Parameter estimation method, system and device of nonlinear diffusion model and storage medium

Abstract

This invention discloses a parameter estimation method, system, device, and storage medium for a nonlinear diffusion model. The parameter estimation method includes: constructing a spatial basis function based on system data snapshots for a nonlinear Fisher-type diffusion system applied to the heating and cooling process of an iron rod, and separating spatiotemporal variables using an orthogonal decomposition method; sparsely sampling the nonlinear terms using a discrete empirical interpolation method to obtain the optimal low-order approximation of the high-order system; initializing a particle swarm to obtain n particles corresponding to the m-dimensional solution vector of the low-order time series model, and calculating the fitness of each particle according to the calculation formula of the parameters to be identified and the objective function; iterating and optimizing each particle, and outputting the position of the particle with the global optimum value; adding a non-Gaussian Levy process to the traditional particle swarm algorithm, and avoiding premature concentration of the particle swarm in the same direction through the random jump of the Levy process, thereby increasing the mutual learning ability between particles.

Description

Technical Field

[0001] This invention relates to the field of automatic control technology, and in particular to a parameter estimation method, system, device, and storage medium for a nonlinear diffusion model. Background Technology

[0002] Nonlinear diffusion systems have been widely used to describe reaction-diffusion phenomena prevalent in nature and engineering, such as heat conduction, thermonuclear reactions, neurophysiological reactions, chemical reactions, and nuclear reactions. Their most prominent characteristic is that changes in the system state depend not only on time but also on spatial location, exhibiting spatiotemporal coupling. Therefore, identifying the coefficients of nonlinear diffusion systems through data-driven methods to establish accurate models that characterize reaction-diffusion processes is of paramount importance.

[0003] Traditional parameter estimation problems mainly focus on the case of ordinary differential equations. However, the coupling characteristics of nonlinear structures and spatial information in nonlinear diffusion systems make it necessary to have huge computing power and precise calculation methods to solve their parameter estimation problems.

[0004] Since parameter estimation is an important part of system modeling, and system modeling is the primary scientific problem to be solved in the study of controlled objects based on system control theory.

[0005] In recent years, the particle swarm optimization (PSO) algorithm has attracted widespread attention due to its advantages such as ease of implementation, high accuracy and fast convergence speed, and has demonstrated its superiority in solving practical problems.

[0006] However, in existing technologies, nonlinear Fisher-type diffusion system models for processes such as heating and cooling of iron bars employ constant inertia coefficients from the classical particle swarm optimization (PSO) algorithm. This can easily cause the algorithm's search strategy to get trapped in local conditions, increasing the complexity of system modeling and the computational complexity of nonlinear terms. Summary of the Invention

[0007] This invention provides a parameter estimation method, system, device, and storage medium for a nonlinear diffusion model. By deriving a low-order model of the nonlinear diffusion process in the iron rod reaction, the complexity of system modeling can be greatly simplified. By adding a Levy stochastic process to the traditional particle swarm optimization algorithm, the search ability of particles can be enhanced, allowing them to escape local optima, and the mutual learning ability between particles can also be enhanced.

[0008] To address the above problems, this invention provides a parameter estimation method for a nonlinear diffusion model, applied to the thermal diffusion process of heating and cooling an iron rod. The parameter estimation method includes:

[0009] Step S 100: Construct a low-order time-series model of the nonlinear thermal diffusion process of an iron rod;

[0010] Step S 200 The parameters of the equivalent low-order time series model of the nonlinear Fisher-type diffusion system are identified. The Levy process is introduced to iterate and optimize each particle. After the iteration termination condition is reached, the position of the particle with the global optimum is output.

[0011] Furthermore, in step S 100 In the text, the low-order time-series model for constructing the nonlinear diffusion model of the iron rod specifically includes:

[0012] Step S 110 : Construct a nonlinear Fisher-type diffusion system model and determine the parameters to be identified;

[0013] The nonlinear Fisher-type diffusion system model is as follows:

[0014]

[0015] Where: α is the axial energy dispersion coefficient, d is the iron rod density, s represents the specific heat capacity, v represents the rate of temperature change, β represents the surface heat transfer coefficient of the iron rod, and T is the spatiotemporal temperature distribution of the iron rod. T represents the change in temperature and time. xx It is the second derivative of temperature with respect to spatial distribution, where x represents the location of the temperature distribution and t represents time.

[0016] The parameters to be identified include the axial energy dispersion coefficient α and the surface heat transfer coefficient β of the iron rod;

[0017] Step S 120 Construct a snapshot of the output data of the nonlinear Fisher-type diffusion system, and construct the optimal spatial basis function of the nonlinear Fisher-type diffusion system;

[0018] The optimal spatial basis function of the nonlinear Fisher-type diffusion system is:

[0019]

[0020] Where: y i This represents a snapshot of the system being built, u i (x) is the feature vector after feature decomposition of the system snapshot, σ i Let φ(x) be the corresponding eigenvalue, and let k be the largest spatial bases selected to reconstruct the system.

[0021] Step S 130 : Projecting nonlinear components onto a subspace to approximate these nonlinear components in order to reconstruct the nonlinear terms;

[0022] The expression for the nonlinear term is as follows:

[0023] F(x,t)=ψc(t);

[0024] Construct an identity matrix to project the nonlinear term, which satisfies:

[0025] E T F(x,t)=E T ψc(t);

[0026] Where: F(x,t) is a snapshot of the system constructed from nonlinear terms, and E represents the identity matrix. T denoted by , c(t) represents the transpose of the identity matrix, c(t) represents the eigenvalues ​​of the nonlinear term, and ψ represents the spatial components of the nonlinear term.

[0027] The optimal weighting coefficients are calculated and expressed as follows:

[0028] c(t)=(E T ψ) -1 E T F(x,t);

[0029] The nonlinear term is reconstructed, and its expression is as follows:

[0030] F(x,t)≈ψ(E T ψ) -1 E T F(x,t);

[0031] Step S 140 The inner product of all terms in the nonlinear Fisher-type diffusion system model with the optimal spatial basis function of the system is used to obtain a low-order approximation of the higher-order system. The low-order approximation of the higher-order system is as follows:

[0032]

[0033] Where: a n (t) represents the time component obtained by decomposing the temperature distribution of the iron rod, φ(x) is the corresponding spatial component, α and β are the unknown parameters of the system, and F(·) represents the nonlinear term of the system.

[0034] Furthermore, in step S 200 The specific steps for parameter identification of the equivalent low-order time series model of the nonlinear Fisher-type diffusion system include:

[0035] Step S 210Initialize the particle swarm to obtain n particles of the m-dimensional solution vector of the corresponding low-order temporal model. Calculate the fitness of each particle according to the calculation formula of the parameters to be identified and the objective function.

[0036] Step S 220 : Calculate the reverse individuals of the particles, and calculate the fitness of each particle. The fitness of the particles The calculation formula is:

[0037]

[0038] Where: N S N T Representing the spatial and temporal dimensions of the data, respectively, and α and β being the undetermined parameter values ​​of the system. It represents the position information corresponding to the optimal fitness during the particle update process; T is the true temperature of the system. It is the estimated temperature of the system;

[0039] Step S 230 Based on the fitness of each particle, n particles with the best fitness values ​​are selected from the current and reversed particles to obtain the global optimum G of the entire particle swarm. best and the individual optimal value P best Each particle updates according to the position update formula;

[0040] Step S 250 Introducing the non-Gaussian process Levy distribution for particle position updating;

[0041] Step S 240 Update the historical individual best fitness and global fitness of the particle, and determine the optimal position of the particle;

[0042] Step S 260 : Perform iterations and optimizations, and output the global optimum G after the iteration termination condition is met. best The location of the particles is the model identification parameter of the nonlinear diffusion model.

[0043] Furthermore, in step S 220 In this context, the calculation method for the reverse individual selection is as follows:

[0044]

[0045] Where: c is a random number within (0,1), i and j are the number of particles and the dimension, respectively, a j (t), b j (t) represent the minimum and maximum values ​​in the j-th dimension, respectively, X ij(t) represents the temperature estimate of the i-th particle in the j-th dimension, and t represents the iteration number. Represents the current particle X ij The reverse individual of (t).

[0046] Furthermore, in step S 250 In the process, the Levy process update formula for each particle, which updates according to the position update formula, is as follows:

[0047]

[0048] Where: S represents a Levy distribution with parameter β, and u and v both satisfy a normal distribution.

[0049] as well as

[0050] σ v =1;

[0051] Where: σ u ,σ v Let u and v represent the variances of random variables u and v, respectively, β be the parameter of the Levy distribution, and Γ be the gamma function.

[0052] Furthermore, in step S 250 In this context, the particle position and velocity update formulas are specifically as follows:

[0053]

[0054] X(t+1)=X(t)+V(t+1);

[0055] Where: a1, a2, and a3 are weighting factors. P is the dot product symbol. best G represents the optimal value for an individual particle during iteration. best Let V(t) be the global optimal value in the particle iteration, V(t) be the particle's velocity in each iteration, X(t) be the particle's position in each iteration, and Levy(·) be the random value that jumps through the Levy process to update the particle's velocity.

[0056] A second objective of this invention is to provide a parameter estimation system for a nonlinear diffusion model, the system comprising:

[0057] The model building module configures a low-order time series model for constructing the nonlinear diffusion process of the iron rod;

[0058] The parameter identification module is configured to perform parameter identification on the equivalent low-order time series model of a nonlinear Fisher-type diffusion system.

[0059] A third objective of this invention is to provide a parameter estimation device for a nonlinear diffusion model, comprising a computer-readable storage medium storing a computer program and a processor, wherein the computer program is read and executed by the processor to implement the parameter estimation method for the nonlinear diffusion model described above.

[0060] A fourth objective of this invention is to provide a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the parameter estimation method for the nonlinear diffusion model as described above.

[0061] Compared with the prior art, the present invention has significant advantages and beneficial effects, specifically reflected in the following aspects:

[0062] This invention proposes to introduce the Levy process, which has long jump and heavy tail characteristics, and to establish a Levy-PSO algorithm to achieve a global solution to the parameter estimation problem of nonlinear Fisher-type diffusion systems. The research results can provide important reference for solving the parameter estimation problem of other nonlinear partial differential equation systems and for further improving the PSO algorithm.

[0063] This invention greatly simplifies the complexity of system modeling by constructing a low-order model of the nonlinear diffusion process. To address the problem of excessive computational complexity of nonlinear terms in the original system, it approximates each nonlinear term by combining projection and interpolation. Based on projecting the nonlinear term onto the low-order model, it evaluates it at specific locations selected by a greedy algorithm. Component values ​​at other locations are interpolated using a reduced basis through an offline pre-computed interpolation matrix. Adding a Levy stochastic process to the traditional particle swarm optimization algorithm enhances the particle's search ability, helps it escape local optima, and also strengthens the mutual learning ability between particles.

[0064] The LevyPSO algorithm proposed in this invention effectively increases the global search capability of particles and completely solves the defect of classical PSO in causing the search strategy to get trapped in local conditions. The parameter estimation method for nonlinear Fisher-type diffusion systems based on the LevyPSO algorithm established in this invention also shows a significant improvement in prediction accuracy compared to the classical PSO algorithm. Attached Figure Description

[0065] Figure 1 This is a flowchart of the parameter estimation method for the nonlinear diffusion model in an embodiment of the present invention;

[0066] Figure 2 Step S in the parameter estimation method of the nonlinear diffusion model in this embodiment of the invention 100 Detailed flowchart;

[0067] Figure 3Step S in the parameter estimation method of the nonlinear diffusion model in this embodiment of the invention 200 Detailed flowchart;

[0068] Figure 4 This is a structural diagram of the nonlinear Fisher-type diffusion system in an embodiment of the present invention;

[0069] Figure 5 This is a data diagram of the nonlinear Fisher-type diffusion system in an embodiment of the present invention;

[0070] Figure 6 This is the fitting function for the optimal fitness in the embodiments of the present invention;

[0071] Figure 7 This is a parameter fitting curve diagram from an embodiment of the present invention;

[0072] Figure 8 This is a schematic diagram illustrating the heat diffusion process, such as heating and cooling, of the iron rod in an embodiment of the present invention. Detailed Implementation

[0073] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0074] Please see Figure 1 As shown, this embodiment of the invention provides a parameter estimation method for a nonlinear diffusion model, which can be applied to thermal diffusion processes such as heating and cooling of iron rods. By constructing a simplified mathematical model, data-driven identification of unknown parameters is achieved. The parameter estimation method includes:

[0075] Step S 100 : Construct a low-order time-series model of the nonlinear thermal diffusion process of an iron rod;

[0076] It should be noted that a class of nonlinear diffusion models based on the LevyPSO algorithm can greatly simplify the complexity of system modeling by constructing a low-order model of the reaction-diffusion process.

[0077] Specifically, please refer to Figure 2 As shown, in step S 100 The low-order time-series model of the nonlinear thermal diffusion process of the iron rod specifically includes:

[0078] Step S 110 : Construct a nonlinear Fisher-type diffusion system model and determine the parameters to be identified;

[0079] The nonlinear Fisher-type diffusion system model is as follows:

[0080]

[0081] Where: α is the axial energy dispersion coefficient, d is the iron rod density, S represents the specific heat capacity, v represents the rate of temperature change, β represents the surface heat transfer coefficient of the iron rod, and T is the spatiotemporal temperature distribution of the iron rod. It refers to the change in temperature over time, T. xx It is the second derivative of temperature with respect to spatial distribution, where x represents the location of the temperature distribution and t represents time.

[0082] The parameters to be identified include the axial energy dispersion coefficient α and the surface heat transfer coefficient β of the iron rod.

[0083] Step S 120 Construct a snapshot of the output data of the nonlinear Fisher-type diffusion system, and construct the optimal spatial basis function of the system;

[0084] The optimal spatial basis function of the nonlinear Fisher-type diffusion system is:

[0085]

[0086] Where: y i This represents a snapshot of the system being built, u i (x) is the feature vector after feature decomposition of the system snapshot, σ i Let φ(x) be the corresponding eigenvalue, and let k be the largest spatial bases selected to reconstruct the system.

[0087] Step S 130 : Projecting nonlinear components onto a subspace to approximate these nonlinear components in order to reconstruct the nonlinear terms;

[0088] The expression for the nonlinear term is as follows:

[0089] F(x,t)=ψc(t);

[0090] Construct an identity matrix to project the nonlinear term, which satisfies:

[0091] E T F(x,t)=E T ψc(t);

[0092] Where: F(x,t) is a snapshot of the system constructed from nonlinear terms, and E represents the identity matrix. T denoted by , c(t) represents the transpose of the identity matrix, c(t) represents the eigenvalues ​​of the nonlinear term, and ψ represents the spatial component of the nonlinear term.

[0093] The optimal weighting coefficients are calculated and expressed as follows:

[0094] c(t)=(E T ψ) -1 ET F(x,t);

[0095] Based on this, the nonlinear term is reconstructed, and the expression for the reconstructed nonlinear term is as follows:

[0096] F(x,t)≈ψ(E T ψ) -1 E T F(x,t);

[0097] Step S 140 The inner product of all terms in the nonlinear Fisher-type diffusion system model with the system's optimal spatial basis functions is used to obtain a low-order approximation of the higher-order system. The low-order approximation of the higher-order system is as follows:

[0098]

[0099] Where: a n (t) represents the temperature distribution of the iron rod, T(x,t) represents the time component obtained from the decomposition, and φ(x) n α and β are the corresponding spatial components, α and β are the unknown parameters of the system, and F(·) represents the nonlinear term of the system.

[0100] Therefore, to address the problem of excessive computational complexity of nonlinear terms in the original system, we approximate each nonlinear term by combining projection and interpolation. Based on projecting the nonlinear term onto a low-order model, we evaluate it at specific locations selected by a greedy algorithm; the component values ​​at other locations are interpolated using a reduced basis through an interpolation matrix that can be pre-computed offline.

[0101] Step S 200 The parameters of the equivalent low-order time series model of the nonlinear Fisher-type diffusion system are identified. The Levy process is introduced to iterate and optimize each particle. After the iteration termination condition is reached, the position of the particle with the global optimum is output.

[0102] Therefore, adding the Levy stochastic process to the traditional particle swarm optimization algorithm can enhance the particle's search ability, help it escape local optima, and also enhance the mutual learning ability between particles.

[0103] Specifically, please refer to Figure 3 As shown, in step S 200 The specific steps for parameter identification of the equivalent low-order time series model of the nonlinear Fisher-type diffusion system include:

[0104] Step S 210 Initialize the particle swarm to obtain n particles of the m-dimensional solution vector of the corresponding low-order temporal model. Calculate the fitness of each particle according to the calculation formula of the parameters to be identified and the objective function.

[0105] Establish the calculation formula for the parameters to be identified and determine the objective function; initialize the particle swarm, assuming there are n particles in the swarm distributed in a solution space of spatial dimension m, and let the position of the i-th particle be X. i =(X i1 ,X i2 ,…,X im ), to obtain n particles of the m-dimensional solution vector of the corresponding nonlinear diffusion model, and calculate the fitness of each particle according to the calculation formula of the parameters to be identified and the objective function.

[0106] Step S 220 : Calculate the reverse individuals of the particles, and calculate the fitness of each particle. The fitness of the particles The calculation formula is:

[0107]

[0108] Where: N S N T These represent the spatial and temporal dimensions of the data, respectively, and α and β are the undetermined parameter values ​​of the system. It is the position information corresponding to the optimal fitness during the particle update process; Let T be the estimated temperature of the system, and T be the actual temperature of the system. By squaring the residual between the actual and estimated values, the cost function can be minimized during the identification process to make the estimated value approximate the actual value.

[0109] Step S 230 Based on the fitness of each particle, n particles with the best fitness values ​​are selected from the current and reversed particles to obtain the global optimum G of the entire particle swarm. best and the individual optimal value P best Each particle updates according to the position update formula;

[0110] Step S 240 Introducing the non-Gaussian process Levy distribution for particle position updating;

[0111] Update the global optimum, and set P best and G best Compare the values ​​of P, if P best The value is greater than G best If the value is better, then update G. best And proceed to the next iteration.

[0112] Step S 250 Update the historical individual optimal fitness and global fitness of the particle, and determine the optimal position of the particle.

[0113] Calculate the fitness of the particles after position update, compare the fitness values ​​before and after the update, select the better particle as the next generation particle, and update the individual optimal value P of the entire particle swarm. best .

[0114] The closer the calculated value based on the formula is to the measured value, the better the parameter identification effect is. Based on this, an objective function can be constructed, and the error function can be selected as the second norm of the error.

[0115] Increase the self-adaptability of particles Equal to the objective optimization function, the better the parameter identification effect, the higher the particle's own fitness. The smaller the value, the worse the parameter identification and the lower the particle's fitness. The larger.

[0116] Step S 260 : Perform iterations and optimizations, and output the global optimum G after the iteration termination condition is met. best The location of the particles is the model identification parameter of the nonlinear diffusion model.

[0117] Specifically, in step S 220 In this context, the calculation method for the reverse individual selection is as follows:

[0118]

[0119] Where: c is a random number within (0,1), i and j are the number of particles and the dimension, respectively, a j (t), b j (t) represent the minimum and maximum values ​​in the j-th dimension, respectively, X ij (t) represents the temperature estimate of the i-th particle in the j-th dimension, and t represents the iteration number. Represents the current particle X ij The reverse individual of (t).

[0120] Specifically, in step S 250 In the process, the Levy process update formula for each particle, which updates according to the position update formula, is as follows:

[0121]

[0122] Where: S represents a Levy distribution with parameter β, and Levy random numbers are calculated using two random variables u and v, both of which satisfy a normal distribution. as well as

[0123] σ v =1;

[0124] Where: σ u ,σv are the variances of random variables u and v, respectively; β is the parameter of the Levy distribution, which is generally in the interval (1,3], and here it is taken as 1.5; Γ is the gamma function, which can be calculated using the gamma function command in Matlab.

[0125] The Levy process is a non-Gaussian stochastic process whose random walk is derived from the Levy stable distribution. It is primarily used to simulate a random walk process in nature where animals forage. It is a type of stochastic process with Markov properties, characterized by long-range jumps, and its step size follows a heavy-tailed stable distribution. Introducing the Levy process into the traditional particle swarm optimization algorithm can increase the random search capability of particles, thereby avoiding getting trapped in local optima.

[0126] Specifically, in step S 250 In this context, the particle position update formula is specifically as follows:

[0127]

[0128] X(t+1)=X(t)+V(t+1);

[0129] Where: a1, a2, and a3 are weighting factors. P is the dot product symbol. best G represents the optimal value for an individual particle during iteration. best Let V(t) be the global optimal value in the particle iteration, V(t) be the particle's flight velocity in each iteration, X(t) be the particle's position in each iteration, and X(t) be the parameter that needs to be identified. Levy(·) represents the random value of the Levy process jump to update the particle's velocity.

[0130] This embodiment selects computational data under different noise levels. The iron rod density is d, the specific heat capacity S, and the rate of temperature change v on the iron rod surface are known parameters. The parameters to be identified are the energy dispersion coefficient α and the heat transfer coefficient β of the iron rod surface. These parameters are identified using the Levy particle swarm optimization algorithm proposed in this embodiment. The parameter fitting curves after identification are shown below. Figure 6 , 7 As shown in Figures 8 and 8, the goodness of fit is shown in Table 1:

[0131] Noise intensity α＝1 β＝-5 No noise 1.0019 -4.9967 5% Levy Noise 0.7588 -4.7439 10% Levt Noise 1.3243 -4.5673

[0132] Table 1

[0133] As shown in Table 1 above, the Levy particle swarm optimization algorithm proposed in this embodiment has relatively accurate identification results for Fisher model parameters, and it can basically converge within 20 iterations, which also shows that the reverse learning process can accelerate the particle convergence speed.

[0134] Figure 6 , 7 The figure shows the numerical solution of the nonlinear diffusion model established in the embodiment of the present invention. As an important model in population dynamics, this equation describes the spatial diffusion of the dominant variant of a specific gene. Later, this one-dimensional diffusion model was used for the diffusion and invasion of insects and animals.

[0135] As a preferred embodiment of this practice, the specific steps for introducing the Levy process as particle position update are as follows:

[0136] Obtain the particle position parameters and its reverse individual;

[0137] Calculate the fitness of each particle. All particles according to their fitness Sort the particles from smallest to largest and select the top n best particles;

[0138] Calculate the fitness of the particles after position update, compare the fitness values ​​before and after the update, select the better particle as the next generation particle, and obtain the global optimum G of the entire particle swarm. best And the individual optimal value;

[0139] Calculate the self-fitness of the particle after position update. Update the fitness of each particle Compared with the historical individual best value, if If the current particle's fitness is less than the historical best value, then the current particle's fitness will be adjusted. The optimal value P of a historical individual best Update using the current particle position;

[0140] Update the global optimum, and set P best and G best Compare the values ​​of P, if P best The value is greater than G best If the value of P is better, then the current individual's optimal P is used. best Update G best And proceed to the next iteration;

[0141] Each particle updates its own position according to the particle position update formula;

[0142] Once the iteration termination condition is met, i.e., the optimization accuracy is satisfied or the maximum number of iterations is reached, the global optimum value G is output. best The location of the particle is the model identification parameter of the Fisher model;

[0143] This embodiment introduces a non-Gaussian Levy process and a reverse learning process into the particle position update formula. Combined with the random jump process, it forms a perturbation factor to affect the particle's motion direction, which can enhance the particle's global learning ability. Introducing the reverse learning process can increase the learning ability between particles and accelerate particle convergence.

[0144] Please see Figure 4 As shown, this embodiment of the invention also provides a parameter estimation system for a nonlinear diffusion model, the system comprising:

[0145] The model building module configures a low-order time series model for constructing the nonlinear diffusion process of the iron rod;

[0146] The parameter identification module is configured to perform parameter identification on the equivalent low-order time series model of a nonlinear Fisher-type diffusion system.

[0147] This invention also provides a parameter estimation apparatus for a nonlinear diffusion model, including a computer-readable storage medium storing a computer program and a processor. The computer program is read and executed by the processor to implement the parameter estimation method for the nonlinear diffusion model as described above.

[0148] This invention also provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the parameter estimation method for the nonlinear diffusion model as described above.

[0149] While the present invention has been disclosed above, its scope of protection is not limited thereto. Those skilled in the art can make various changes and modifications without departing from the spirit and scope of this disclosure, and all such changes and modifications will fall within the scope of protection of this invention.

Claims

1. A parameter estimation method for a nonlinear diffusion model, applied to the thermal diffusion process of heating and cooling iron rods, characterized in that... The parameter estimation method includes: Step S 100 : Construct a low-order time-series model of the nonlinear thermal diffusion process of an iron rod; Step S 110 : Construct a nonlinear Fisher-type diffusion system model and determine the parameters to be identified; The nonlinear Fisher-type diffusion system model is as follows: in: The axial energy dispersion coefficient is... For the density of the iron rod, Represents specific heat capacity. Indicates temperature The rate of change This represents the heat transfer coefficient of the iron rod surface. The spatiotemporal temperature distribution of the iron rod, Represents the change in temperature and time. It is the second derivative of temperature with respect to spatial distribution. This represents the location of temperature distribution. t Represents time; The parameters to be identified include the axial energy dispersion coefficient. The heat transfer coefficient of the iron rod surface ; Step S 120 Construct a snapshot of the output data of the nonlinear Fisher-type diffusion system, and construct the optimal spatial basis function of the nonlinear Fisher-type diffusion system; The optimal spatial basis function of the nonlinear Fisher-type diffusion system is: in: This represents a snapshot of the system being built. It is the feature vector after feature decomposition of the system snapshot. For the corresponding eigenvalues, This indicates selecting the largest one. The system is reconstructed using spatial bases; Step S 130 The nonlinear components are projected onto a subspace to approximate the nonlinear components in order to reconstruct the nonlinear terms; The expression for the nonlinear term is as follows: Construct an identity matrix projected onto the nonlinear term, satisfying: in: It is a snapshot of the system constructed from nonlinear terms. Represents the identity matrix. Represents the transpose of the identity matrix. Characteristic quantities representing nonlinear terms, Spatial components representing nonlinear terms ； The optimal weighting coefficients are calculated and expressed as follows: The nonlinear term is reconstructed, and its expression is as follows: Step S 140 The inner product of all terms in the nonlinear Fisher-type diffusion system model with the system's optimal spatial basis functions is used to obtain a low-order approximation of the higher-order system. The low-order approximation of the higher-order system is as follows: in: This represents the time component obtained from the decomposition of the temperature distribution in the iron rod. These are the corresponding spatial components. and For unknown parameters of the system, Represents the nonlinear terms of the system; Step S 200 The parameters of the equivalent low-order time series model of the nonlinear Fisher-type diffusion system are identified. The Levy process is introduced to iterate and optimize each particle. After the iteration termination condition is reached, the position of the particle with the global optimum is output.

2. The parameter estimation method for the nonlinear diffusion model according to claim 1, characterized in that, In step S 200 The specific steps for parameter identification of the equivalent low-order time series model of the nonlinear Fisher-type diffusion system include: Step S 210 Initialize the particle swarm optimization to obtain the corresponding low-order temporal model. dimensional solution vector For each particle, the fitness of each particle is calculated based on the calculation formula of the parameters to be identified and the objective function. Step S 220 : Obtain the corresponding reverse individual of the particle and calculate the fitness of each particle. The fitness of the particles The calculation formula is: in: These represent the spatial and temporal dimensions of the data, respectively. These are the undetermined parameter values ​​for the system. It is the position information corresponding to the optimal fitness during the particle update process; This is the system's actual temperature. It is the estimated temperature of the system; Step S 230 Based on a comparison of each particle's own fitness, the particle with the best fitness value is selected from both the current and reversed particles. Individual particles are used to obtain the global optimum of the entire particle swarm. and individual optimal value Each particle updates according to the position update formula; Step S 240 Introducing the non-Gaussian process Levy distribution for particle position updating; Step S 250 Update the historical individual best fitness and global fitness of the particle, and determine the optimal position of the particle; Step S 260 : Perform iteration and optimization, and output the global optimum value after the iteration termination condition is met. The location of the particles is the model identification parameter of the nonlinear diffusion model.

3. The parameter estimation method for the nonlinear diffusion model according to claim 2, characterized in that, In step S 220 In this context, the calculation method for the reverse individual selection is as follows: in: c It is a random number within (0,1). These are the number of particles and the dimension. , They are the first Minimum and maximum values ​​in dimension Represents the current number The particle in the first Temperature estimates in each dimension For the number of iterations, Represents the current particle .

4. The parameter estimation method for the nonlinear diffusion model according to claim 3, characterized in that, In step S 250 In the process, the Levy process update formula for each particle, which updates according to the position update formula, is as follows: in: S The representative parameter is Levy distribution, and All satisfy a normal distribution. , as well as in: These are random variables and variance These are the parameters of the Levy distribution. This is a gamma function.

5. The parameter estimation method for the nonlinear diffusion model according to claim 4, characterized in that, In step S 250 In this context, the particle position and velocity update formulas are specifically as follows: in: , , As a weighting factor, The dot product symbol. This represents the optimal value for an individual particle during iteration. This represents the global optimal value in particle iteration. It is the particle's flight speed in each iteration. It represents the position of the particle in each iteration. The Levy process uses random values ​​to update the particle's velocity.

6. A parameter estimation system for a nonlinear diffusion model, employing the parameter estimation method for the nonlinear diffusion model as described in any one of claims 1-5, characterized in that, The system includes: The model building module configures a low-order time series model for constructing the nonlinear diffusion process of the iron rod; The parameter identification module is configured to perform parameter identification on the equivalent low-order time series model of a nonlinear Fisher-type diffusion system.

7. A parameter estimation device for a nonlinear diffusion model, characterized in that, The method includes a computer-readable storage medium storing a computer program and a processor, the computer program being read and executed by the processor to implement the parameter estimation method for the nonlinear diffusion model as described in any one of claims 1-5.

8. A computer-readable storage medium having a computer program stored thereon, characterized in that, When executed by a processor, the program implements the parameter estimation method for the nonlinear diffusion model as described in any one of claims 1-5.

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