Time-varying matrix inversion method based on double integral recurrent neural network
By introducing the IA-DTD-RNN model with double integral term and discretization processing, the problem of insufficient perturbation resistance during the time-varying matrix inversion process of RNN model is solved, and fast convergence and high-precision solution under complex perturbation are achieved.
Patent Information
- Application Number
- CN202510648810.6
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-05-20
- Publication Date
- 2025-08-19
AI Technical Summary
The existing RNN models lack the ability to resist disturbances in solving time-varying matrix inverses, especially the lack of processing capabilities for complex perturbations such as discrete time-varying secondary perturbations, which affects the model accuracy and stability.
The double integral term is introduced, the recurrent neural network is redesigned and discrete the 5-instantaneous discretization formula is used to form the IA-DTD-RNN model to improve the perturbation resistance and solution accuracy.
Under complex perturbations, the IA-DTD-RNN model converges rapidly, has good anti-perturbation performance and high precision, and is suitable for complex production environments.
Smart Images
Figure CN120508738A_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the field of neural network technology, and in particular to a time-varying matrix inversion method. Background Art
[0002] Time-varying matrix inversion is crucial in numerous engineering fields, including robotic control and signal processing. In practical applications, recurrent neural networks (RNNs), due to their ability to handle dynamic problems, have become an effective tool for solving this problem. However, the reality is that a wide variety of perturbations (such as constants, linear time-varying perturbations, and even more complex quadratic time-varying perturbations) are common in real-world engineering applications. These perturbations can severely impact model accuracy and stability, and can even lead to model failure. While existing research (such as integral-enhanced RNNs) has attempted to suppress some perturbations, their ability to handle complex perturbations (especially quadratic perturbations) remains insufficient. Currently, research on RNN models for discrete time-varying matrix inversion is still limited in its ability to effectively suppress multiple perturbations.
[0003] Therefore, the existing technology lacks an advanced RNN inversion model that can simultaneously and efficiently solve the discrete time-varying matrix inverse and effectively suppress multiple disturbances (especially discrete time-varying quadratic disturbances). Therefore, the development of such a model with both high precision and strong disturbance suppression capabilities has important theoretical and application value. Summary of the Invention
[0004] In response to the deficiencies in the prior art, the present invention provides a time-varying matrix inversion method based on a double integral recursive neural network, which solves the problem of insufficient anti-disturbance ability of the existing RNN inversion model in the process of solving the time-varying matrix. While improving the anti-disturbance ability, it can also improve the solution accuracy of the model, thereby making the model have good versatility.
[0005] The object of the present invention is achieved by: a time-varying matrix inversion method based on a double integral recursive neural network, comprising the following steps:
[0006] Step S1: define the problem model of time-varying matrix inversion;
[0007] Step S2: Introduce the double integral term to obtain a new recurrent neural network design formula and establish the IA-CTD-RNN model;
[0008] Step S3: Propose and rearrange the 5-instantaneous discretization formula;
[0009] Step S4: Using the 5-instantaneous discretization formula, discretize the IA-CTD-RNN model to obtain the IA-DTD-RNN model, completing the time-varying matrix inversion method.
[0010] Furthermore, the step S1 is specifically as follows:
[0011] The problem formula for inverting a continuous time-varying matrix is defined as:
[0012] ;
[0013] in is an unknown matrix to be solved, is the identity matrix, is a non-singular smooth time-varying matrix;
[0014] The problem formula for inverting a discrete time-varying matrix is defined as:
[0015] ;
[0016] in, represent , Represents the step length.
[0017] Furthermore, the step S2 is specifically as follows:
[0018] The continuous time-varying matrix inversion process is monitored by the following error function:
[0019] ;
[0020] According to the existing recurrent neural network design formula, the double integral term is introduced to obtain a new recurrent neural network design formula:
[0021] ;
[0022] in , and These are three design parameters that are closely related to the convergence speed of the IA-CTD-RNN model. Combining the error function with the new recurrent neural network design formula, we get the IA-CTD-RNN model:
[0023] Furthermore, the step S3 is specifically as follows:
[0024] 5-The instantaneous discretization formula is:
[0025] ;
[0026] By rearranging the formula, we get the formula:
[0027]
[0028] The rearranged formula is used to discretize the IA-CTD-RNN model to obtain the IA-DTD-RNN model.
[0029] Furthermore, the step S4 is specifically as follows:
[0030] For the double integral term in the IA-CTD-RNN model, a new single-step forward difference formula for approximate calculation of the double integral is defined based on Taylor expansion:
[0031] ;
[0032] in represent The second derivative of
[0033] Using the 5-instantaneous discretization formula as the basis for the discretization of the IA-CTD-RNN model, the IA-DTD-RNN model is obtained as follows:
[0034] ;
[0035] In the above formula, and The terms represent the single integral term and the double integral term respectively. The single integral term is approximately calculated using the 5-instantaneous discretization formula, and the result is:
[0036] ;
[0037] Using the forward difference formula to approximate the double integral term, we can get the following The central difference formula of order accuracy is:
[0038] ;
[0039] Afterwards, we can get the formula:
[0040] .
[0041] Compared with the existing technology, the beneficial effect of the present invention is that the double integral term is introduced into the traditional RNN inversion model, which effectively improves the accuracy of the inversion result, and the model can still converge quickly under complex interference conditions, proving that the model has good anti-disturbance performance, and can therefore be extended to complex production environments. BRIEF DESCRIPTION OF THE DRAWINGS
[0042] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are merely embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on the provided drawings without paying any creative work.
[0043] Figure 1 The figure is a flow chart of the model implementation method of the present invention.
[0044] Figure 2 This is a comparison chart of the state solution of the IA-DTD-RNN model in the present invention without disturbance and the theoretical solution.
[0045] Figure 3 This is the convergence curve of the IA-DTD-RNN model under constant disturbance of the present invention.
[0046] Figure 4 This is the convergence curve diagram of the IA-DTD-RNN model under discrete time-varying linear perturbations of the present invention.
[0047] Figure 5 This is the convergence curve diagram of the IA-DTD-RNN model under discrete time-varying quadratic disturbance of the present invention. DETAILED DESCRIPTION
[0048] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.
[0049] The present invention proposes a time-varying matrix inversion method based on a double integral RNN, comprising the following steps:
[0050] Step S1, first define the problem model of time-varying matrix inversion, specifically:
[0051]
[0052] in is an unknown matrix to be solved, is the identity matrix, is a non-singular smooth time-varying matrix. The purpose of this equation is to and Get the given value in real time and efficiently The inverse of ; In addition, the discrete time-varying matrix inversion problem formula is defined as:
[0053]
[0054] in, represent , Represents the step length.
[0055] Step S2: propose a new RNN design formula and establish the IA-CTD-RNN model, specifically:
[0056] The continuous time-varying matrix inversion process can be monitored by the following error function:
[0057]
[0058] Time the error function The first-order derivative of , we get:
[0059]
[0060] According to the RNN design formula proposed in previous studies, a new RNN design formula can be obtained by introducing the double integral term:
[0061]
[0062] in , and are three design parameters that are closely related to the convergence speed of the IA-CTD-RNN model.
[0063] Combining the time derivative of the error function with the new RNN design formula, we can derive the following implicit neural network model, namely the IA-CTD-RNN model:
[0064]
[0065] Step S3: Propose and rearrange the 5-instantaneous discretization formula, specifically:
[0066]
[0067] By rearranging the formula, we get:
[0068]
[0069] This formula is used to discretize the IA-CTD-RNN model to obtain the IA-DTD-RNN model.
[0070] Step S4: discretize the IA-CTD-RNN model using the 5-instantaneous discretization formula to obtain the IA-DTD-RNN model, specifically:
[0071] For the double integral term in the IA-CTD-RNN model, a new single-step forward difference formula for approximate calculation of the double integral is defined based on Taylor expansion:
[0072]
[0073] in represent The second derivative of .
[0074] Using the 5-instantaneous discretization formula and the forward difference formula as the basis for the discretization of the IA-CTD-RNN model, perform the following steps:
[0075] The first step is to combine the time derivative of the error function with the new RNN design formula to obtain the formula:
[0076]
[0077] The items in the brackets are , we can get the formula:
[0078]
[0079] In the second step, the single integral term is discretized using the 5-instantaneous discretization formula, and the formula is obtained:
[0080]
[0081] The third step is to discretize the double integral term using the forward difference formula, and the formula can be obtained:
[0082]
[0083] Rearranging the formula, we get the formula:
[0084]
[0085]
[0086] The fourth step is to After discretization as a whole, we can get the formula:
[0087]
[0088] Following the method in the first step, record the items in the brackets as , and remove the negative sign in the brackets, we can get the formula:
[0089]
[0090] Will Substituting the rearranged 5-instantaneous discretization formula, we can obtain the formula:
[0091]
[0092] The above is the required IA-CTD-RNN model.
[0093] The present invention is further described below by way of examples, but the present invention is not limited to the scope of the examples.
[0094] Example 1
[0095] In order to verify the convergence and good disturbance suppression ability of the IA-DTD-RNN model, a discrete time-varying matrix is given as an example, specifically:
[0096]
[0097] The theoretical solution of this discrete time-varying matrix is:
[0098]
[0099] Use the IA-DTD-RNN model to solve the matrix in Example 1 The inverse of , sets the model parameter values to:
[0100] , , , ;
[0101] Using four randomly generated initial states as starting points, we compare the state solution of the IA-DTD-RNN model with the theoretical solution. The experimental results are shown in Figure 2. Figure 2 As shown, the state solution of the IA-DTD-RNN model (the blue dotted line in the figure) gradually approaches the theoretical solution of Example 1, which indicates that in the absence of external disturbances, the state solution of the IA-DTD-RNN model can quickly and accurately converge to the theoretical solution of the discrete time-varying matrix inverse in the embodiment, which preliminarily verifies the effectiveness and convergence of the model itself.
[0102] Three different types of perturbations are applied to the IA-DTD-RNN model: constant perturbation, discrete time-varying linear perturbation, and discrete time-varying quadratic perturbation. The model is then compared with two previously proposed RNN inversion models (the T-DT-RNN model and the IE-DT-RNN model) to test the perturbation suppression capability of the IA-DTD-RNN model. Specifically,
[0103] Apply constant perturbation to the IA-DTD-RNN model , the model parameters are:
[0104] , , , ;
[0105] The two RNN inversion models used for comparison have the same model parameter settings (the same settings are used in the following two experiments);
[0106] The experimental results are as follows Figure 3 As shown in the figure, it can be observed that the final steady-state errors of the IE-DT-RNN model and the IA-DTD-RNN model are significantly lower than those of the T-DT-RNN model, indicating that they have stronger constant disturbance suppression capabilities. In order to further illustrate the disturbance suppression capability of the IA-DTD-RNN model, discrete time-varying linear perturbations are then applied to the experimental model.
[0107] Apply discrete time-varying linear perturbations to the IA-DTD-RNN model , , , the model parameters are:
[0108] , , , ;
[0109] The experimental results are as follows Figure 4 As shown in Figure 2, it can be observed that the final steady-state error of the IA-DTD-RNN model is significantly lower than that of the IE-DT-RNN model and the T-DT-RNN model, indicating that the IA-DTD-RNN model has better linear perturbation suppression ability.
[0110] Apply discrete time-varying quadratic perturbations to the IA-DTD-RNN model , , , the model parameters are:
[0111] , , , ;
[0112] The experimental results are as follows Figure 5 As shown in the figure, it can be observed that the errors of the T-DT-RNN model and the IE-DT-RNN model cannot converge under discrete time-varying quadratic perturbations, and increase significantly over time, indicating that they cannot suppress discrete time-varying quadratic perturbations. Only the error of the IA-DTD-RNN model converges quickly to a stable and low level, which strongly proves that the IA-DTD-RNN model has significant superiority over the existing RNN inversion model when dealing with discrete time-varying quadratic perturbations.
[0113] The above embodiments are only intended to help understand the method and core concept of the present invention. It should be noted that, without departing from the principles of the present invention, a number of improvements and modifications may be made to the present invention by those skilled in the art, and such improvements and modifications also fall within the scope of protection of the claims of the present invention.
Claims
1. A time-varying matrix inversion method based on double integral recursive neural network, characterized in that: The following steps are involved: Step S1: define the problem model of time-varying matrix inversion; Step S2: Introduce the double integral term to obtain a new recurrent neural network design formula and establish the IA-CTD-RNN model; Step S3: Propose and rearrange the 5-instantaneous discretization formula; Step S4: Using the 5-instantaneous discretization formula, discretize the IA-CTD-RNN model to obtain the IA-DTD-RNN model, completing the time-varying matrix inversion method.
2. The method for inverting a time-varying matrix based on a double integral recursive neural network according to claim 1, wherein: The step S1 is specifically as follows: The problem formula for inverting a continuous time-varying matrix is defined as: ； in is an unknown matrix to be solved, is the identity matrix, is a non-singular smooth time-varying matrix; The problem formula for inverting a discrete time-varying matrix is defined as: ； in, represent , Represents the step length.
3. The method for inverting a time-varying matrix based on a double integral recursive neural network according to claim 1, wherein: The step S2 is specifically as follows: The continuous time-varying matrix inversion process is monitored by the following error function: ； According to the existing recurrent neural network design formula, the double integral term is introduced to obtain a new recurrent neural network design formula: ； in , and These are three design parameters that are closely related to the convergence speed of the IA-CTD-RNN model. Combining the error function with the new recurrent neural network design formula, we get the IA-CTD-RNN model: 。 4. The method for inverting a time-varying matrix based on a double integral recursive neural network according to claim 1, wherein: The step S3 is specifically as follows: 5- The instantaneous discretization formula is: ； By rearranging the formula, we get the formula: ； The rearranged formula is used to discretize the IA-CTD-RNN model to obtain the IA-DTD-RNN model.
5. The method for inverting a time-varying matrix based on a double integral recursive neural network according to claim 1, wherein: The step S4 is specifically as follows: For the double integral term in the IA-CTD-RNN model, a new single-step forward difference formula for approximate calculation of the double integral is defined based on Taylor expansion: ； in represent The second derivative of Using the 5-instantaneous discretization formula as the basis for the discretization of the IA-CTD-RNN model, the IA-DTD-RNN model is obtained as follows: ； In the above formula, and The terms represent the single integral term and the double integral term respectively. The single integral term is approximately calculated using the 5-instantaneous discretization formula, and the result is: ； Using the forward difference formula to approximate the double integral term, we can get the following The central difference formula of order accuracy is: ； Afterwards, we can get the formula: 。