behind complex systems. The data-driven method has great limitations, deep learning methods require a large amount of data and have high hardware costs, and run it. Inefficient.
Use attention mechanism to sample time-space data, use prior knowledge and basis function notation to build a differential term alternative library, learn the coefficients of differential terms in the alternative library through multiple single-layer regression networks, and add regularity to the loss function To reduce the interference terms using sparse regression, we obtain the final partial differential equation.
It improves learning accuracy and efficiency, reduces the amount of data required by the model, improves the accuracy of partial differential equations, reduces hardware requirements, and is suitable for complex system learning under small data volumes.
Smart Images
Figure CN112784205B_ABST
Abstract
Description
Technical Field
[0001] The present invention belongs to the technical field of partial differential equation processing implicit in learning a system from data, and in particular, relates to a partial differential equation data processing method, system, storage medium, device and application. Background Art
[0002] At present: Differential equations, especially partial differential equations, play an important role in many disciplines, and can be used to describe the physical laws behind a given system. Traditionally, partial differential equations are derived mathematically or physically based on some basic principles, such as from the Schrödinger equation in quantum mechanics to molecular dynamics models, from the Boltzmann equation to the Navier-Stokes equations, etc. However, the mechanisms behind many complex systems in modern applications (such as multiphase flow, neuroscience, finance, biological sciences, and many other problems) are generally unclear, and the partial differential equations of these systems are usually obtained by empirical formulas. In the past decade, with the rapid development of sensors, computing power, and data storage technology, it is now easy to collect, store, and process large amounts of data. Such a large amount of data provides new opportunities to discover (possibly new) physical laws from the data. Therefore, building a model that learns partial differential equations from data to approximate the observed complex dynamic data will be of great significance to people's analysis and understanding of the mechanisms behind complex systems in modern applications.
[0003] In the early days, summarizing the implicit mechanism from the system often relied on people's experience, or first proposed a hypothesis and then conducted experimental verification. This method is greatly influenced by human subjectivity and is inefficient. In addition, it is difficult to rely on empirical formulas to obtain the implicit partial differential equations of complex or newly emerging systems. Therefore, with the development and progress of computing power and data storage technology, solutions to summarize the implicit partial differential equations of the system through computer technology have gradually emerged. Josh Bongard and Michael Schmidt made preliminary attempts on how to learn the implicit partial differential equations in the system under data-driven in 2007 and 2009, respectively. Their main idea was to compare the numerical differential of the input data with the analytical differential of the candidate function, and use symbolic regression and evolutionary algorithms to determine the nonlinear dynamic system. In 2017, Emmanuel deBezenac used a sparse identification method for nonlinear dynamics (SINDy) to model the partial differential equations of sea surface temperature. The main idea of SINDy is to first build a sufficiently large candidate library of partial differential equation terms, and then learn the implicit partial differential equations from the sea surface temperature data to predict the sea surface temperature. In 2018, Marizar Raissi proposed a model for learning unknown parameters under the premise that the nonlinear response function form of the partial differential equation is known. His main idea is to introduce the idea of regularization through Gaussian process between two consecutive time steps. Generally speaking, the method of learning partial differential equations from data focuses on representing the observed data with a relatively simple model and obtaining the analytical form of the partial differential equation.
[0004] There are several problems and defects in the existing methods of learning partial differential equations implicit in a system from data: (1) The traditional method of learning partial differential equations implicit in a system through artificial experience and experimental verification is too inefficient, easily affected by human subjective factors, and it is difficult to find the mechanism behind the complex system. (2) The existing methods of learning partial differential equations implicit in a system under data-driven learning have great limitations, with many restrictions on the terms of partial differential equations that can be learned, and the learning effect is easily affected by noise. (3) The existing methods of learning partial differential equations implicit in a system through deep learning require a large amount of sample data, but it is difficult for some systems to collect enough data for training. In addition, the hardware cost of deep learning is high, and the operating efficiency is not high, which limits the use of these methods.
[0005] Through the above analysis, the problems and defects of the prior art are as follows:
[0006] (1) The traditional method of learning the partial differential equations implicit in the system through artificial experience and experimental verification is too inefficient, easily affected by human subjective factors, and difficult to find the mechanism behind the complex system, resulting in the partial differential equations obtained in the end having large errors and unstable effects.
[0007] (2) Existing methods for learning the partial differential equations implicit in the system under data-driven conditions have great limitations. There are many restrictions on the terms of the partial differential equations that can be learned, resulting in the learned partial differential equations not conforming to the analytical form of the true equations. In addition, the learning effect is easily affected by noise, resulting in excessive errors in the final partial differential equations.
[0008] (3) Existing methods for learning the partial differential equations implicit in the system through deep learning require a large amount of sample data, but it is difficult to collect enough data for training in some systems. In addition, the hardware cost of deep learning is high and the operating efficiency is low, which limits the use of these methods.
[0009] The difficulty of solving the above problems and defects is: how to learn the partial differential equations implicit in the system with a small amount of data, and how to improve learning efficiency and accuracy.
[0010] The significance of solving the above problems and defects is that exploring the partial differential equations implicit in the system through data-driven methods can save a lot of labor costs, help people understand the operating laws of complex systems, and promote progress in the field of learning the partial differential equations implicit in the system from data. Summary of the invention
[0011] In view of the problems existing in the prior art, the present invention provides a partial differential equation data processing method, system, storage medium, device and application.
[0012] The present invention is implemented in a partial differential equation data processing method, the partial differential equation data processing method comprising:
[0013] Use the attention mechanism to sample spatiotemporal data to improve learning accuracy and initial value robustness;
[0014] Using prior knowledge and basis function representation to construct a candidate library of differential terms to obtain a reasonable and complete candidate library;
[0015] A deep network consisting of multiple single-layer regression networks learns the corresponding coefficients of differential terms in the candidate library to complete the preliminary partial differential equation learning;
[0016] Regularization is added to the loss function, and interference terms are reduced through sparse regression to obtain the final partial differential equation.
[0017] The present invention utilizes the attention mechanism to sample spatiotemporal data. Because some systems controlled by partial differential equations (such as image denoising and image segmentation tasks based on partial differential equations) are more concerned about the final value of the partial differential equations, the attention mechanism is introduced during time sampling to improve the accuracy and initial value robustness, making it more effective in tasks that focus on the final value results of the partial differential equations. Prior knowledge and basis function representation are used to construct a candidate library of differential terms, so that the candidate library is complete and reasonable, and the amount of calculation is reduced. A deep network composed of multiple single-layer regression networks is then used to learn the coefficients corresponding to the candidate items in the candidate library to complete preliminary partial differential equation learning. Finally, a regularization term is added to the loss function to make the network a partial differential equation learner that utilizes a sparse regression method to reduce irrelevant interference terms.
[0018] Furthermore, the data sampling of the partial differential equation data processing method specifically includes: firstly, spatial sampling is performed on the input time series data, the spatial sampling uses random sampling, and some points in the input space are selected as observation objects; then, time sampling is performed, the time sampling uses uniform sampling, and new sample values are taken after a certain period of time based on the observation points selected by the spatial sampling, and an attention mechanism is introduced during time sampling, and the weight of the second half of the data is increased by controlling the time sampling rate;
[0019] Furthermore, the network parameters of the partial differential equation data processing method during data sampling are set as follows: boundary width: 5; spatial sampling rate: 2%; total time sampling rate: 25%; number of time sampling equal segments: 5; attention mechanism correction: the sampling rate of each segment increases by 5%; segmented time sampling rate: 15%, 20%, 25%, 30%, 35%.
[0020] Furthermore, the partial differential equation data processing method constructs a candidate library of differential terms: an equation containing partial derivatives or partial differentials of an unknown function is called a partial differential equation, and the general form of a partial differential equation is expressed as:
[0021] u t (t,x,y)=F(x,y,u,u x ,u y ,u xx ,u xy ,u yy ,...),(x,y)∈R 2 ,t∈[0,T].;
[0022] Where t represents time, x and y represent space, and the partial differential equation u t are the differential terms x, y, u, u x ,u y ,u xx ,u xy ,u yy, ... function; the partial differential equation is expressed as follows:
[0023] ω t =Θ(ω,u,v)ξ;
[0024] Where Θ represents the matrix composed of each differential term ω, u, v, and ξ represents the coefficient of the differential term. That is, first construct a candidate library of differential terms, then learn the coefficient ξ corresponding to each differential term to obtain the final partial differential equation. The construction of the candidate library is based on the basis function representation method to construct x, y, u, u x ,u y ,u xx ,u xy ,u yy Basic differential terms such as , and when targeting a specific system, some complex differential terms closely related to the system can be added to the candidate library based on prior knowledge; the candidate library should not be too small, otherwise it will affect the accuracy of the partial differential equations finally learned; the candidate library should not be too large, otherwise it will affect the operating efficiency;
[0025] Compress the candidate library. After obtaining the original differential candidate library, the candidate library matrix should be compressed according to the data sampling results of the previous step to reduce the amount of calculation:
[0026]
[0027] in Indicates matrix compression, and the compressed new matrix is used as the final candidate library for differential terms;
[0028] Among them, the methods of constructing candidates selected when constructing the differential term candidate library include forward difference, backward difference, central difference, and Chebyshev polynomial interpolation method; when the partial differential equation obtained later needs to be solved iteratively, the Chebyshev polynomial interpolation method is used to construct the differential term candidate library.
[0029] Furthermore, the differential term coefficient learning of the partial differential equation data processing method is as follows: a deep network is constructed to obtain the corresponding coefficient of each differential term in the candidate library. The deep network is composed of multiple single-layer regression networks stacked to improve the accuracy of the network training results layer by layer. The composition of a single regression network is as follows: the coefficient ξ of the differential term is obtained by using a regression method for the compressed differential term candidate library. The loss function to be minimized is as follows:
[0030]
[0031] in Represents the estimated value, and the differential term coefficient obtained at the i-th layer is ξ i , the coefficients of the final output of the single-layer regression network are used as the coefficients ξ of the partial differential equation;
[0032] The regression method of the network is least squares regression, and the lstsq tool in Numpy is used to complete the least squares regression task.
[0033] Furthermore, the regularization of the partial differential equation data processing method includes: modifying the loss function to:
[0034]
[0035] Where λ represents the regularization coefficient. The partial differential equation after sparse regression will ignore the interference terms with very small coefficients, making the final result closer to the real partial differential equation.
[0036] When regularizing, L1 regularization is selected for sparse regression. The regularization coefficient λ of the network increases from 10 to -6 to 10 -5 The gradually changing terms of the partial differential equation and the coefficients corresponding to each differential term constitute the partial differential equation that the model finally learns.
[0037] Another object of the present invention is to provide a computer device, the computer device comprising a memory and a processor, the memory storing a computer program, and when the computer program is executed by the processor, the processor performs the following steps:
[0038] Use attention mechanism to sample spatiotemporal data;
[0039] Using prior knowledge and basis function representation to construct a library of candidate differential terms;
[0040] The corresponding coefficients of the differential terms in the candidate library of deep network learning composed of multiple single-layer regression networks;
[0041] Regularization is added to the loss function, and interference terms are reduced through sparse regression to obtain the final partial differential equation.
[0042] Another object of the present invention is to provide a computer-readable storage medium storing a computer program, wherein when the computer program is executed by a processor, the processor executes the following steps:
[0043] Use attention mechanism to sample spatiotemporal data;
[0044] Using prior knowledge and basis function representation to construct a library of candidate differential terms;
[0045] The corresponding coefficients of the differential terms in the candidate library of deep network learning composed of multiple single-layer regression networks;
[0046] Regularization is added to the loss function, and interference terms are reduced through sparse regression to obtain the final partial differential equation.
[0047] Another object of the present invention is to provide a partial differential equation data processing system for implementing the partial differential equation data processing method, the partial differential equation data processing system comprising:
[0048] The data sampling module is used to sample the data first, and the data sampling is performed in both space and time;
[0049] A differential term candidate library construction module is used to construct a candidate library, and uses prior knowledge and basis function representation to select differential terms to construct the differential term candidate library;
[0050] The differential term coefficient learning module is used to learn the coefficients corresponding to the candidate items in the candidate library using a deep network composed of multiple single-layer regression networks;
[0051] The regularization screening differential term module is used to add regularization terms to the loss function, making the network a partial differential equation learner that uses a sparse regression method to reduce irrelevant interference terms. The final partial differential equation terms and the coefficients corresponding to each differential term constitute the partial differential equation finally learned by the network.
[0052] Another object of the present invention is to provide a data learning terminal, which is used to implement the partial differential equation data processing method.
[0053] Combining all the above technical solutions, the advantages and positive effects of the present invention are as follows: the present invention first samples the data, and the data sampling is performed in space and time respectively. Because some systems controlled by partial differential equations (such as image denoising and image segmentation tasks based on partial differential equations) are more concerned about the final value of the partial differential equation, the attention mechanism is introduced during time sampling to improve the accuracy and initial value robustness, so that it is more effective in tasks that focus on the final value results of the partial differential equation; then the candidate library is constructed, and reasonable differential terms are selected to construct the differential term candidate library using mathematical methods such as prior knowledge and basis function representation; then a deep network composed of multiple single-layer regression networks is used to learn the coefficients corresponding to the candidate items in the candidate library; finally, a regularization term is added to the loss function to make the network a partial differential equation learner that uses a sparse regression method to reduce irrelevant interference terms, and the final partial differential equation terms and the coefficients corresponding to each differential term constitute the partial differential equation finally learned by the network.
[0054] Compared with other existing methods of learning partial differential equations from the system, the present invention can make full use of the spatial and temporal information of the observed data, reduce the amount of data required for the model, and improve the accuracy of the learned partial differential equations while ensuring efficiency. The present invention designs a deep differential equation network to learn the partial differential equations implicit in the system, which is used to solve the problem that the traditional method of learning the partial differential equations implicit in the system through artificial experience and experimental verification is too inefficient, easily affected by human subjective factors, and difficult to find the mechanism behind the complex system. The deep differential equation network can learn a variety of differential terms of partial differential equations, and has stronger initial value robustness in processing tasks that are more sensitive to final value data. The deep differential equation network requires a small amount of data, has low hardware requirements, and has high operating efficiency.
[0055] Table 1 Effect of the method of the present invention on learning classical partial differential equations:
[0056]
[0057] Table 2 Effect of the method of the present invention on learning image processing partial differential equations:
[0058]
[0059] Average running time of the algorithm: 5.018s. BRIEF DESCRIPTION OF THE DRAWINGS
[0060] In order to more clearly illustrate the technical solutions of the embodiments of the present application, the following briefly introduces the drawings required for use in the embodiments of the present application. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.
[0061] Figure 1 It is a flow chart of a partial differential equation data processing method provided by an embodiment of the present invention.
[0062] Figure 2 It is a structural diagram of a partial differential equation data processing system provided by an embodiment of the present invention.
[0063] Figure 3 It is a flow chart for implementing the partial differential equation data processing method provided by an embodiment of the present invention. DETAILED DESCRIPTION
[0064] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.
[0065] In view of the problems existing in the prior art, the present invention provides a partial differential equation data processing method, system, storage medium, device and application. The present invention is described in detail below in conjunction with the accompanying drawings.
[0066] like Figure 1 As shown, the partial differential equation data processing method provided by the present invention comprises the following steps:
[0067] S101: Sampling spatiotemporal data using attention mechanism;
[0068] S102: Construct a library of candidate differential terms using prior knowledge and basis function representation;
[0069] S103: The corresponding coefficients of the differential terms in the candidate library of deep network learning composed of multiple single-layer regression networks;
[0070] S104: Regularization is added to the loss function, and interference terms are reduced through sparse regression to obtain the final partial differential equation.
[0071] Ordinary technicians in the industry can also use other steps to implement the partial differential equation data processing method provided by the present invention. Figure 1 The partial differential equation data processing method provided by the present invention is only a specific embodiment.
[0072] like Figure 2 As shown, the partial differential equation data processing system provided by the present invention includes:
[0073] The data sampling module is used to sample the data first. The data sampling is performed in both space and time. Because some systems controlled by partial differential equations (such as image denoising and image segmentation tasks based on partial differential equations) are more concerned about the final value of partial differential equations, the attention mechanism is introduced during time sampling to improve the accuracy and initial value robustness, making it more effective in tasks that focus on the final value results of partial differential equations.
[0074] The differential term candidate library construction module is used to construct the candidate library, and uses mathematical methods such as prior knowledge and basis function representation to select reasonable differential terms to construct the differential term candidate library;
[0075] The differential term coefficient learning module is used to learn the coefficients corresponding to the candidate items in the candidate library using a deep network composed of multiple single-layer regression networks;
[0076] The regularization screening differential term module is used to add regularization terms to the loss function, making the network a partial differential equation learner that uses a sparse regression method to reduce irrelevant interference terms. The final partial differential equation terms and the coefficients corresponding to each differential term constitute the partial differential equation finally learned by the network.
[0077] The technical solution of the present invention is further described below in conjunction with the accompanying drawings.
[0078] like Figure 3 As shown, the partial differential equation data processing method provided by the present invention comprises the following steps:
[0079] Step 1: Data sampling. First, spatial sampling is performed on the input time series data. Spatial sampling uses random sampling and selects some points in the input space as observation objects. Then temporal sampling is performed. Temporal sampling uses uniform sampling and takes new sample values after a certain period of time based on the observation points selected by spatial sampling. The attention mechanism is introduced during temporal sampling to increase the weight of the second half of the data by controlling the temporal sampling rate, so as to increase the accuracy of the model and the robustness of the initial value.
[0080] The effect of the attention mechanism on the time sampling of the input is as follows: the total time sampling rate is 25% of all time series, and all input time series are divided into five parts. The time sampling rate of the first part is 15%, the time sampling rate of the second part is 20%, and so on. The attention mechanism correction value of each sampling rate is 5%.
[0081] The network parameters for data sampling are as follows: Boundary width: 5; Spatial sampling rate: 2%; Total time sampling rate: 25%; Number of time sampling equal segments: 5; Attention mechanism correction: Sampling rate increases by 5% per segment. Segmented time sampling rates: 15%, 20%, 25%, 30%, 35%.
[0082] Step 2: Build a library of candidate differential terms: Equations containing partial derivatives (or partial differentials) of unknown functions are called partial differential equations. The general form of partial differential equations can be expressed as:
[0083] u t (t,x,y)=F(x,y,u,u x ,u y ,u xx ,u xy ,u yy , ...), (x, y)∈R 2 , t∈[0,T].
[0084] Where t represents time, x and y represent space, and the partial differential equation u t are the differential terms x, y, u, u x ,u y ,u xx ,u xy ,u yy , ... is a function of. In the present invention, the partial differential equation is expressed as follows:
[0085] ω t=Θ(ω,u,v)ξ;
[0086] Where Θ represents the matrix composed of each differential term ω, u, v, and ξ represents the coefficient of the differential term. That is, first construct a candidate library of differential terms, then learn the coefficient ξ corresponding to each differential term to obtain the final partial differential equation. The construction of the candidate library is based on the basis function representation method to construct x, y, u, u x ,u y ,u xx ,u xy ,u yy Basic differential terms such as , and when targeting a specific system, some complex differential terms closely related to the system can be added to the candidate library based on prior knowledge. The candidate library should not be too small, otherwise it will affect the accuracy of the partial differential equations finally learned; the candidate library should not be too large, otherwise it will affect the operating efficiency.
[0087] Next, the candidate library is compressed. After obtaining the original differential candidate library, the candidate library matrix should be compressed according to the data sampling results of the previous step to reduce the amount of calculation:
[0088]
[0089] in Indicates matrix compression, and the compressed new matrix serves as the final candidate library for differential terms.
[0090] Among them, the methods of constructing candidates that can be selected when constructing the differential term candidate library include forward difference, backward difference, central difference, Chebyshev polynomial interpolation method, etc. When the partial differential equation obtained later needs to be solved iteratively, if the numerical solution method and the construction method are consistent, the solution accuracy is the highest. In this specific implementation plan, the Chebyshev polynomial interpolation method is used to construct the differential term candidate library.
[0091] Step 3: Learning the coefficient of differential term: Construct a deep network to find the corresponding coefficient of each differential term in the candidate library. The deep network is composed of multiple single-layer regression networks stacked to improve the accuracy of network training results layer by layer. The composition of a single regression network is: the coefficient ξ of the differential term is obtained by regression method for the compressed differential term candidate library. The loss function to be minimized is as follows:
[0092]
[0093] in Represents the estimated value, and the differential term coefficient obtained at the i-th layer is ξ i , the coefficients of the final single-layer regression network output are used as the coefficients ξ of the partial differential equation.
[0094] The regression method of the network is least squares regression, and the lstsq tool in Numpy is used to complete the least squares regression task.
[0095] Step 4: Regularization: When constructing a candidate library of differential terms, an overcomplete library is generally constructed in order to represent complex partial differential equations, so it contains some interference terms that are not included in the real partial differential equations. In order to eliminate these interferences and make the learned partial differential equations closer to the real equations, the present invention introduces a regularization term and modifies the loss function to:
[0096]
[0097] Where λ represents the regularization coefficient. In this way, the partial differential equation after sparse regression will ignore the interference terms with very small coefficients, making the final result closer to the real partial differential equation.
[0098] L1 regularization is selected for sparse regression during regularization. Considering that the regularization coefficient should not be set too large at the beginning when training a deep network, otherwise some terms of the real partial differential equations may be ignored in the first few layers of the network due to their small coefficients, the regularization coefficient λ of the network is increased from 10 to -6 to 10 -5 The final partial differential equation terms and the coefficients corresponding to each differential term constitute the partial differential equation that the model finally learns.
[0099] The technical effects of the present invention are described in detail below in conjunction with simulation.
[0100] 1. Simulation conditions. The present invention is carried out using Python language on a PC with an Intel(R) Core(TM) i7-9700 CPU, CPU 3.60 GHz, RAM 16.00 GB, and Ubuntu 16.04 operating system to complete the simulation experiment of the present invention.
[0101] 2. Simulation experiment content and result analysis. The experimental data of the present invention are data generated by various classical partial differential equations and real noisy data generated when partial differential equations are used for image processing. The classical partial differential equations include Burgers equation, KdV equation, NLS equation and Navier Stokes equation; the image processing performed by partial differential equations includes image denoising data using heat diffusion equation and level set image segmentation data using GAC model. The experimental results are shown in Table 1 and Table 2.
[0102] Table 1 Effect of the method of the present invention on learning classical partial differential equations:
[0103]
[0104]
[0105] Table 2 Effect of the method of the present invention on learning image processing partial differential equations:
[0106]
[0107] Average running time of the algorithm: 5.018s.
[0108] According to the above results, the differential terms of partial differential equations that can be learned by the deep differential equation network are rich in types, and high accuracy and efficiency can be obtained when the required data volume is small, and good learning effect can still be achieved when processing real data with noise interference generated when processing images through partial differential equations. The hardware part can be implemented using dedicated logic; the software part can be stored in a memory and executed by an appropriate instruction execution system, such as a microprocessor or dedicated design hardware. A person of ordinary skill in the art can understand that the above-mentioned devices and methods can be implemented using computer executable instructions and / or included in a processor control code, such as a carrier medium such as a disk, CD or DVD-ROM, a programmable memory such as a read-only memory (firmware), or a data carrier such as an optical or electronic signal carrier. Such code is provided on the carrier medium. The device and its modules of the present invention can be implemented by hardware circuits such as ultra-large-scale integrated circuits or gate arrays, semiconductors such as logic chips, transistors, etc., or programmable hardware devices such as field programmable gate arrays, programmable logic devices, etc., can also be implemented by software executed by various types of processors, or can be implemented by a combination of the above-mentioned hardware circuits and software, such as firmware.
[0109] The above description is only a specific implementation mode of the present invention, but the protection scope of the present invention is not limited thereto. Any modifications, equivalent substitutions and improvements made by any technician familiar with the technical field within the technical scope disclosed by the present invention and within the spirit and principle of the present invention should be covered by the protection scope of the present invention.
Claims
1. A partial differential equation data processing method, characterized in that: The partial differential equation data processing method comprises: Use attention mechanism to sample spatiotemporal data; Using prior knowledge and basis function representation to construct a library of candidate differential terms; The corresponding coefficients of the differential terms in the candidate library of deep network learning composed of multiple single-layer regression networks; Regularization is added to the loss function, and interference terms are reduced through sparse regression to obtain the final partial differential equation; Image processing by partial differential equations includes image denoising data using the heat diffusion equation and level set image segmentation data using the GAC model; The partial differential equation data processing method constructs a differential term candidate library. The equation containing the partial derivative or partial differential of the unknown function is called a partial differential equation. The general form of the partial differential equation is expressed as: u t (t,x,y)=F(x,y,u,u x ,u y ,u xx ,u xy ,u yy ,…),(x,y)∈R 2 ,t∈[0,T]; Where t represents time, x and y represent space, and the partial differential equation u t are the differential terms x, y, u, u x ,u y ,u xx ,u xy ,u yy , ... of functions; the partial differential equation is expressed as follows: oh t =Θ(ω,u,v)ξ; Where Θ represents the matrix composed of each differential term ω, u, u, and ξ represents the coefficient of the differential term. That is, the candidate library of differential terms is first constructed, and then the coefficient ξ corresponding to each differential term is learned to obtain the final partial differential equation. The construction of the candidate library is based on the basis function representation method to construct x, y, u, u x ,u y ,u xx ,u xy ,u yy Basic differential terms such as , and when targeting a specific system, some complex differential terms closely related to the system can be added to the candidate library based on prior knowledge; the candidate library should not be too small, otherwise it will affect the accuracy of the partial differential equations finally learned; the candidate library should not be too large, otherwise it will affect the operating efficiency; After compressing the candidate library and obtaining the original differential candidate library, the candidate library matrix should be compressed according to the data sampling results of the previous step to reduce the amount of calculation: in Indicates matrix compression, and the compressed new matrix is used as the final candidate library for differential terms; The methods of constructing candidate items when constructing the differential term candidate library include forward difference, backward difference, central difference, and Chebyshev polynomial interpolation method. When the partial differential equation obtained later needs to be solved iteratively, the Chebyshev polynomial interpolation method is used to construct the differential term candidate library. The differential term coefficient learning of the partial differential equation data processing method is as follows: a deep network is constructed to obtain the corresponding coefficient of each differential term in the candidate library. The deep network is composed of multiple single-layer regression networks stacked to improve the accuracy of the network training results layer by layer. The composition of a single regression network is as follows: the coefficient ξ of the differential term is obtained by regression method for the compressed differential term candidate library. The loss function to be minimized is as follows: in Represents the estimated value, and the differential term coefficient obtained at the i-th layer is ξ i , the coefficients of the final output of the single-layer regression network are used as the coefficients ξ of the partial differential equation; The regression method of the network is least squares regression, and the lstsq tool in Numpy is used to complete the least squares regression task.
2. The partial differential equation data processing method according to claim 1, characterized in that: The data sampling of the partial differential equation data processing method specifically includes: firstly spatially sampling the input time series data, using random sampling for spatial sampling, and selecting some points in the input space as observation objects; then performing time sampling, using uniform sampling for time sampling, taking new sample values after a certain period of time based on the observation points selected by spatial sampling, introducing an attention mechanism during time sampling, and increasing the weight of the second half of the data by controlling the time sampling rate.
3. The partial differential equation data processing method according to claim 2, characterized in that: The network parameters of the partial differential equation data processing method during data sampling are set as follows: boundary width: 5; spatial sampling rate: 2%; total time sampling rate: 25%; number of time sampling equal segments: 5; attention mechanism correction: the sampling rate of each segment increases by 5%; segmented time sampling rate: 15%, 20%, 25%, 30%, 35%.
4. The partial differential equation data processing method according to claim 1, characterized in that: The regularization of the partial differential equation data processing method includes: modifying the loss function to: Where λ represents the regularization coefficient. The partial differential equation after sparse regression will ignore the interference terms with very small coefficients, making the final result closer to the real partial differential equation. When regularizing, L1 regularization is selected for sparse regression. The regularization coefficient λ of the network increases from 10 to -6 to 10 -5 The gradually changing terms of the partial differential equation and the coefficients corresponding to each differential term constitute the partial differential equation that the model finally learns.
5. A computer-readable storage medium storing a computer program, wherein when the computer program is executed by a processor, the processor executes the steps of the partial differential equation data processing method as described in any one of claims 1 to 4.
6. A partial differential equation data processing system for implementing the partial differential equation data processing method according to any one of claims 1 to 4, characterized in that: The partial differential equation data processing system comprises: The data sampling module is used to sample the data first, and the data sampling is performed in both space and time; A differential term candidate library construction module is used to construct a candidate library, and uses prior knowledge and basis function representation to select differential terms to construct the differential term candidate library; The differential term coefficient learning module is used to learn the coefficients corresponding to the candidate items in the candidate library using a deep network composed of multiple single-layer regression networks; The regularization screening differential term module is used to add regularization terms to the loss function, making the network a partial differential equation learner that uses a sparse regression method to reduce irrelevant interference terms. The final partial differential equation terms and the coefficients corresponding to each differential term constitute the partial differential equation finally learned by the network.
7. A data learning terminal, characterized in that: The data learning terminal is used to implement the partial differential equation data processing method described in any one of claims 1 to 4.