A method for optimizing parameters of hydrological models based on Neural Ordinary Differential Equations

Abstract

The present invention belongs to the technical field of hydrological model parameter optimization, and specifically discloses a hydrological model parameter optimization method based on Neural Ordinary Differential Equations, the steps of which include: parameter learning neural network construction, forward numerical solution of ordinary differential equations, back propagation update of hydrological model parameters, and iterative optimization of hydrological model parameters. The method described in the present invention embeds a neural network into a differential form hydrological model structure, combines the physical mechanism of the basin hydrological process with the data mining capability of a deep learning algorithm, and forms a hydrological model parameter optimization method based on Neural Ordinary Differential Equations, which reduces the uncertainty of model parameters and improves the model simulation accuracy. The present invention solves the difficult problem of obtaining global optimal and stable solutions for model parameters in differential form hydrological models.

Description

Technical Field

[0001] The invention belongs to the technical field of hydrological model parameter optimization, and in particular relates to a hydrological model parameter optimization method based on Neural Ordinary Differential Equations. Background Art

[0002] Hydrological model is an important means of flood simulation and forecasting. Structure and parameters are the core of hydrological model. Parameters are used to describe the characteristics of the basin's underlying surface and are one of the key factors to ensure the accuracy of the model. In theory, the parameters of hydrological model can be determined according to the physical characteristics of the basin's underlying surface. However, in actual work, it is usually necessary to use measured hydrological data and determine the model parameters through parameter optimization methods in order to achieve the best match between the model simulation results and the observed data. Therefore, improving parameter optimization performance is of great significance for accurately simulating the basin's hydrological process.

[0003] At present, heuristic algorithms are commonly used in parameter optimization, such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO) and Shuffled Complex Evolution (SCE-UA). They adopt random or probabilistic search strategies inspired by natural or social phenomena, and perform global search by simulating biological genetics, particle swarm behavior, shuffled complex evolution and other processes. They perform well in the optimization of complex problems and effectively avoid subjective bias and errors. However, these algorithms also have some limitations. For example, due to insufficient crossover and mutation operations or small population size, the GA algorithm may face premature convergence problems in multi-objective optimization, making it difficult to obtain the global optimal solution; the PSO algorithm is prone to fall into the local optimal solution when dealing with complex and multi-peak functions, and is sensitive to the selection of algorithm parameters (such as inertia weight and acceleration constant); the SCE-UA algorithm requires parallel evolution and interaction of multiple complex shapes. Due to the inherent randomness of the algorithm, its parameter optimization results may not be unique, which increases the difficulty of obtaining the global optimal solution and stable solution of the hydrological model. In addition, the theory of watershed hydrological models has been continuously developing in recent years. In order to reduce the numerical error in solving hydrological models, the mathematical description of hydrological models has gradually developed from differential form to differential form. The parameter optimization of differential form hydrological models is an issue worthy of study. Summary of the invention

[0004] The purpose of the present invention is to overcome the above technical defects and provide a hydrological model parameter optimization method based on Neural Ordinary Differential Equations to solve the problem of obtaining global optimal solutions and stable solutions of model parameters in differential form hydrological models.

[0005] In order to solve the above technical problems, the technical solution adopted by the present invention is:

[0006] A method for optimizing hydrological model parameters based on Neural Ordinary Differential Equations comprises the following steps:

[0007] Step 1, parameter learning neural network construction:

[0008] Construct a neural network N for learning the relationship between the hydrological model parameters and the underlying surface attributes of the watershed, and denote the number of iterations as k;

[0009] At the kth iteration, the initial value of the neural network parameter θ is denoted by θ k,0 Based on the neural network N and the underlying surface attribute data A of the watershed, the parameter value of the hydrological model parameter δ is converted and recorded as δ k ;

[0010] Use 0 to initialize the number of iterations k, and the initial value of the neural network parameter θ is θ 0,0 , initialize θ 0,0 , the corresponding converted hydrological model parameter value is δ0;

[0011] Step 2, forward numerical solution of the ordinary differential equations:

[0012] Substitute the hydrological model parameter value δ0 obtained in step 1 into the ordinary differential equations of the hydrological model, set the initial values ​​of the hydrological model state variables and the basin hydrological and meteorological input data, use the ordinary differential equation solver to perform forward numerical solution on the ordinary differential equations of the hydrological model, and obtain the flow simulation data;

[0013] Step 3, back propagation update of neural network parameters:

[0014] Based on the flow observation data of the basin outlet section and the flow simulation data obtained in step 2, the loss function value is calculated, and then the gradient of the loss function value relative to the neural network parameter θ in step 1 is calculated. The neural network parameter θ is updated by back propagation to obtain the updated value of the neural network parameter θ k+1,0 ;

[0015] Step 4: Iterative optimization of hydrological model parameters:

[0016] Based on the neural network N in step 1 and the underlying surface attribute data A of the watershed, the neural network parameter update value θ obtained in step 3 k+1,0 , calculate the updated value of the hydrological model parameter δ k+1 ;

[0017] Using Delta k+1 Replace step 1 to obtain the hydrological model parameter value δ0 or the hydrological model parameter update value δ calculated in the last iteration k, as the input of step 2, the forward numerical solution process of the ordinary differential equation group described in step 2 and the back propagation update process of the neural network parameters described in step 3 are performed again in sequence, based on the neural network N in step 1 and the underlying surface attribute data A of the watershed, and the neural network parameter update value θ obtained in step 3 k+2,0 , calculate the updated value of the hydrological model parameter δ k+2 , add 1 to the number of iterations k to complete a complete iteration process;

[0018] Repeat the iterative process until the loss function converges, and the updated value of the hydrological model parameter δ k+2 This is the optimal parameter value of the hydrological model.

[0019] The technologies not mentioned in the present invention are all referred to the prior art.

[0020] The advantages and beneficial effects of the present invention are:

[0021] The present invention provides a method for optimizing hydrological model parameters based on neural ordinary differential equations. By embedding a neural network into the differential form hydrological model structure, combining the physical mechanism of the basin hydrological process with the data mining capability of deep learning, the optimal solution of the hydrological model parameters is obtained through a smaller number of iterations, thereby reducing the uncertainty of the model parameters and improving the simulation accuracy of the hydrological model. It solves the problem of obtaining the global optimal and stable solution of the model parameters of the differential form hydrological model, and has a strong engineering significance. BRIEF DESCRIPTION OF THE DRAWINGS

[0022] The present invention will be further described below in conjunction with the accompanying drawings and embodiments.

[0023] Figure 1 is a flow chart of a hydrological model parameter optimization method based on the Godly ordinary differential equation described in an embodiment;

[0024] Figure 2 is the change of loss function of different fragments of the embodiment;

[0025] Figure 3 2 is a flow process diagram of a simulation example. DETAILED DESCRIPTION

[0026] The present invention will be further described below in conjunction with the accompanying drawings. The following embodiments are only used to more clearly illustrate the technical solution of the present invention, and cannot be used to limit the protection scope of the present invention.

[0027] Example:

[0028] like Figure 1 As shown, this embodiment takes a river basin in the southern humid area as an example, and provides a method for optimizing hydrological model parameters based on the Neural Ordinary Differential Equation, including the following steps:

[0029] Step 1, parameter learning neural network construction:

[0030] Construct a neural network N for learning the relationship between the hydrological model parameters and the underlying surface attributes of the watershed. The number of iterations is k. At the kth iteration, the initial values ​​of the neural network parameters θ (weights and biases) are θ k,0 Based on the neural network N and the underlying surface attribute data A of the watershed, the parameter value of the hydrological model parameter δ is converted and recorded as δ k ; Use 0 to initialize the number of iterations k, and the initial value of the neural network parameter θ is θ 0,0 , using KaiMing normal distribution initialization technology to initialize θ 0,0 , the corresponding converted hydrological model parameter value is δ0. The specific steps are:

[0031] S11, select a watershed in the southern humid area as the application watershed, and collect the underlying surface attribute data of the application watershed, specifically:

[0032] A={a i | i=1, 2, ..., n} (1)

[0033] In formula (1): A is the underlying surface attribute data of the watershed; a i is the i-th attribute value of the basin, including the basin area, basin land use type and its area proportion, etc.; n is the number of underlying surface attributes of the basin.

[0034] S12, the hydrological model used is the differential form Xin'anjiang model (patent number: ZL 202211231625.X), and the physical meanings and upper and lower limits of its model parameters are shown in the following table:

[0035] Table 1 Physical meaning and value range of parameters of Xinanjiang model in differential form

[0036]

[0037] A multilayer perceptron neural network N is constructed. The neural network includes two hidden layers, each with five neurons. The activation function is the tanh function. It is used to learn the relationship between the hydrological model parameters and the underlying surface properties of the watershed. Its input is the underlying surface property data A of the watershed, and the output value ranges from 0 to 1:

[0038] The number of iterations is recorded as k. At the kth iteration, the initial value of the neural network parameter θ (weight and bias) is recorded as θ k,0 ; δ is any hydrological model parameter to be optimized in Table 1, except the number of linear reservoirs in series n f , underground runoff coefficient K gIn addition to the fixed values, the other 13 parameters in Table 1 are all involved in the optimization. For each parameter, a multilayer perceptron neural network needs to be constructed respectively. Based on the neural network N and the underlying surface attribute data A of the watershed, the output of the neural network N is converted into the parameter value δ of the hydrological model parameter δ by the following formula: k :

[0039] δ k =(δ max -δ min )·N(A; θ k,0 )+δ min (2)

[0040] In formula (2), k is the number of iterations, which is a non-negative integer; δ k is the hydrological model parameter value; δ max is the upper limit of the hydrological model parameters; δ min is the lower limit of the hydrological model parameters; N is the neural network; θ k,0 is the initial value of the neural network parameters at the kth iteration.

[0041] Use 0 to initialize the number of iterations k, and the initial value of the neural network parameter θ is θ 0,0 , using KaiMing normal distribution initialization technology to initialize θ 0,0 , the corresponding converted hydrological model parameter value is δ0.

[0042] Step 2, forward numerical solution of the ordinary differential equations:

[0043] Substitute the hydrological model parameter value δ0 obtained in step 1 into the ordinary differential equation group of the differential form Xin'anjiang model, set the initial values ​​of the state variables of the differential form Xin'anjiang model (i.e., the initial conditions of the hydrological model) and the basin hydrometeorological input data (i.e., the hydrological model input), and use the ordinary differential equation solver to perform forward numerical solutions to the ordinary differential equation group of the differential form Xin'anjiang model to obtain the flow simulation data.

[0044] The specific steps are:

[0045] S21, collect the hydrological and meteorological input data of the application basin (as the input of the hydrological model) and the flow observation data of the basin outlet section (for calculating the loss function), and unify the data period length and total number of periods of hydrological and meteorological and flow observation data. Specifically, the period length is unified as 1 day, and the total number of periods is 2557 (7 years), which can be expressed as:

[0046]

[0047] Y={Q i | i=1, 2, ..., T} (4)

[0048] In formulas (3) and (4), X is the hydrological and meteorological input data; P i is the rainfall in the ith period, which is calculated by the arithmetic mean method based on all rainfall observations of the rainfall station network in the basin during the period, and the period is 1 day; P T is the rainfall in the Tth period; E i is the evapotranspiration in the i-th period; E T is the evapotranspiration in the Tth period; T is the total number of periods, specifically 2557 (7 years); Y is the flow observation data of the basin outlet section, and the total number of periods containing data in Y is the same as that in X; Q i is the observed flow value of the basin outlet section in the i-th period.

[0049] S22, the hydrological and meteorological input data X and the flow observation data Y of the watershed outlet section are divided into 7 segments, X = {X j |j=1,2,…,7}、Y={Y j |j=1,2,…,7}, the hydrological and meteorological input data X of the jth segment j and Y j Contains data of equal number of time periods, denoted by X j and Y j The number of data periods is c j , corresponding to The number of data periods in the seven segments are 365, 366, 365, 365, 365, 366, and 365 respectively;

[0050] The initial time of the i-th time period in the j-th segment is t j,i,start , the end time is t j,i,end , the initial time of the first period of this segment is t j,0,start , the last period ends at

[0051] S23, substituting the hydrological model parameter value δ0 obtained in step 1 into the ordinary differential equation system of the differential form Xin'anjiang model:

[0052]

[0053] In formula (5): t is time; U = (W u W l W d S0O i O g F1 F2 F3) T is the state variable of the Xinanjiang model in differential form, W u , W l , W dare the average tensile water storage of the upper, lower and deep layers of the basin, S0 is the average free water storage of the basin, O i and O g is the storage capacity of the linear reservoir for subsoil flow and groundwater runoff, F1, F2, and F3 are the storage capacities of the first, second, and third reservoirs of the Nash unit line; the variables on the right side of the equal sign are all the fluxes of the Xin'anjiang model in differential form, and their meanings and flux equations are shown in Table 2; P and E in Table 2 are hydrological and meteorological data of rainfall and evaporation, specifically from X; the physical meanings and value ranges of the model parameters are shown in Table 1, and δ is the number of linear reservoirs in series in Table 1 except for n f Any parameter of .

[0054] Table 2 Flux and flux equation of the Xinanjiang model in differential form

[0055]

[0056]

[0057] Starting from j = 1, we take m = 7 in order and perform a forward numerical solution for each segment. For the jth segment, we set the initial value of the hydrological model state variable U(t j,0,start ), the initial values ​​of all state variables in the first segment are set to 0, and the initial values ​​of state variables in subsequent segments are set to the final values ​​of the state variables in the last period of the previous segment, specifically:

[0058]

[0059] According to the hydrological and meteorological input data X of the jth segment j and the initial value of the hydrological model state variable U(t j,0,start ), and the forward numerical solution was performed using an ordinary differential equation solver (based on the third-order Runge-Kutta method of Bogacki-Shampine, BS3). The relative error and absolute error of the ordinary differential equation solver were set to 1×10 -4 , and obtain the state variable continuous process U(t) corresponding to the segment, where The solution process can be expressed as:

[0060] U(t)=solver[f, U(t j,0, start ), t, X j (t); δ0] (7)

[0061] In formula (7), solver is the ordinary differential equation solver, specifically BS3; U(t j,0,start ) is the initial value of the hydrological model state variable; t is the time, specifically According to U(t) and δ0, the flow simulation data of the jth segment is calculated Specifically:

[0062]

[0063] In formulas (8) and (9): is the flow simulation value of the ith period in the jth segment. Formula (8) is the specific calculation function of the flow simulation value in the differential form of the Xinanjiang model for the scenario with a period length of 1 day.

[0064] Step 3, back propagation update of neural network parameters:

[0065] The loss function value is calculated based on the flow observation data of the basin outlet section and the flow simulation data obtained in step 2, and then the gradient of the loss function value relative to the neural network parameter θ in step 1 is calculated. The neural network parameter θ is updated by back propagation to obtain the updated value of the neural network parameter θ k+1,0 The specific steps are:

[0066] Starting from j=1, we take the values ​​from 1 to m in order, and then calculate the flow observation data Y of the outlet section of the basin corresponding to each segment. j With simulated data Calculate the loss function L corresponding to each fragment j :

[0067]

[0068] In formula (10): L j is the loss function of the jth segment, j = 1, 2, ..., m; loss is the loss function evaluation index, specifically the mean square error (E MS );Y j and are the observed and simulated flow data of the watershed outlet section of the jth segment, respectively;

[0069] Using the adjoint sensitivity method, based on the loss function L calculated using the j-th segment j and the final value of the state variable of the fragment Calculate the loss function L for this segment j The gradient of the neural network parameter θ According to the gradient update step 1, the neural network parameter θ is:

[0070]

[0071] In formula (11), k is the number of iterations, which is a non-negative integer; η is the learning rate, which is 1×10 -3Based on the forward numerical solution results of m fragments, the neural network parameter θ in step 1 can be updated m times, and the parameter value of the neural network parameter θ after updating m times is used as the initial value of the neural network parameter for the next iteration, specifically:

[0072] θ k+1,0 =θ k,m (12).

[0073] Step 4: Iterative optimization of hydrological model parameters:

[0074] Set the number of iterations k to 0;

[0075] The forward numerical solution process of the ordinary differential equation group described in step 2 and the back propagation update process of the hydrological model parameters described in step 3 are performed in sequence. Based on the neural network N and the basin underlying surface attribute data A in step 1 and the neural network update value θ obtained in step 3, k+1,0 , according to formula (2) in step 1, the updated value of the hydrological model parameter δ is calculated k+1 , using δ k+1 Replace step 1 to obtain the hydrological model parameter value δ0 or the hydrological model parameter update value δ used in the last iteration k , as the input of step 2 of the next iteration process, the number of iterations k is increased by 1, which is a complete iteration process;

[0076] Repeat the iterative process, different fragment loss functions (L j |j=1,2,...,m) changes as follows Figure 2 As shown; when the number of iterations is 80, the loss function value (L j |j=1,2,...,m) converges, and the hydrological model parameter value δ 81 That is the optimal value of the hydrological model parameters. The optimal parameter values ​​of the hydrological model are shown in Table 3.

[0077] Table 3 Optimal values ​​of hydrological model parameters calibrated based on the Neural Ordinary Differential Equation method

[0078]

[0079] The differential form Xinanjiang model described in step 1, the hydrological and meteorological input data X described in step 2, and the optimal hydrological model parameter values ​​in Table 3 are used to obtain the flow simulation data of the basin outlet section. Based on the flow observation data Y of the basin outlet section, the Nash-Sutcliffe efficiency coefficient (E NS ), Kling-Gupta efficiency coefficient (E KG ) and Flood Volume Relative Error (E FVR) The flow simulation accuracy is evaluated year by year, and the results are shown in Table 4.

[0080] Table 4 Accuracy evaluation table of application basin flow simulation results

[0081]

[0082] As shown in Table 4, the optimal values ​​of the hydrological model parameters obtained by the parameter optimization method based on the Neural Ordinary Differential Equation have a high accuracy in the corresponding flow simulation data. NS The maximum value is 0.96, the minimum value is 0.86, and the average value is 0.91; KG The maximum value is 0.91, the minimum value is 0.84, and the average value is 0.89; E FVR The absolute maximum value is 14.61%, and the absolute minimum value is 0.44%, both within ±20%. As an example, the observation and simulation data of the flow at the outlet section of the basin for one year are selected, and its process line is as follows Figure 3 shown.

[0083] Finally, it should be noted that the above is only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solution of the present invention (such as the use of various formulas, the sequence of steps, etc.) can be modified or replaced by equivalents without departing from the spirit and scope of the technical solution of the present invention.

Claims

1. A method for optimizing hydrological model parameters based on Neural Ordinary Differential Equations. It is characterized in that The method comprises the following steps: Step 1, parameter learning neural network construction: Collect the underlying surface attribute data of the application basin, specifically: A={a i | i=1, 2, …, n} (1) In formula (1): A is the underlying surface attribute data of the watershed; a i is the i-th attribute value of the watershed; n is the number of underlying surface attributes of the watershed; Construct a neural network N for learning the relationship between the hydrological model parameters and the underlying surface attributes of the watershed, and use it to learn the relationship between the hydrological model parameters and the underlying surface attributes of the watershed. Its input is the underlying surface attribute data A of the watershed, and the output value ranges from 0 to 1. The number of iterations is recorded as k. At the kth iteration, the initial value of the neural network parameter θ is recorded as θ k,0 Based on the neural network N and the underlying surface attribute data A of the watershed, the output of the neural network N is converted into the parameter value δ of the hydrological model parameter δ by the following formula (2): k : d k =(δ max -d min )·N(A; θ k,0 )+d min (2) In formula (2), k is the number of iterations, which is a non-negative integer; δ k is the hydrological model parameter value; δ max is the upper limit of the hydrological model parameters; δ min is the lower limit of the hydrological model parameters; N is the neural network; θ k,0 is the initial value of the neural network parameters at the kth iteration; Step 2, forward numerical solution of the ordinary differential equations: The hydrological model parameter value δ obtained in step 1 k Substitute into the ordinary differential equations of the hydrological model, set the initial values ​​of the hydrological model state variables and the basin hydrological and meteorological input data, use the ordinary differential equation solver to perform forward numerical solution on the ordinary differential equations of the hydrological model, and obtain the flow simulation data; Step 3, back propagation update of neural network parameters: Based on the flow observation data of the basin outlet section and the flow simulation data obtained in step 2, the loss function value is calculated, and then the gradient of the loss function value relative to the neural network parameter θ in step 1 is calculated. The neural network parameter θ is updated by back propagation to obtain the updated value of the neural network parameter θ k,0 ; Step 4: Iterative optimization of hydrological model parameters: Based on the neural network N in step 1 and the underlying surface attribute data A of the watershed, the neural network parameter update value θ obtained in step 3 k+1,0 , calculate the updated value of the hydrological model parameter δ k+1 ; Using Delta k+1 Replace the updated value of the hydrological model parameter δ calculated in the last iteration k , as the input of step 2, add 1 to the number of iterations k and complete a complete iteration process; Repeat the iterative process until the loss function converges, and the updated value of the hydrological model parameter δ k This is the optimal parameter value of the hydrological model.

2. The hydrological model parameter optimization method based on the Neural Ordinary Differential Equation according to claim 1, It is characterized in that The hydrological model includes a lumped hydrological model in differential form, a semi-distributed hydrological model and a distributed hydrological model.

3. The hydrological model parameter optimization method based on the Neural Ordinary Differential Equation according to claim 1 or 2, It is characterized in that The specific steps of the forward numerical solution process of the ordinary differential equation system in step 2 include: S21, collect the hydrological and meteorological input data of the application basin and the flow observation data of the basin outlet section, unify the data period length and total number of periods of hydrological and meteorological and flow observation data, specifically: Y={Q i | i=1, 2, …, T} (4) In formulas (3) and (4), X is the hydrological and meteorological input data; P i is the rainfall in the ith period; P T is the rainfall in the Tth period; E i is the evapotranspiration in the i-th period; E T is the evapotranspiration in the Tth period; T is the total number of periods; Y is the flow observation data of the basin outlet section, and the total number of periods containing data in Y is the same as that in X; Q i is the observed flow value of the i-th period at the outlet section of the basin; S22, the hydrological and meteorological input data X and the flow observation data Y of the watershed outlet section are divided into m segments, specifically X = {Xj|j = 1, 2, ..., m}, Y = {Yj|j = 1, 2, ..., m}, the hydrological and meteorological input data X of the jth segment j and Y j Contains data of equal number of time periods, denoted by X j and Y j The number of data periods is c j , corresponding to The initial time of the i-th time period in the j-th segment is t j,i,start , the end time is t j,i,end , the initial time of the first period of this segment is t j,0,start , the last period ends at S23, the hydrological model parameter value δ obtained in step 1 k Substitute into the ordinary differential equations of the hydrological model: In formula (5), t is time; U is the state variable of the hydrological model; f is the flux equation group of the hydrological model; X is the hydrological and meteorological input data; Starting from j = 1, we take the values ​​from 1 to m in order and perform a forward numerical solution for each segment. For the jth segment, we set the initial value of the hydrological model state variable U(t j,0,start ), the initial values ​​of all state variables in the first segment are set to 0, and the initial values ​​of state variables in subsequent segments are set to the final values ​​of the state variables in the last period of the previous segment, specifically: According to the hydrological and meteorological input data X of the jth segment j and the initial value of the hydrological model state variable U(t j,0,start ), use the ordinary differential equation solver to perform forward numerical solution and obtain the state variable continuous process U(t) corresponding to the segment, where The solution process can be expressed as: U(t)=solver[f, U(t j,0,start ), t, X j (t); δ k ] (7) In formula (7), solver is the ordinary differential equation solver; U(t j,0,start ) is the initial value of the hydrological model state variable; t is the time, specifically According to U(t) and δ 0 , calculate the flow simulation data of the jth segment Specifically: In formulas (8) and (9): is the flow simulation value of the i-th time period in the j-th segment; G is the function for calculating the flow simulation value of the time period.

4. The hydrological model parameter optimization method based on the Neural Ordinary Differential Equation according to claim 1 or 2, It is characterized in that In step 3, the back propagation update of the neural network parameters includes the following steps: Starting from j=1, we take the values ​​from 1 to m in order, and calculate the flow observation data Y of the outlet section of the basin corresponding to each segment. j With simulated data Calculate the loss function L corresponding to each fragment j : In formula (10): L j is the loss function of the jth segment, j = 1, 2, ..., m; loss is the loss function evaluation index; Y j and are the observed and simulated flow data of the watershed outlet section of the jth segment, respectively; Using the adjoint sensitivity method, based on the loss function L calculated using the j-th segment j and the final value of the state variable of the fragment Calculate the loss function L for this segment j The gradient of the neural network parameter θ According to the gradient update step 1, the neural network parameter θ is: In formula (11), k is the number of iterations, which is a non-negative integer; η is the learning rate; based on the forward numerical solution results of m fragments, the neural network parameter θ in step 1 can be updated m times, and the parameter value of the neural network parameter θ after updating m times is used as the initial value of the neural network parameter for the next iteration, specifically: i k+1,0 =θ k,m (12)。

Images