Accumulated snow coverage rate determination method based on divide-and-conquer strategy and symbolic regression

Abstract

The invention discloses an accumulated snow coverage rate determination method based on a divide-and-conquer strategy and symbolic regression, and relates to the technical field of environment monitoring, and the method comprises the steps: carrying out the pre-training of a full-connection feedforward neural network through a data set according to a plurality of target environment parameters, and obtaining a black-box accumulated snow coverage rate determination model; determining a model based on the black box accumulated snow coverage rate, and constructing a symbolic neural network; optimizing the symbolic neural network based on the weight importance score and the white-box loss function to obtain an accumulated snow coverage rate determination equation; acquiring real-time measurement values of various target environment parameters of the to-be-measured area; and inputting the real-time measurement values of the multiple target environment parameters of the to-be-measured area into the snow coverage rate determination equation to obtain the snow coverage rate of the to-be-measured area. According to the application, the symbolic neural network is established based on the divide-and-conquer strategy, and path optimization is completed based on the weight importance score and the white-box loss function; and thus, the determination precision and efficiency of the snow coverage rate are improved.

Description

Technical Field

[0001] The present application relates to the technical field of environmental monitoring, and in particular to a method for determining snow cover based on a divide-and-conquer strategy and symbolic regression. Background Art

[0002] Snow cover rate refers to the percentage of snow-covered area to the total area of ​​a region, and is an important indicator for studying climate change. The existing method for determining snow cover rate is to model complex systems to simulate system changes, optimize and control system behavior, and then complete the snow cover rate determination. As the main behavior of the system becomes more complicated, the use of black box models for modeling can no longer meet the task requirements. Symbolic regression is a white box modeling method that uses system observation data to find the symbolic expression that best fits the data features, which can better mine the inherent laws of the data and has stronger extrapolation than black box modeling. The symbolic regression method achieves symbolic expression fitting by simulating system behavior, collecting data, and fitting equations. It mainly includes two types of methods: search-based symbolic regression methods and symbolic regression models based on large-scale pre-training. In order to save computing resources and better adapt to the divide-and-conquer strategy, the search-based symbolic regression method can use multiple search strategies to learn equations with less resource consumption. In the search-based symbolic regression method, the two problems of equation structure search and equation coefficient fitting must be solved at the same time. The general search method uses brute force search, evolutionary computing or reinforcement learning to model the symbolic expression structure in a tree structure, and uses common equation coefficient optimization methods to fit the coefficients, and finally selects the optimal solution by calculating the expression fitness. Another method is based on neural networks, using equation basis functions with gradient protection as activation functions, treating network training as a global search process, and adding weight regularization to balance the accuracy of the solution and the complexity of the equation.

[0003] The survey found that the existing search-based symbolic regression methods still have some problems: most methods tend to ignore the relationship between equation variables or the internal properties of the equation itself, and only judge the symbolic regression effect from the degree of fit between the expression calculation results and the collected data. This leads to the situation where the error calculation effect is good, but the extrapolation is poor and the form of the equation is unreasonable when facing equation fitting scenarios with many input variables. At the same time, the symbolic regression method based on neural networks often only uses regularization to achieve network sparsity to ensure that the equation meets the Occam's razor principle, but such a sparse method requires huge computing resources and the final effect is still redundant. Summary of the invention

[0004] The purpose of this application is to provide a snow cover determination method based on a divide-and-conquer strategy and symbolic regression, which can improve the accuracy and efficiency of snow cover determination.

[0005] To achieve the above objectives, this application provides the following solutions:

[0006] This application provides a method for determining snow cover based on a divide-and-conquer strategy and symbolic regression, including:

[0007] Selecting a first preset number of environmental parameters from the data set as target environmental parameters;

[0008] According to a variety of target environmental parameters, a fully connected feedforward neural network is pre-trained using the data set to obtain a black box snow cover determination model;

[0009] Based on the black box snow cover determination model, a symbolic neural network is constructed;

[0010] Optimizing the symbolic neural network based on the weight importance score and the white box loss function to obtain a snow cover determination equation;

[0011] Obtain real-time measurement values ​​of multiple target environmental parameters in the area to be tested;

[0012] The real-time measured values ​​of various target environmental parameters of the area to be measured are input into the snow cover ratio determination equation to obtain the snow cover ratio of the area to be measured.

[0013] According to the specific embodiments provided in this application, this application discloses the following technical effects:

[0014] The present application provides a method for determining snow cover based on a divide-and-conquer strategy and symbolic regression, and a method for adaptively constructing a neural network based on a divide-and-conquer strategy. By establishing an agent model (black-box snow cover determination model), the observed data interpolation function is fitted, and the intrinsic mathematical properties of the symbolic expression are discovered through its fitting effect on the observed data and the back-propagation characteristics of the neural network. By using the method of recursive recognition and establishment of a symbolic neural network model, the modular nature of the equation graph detected by the agent model is utilized to construct an end-to-end symbolic neural network, and the semantic correctness constraints of the symbolic expression are realized through special basic building block improvements and constraint-based regularization. The weight importance and candidate neuron loss based on the first-order Taylor expansion are used as two indicators to guide the sparsity optimization of the symbolic neural network. On the one hand, the weight importance score is used to save computing resources, and on the other hand, the candidate neuron loss is used to solve some uncomputable problems, and taboo search is added at the same time to avoid repeated searches, and finally the redundant coefficients are processed by coefficient simplification and permutation and combination. The divide-and-conquer strategy is used to effectively reduce the dimensionality of the problem search space and better fit the symbolic expression of higher-dimensional input variables. Based on the modular properties of the various equation graphs identified by the divide-and-conquer strategy, an end-to-end trainable symbolic neural network is adaptively constructed. The equation substructure can be explicitly determined during the construction process. At the same time, for unknown structures, there is no need to consume a lot of computing resources to calculate the intermediate process through the brute force method. It only needs to be built into the network by the improved EQL module (constructing an improved symbolic regression module), which effectively solves the problem that the time complexity of equation fitting increases exponentially with the increase of the input variable dimension. Combining the advantages of various heuristic strategies and search strategies, while ensuring the full exploration of the search space, the waste of computing resources is effectively avoided. Based on only the prior knowledge of the general equation properties and the collected data, the modular properties of the internal graph of the expression can be identified, and a trainable end-to-end symbolic neural network can be constructed according to the corresponding properties. While ensuring the semantic correctness of the equation, it can be sparsely distributed to the target equation with the optimal form and error. BRIEF DESCRIPTION OF THE DRAWINGS

[0015] In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the drawings required for use in the embodiments will be briefly introduced below. 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 paying creative work.

[0016] Figure 1 This is a flow chart of a method for determining snow cover ratio based on a divide-and-conquer strategy and symbolic regression in one embodiment of the present application;

[0017] Figure 2A schematic diagram of translational symmetry, separability and generalized symmetry in an embodiment of the present application;

[0018] Figure 3 A schematic diagram of constructing an improved symbolic regression module in an embodiment of the present application;

[0019] Figure 4 A symbolic neural network diagram in one embodiment of the present application;

[0020] Figure 5 A flow chart for determining the form of an undetermined equation in an embodiment of the present application;

[0021] Figure 6 A schematic diagram of the structure of a computer device provided in one embodiment of the present application. DETAILED DESCRIPTION

[0022] The following will be combined with the drawings in the embodiments of the present application to clearly and completely describe the technical solutions in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, not all of the embodiments. Based on the embodiments in the present application, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of this application.

[0023] In order to make the above-mentioned objects, features and advantages of the present application more obvious and easy to understand, the present application is further described in detail below with reference to the accompanying drawings and specific implementation methods.

[0024] In an exemplary embodiment, Figure 1 As shown in FIG. 1 , a method for determining snow cover based on a divide-and-conquer strategy and symbolic regression is provided, including:

[0025] Step 101: Select a first preset number of environmental parameters from a data set as target environmental parameters.

[0026] Step 102: Based on a variety of target environmental parameters, a fully connected feedforward neural network is pre-trained using a data set to obtain a black box snow cover determination model.

[0027] Since the equation is unknown, a black box agent model (black box snow cover determination model) is used to fit the obtained equation data. When the input data variable dimension is N-dimensional, the input data point can be expressed as {x1,...,x N}, now use the trained agent model f agent (x1,...,x N ), the value of the test function f can be calculated. At the same time, due to the characteristics of neural network training using back propagation, the partial derivative of f for each variable can be calculated according to the agent model To determine the simplified properties of equations.

[0028] Step 103: Construct a symbolic neural network based on the black box snow cover determination model.

[0029] Step 104: Optimize the symbolic neural network based on the weight importance score and the white box loss function to obtain a snow cover determination equation.

[0030] Step 105: Acquire real-time measurement values ​​of multiple target environmental parameters of the area to be measured.

[0031] Step 106: input the real-time measured values ​​of the various target environmental parameters of the area to be measured into the snow coverage ratio determination equation to obtain the snow coverage ratio of the area to be measured.

[0032] Step 102 includes:

[0033] The target environmental parameters are extracted from each environmental parameter group in the dataset to construct a training environmental parameter group, and the snow cover rate corresponding to each training environmental parameter group is determined. The types of environmental parameters in an environmental parameter group in the dataset include: snow depth, snow water equivalent, temperature, shortwave radiation, longwave radiation, air pressure, precipitation, specific humidity, subgrid terrain standard deviation, terrain slope sine and terrain aspect cosine.

[0034] Taking the training environment parameter group as input and the snow cover percentage corresponding to the training environment parameter group as output, the black box loss function is used to pre-train the fully connected feedforward neural network to obtain the black box snow cover percentage determination model.

[0035] Step 103 includes:

[0036] Determine all target environmental parameters as independent variables of the snow cover determination equation.

[0037] Based on the black box snow cover determination model, translational symmetry detection is performed on all independent variables to obtain the translational symmetry detection results.

[0038] Based on the black box snow cover determination model, separability tests are performed on all independent variables to obtain the separability test results.

[0039] Based on the black box snow cover determination model, all independent variables are tested for generalized symmetry and the generalized symmetry test results are obtained. Translational symmetry, separability and generalized symmetry are as follows Figure 2 .

[0040] In combination with the actual problem of complex system behavior modeling, the agent model f agent (x1,...,x N ) tests the following modular properties of equation graphs: translation symmetry, f(x1,x2,...,xN )=g(x1⊙x2,..,x N ),⊙∈{+,-,×,÷}) can be used using the method f(x1,x2,...,x N )=g(x1+a,x2+a,...,x N ) to identify. Separability of equations (Separability, f(x1,..,x k ,x k+1 ,...,x N )=g(x1,...,x k )⊙h(x k+1 ,...,x N ),⊙∈{+,×}) can be calculated by the indivisibility △ sep (x1, x2) to detect. The generalized symmetry of the equation (Generalized symmetry, f(x1,...,x k ,x k+1 ,...,x N )=g[h(x1,...,x k ),x k+1 ,...,x N ]) by finding a satisfying The h function is used to identify.

[0041] Construct an improved symbolic regression module. Figure 3 The improved symbolic regression module includes multiple symbolic regression layers. The multiple symbolic regression layers are connected by residuals. Each symbolic regression layer includes multiple neurons. The neurons correspond to the basis functions in the basis function library one by one. The basis functions in the basis function library include: addition, subtraction, multiplication, division, sine function, cosine function, exponential function, logarithmic function and square operation.

[0042] The traditional method of symbolic regression based on neural network (equation learner, EQL) is improved. First, the traditional linear layer is changed into a sub-neuron management method to more conveniently control the construction of the computational graph of the entire symbolic neural network. For the symbolic regression base function library funcs = {+, -, ×, ÷, sin, cos, exp, log, square}, the neuron dictionary W of each layer is defined according to the neurons corresponding to the input parameters of each basis function. (l) ={k (l) ,w (l) |k (l) ∈funcs (l)}, in order to correctly add semantic correctness constraints and corresponding regularization, define the weight mask dictionary M corresponding to each neuron (l) ={k (l),m (l) |k (l) ∈funcs (l)}, then for each layer of input x (l) , the neuron can be represented as In order to prevent the gradient from disappearing and ensure the reusability of shallow neurons, a residual connection is added to each layer. At this time, each layer of the module can be expressed as Represents the basis function (improved activation function) corresponding to the neuron cell in the lth layer.

[0043] Based on the translation symmetry test results, separability test results and generalized symmetry test results, the improved symbolic regression module is called to build a symbolic neural network. The symbolic neural network is a tree structure, such as Figure 4 .

[0044] Since the identified properties will affect the unidentified properties, a recursive construction method is adopted. While iteratively identifying the properties of the equations for different properties, the overall model is constructed based on the improved symbolic regression module. For translational symmetry and generalized symmetry {p,I|p∈(+,-,×,÷)}, the two properties containing variables without intersection are constructed in the same layer. At this time, the constructed layer can be expressed as For the separability {p,I1,I2|p∈(+,×),I1+I2=N}, the output layer can be directly constructed based on it. The model output can be expressed as For the non-separable parts, the improved symbol regression module is directly used as the output layer, with z (L) As the model output, the trivial term α, the approximately trivial term β and the complex constant term γ are recorded during the network forward propagation process to further implement the semantic correctness constraint of the equation through regularization.

[0045] Among them, based on the black box snow cover determination model, the translation symmetry test is performed on all independent variables to obtain the translation symmetry test results, including:

[0046] Determine the snow cover percentage corresponding to the training environment parameter group in the data set as the snow cover percentage verification value.

[0047] Get the translational symmetry factor.

[0048] Let the independent variable number n=1.

[0049] Let the iteration amount i=1.

[0050] Identify the nth independent variable as the left independent variable.

[0051] Determine the n+ith independent variable as the right independent variable.

[0052] Based on the translational symmetry factor and multiple snow cover verification values, the translational symmetry detection results of the left independent variable and the right independent variable are determined.

[0053] It is determined whether n+i is equal to a first preset number N to obtain a first determination result.

[0054] If the result of the first judgment is no, the value of the iteration amount i is increased by 1, and the process returns to the step of "determining that the n+ith independent variable is the right independent variable".

[0055] If the first judgment result is yes, then it is determined whether the variable number n is equal to N-1 to obtain the second judgment result.

[0056] If the second judgment result is no, the value of the independent variable number n is increased by 1, and the process returns to the step of "setting the iteration amount i=1".

[0057] If the second judgment result is yes, the translation symmetry detection is completed.

[0058] Among them, based on the translational symmetry factor and multiple snow cover verification values, the translational symmetry test results of the left independent variable and the right independent variable are determined, including:

[0059] Any translational symmetry index that determines the translational symmetry is the current translational symmetry index. The translational symmetry index includes addition, subtraction, multiplication, and division.

[0060] Based on the translation symmetry factor, the left independent variables and the right independent variables in all training environment parameter groups are updated according to the simplified rule corresponding to the current translation symmetry index to obtain multiple deformation training environment parameter groups.

[0061] Multiple deformed training environment parameter groups are respectively input into the black box snow cover determination model to obtain the snow cover prediction value.

[0062] The absolute value of the difference between the predicted snow cover value and the verified snow cover value corresponding to the same deformation training environment parameter group is determined as the initial judgment value.

[0063] The average of all initial determination quantities is determined as the first determination quantity.

[0064] It is determined whether the first determination value is less than the first determination value threshold value to obtain a third determination result.

[0065] If the third judgment result is yes, it is determined that there is translation symmetry between the left independent variable and the right independent variable, and the translation symmetry index of the left independent variable and the right independent variable is determined as the current translation symmetry index.

[0066] If the third judgment result is no, then it is determined whether all translational symmetry indexes are traversed to obtain a fourth judgment result.

[0067] If the result of the fourth judgment is no, update the current translational symmetry index and return to the step "based on the translational symmetry factor, update the left and right independent variables in all training environment parameter groups according to the simplified rules corresponding to the current translational symmetry index to obtain multiple deformation training environment parameter groups."

[0068] If the fourth judgment result is yes, it is determined that there is no translational symmetry between the left independent variable and the right independent variable.

[0069] The simplified rule for addition is: let the value of the left independent variable plus the translation symmetry factor be the updated value of the left independent variable. Let the value of the right independent variable minus the translation symmetry factor be the updated value of the right independent variable.

[0070] The simplification rule corresponding to subtraction is: let the value of the left independent variable plus the translation symmetry factor be the updated value of the left independent variable. Let the value of the right independent variable plus the translation symmetry factor be the updated value of the right independent variable.

[0071] The simplified rule for multiplication is: let the value of the left independent variable be multiplied by the translation symmetry factor and used as the updated value of the right independent variable. Let the value of the right independent variable be divided by the translation symmetry factor and used as the updated value of the right independent variable.

[0072] The simplified rule for division is: let the value of the left independent variable be multiplied by the translation symmetry factor and then be the updated value of the right independent variable. Let the value of the right independent variable be multiplied by the translation symmetry factor and then be the updated value of the right independent variable.

[0073] Among them, based on the black box snow cover determination model, all independent variables are tested for separability, and the separability test results are obtained, including:

[0074] Constructs a sequence of independent variables.

[0075] Let the divisible sequence number k=1.

[0076] Determine the first k variables of the training environment parameter group as left variables.

[0077] Determine any previous variable as the current left variable.

[0078] Determine the mean of the current left variable in all training environment parameter groups as the variable mean of the current left variable.

[0079] Update the current left variable and return to step "Determine that the mean of the current left variable in all training environment parameter groups is the variable mean of the current left variable".

[0080] After determining the training environment parameter group, Nk variables are right variables.

[0081] Determine any previous variable as the current right variable.

[0082] Determine the mean of the current right variable in all training environment parameter groups as the variable mean of the current right variable.

[0083] Update the current right variable and return to step "determine that the mean of the current right variable in all training environment parameter groups is the variable mean of the current right variable".

[0084] Determine any training environment parameter group as the current training environment parameter group.

[0085] The values ​​of the last Nk variables in the current training environment parameter group are replaced with the corresponding variable means to obtain the first variable group.

[0086] The values ​​of the first k variables in the current training environment parameter group are replaced with the corresponding variable means to obtain the second variable group.

[0087] The values ​​of all variables in the current training environment parameter group are replaced with the corresponding variable means to obtain the third variable group.

[0088] Get the hidden variable values ​​of the black box snow cover determination model.

[0089] The first variable group is input into the black box snow cover determination model to obtain the first predicted quantity.

[0090] The second variable group is input into the black box snow cover determination model to obtain a second predicted quantity.

[0091] The third variable group is input into the black box snow cover determination model to obtain the third predicted quantity.

[0092] The product of the first predicted amount and the second predicted amount is determined as a first intermediate amount.

[0093] The ratio of the first intermediate quantity to the third predicted quantity is determined as the second intermediate quantity.

[0094] The absolute value of the difference between the latent variable value and the second intermediate value is determined as the second determination value.

[0095] Update the current training environment parameter group, and return to the step of "determining that the mean of the first k variables of the current training environment parameter group is the previous mean" until all training environment parameter groups are traversed, and determine that the mean of the second judgment value of all training environment parameter groups is the integrated judgment value.

[0096] According to the integrated judgment value and the indivisibility threshold, the separability test result of the independent variable sequence at k is determined.

[0097] It is determined whether the divisible sequence number k is equal to the first preset number N to obtain a fifth determination result.

[0098] If the fifth judgment result is no, the value of the divisible sequence number k is increased by 1, and the process returns to the step of "determining any training environment parameter group as the current training environment parameter group".

[0099] If the fifth judgment result is yes, the separability test is completed.

[0100] Among them, according to the integrated judgment amount and the indivisibility threshold, the separability test result of the independent variable sequence at k is determined, including:

[0101] It is determined whether the integrated determination amount is less than the indivisible threshold value to obtain a sixth determination result.

[0102] If the sixth judgment result is no, it is determined that the independent variable sequence is indivisible at k.

[0103] If the sixth judgment result is yes, it is determined that the independent variable sequence is divisible at k, and k is determined as a divisible index. Wherein, a generalized symmetry test is performed on all independent variables based on the black box snow cover determination model to obtain a generalized symmetry test result, including:

[0104] Constructs a sequence of independent variables.

[0105] Let the divisible sequence number k=1.

[0106] The first k variables are used as independent variables to construct the intermediate function h. The intermediate function h has no specific form and refers to the sub-function part of the composite function that contains the first k variables and is independent of other variables. During the construction process, its maximum length is set and it is constructed in a brute force manner starting from 1, that is, from length 1 to the maximum length, all possible h are constructed, and each one is tested to see if it meets the generalized symmetry condition.

[0107] Determine the partial derivative of the intermediate function h as the intermediate function partial derivative.

[0108] The first k variables of each training environment parameter group are input into the intermediate function h to obtain the fitting value of each training environment parameter group.

[0109] The values ​​of the first k variables in each training environment parameter group are replaced with the fitting values ​​corresponding to the training environment parameter group to obtain multiple fourth variable groups.

[0110] Based on the partial derivative of the intermediate function and multiple fourth variable groups, the generalized symmetry detection result of the independent variable sequence at k is determined.

[0111] It is determined whether the divisible sequence number k is equal to the first preset number N to obtain a seventh determination result.

[0112] If the result of the seventh judgment is no, the value of the divisible sequence number k is increased by 1, and the process returns to the step of "determining any training environment parameter group as the current training environment parameter group".

[0113] If the seventh judgment result is yes, the generalized symmetry detection is completed.

[0114] Wherein, based on the partial derivative of the intermediate function and multiple fourth variable groups, the generalized symmetry test result of the independent variable sequence at k is determined, including:

[0115] The eighth judgment result is obtained by judging whether the black box snow cover determination model has a linear relationship with the partial derivative of the intermediate function when the fourth variable group is used as a variable.

[0116] If the result of the eighth judgment is no, it is determined that the independent variable sequence has no generalized symmetry at k.

[0117] If the result of the eighth judgment is yes, it is determined that the independent variable sequence has generalized symmetry at k, and k is determined to be a generalized symmetric index.

[0118] Specifically, based on the translational symmetry detection results, the separability detection results and the generalized symmetry detection results, the improved symbolic regression module is called to build a symbolic neural network, including: in the process of automatically building the symbolic neural network, three properties need to be tested at the same time, the satisfaction degree of the three properties is compared, and the most satisfactory one is selected as the currently identified property. The testing of all properties and the construction of the symbolic neural network are carried out alternately. A part of the network is constructed when one property is identified. Before the separability identification, translational symmetry and generalized symmetry can be identified, both of which can achieve dimensionality reduction for variables. If the separable property is identified in a certain step, the constructed part of the network is divided into two from the root node, and each part enters the cyclic identification-construction process as two separate parts. At the same time, separability includes addition and multiplication, both of which can be used as child nodes of the output node. If the index identified during the symmetry test contains the parent node generated by the previous construction, a new layer of tree structure is added. If not, the parent node continues to be constructed in the same layer.

[0119] Among them, based on the translation symmetry detection results, the separability detection results and the generalized symmetry detection results, an implementation method of calling the improved symbolic regression module to build a symbolic neural network includes:

[0120] Identify any pair of independent variables in the independent variable sequence that has translational symmetry.

[0121] Use the translational symmetry index as the parent node of both independent variables in the independent variable pair.

[0122] All parent nodes and all independent variables without parent nodes are identified as input nodes.

[0123] When there are no independent variable pairs with translational symmetry, all independent variables are determined to be input nodes.

[0124] When the independent variable sequence is separable, the multiplication node is used as the child node of the output node, and the independent variable sequence is divided into multiple primary variable groups according to the separable index.

[0125] For each primary variable group, an improved symbolic regression module is called as a child node of the multiplication node.

[0126] Determine any primary variable group as the current primary variable group.

[0127] The improved symbolic regression module corresponding to the current primary variable group is determined as the current intermediate node.

[0128] When the current first-level variable group has generalized symmetry, the current first-level variable group is divided into multiple second-level variable groups according to the generalized symmetry index.

[0129] Determine any secondary variable group as the current secondary variable group.

[0130] When the number of variables in the current secondary variable group is 1, the variables in the current secondary variable group are connected to the current intermediate node.

[0131] When the number of variables in the current secondary variable group is not 1, an improved symbolic regression module is called to use all input nodes in the current secondary variable group as child nodes of the current intermediate node.

[0132] Update the current secondary variable group and return to the step "when the number of variables in the current secondary variable group is 1, connect the variables in the current secondary variable group to the current intermediate node" until all secondary variable groups are traversed.

[0133] When the current first-level variable group does not have generalized symmetry, all input nodes in the current first-level variable group are used as child nodes of the current intermediate node.

[0134] Update the current primary variable group, and return to the step of "determining the improved symbolic regression module corresponding to the current primary variable group as the current intermediate node" until all primary variable groups are traversed to obtain the symbolic neural network.

[0135] Step 104 includes:

[0136] The symbolic neural network is searched based on the weight importance score and the white-box loss function, and multiple undetermined equation forms are obtained.

[0137] The weight importance score of the selected layer is calculated based on the first-order Taylor expansion method, which can be expressed as T(x, y, w (l) ), this indicator approximately calculates the impact on the model loss when the corresponding weight is reset to 0. For the selected candidate sub-models, the loss of each candidate model c can be described as L c (x,y,w c), and then search for a minimalist subnetwork based on the weight importance score and the candidate model loss two heuristic criteria, so as to obtain the optimal equation structure containing the coefficients to be optimized

[0138] Initialize the objective function value of the previous objective equation to infinity.

[0139] Determine the form of any undetermined equation as the current undetermined equation form.

[0140] Use the quasi-Newton method (Broyden-Fletcher-Goldfarb-Shanno, BFGS) to fit and reduce the dimensions of the coefficients of the current undetermined equation to obtain the current coefficient matrix A M×P . Where M represents the precision of the coefficient. P represents the position of the coefficient. The element a in the current coefficient matrix m×p Indicates the value of the pth coefficient in the current undetermined equation when the precision is m. The BFGS method is a quasi-Newton method.

[0141] Construct the current coefficient group by taking an element from each column of the current coefficient matrix.

[0142] Substitute the current coefficient group into the current undetermined equation form to obtain the current target equation.

[0143] Multiple training environment parameter groups are respectively input into the current target equation to obtain a fourth prediction quantity corresponding to each training environment parameter group.

[0144] The objective function value of the current objective equation is determined based on the snow cover verification value corresponding to the training environment parameter group and the fourth prediction value.

[0145] The equation with the smallest objective function value between the previous objective equation and the current objective equation is determined as the optimal equation to be determined.

[0146] It is determined whether all coefficient groups are traversed to obtain a ninth determination result.

[0147] If the result of the ninth judgment is no, the undetermined optimal equation is used as the previous target equation, and the process returns to the step of "taking one element from each column of the current coefficient matrix to construct the current coefficient group".

[0148] If the ninth judgment result is yes, then it is determined whether all the pending equation forms are traversed to obtain the tenth judgment result.

[0149] If the result of the tenth judgment is no, then update the current undetermined equation form, take the undetermined optimal equation as the previous target equation, and return to step "Use the BFGS method to fit and reduce the coefficients of the current undetermined equation form to obtain the current coefficient matrix A M×P ”.

[0150] If the tenth judgment result is yes, the undetermined optimal equation is determined to be the snow cover determination equation.

[0151] Use the quasi-Newton method to optimize the equation coefficients and further expand the equation constant list, simplify the redundant constants to the corresponding number of digits, and for a specific number of simplification digits B and each coefficient c i , construct a two-dimensional table to store the equation coefficients expressed as After permutations and combinations, a fusion coefficient table is formed, and the effects of different coefficient combinations on the model loss are tested one by one, and finally the optimal objective equation f is obtained. * (x1,...,x N ).

[0152] like Figure 5 , based on the weight importance score and the white-box loss function, the symbolic neural network is searched, and multiple equations to be determined are obtained, including:

[0153] Get the number Q of improved symbolic regression modules in the symbolic neural network.

[0154] Get the preset number H of the form of the equation to be determined.

[0155] Let the improved symbolic regression module be numbered q=Q.

[0156] It is determined whether the improved symbolic regression module serial number q is equal to 0 to obtain an eleventh determination result.

[0157] If the eleventh judgment result is no, the qth improved symbol regression module is determined as the current improved symbol regression module.

[0158] Get the number of symbolic regression layers D in the current improved symbolic regression module.

[0159] Let the symbol regression layer number d=D.

[0160] It is determined whether the symbol regression layer number d is equal to 0 to obtain a twelfth determination result.

[0161] If the twelfth judgment result is yes, the value of the improved symbol regression module number q is reduced by 1, and the process returns to the step of "judging whether the improved symbol regression module number q is equal to 0 to obtain the eleventh judgment result".

[0162] If the twelfth judgment result is no, then according to the indexes of all neurons in the (q, d)th symbol regression layer, it is judged whether there is a neuron in the (q, d)th symbol regression layer that is selected by the next symbol regression layer, and the thirteenth judgment result is obtained. When the symbol regression layer sequence number d = D, the next symbol regression layer of the (q, d)th symbol regression layer is the (q-1, 1)th symbol regression layer. When the symbol regression layer sequence number d is not equal to D, the next symbol regression layer of the (q, d)th symbol regression layer is the (q, d+1)th symbol regression layer.

[0163] If the thirteenth judgment result is yes, the special candidate is stored.

[0164] If the thirteenth judgment result is no, the neurons in the (q, d)th symbol regression layer that are selected by the next symbol regression layer are determined as selected neurons, and a selected neuron list is constructed.

[0165] Determine any selected neuron as the current neuron.

[0166] Determine all neurons in the previous symbol regression layer of the (q, d)th symbol regression layer as candidate neurons.

[0167] Determine the importance score of each candidate neuron to the current neuron.

[0168] All candidate neurons are sorted in descending order according to their importance scores, and the first H candidate neurons are obtained as candidate neurons for the current neuron.

[0169] Update the current neuron and return to step "determine the importance score of each candidate neuron to the current neuron" until the list of selected neurons is traversed.

[0170] Select one of the multiple candidate neurons corresponding to each selected neuron as the current candidate neuron.

[0171] Connect the current candidate neuron with the corresponding selected neuron to form an alternative path.

[0172] Determine the white-box loss function value of the current candidate path.

[0173] Return to step "selecting one of the multiple candidate neurons corresponding to each selected neuron as the current candidate neuron" until all candidate paths are traversed.

[0174] Arrange the candidate paths in descending order according to the white-box loss function value, and obtain the first H candidate paths as the selected paths.

[0175] The candidate neuron corresponding to the selection path is determined to be the neuron selected by the (q, d)th symbol regression layer in the previous symbol regression layer as the selected neuron.

[0176] Subtract 1 from the value of the symbol regression layer number d, and return to the step of "determining whether the symbol regression layer number d is equal to 0, and obtaining the twelfth determination result".

[0177] If the eleventh judgment result is yes, it is determined to obtain the H selected neurons of the (1,1)th sign regression layer as the starting neurons.

[0178] The total path corresponding to each starting point neuron is determined as the undetermined path.

[0179] The equation form corresponding to the undetermined path is determined to be the undetermined equation form.

[0180] Among them, the total path corresponding to each starting point neuron is determined as the undetermined path, including:

[0181] Determine any starting neuron as the current starting neuron.

[0182] Let the improved symbolic regression module serial number q=1.

[0183] Let the symbol regression layer number d=1.

[0184] It is determined whether the symbol regression layer sequence number d is equal to the number of symbol regression layers D, and a fourteenth determination result is obtained.

[0185] If the result of the fourteenth judgment is no, the current starting point neuron is selected in the (q, d+1)th symbol regression layer and connected to the current starting point neuron.

[0186] Select the current starting neuron in the (q, d+1)th symbol regression layer, add 1 to the value of the symbol regression layer serial number d, and return to the step "determine whether the symbol regression layer serial number d is equal to the number of symbol regression layers D, and obtain the fourteenth judgment result".

[0187] If the fourteenth judgment result is yes, then determine whether the improved symbol regression module serial number q is equal to the improved symbol regression module quantity Q to obtain the fifteenth judgment result.

[0188] If the fifteenth judgment result is no, the value of the improved symbol regression module number q is increased by 1, and the process returns to the step of "setting the symbol regression layer number d=1".

[0189] If the fifteenth judgment result is yes, a pending path is obtained.

[0190] Update the starting neuron and return to step "set the improved symbolic regression module serial number q=1" until all starting neurons are traversed to obtain multiple pending paths.

[0191] The dataset is constructed by obtaining the daily snowfall reanalysis data from UCLA in Alpine Asia for the wet years from 1999 to 2012 and 2013 to 2016. The independent variables of the dataset include: snow depth (unit: meter, m), snow water equivalent (unit: meter, m), temperature (unit: Kelvin, K), shortwave radiation (unit: watt per square meter, W / m 2 ), long-wave radiation (unit: watt per square meter, W / m 2 ), air pressure (unit: hectopascal, hPa), precipitation (unit: millimeter per day, mm / day), specific humidity (unit: kilogram per kilogram, kg / kg), subgrid terrain standard deviation (unit: meter, m), terrain slope sine and terrain slope cosine, dependent variable: snow cover rate (unit: %). Below, taking snow depth, air temperature, snow water equivalent, air temperature, specific humidity, and subgrid terrain standard deviation as an example, the snow cover rate determination method based on the divide and conquer strategy and symbolic regression disclosed in this embodiment is specifically described. The snow cover rate determination method based on the divide and conquer strategy and symbolic regression includes:

[0192] Step 1: Identification of simplified properties of equations based on agent networks

[0193] 1.1 In order to make the agent model better approximate the target equation, so that it can better serve as the interpolation function corresponding to the sampled data, while ensuring the accuracy of property detection. Suppose the input and output data sets are {x1,...,x N ,y}, using a fully connected feedforward neural network as the agent model, the interpolation function can be calculated and output as: y * =f agent (x1,...,x N ), you can use The agent model is trained as the loss function, and its weights are optimized using the Adam optimizer. * represents the predicted value of the neural network; y represents the true value, Represents the network prediction value of the i-th group of data in the dataset; y i Represents the true value of the i-th group of data in the dataset.

[0194] 1.2 The agent model that has been trained to meet the convergence conditions can be used as the interpolation function of the sampled data. At this time, the modular properties of each graph of the target symbolic expression can be tested through corresponding calculations.

[0195] 1.2.1 Translational symmetry.

[0196] For the input data {x1,...,x N}, assuming that the target variable is {x1,x2}, the translation symmetry of the equation can be expressed as:

[0197] f(x1,x2,...,x N )=g(x1⊙x2,..,x N ),⊙∈{+,-,×,÷}

[0198] Among them, f is the target equation for fitting, and g is the equation after reducing the dimension of the data using translation symmetry.

[0199] For different translational symmetries, the target variables to be detected are modified accordingly as follows to judge different translational symmetries:

[0200]

[0201] Among them, f agent is the trained agent network model, and a is the constant required to simulate the translation transformation of the detected variable. At this time, after calculating the translation symmetry transformation, whether the calculation results on both sides of the equation are less than a certain threshold δ TS , so as to determine whether the equation satisfies the translation symmetry. If the equation satisfies the simplified property, replace {x1,x2} with x1⊙x2 as the input variable of the next layer of the model.

[0202] 1.2.2 Separability.

[0203] With input data {x1,...,x k ,x k+1 ,...,x N}, for example, to determine {x1,...,x k} and {x k+1 ,...,x N The separability of the two parts, i.e. the judgment:

[0204] f(x1,..,x k ,x k+1 ,...,x N )=g(x1,...,x k )⊙h(x k+1 ,...,x N ),⊙∈{+,×}

[0205] The equation inseparability can be defined as follows:

[0206]

[0207] The mean C1 of each variable in the left variable group is {x1,...,x k}, that is, the left mean; the mean C2 of each variable in the right variable group is {x k+1 ,...,x N}, that is, the right mean; this formula is the indivisibility calculation formula for the multiplicative separability test, where k starts from 1, To represent the second half, calculate rms, f(x1,...,x N ) is the hidden variable value in the training data set, f agent To train the agent network, C i is the average value calculated by the data set for each variable in the i-th part. When the inseparability calculation result is less than 10 times the final training error of the agent model, the target equation is judged to be separable. When the calculated value is less than the inseparability threshold δ sep When , the two separable variables are divided according to the index, and the separable layer is defined, and the simplified properties of the two independent parts in the layer are recursively detected.

[0208] 1.2.3 Generalized symmetry.

[0209] When the input data is {x1,...,x k ,x k+1 ,...,x N}, the variables {x1,...,x k The generalized symmetry of} can be expressed as:

[0210] f(x1,...,x k ,x k+1 ,...,x N )=g[h(x1,...,x k ),x k+1 ,...,x N ]

[0211] For generalized symmetry, the agent model can be calculated by the chain rule because its neural network is trainable, that is, the model calculation value is differentiable with respect to the model input variable:

[0212]

[0213] in, is the derivative of the generalized symmetry detection part of the trained agent model, is the derivative of the equation after generalized symmetry transformation for the detection part. g1 represents the inverse of the identified partial variable, so at this time, we can judge whether the corresponding partial variable has generalized symmetry based on the linear correlation between the agent model and the derivative of the intermediate function h.

[0214] Step 2: Adaptive construction of symbolic neural network based on divide-and-conquer.

[0215] 2.1 In order to meet the sparsity requirements of symbolic neural networks and the semantic correctness of equations, the symbolic regression layer is rebuilt. To meet the needs of symbolic neural networks, the model weights are guaranteed not to be frequently modified during the sparsity optimization of equations. The corresponding masks are assigned to the model weights. At the same time, the formal prior constraints on the semantic correctness of equations are implemented, that is, the mutual exclusivity between the basis functions {sin, cos, log, exp}, and the way of modifying weights is avoided to affect the model calculation graph, thereby affecting model training. The corresponding transformation of the linear layer is converted into a sub-neuron management method to flexibly construct the model calculation graph. Finally, in order to prevent the gradient from disappearing and ensure the reusability of shallow neurons, residual connections are added. In summary, the hierarchical elements of the improved EQL model can be expressed as:

[0216] W (l) ={k (l) ,w (l) |k (l) ∈funcs (l)},

[0217] M (l) ={k (l) ,m (l) |k (l) ∈funcs (l)},

[0218]

[0219] Among them, W (l) is the weight dictionary of the lth layer, k (l) is the key corresponding to each neuron, w (l) For each neuron weight, M (l) is the mask dictionary of the lth layer, m (l) is the mask corresponding to a single neuron, is the calculation result of the linear transformation of the corresponding neuron in the lth layer, z (l) is the output of the lth layer containing the residual connection. Represents the weight of the cell neuron in the lth layer; Represents the weight mask of the cell neuron in the lth layer; x (l) represents the input of layer l; z (0) represents the output of the l-1 layer before the lth layer; z represents the residual connection Represents the basis function (improved activation function) corresponding to the neuron cell in the lth layer.

[0220] 2.2 Based on the improved EQL module in step 2.1, according to the modular properties of the equation graph identified in 1.2, the neural symbolic network is adaptively constructed according to different situations.

[0221] 2.2.1 When the number of variables is small or the simplified properties of the equations are all greater than the corresponding threshold, the improved EQL module is directly used as the overall model. At this time, the model can be expressed as:

[0222] f * (x1,...,x N )=z (L) (x1,...,x N )

[0223] Among them, L is the last layer of the EQL module. * (x1,...,x N ) represents the overall output of the improved symbolic regression block.

[0224] 2.2.2 For the identified symmetry, the input variable index is judged. If the index corresponding to the input variable is generated in the previous recursive construction process, or the simplified property of the equation is identified for the first time, the symbolic network is constructed from the shallow to the deep layer. Assuming that the identified symmetry and the target variable index are {p,I|p∈(+,-,×,÷)}, the corresponding layer of the symbolic neural network can be expressed as:

[0225]

[0226] Among them, Z (l) is the output of layer l, is the variable part of the newly constructed layer l that is reduced in dimension after the identified symmetry transformation, x j ,...,x m are the identified partial variables with symmetry, [j,m] and [k,n] are two non-intersecting index sets. Represents the index set of the first parent node of layer l. j, m, k, and n have no actual meaning. They are used to represent the set boundary defined by two disjoint sets.

[0227] If the index corresponding to the input variable does not intersect with the target variable index of the symmetric part identified by the current layer, the symmetric module will continue to be constructed in the same layer. At this time, the corresponding layer of the symbolic neural network can be expressed as:

[0228]

[0229] When the variables contained in the recursively identified new symmetry are not repeated in the variables generated in the previous recursion, the intermediate process formula for symmetry transformation and network construction is performed at the same level as the previously identified property, where is the variable dimension reduction operation for the currently identified symmetry transformation, is the symmetry transformation operation identified previously. It represents the first identified symmetry module in layer l and its corresponding output, and the variable group with this property is in brackets; It represents the i-1th identified symmetry module in the lth layer and its corresponding output, and the variable group with this property is in brackets; represents the ith symmetry module identified in the lth layer and its corresponding output, and the variable group with this property is in brackets; x ij represents the index j-th variable of the i-th index set; x im Represents the index m-th variable of the i-th index set.

[0230] The form of the equation corresponding to the generalized symmetry module is unknown, so the operator corresponding to the generalized symmetry needs to be determined by fitting the interpolation function using the improved EQL module. Therefore, it is specific to the generalized symmetry and can be expressed as:

[0231] p(x j ,...,x m )=z (L) (x j ,...,x m ),[j,m]∈I

[0232] This formula is a symmetry transformation operation specific to generalized symmetry, z (L) To improve the symbolic regression module for the identified variable x j ,...,x m The output of . I represents the index group with the symmetry variable.

[0233] 2.2.3 The determination of separability will directly affect the overall network structure. Since it directly divides the complete input variable into two parts, the separability module is directly constructed as the symbolic neural network output layer. At this time, the symbolic neural network output layer can be represented as follows based on the identified separability and the corresponding indexes of the two target variables {p, I1, I2|p∈(+,×), I1+I2=N}:

[0234]

[0235] Since the separability transformation requires dividing the variables into two parts starting from the output layer, the formula f * is the overall output of the model, is the output of the i-th part after division, [i,m] and [j,n] are two non-intersecting index sets. and They are the partial module structures that need to be recursively and adaptively constructed for the two separable modules after division.

[0236] 2.2.4 For the overall symbolic neural network and the recursive adaptive construction part of a separable module, if the corresponding layer has only less than the threshold δ varNum For the number of variables, an improved EQL module is used to construct the model output layer, and its corresponding representation is the same as 2.2.1.

[0237] Step 3: Symbolic neural network optimization based on dual-index inspiration.

[0238] 3.1 Based on the symbolic neural network constructed in step 2.2, MSE is used as the model error, and smoothL is used 0.5 Regularization constrains the model weights from a global perspective, and its regularization term is described as:

[0239]

[0240] in, is the corresponding threshold, is the symbol for smoothL0.5 regularization; w is the penalized weight.

[0241] Based on this regularization term, the trivial terms (x / x), approximately trivial terms (cos(x / 2)), and complex constant terms (2.1+1) that may appear in the equation are recorded, and the weights that cause these semantic errors are penalized. The semantic correctness regularization term is as follows:

[0242]

[0243] Among them, L constrain (w) is the loss function regularization term based on semantic correctness constraints, α is the penalty term generated when the weight causes the equation to produce trivial terms, β is the penalty term generated when the weight causes approximately trivial terms, and γ is the penalty term generated when the weight causes the basis function parameters to be all constants. α , w β , w γ are the weights corresponding to the three weight penalty terms. In summary, the final symbolic neural network training loss function is as follows:

[0244]

[0245] In the formula, f * This is the model calculation result. The overall formula is the loss function formula generated by combining the MSE loss and the two regularization terms in the briefing book.

[0246] At this point, the network can be trained, that is, a global search of the expression structure can be performed.

[0247] 3.2 For a certain weight in the model (weight in the weight dictionary), when it is set to 0, the loss function of symbolic neural network training can be obtained by the first-order Taylor expansion:

[0248]

[0249] Ignoring higher-order terms, we get:

[0250]

[0251] where Θ w←0 is the model parameter when the target weight to be scored is reset to 0, and the remainder is the standard Taylor expansion.

[0252] Taylor's Middle Value Theorem 1 If a function f(x) has an nth-order derivative at x0, then there exists a neighborhood of x0 such that, for any x in the neighborhood,

[0253]

[0254] in:

[0255] R n (x) = o((x-x0) n )

[0256] In the formula, f (n) (x) represents the first-order derivative of f(x), R n (x) is the remainder of Taylor's formula, o((x-x0) n ) is (x-x0) n Higher-order infinitesimals.

[0257] Here is the case of n=1, that is, the first-order Taylor expansion:

[0258]

[0259] Move the right side of the first-order Taylor expansion formula L(x, y, Θ) to the right and take the absolute value transformation. When a certain weight disappears, the degree of influence on the overall loss of the model, that is, the weight importance score:

[0260]

[0261] Among them, |L(x,y,Θ w←0 The -L(x,y,Θ)| part is the weight importance score required by this application, expressed as T(x,y,w), which describes the impact on the model loss when a certain weight is reset to 0.

[0262] Through the above calculations, neurons can be searched layer by layer from deep to shallow, that is, the symbolic expression tree search process from the root node to the leaf node. Beam search is used for each layer to ensure sufficient search space. At the same time, as a dynamic search, that is, a method of searching while training, a taboo list is defined for each improved EQL module. When a candidate expression form is searched, it is stored in the taboo list and given a certain taboo period. If the number of repeated searches exceeds the taboo period, it is put back into the searchable individuals. For weights that are not in the same neuron in the network, the importance relationship between weights cannot be directly calculated through gradients. Therefore, it is necessary to calculate the loss for these individual candidate individuals, that is, the second heuristic indicator:

[0263] L cand (x,y,w i ,...,w j )=MSE(f * (x,w i ,...,w j ),t),[i,j]∈candList

[0264] where w i ,...,w j is the selection weight among the current candidate individuals, f * (x,w i ,...,w j ) is the output of the symbolic neural network for the weight calculation of the current candidate individual, candList represents the weight index list of the candidate cand, L cand is the white box loss of candidate cand,.

[0265] 3.3 Based on the target equation form searched in step 3.2, the BFGS method is first used to perform a preliminary fit on the equation coefficients. Then, for the redundant coefficient terms, all coefficients are first converted into a list and simplified by a fixed number of bits to increase the list dimension to a coefficient matrix. Then, the corresponding position coefficients are arranged and combined according to different simplification digits, and the optimization objective is achieved:

[0266]

[0267] in Indicates parameter c i The minimum value of f * (x,c i ) represents the symbolic neural network output; f(x) represents the true value; MSE(,) represents the mean square error

[0268] The optimization goal is to select the parameter combination that minimizes the MSE loss. By using this formula for optimization, the optimal coefficient combination can be screened and the optimal target equation can be finally searched.

[0269] In an exemplary embodiment, a computer device is provided. The computer device may be a server or a terminal. The internal structure diagram thereof may be as follows: Figure 6 As shown. The computer device includes a processor, a memory, an input / output interface (Input / Output, referred to as I / O) and a communication interface. The processor, the memory and the input / output interface are connected through a system bus, and the communication interface is connected to the system bus through the input / output interface. The processor of the computer device is used to provide computing and control capabilities. The memory of the computer device includes a non-volatile storage medium and an internal memory. The non-volatile storage medium stores an operating system, a computer program and a database. The internal memory provides an environment for the operation of the operating system and the computer program in the non-volatile storage medium. The input / output interface of the computer device is used to exchange information between the processor and an external device. The communication interface of the computer device is used to communicate with an external terminal through a network connection. When the computer program is executed by the processor, a method for determining snow cover based on a divide-and-conquer strategy and symbolic regression is implemented.

[0270] Those skilled in the art will understand that Figure 6 The structure shown in the figure is only a block diagram of a part of the structure related to the solution of the present application, and does not constitute a limitation on the computer device to which the solution of the present application is applied. The specific computer device may include more or fewer components than those shown in the figure, or combine certain components, or have a different arrangement of components. In an exemplary embodiment, a computer device is provided, including a memory and a processor, wherein a computer program is stored in the memory, and the processor implements the steps in the above-mentioned method embodiments when executing the computer program.

[0271] In an exemplary embodiment, a computer-readable storage medium is provided, storing a computer program, and when the computer program is executed by a processor, the steps in the above method embodiments are implemented.

[0272] In an exemplary embodiment, a computer program product is provided, including a computer program, and when the computer program is executed by a processor, the steps in the above method embodiments are implemented.

[0273] It should be noted that the user information (including but not limited to user device information, user personal information, etc.) and data (including but not limited to data used for analysis, stored data, displayed data, etc.) involved in this application are all information and data authorized by the user or fully authorized by all parties, and the collection, use and processing of relevant data must comply with relevant regulations.

[0274] Those of ordinary skill in the art can understand that all or part of the processes in the above-mentioned embodiment methods can be completed by instructing the relevant hardware through a computer program, and the computer program can be stored in a non-volatile computer-readable storage medium. When the computer program is executed, it can include the processes of the embodiments of the above-mentioned methods. Among them, any reference to the memory, database or other medium used in the embodiments provided in the present application can include at least one of non-volatile and volatile memory. Non-volatile memory can include read-only memory (ROM), magnetic tape, floppy disk, flash memory, optical memory, high-density embedded non-volatile memory, resistive random access memory (ReRAM), magnetic random access memory (MRAM), ferroelectric random access memory (FRAM), phase change memory (PCM), graphene memory, etc. Volatile memory can include random access memory (RAM) or external cache memory, etc. By way of illustration and not limitation, RAM may be in various forms, such as static random access memory (SRAM) or dynamic random access memory (DRAM).

[0275] The database involved in each embodiment provided in this application may include at least one of a relational database and a non-relational database. The non-relational database may include a distributed database based on blockchain, etc., but is not limited thereto. The processor involved in each embodiment provided in this application may be a general-purpose processor, a central processing unit, a graphics processor, a digital signal processor, a programmable logic device, a data processing logic device based on quantum computing, etc., but is not limited thereto.

[0276] The technical features of the above embodiments may be combined arbitrarily. To make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0277] This article uses specific examples to illustrate the principles and implementation methods of this application. The description of the above embodiments is only used to help understand the method and core ideas of this application. At the same time, for those skilled in the art, according to the ideas of this application, there will be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as limiting this application.

Claims

1. A method for determining snow cover based on a divide-and-conquer strategy and symbolic regression, characterized in that: include: Selecting a first preset number of environmental parameters from the data set as target environmental parameters; According to a variety of target environmental parameters, a fully connected feedforward neural network is pre-trained using the data set to obtain a black box snow cover determination model; Based on the black box snow cover determination model, a symbolic neural network is constructed; Optimizing the symbolic neural network based on the weight importance score and the white box loss function to obtain a snow cover determination equation; Obtain real-time measurement values ​​of multiple target environmental parameters in the area to be tested; The real-time measured values ​​of various target environmental parameters of the area to be measured are input into the snow cover ratio determination equation to obtain the snow cover ratio of the area to be measured.

2. The method for determining snow cover rate based on divide-and-conquer strategy and symbolic regression according to claim 1, characterized in that: According to a variety of target environmental parameters, the fully connected feedforward neural network is pre-trained using the data set to obtain a black box snow cover determination model, including: Extract target environmental parameters from each environmental parameter group in the data set to construct a training environmental parameter group, and determine the snow cover rate corresponding to each training environmental parameter group; the types of environmental parameters in an environmental parameter group in the data set include: snow depth, snow water equivalent, temperature, shortwave radiation, longwave radiation, air pressure, precipitation, specific humidity, subgrid terrain standard deviation, terrain slope sine and terrain aspect cosine; Taking the training environment parameter group as input and the snow cover percentage corresponding to the training environment parameter group as output, the fully connected feedforward neural network is pre-trained using the black box loss function to obtain the black box snow cover percentage determination model.

3. The method for determining snow cover rate based on divide-and-conquer strategy and symbolic regression according to claim 1, characterized in that: Based on the black box snow cover determination model, a symbolic neural network is constructed, including: Determine all target environmental parameters as independent variables of the snow cover determination equation; Based on the black box snow cover determination model, a translational symmetry test is performed on all independent variables to obtain a translational symmetry test result; Based on the black box snow cover determination model, a separability test is performed on all independent variables to obtain a separability test result; Performing a generalized symmetry test on all independent variables based on the black box snow cover determination model to obtain a generalized symmetry test result; Constructing an improved symbolic regression module; the improved symbolic regression module comprises a plurality of symbolic regression layers; a plurality of symbolic regression layers are connected by residuals; and each symbolic regression layer comprises a plurality of neurons; the neurons correspond one to one with the basis functions in the basis function library; the types of basis functions in the basis function library include: addition operation, subtraction operation, multiplication operation, division operation, sine function, cosine function, exponential function, logarithmic function and square operation; Based on the translational symmetry detection result, the separability detection result and the generalized symmetry detection result, the improved symbolic regression module is called to build a symbolic neural network; the symbolic neural network is a tree structure.

4. The method for determining snow cover rate based on divide-and-conquer strategy and symbolic regression according to claim 3, characterized in that: Based on the black box snow cover determination model, translational symmetry detection is performed on all independent variables to obtain translational symmetry detection results, including: Determine the snow cover rate corresponding to the training environment parameter group in the data set as the snow cover rate verification value; Get the translational symmetry factor; Let the independent variable number n = 1; Let iteration amount i = 1; Determine the nth independent variable as the left independent variable; Determine the n+ith independent variable as the right independent variable; Determine the translational symmetry detection results of the left independent variable and the right independent variable based on the translational symmetry factor and the plurality of snow cover verification values; Determine whether n+i is equal to a first preset number N, and obtain a first determination result; If the first judgment result is no, the value of the iteration amount i is increased by 1, and the process returns to the step of "determining that the n+ith independent variable is the right independent variable"; If the first judgment result is yes, then judging whether the variable number n is equal to N-1, and obtaining a second judgment result; If the second judgment result is no, the value of the independent variable number n is increased by 1, and the process returns to the step of "setting the iteration amount i=1"; If the second judgment result is yes, the translation symmetry detection is completed.

5. The method for determining snow cover rate based on divide-and-conquer strategy and symbolic regression according to claim 4, characterized in that: Based on the translational symmetry factor and the plurality of snow cover verification values, the translational symmetry detection results of the left independent variable and the right independent variable are determined, including: Determine any translational symmetry index of the translational symmetry as a current translational symmetry index; the translational symmetry index includes addition, subtraction, multiplication and division; Based on the translation symmetry factor, the left independent variables and the right independent variables in all training environment parameter groups are updated according to the simplified rule corresponding to the current translation symmetry index to obtain multiple deformation training environment parameter groups; Inputting a plurality of deformation training environment parameter groups into the black box snow cover determination model respectively to obtain a snow cover prediction value; Determine the absolute value of the difference between the snow cover prediction value and the snow cover verification value corresponding to the same deformation training environment parameter group as the initial judgment value; Determine the average of all initial determination quantities as the first determination quantity; Determine whether the first determination value is less than a first determination value threshold value, and obtain a third determination result; If the third judgment result is yes, it is determined that there is translation symmetry between the left independent variable and the right independent variable, and the translation symmetry index of the left independent variable and the right independent variable is determined as the current translation symmetry index; If the third judgment result is no, then determine whether to traverse all translational symmetry indexes to obtain a fourth judgment result; If the fourth judgment result is no, the current translation symmetry index is updated, and the process returns to step "based on the translation symmetry factor, the left independent variables and the right independent variables in all training environment parameter groups are updated according to the simplified rule corresponding to the current translation symmetry index to obtain multiple deformation training environment parameter groups"; If the fourth judgment result is yes, it is determined that there is no translation symmetry between the left independent variable and the right independent variable; The simplified rule for addition is: let the value of the left independent variable plus the translation symmetry factor be the updated value of the left independent variable; let the value of the right independent variable minus the translation symmetry factor be the updated value of the right independent variable; The simplified rule corresponding to subtraction is: let the value of the left independent variable plus the translation symmetry factor be the updated value of the left independent variable; let the value of the right independent variable plus the translation symmetry factor be the updated value of the right independent variable; The simplified rule for multiplication is: let the value of the left independent variable be multiplied by the translation symmetry factor as the updated value of the right independent variable; let the value of the right independent variable be divided by the translation symmetry factor as the updated value of the right independent variable; The simplified rule corresponding to division is: let the value of the left independent variable be multiplied by the translation symmetry factor and used as the updated value of the right independent variable; let the value of the right independent variable be multiplied by the translation symmetry factor and used as the updated value of the right independent variable.

6. The method for determining snow cover rate based on divide-and-conquer strategy and symbolic regression according to claim 5, characterized in that: Based on the black box snow cover determination model, all independent variables are tested for separability, and the separability test results are obtained, including: Construct the sequence of independent variables; Let the divisible sequence number k = 1; Determine the first k variables of the training environment parameter group as left variables; Determine any previous variable as the current left variable; Determine that the mean of the current left variable in all training environment parameter groups is the variable mean of the current left variable; Update the current left variable and return to step "determine that the mean of the current left variable in all training environment parameter groups is the variable mean of the current left variable"; After determining the training environment parameter group, Nk variables are right variables; Determine any previous variable as the current right variable; Determine that the mean of the current right variable in all training environment parameter groups is the variable mean of the current right variable; Update the current right variable and return to step "determine that the mean of the current right variable in all training environment parameter groups is the variable mean of the current right variable"; Determine any training environment parameter group as the current training environment parameter group; Replace the values ​​of the last Nk variables in the current training environment parameter group with the corresponding variable means to obtain the first variable group; Replace the values ​​of the first k variables in the current training environment parameter group with the corresponding variable means to obtain the second variable group; Replace the values ​​of all variables in the current training environment parameter group with the corresponding variable means to obtain a third variable group; Obtain the hidden variable values ​​of the black box snow cover determination model; Inputting the first variable group into the black box snow cover determination model to obtain a first predicted quantity; Inputting the second variable group into the black box snow cover determination model to obtain a second predicted quantity; Inputting the third variable group into the black box snow cover determination model to obtain a third prediction quantity; Determine the product of the first predicted amount and the second predicted amount as a first intermediate amount; Determine the ratio of the first intermediate quantity to the third predicted quantity as the second intermediate quantity; Determine the absolute value of the difference between the latent variable value and the second intermediate value as the second determination value; Update the current training environment parameter group, and return to step "determine that the mean of the first k variables of the current training environment parameter group is the previous mean" until all training environment parameter groups are traversed, and determine that the mean of the second determination amount of all training environment parameter groups is the integrated determination amount; According to the integrated judgment amount and the indivisibility threshold, the separability test result of the independent variable sequence at k is determined; Determine whether the divisible sequence number k is equal to the first preset number N, and obtain a fifth determination result; If the result of the fifth judgment is no, the value of the divisible sequence number k is increased by 1, and the process returns to the step of "determining any training environment parameter group as the current training environment parameter group"; If the fifth judgment result is yes, the separability test is completed; Among them, according to the integrated judgment amount and the indivisibility threshold, the separability test result of the independent variable sequence at k is determined, including: Determine whether the integrated determination amount is less than the indivisible threshold value to obtain a sixth determination result; If the sixth judgment result is no, it is determined that the independent variable sequence is indivisible at k; If the sixth judgment result is yes, it is determined that the independent variable sequence is divisible at k, and k is determined to be a divisible index.

7. The method for determining snow cover rate based on divide-and-conquer strategy and symbolic regression according to claim 5, characterized in that: Based on the black box snow cover determination model, a generalized symmetry test is performed on all independent variables to obtain the generalized symmetry test results, including: Construct the sequence of independent variables; Let the divisible sequence number k = 1; Construct the intermediate function h with the previous k variables as independent variables; Determine the partial derivative of the intermediate function h as the intermediate function partial derivative; Input the first k variables of each training environment parameter group into the intermediate function h to obtain the fitting value of each training environment parameter group; The values ​​of the first k variables in each training environment parameter group are replaced by the fitting values ​​corresponding to the training environment parameter group to obtain multiple fourth variable groups; Based on the partial derivative of the intermediate function and multiple fourth variable groups, the generalized symmetry test result of the independent variable sequence at k is determined; Determine whether the divisible sequence number k is equal to the first preset number N, and obtain a seventh determination result; If the seventh judgment result is no, the value of the divisible sequence number k is increased by 1, and the process returns to the step of "determining any training environment parameter group as the current training environment parameter group"; If the seventh judgment result is yes, then the generalized symmetry detection is completed; Wherein, based on the partial derivative of the intermediate function and multiple fourth variable groups, the generalized symmetry test result of the independent variable sequence at k is determined, including: It is determined whether the black box snow cover determination model has a linear relationship with the partial derivative of the intermediate function when the fourth variable group is used as a variable, and the eighth determination result is obtained; If the eighth judgment result is no, it is determined that the independent variable sequence has no generalized symmetry at k; If the eighth judgment result is yes, it is determined that the independent variable sequence has generalized symmetry at k, and k is determined to be a generalized symmetric index.

8. The method for determining snow cover rate based on divide-and-conquer strategy and symbolic regression according to claim 4, characterized in that: Based on the translational symmetry detection result, the separability detection result and the generalized symmetry detection result, calling the improved symbolic regression module to build a symbolic neural network includes: Determine any pair of independent variables in the independent variable sequence that has translational symmetry; The translation symmetry index is used as the parent node of the two independent variables in the independent variable pair; Determine all parent nodes and all independent variables without parent nodes as input nodes; When there is no pair of independent variables with translational symmetry, all independent variables are determined to be input nodes; When the independent variable sequence is separable, the multiplication node is used as the child node of the output node, and the independent variable sequence is divided into multiple primary variable groups according to the separable index; For each primary variable group, call an improved symbolic regression module as a child node of the multiplication node; Determine any first-level variable group as the current first-level variable group; Determine the improved symbolic regression module corresponding to the current primary variable group as the current intermediate node; When the current first-level variable group has generalized symmetry, the current first-level variable group is divided into multiple second-level variable groups according to the generalized symmetry index; Determine any secondary variable group as the current secondary variable group; When the number of variables in the current secondary variable group is 1, connect the variables in the current secondary variable group to the current intermediate node; When the number of variables in the current secondary variable group is not 1, an improved symbolic regression module is called to take all input nodes in the current secondary variable group as child nodes of the current intermediate node; Update the current secondary variable group and return to step "when the number of variables in the current secondary variable group is 1, connect the variables in the current secondary variable group to the current intermediate node" until all secondary variable groups are traversed; When the current first-level variable group does not have generalized symmetry, all input nodes in the current first-level variable group are used as child nodes of the current intermediate node; Update the current primary variable group, and return to step "determine the improved symbolic regression module corresponding to the current primary variable group as the current intermediate node" until all primary variable groups are traversed to obtain the symbolic neural network.

9. The method for determining snow cover ratio based on divide-and-conquer strategy and symbolic regression according to claim 4, characterized in that: The symbolic neural network is optimized based on the weight importance score and the white box loss function to obtain the snow cover determination equation, including Searching the symbolic neural network based on the weight importance score and the white box loss function to obtain a plurality of undetermined equation forms; Initialize the objective function value of the previous objective equation to infinity; Determine the form of any undetermined equation as the current undetermined equation form; Use the quasi-Newton method to fit and reduce the dimensions of the coefficients of the current undetermined equation to obtain the current coefficient matrix A M×P ; Where M represents the precision of the coefficient; P represents the position of the coefficient; element a in the current coefficient matrix m×p Indicates the value of the pth coefficient in the current undetermined equation form when the precision is m; Take one element from each column of the current coefficient matrix to construct the current coefficient group; Substitute the current coefficient group into the current undetermined equation form to obtain the current target equation; Inputting a plurality of training environment parameter groups into the current target equation respectively to obtain a fourth predicted quantity corresponding to each training environment parameter group; Determine the objective function value of the current objective equation according to the snow cover verification value corresponding to the training environment parameter group and the fourth prediction quantity; Determine the equation with the smallest objective function value among the previous objective equation and the current objective equation as the optimal equation to be determined; Determine whether to traverse all coefficient groups to obtain a ninth determination result; If the result of the ninth judgment is no, the undetermined optimal equation is used as the previous target equation, and the process returns to the step of "taking one element from each column of the current coefficient matrix to construct the current coefficient group"; If the ninth judgment result is yes, then determine whether to traverse all the pending equation forms to obtain a tenth judgment result; If the result of the tenth judgment is no, then the current undetermined equation form is updated, the undetermined optimal equation is used as the previous target equation, and the process returns to step "using the BFGS method to perform fitting and dimensionality reduction processing on the coefficients of the current undetermined equation form to obtain the current coefficient matrix A M×P ”; If the tenth judgment result is yes, the undetermined optimal equation is determined to be the snow cover ratio determination equation.

10. The method for determining snow cover rate based on divide-and-conquer strategy and symbolic regression according to claim 9, characterized in that: The symbolic neural network is searched based on the weight importance score and the white box loss function to obtain a plurality of undetermined equation forms, including: Obtaining the number Q of improved symbolic regression modules in the symbolic neural network; Get the preset number H of the form of the undetermined equation; Let the improved symbolic regression module serial number q = Q; Determine whether the improved symbolic regression module serial number q is equal to 0, and obtain an eleventh determination result; If the eleventh judgment result is no, determining the qth improved symbol regression module as the current improved symbol regression module; Get the number of symbolic regression layers D in the current improved symbolic regression module; Let the symbol regression layer number d = D; Determine whether the symbol regression layer number d is equal to 0, and obtain a twelfth determination result; If the twelfth judgment result is yes, the value of the improved symbol regression module number q is reduced by 1, and the process returns to the step of "determining whether the improved symbol regression module number q is equal to 0 to obtain the eleventh judgment result"; If the twelfth judgment result is no, then according to the indexes of all neurons in the (q, d)th symbol regression layer, it is judged whether the (q, d)th symbol regression layer has neurons selected by the subsequent symbol regression layer, and the thirteenth judgment result is obtained; when the symbol regression layer sequence number d=D, the subsequent symbol regression layer of the (q, d)th symbol regression layer is the (q-1, 1)th symbol regression layer; when the symbol regression layer sequence number d is not equal to D, the subsequent symbol regression layer of the (q, d)th symbol regression layer is the (q, d+1)th symbol regression layer; If the thirteenth judgment result is yes, storing the special candidate; If the thirteenth judgment result is no, the neuron selected by the next symbol regression layer in the (q, d)th symbol regression layer is determined as the selected neuron, and a selected neuron list is constructed; Determine any selected neuron as the current neuron; Determine all neurons in the previous symbol regression layer of the (q, d)th symbol regression layer as candidate neurons; Determine the importance score of each candidate neuron to the current neuron; Arrange all candidate neurons in descending order according to their importance scores, and obtain the first H candidate neurons as candidate neurons for the current neuron; Update the current neuron and return to step "determine the importance score of each candidate neuron to the current neuron" until the list of selected neurons is traversed; Select one of the multiple candidate neurons corresponding to each selected neuron as the current candidate neuron; Connect the current candidate neuron with the corresponding selected neuron to form an alternative path; Determine the white box loss function value of the current alternative path; Return to step "selecting one of the multiple candidate neurons corresponding to each selected neuron as the current candidate neuron" until all candidate paths are traversed; Arrange the candidate paths in descending order according to the white box loss function value, and obtain the first H candidate paths as the selected paths; Determine that the candidate neuron corresponding to the selection path is the neuron selected by the (q, d)th symbol regression layer in the previous symbol regression layer as the selected neuron; Subtract 1 from the value of the symbol regression layer number d, and return to the step of "determining whether the symbol regression layer number d is equal to 0, and obtaining the twelfth determination result"; If the eleventh judgment result is yes, determine to obtain H selected neurons of the (1,1)th sign regression layer as the starting neurons; Determine the total path corresponding to each starting point neuron as the undetermined path; Determine that the equation form corresponding to the undetermined path is the undetermined equation form; Among them, the total path corresponding to each starting point neuron is determined as the undetermined path, including: Determine any starting neuron as the current starting neuron; Let the improved symbolic regression module serial number q = 1; Let the symbol regression layer number d = 1; Determine whether the symbol regression layer number d is equal to the number of symbol regression layers D, and obtain a fourteenth determination result; If the fourteenth judgment result is no, then the current starting point neuron is selected in the (q, d+1)th symbol regression layer and connected to the current starting point neuron; Select the current starting neuron in the (q, d+1)th symbolic regression layer, add 1 to the value of the symbolic regression layer number d, and return to step "determine whether the symbolic regression layer number d is equal to the number of symbolic regression layers D, and obtain the fourteenth determination result"; If the fourteenth judgment result is yes, then determine whether the improved symbol regression module sequence number q is equal to the improved symbol regression module quantity Q, and obtain the fifteenth judgment result; If the fifteenth judgment result is no, the value of the improved symbol regression module number q is increased by 1, and the process returns to the step of "setting the symbol regression layer number d=1"; If the fifteenth judgment result is yes, a pending path is obtained; Update the starting neuron and return to step "set the improved symbolic regression module serial number q = 1" until all starting neurons are traversed to obtain multiple pending paths.

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