CN108959182B - Small celestial body gravitational field modeling method based on Gaussian process regression - Google Patents

Small celestial body gravitational field modeling method based on Gaussian process regression Download PDF

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CN108959182B
CN108959182B CN201810485427.3A CN201810485427A CN108959182B CN 108959182 B CN108959182 B CN 108959182B CN 201810485427 A CN201810485427 A CN 201810485427A CN 108959182 B CN108959182 B CN 108959182B
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高艾
廖文韬
王高岳
贺佳文
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Beijing Institute of Technology BIT
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Abstract

The invention discloses a small celestial body gravitational field modeling method based on Gaussian process regression GPR, and belongs to the technical field of deep space exploration. The realization method of the invention is as follows: acquiring a training set of gravitational field data by a polyhedral method by using the spherical coordinates of field points near the small celestial bodies; and establishing a Gaussian process regression GPR model by using the obtained training set. And predicting the gravitational field at the inspection point to obtain the mapping relation between the field point and the gravitational acceleration, namely, rapidly and accurately performing modeling calculation on the gravitational field near the small celestial body by using a Gaussian process regression GPR method, reducing the calculated amount, improving the modeling speed and meeting the requirement of online operation. The invention can be applied to the field of deep space exploration, provides technical support and reference for establishing the dynamic environment around the small celestial body in the small celestial body exploration task, and solves the related engineering problem.

Description

Small celestial body gravitational field modeling method based on Gaussian process regression
Technical Field
The invention relates to a small celestial body gravitational field modeling method based on Gaussian process regression GPR, and belongs to the technical field of deep space exploration.
Background
The detection task of the small celestial body has many requirements, wherein the determination of the orbit environment is a crucial part, and whether the accurate determination of the gravitational field of the small celestial body directly influences the whole detection task. Therefore, the efficient gravity field modeling is not only the first problem to be solved for researching and designing the satellite orbit of the small celestial body, but also one of the main scientific targets of the small celestial body detection task.
The traditional small celestial body modeling method has three main types. The first method is a spherical harmonic method, which mainly uses a series expansion formula to directly approximate the gravitational potential energy. Such methods do not yield an accurate solution in the brillouin sphere. The second method is a polyhedral method, and the main idea is to divide volumes in the gravitational potential into line integrals of polyhedral edges by using a Gaussian formula and a Green formula. The third method is a particle group method, the principle of the particle group method is simple, a polyhedron model is mainly replaced by a particle group, and the gravitational field of a small celestial body is modeled by calculating the gravitational field generated by each particle, but the particle group method and the polyhedron method cannot avoid a large amount of complex calculation.
Disclosure of Invention
Aiming at the problems of complex calculation process, incapability of obtaining accurate solution and the like of the traditional gravitational field modeling method, the small celestial body gravitational field modeling method based on the Gaussian process regression GPR disclosed by the invention aims to solve the technical problems that: the gravitational field near the small celestial body is quickly and accurately modeled and calculated by using a Gaussian process regression GPR method, the calculated amount can be reduced, and the modeling speed is increased. The invention can be applied to the field of deep space exploration, provides technical support and reference for establishing the dynamic environment around the small celestial body in the small celestial body exploration task, and solves the related engineering problem.
The purpose of the invention is realized by the following technical scheme.
The invention discloses a small celestial body gravitational field modeling method based on Gaussian process regression GPR, which utilizes the spherical coordinates of field points near a small celestial body to obtain a training set of gravitational field data through a polyhedral method. And establishing a Gaussian process regression GPR model by using the obtained training set. And predicting the gravitational field at the inspection point to obtain the mapping relation between the field point and the gravitational acceleration, namely, the gravitational field near the small celestial body is quickly and accurately modeled and calculated by using a Gaussian process regression GPR method, the calculated amount can be reduced, and the modeling speed is increased.
The invention discloses a small celestial body gravitational field modeling method based on Gaussian process regression GPR, which comprises the following steps of:
step 1: and acquiring a training set of the gravitational field data by a polyhedral method.
In the training process of the training set, firstly, random point taking is carried out in a preset range around the small celestial body, the sphere coordinate position data lambda of the field point is calculated,
Figure BDA0001666605650000021
r is used as an input vector of a training set, the output of the training set is gravity acceleration g at a field point, and the gravity acceleration g is calculated by a polyhedral method.
The specific implementation method of the step 1 is as follows:
in the polyhedral method, the coordinate of an arbitrary point P is P (x, y, z), and the coordinate of P in a spherical coordinate system:
Figure BDA0001666605650000022
the training set will be trained using the ball coordinate position data, the conversion of the ball coordinates to cartesian coordinates being:
Figure BDA0001666605650000023
modeling of the small celestial body gravity field is essentially a process of finding the gravitational acceleration function g (x, y, z) at the gravity field midpoint P. The gravitational acceleration g is obtained from gravitational potential energy V (x, y, z), and the relationship between the gravitational acceleration g and the gravitational potential energy V (x, y, z) is as follows:
Figure BDA0001666605650000024
in the process of solving the gravitational acceleration g in the training set, the small celestial body is divided into a plurality of volume infinitesimals, S is a volume infinitesimal with mass dm in the small celestial body, and r is the distance from S to the check point P. The gravitational potential at the checkpoint is defined by a triple integral:
Figure BDA0001666605650000025
and finally, applying a Gaussian formula and a Green formula to derive the gravitational acceleration g as:
Figure BDA0001666605650000026
wherein R is the position vector of the inspection point P under the asteroid fixed connection coordinate system, eedgeRepresents an edge, reIs a vector from any point on the edge e of the polyhedron to the inspection point, and
Figure BDA0001666605650000031
Figure BDA0001666605650000032
Figure BDA0001666605650000033
Figure BDA0001666605650000034
Figure BDA0001666605650000035
wherein EeIs a 3 x 3 matrix;
Figure BDA0001666605650000036
an outer normal direction vector of edge e within plane A;
Figure BDA0001666605650000037
is the outer normal direction vector of the face A; re1, re2 is the distance from the check point to the two end points of the edge; e12 is the length of edge e;
Figure BDA0001666605650000038
is the outer normal direction vector of the face f.
In addition, the inspection point can be directly judged not to be outside the celestial body, in the formula
Figure BDA0001666605650000039
And judging the position of the check point for the Laplacian operator of the gravitational field, wherein the criterion is as follows:
Figure BDA00016666056500000310
and (4) obtaining an output value of the training set, namely the gravitational acceleration g at the input point, by calculation of a polyhedron method and combination of input point-sphere coordinates in the training set.
The input and output of the training set are:
Figure BDA00016666056500000311
wherein, giIs the gravitational acceleration g at the ith data point in the input vector.
Formula (12) is a training set of gravitational field data obtained by the polyhedral method.
Step 2: and (3) establishing a Gaussian process regression GPR model by using the training set obtained in the step 1.
The step 2 is realized by the following specific method:
training set D { (X)i,Yi) I | (1, 2 …, n } ═ X, Y), input vector
Figure BDA00016666056500000312
I.e. the spherical coordinate data described in step 1. Output scalar YiIs the gravitational acceleration g at the ith point. The regression task is to learn according to a given training set to obtain an output scalar YiAnd the input vector XiThe mapping relation between the test points is finally determined according to the test point X*Calculating the output value Y with the maximum possibility*。
When a regression GPR model of the Gaussian process is established by using the training set obtained in the step 1, observed values Y of n training functionsi∈R，i＝1,2…n。
Given the training set, the noise ε -N (0, σ) is due to the noisy observations2) Then, there is an observed value:
Y＝f(X)+ε (13)
wherein X is an input vector, Y is an observed value containing noise, and f (X) is a function value. (x) the gaussian process is given a priori, i.e.:
f(X)～GP(m(X),k(X,X')) (14)
for the sake of symbol simplicity, the mean function m (x) is defined as 0, and the covariance function k is chosen differently.
The establishment of a regression GPR model of the Gaussian process is completed, and for the problem of the regression GPR of the Gaussian process, the target is to a new detection point X*Can obtain a corresponding Y*。
And step 3: and (3) predicting the gravitational acceleration at the inspection point by using the Gaussian process regression GPR model in the step (2), namely, rapidly and accurately performing modeling calculation on the gravitational field near the small celestial body by using the Gaussian process regression GPR method, reducing the calculation amount and improving the modeling speed.
The concrete implementation method of the step 3 is as follows:
due to test data (X)*,Y*) And training data (X, Y) are all from the same distribution, and training data (X, Y) and test data (X) are obtained*,Y*) The joint distribution of (A) is:
Figure BDA0001666605650000041
where m is the mean function and K is the covariance function, also known as the kernel function.
Figure BDA0001666605650000042
By using kernel function, Gaussian process can be obtainedBy combining the input/output relationship of the GPR prediction model with the formula (15), Y can be obtained*The predicted value of (2).
After the training set shown in formula (12) is obtained in step 1, a square exponential form (EQ) is selected as a kernel function form, a zero mean value is used as a mean function, and the expression is as follows:
Figure BDA0001666605650000051
after adding noise, k (X, X') is:
Figure BDA0001666605650000052
Y*the condition distribution of (1):
Figure BDA0001666605650000053
selecting the mean value of the distribution as Y*I.e.:
Figure BDA0001666605650000054
thus, Y is obtained*The parameter θ referred to in the kernel function k (X, X') shown in equation (18) [ < l >, σ >, is [ < l >, σ >f,σn]And l is a variance measure,
Figure BDA0001666605650000055
as signal variance, σnIs the variance of the noise. The variance scale l, the signal variance
Figure BDA0001666605650000056
Variance parameter sigma of noisenReferred to as a hyper-parameter. The maximum likelihood method is applied to solve the hyper-parameters, firstly, a negative log-likelihood function of prior probability distribution is obtained, then, a partial derivative is solved for the likelihood function, and the partial derivative is obtained through the common useAnd solving the minimum value of the likelihood function by a yoke gradient method to obtain the optimal solution of the hyper-parameter.
The likelihood function L (θ) is:
Figure BDA0001666605650000057
and (3) performing partial derivation on the likelihood function to obtain:
Figure BDA0001666605650000058
after the partial derivative of the likelihood function is obtained, the optimal hyper-parameter is obtained through a conjugate gradient method, and the obtained hyper-parameter is substituted into the formula (18) and the formula (20) to obtain the predicted value of the gravitational acceleration g, so that the gravitational field near the celestial body can be rapidly and accurately modeled and calculated through a Gaussian process regression GPR method, the calculated amount can be reduced, and the modeling speed can be improved.
Further comprising the application step 4: the method from the step 1 to the step 3 is applied to the field of deep space exploration, provides technical support and reference for establishment of the dynamic environment around the celestial body in the small celestial body exploration task, and solves the related engineering problem.
Has the advantages that:
1. the invention discloses a gravitational field modeling method based on a Gaussian process regression GPR algorithm, which utilizes the spherical coordinates of field points near a small celestial body to obtain a training set of gravitational field data through a polyhedral method, utilizes the obtained training set to establish a Gaussian process regression GPR model, is a machine learning method, can avoid the complex calculation process of the traditional gravitational field modeling by utilizing the Gaussian process regression GPR model, obtains a check point spherical coordinate lambda from the data statistics angle,
Figure BDA0001666605650000062
the mapping relation between r and the gravitational acceleration g realizes the rapid and accurate modeling calculation of the gravitational field near the small celestial body by using a Gaussian process regression GPR method, and can reduce the calculated amount and improve the modeling speedAnd the requirement of online operation is met.
2. The gravitational field modeling method based on the Gaussian process regression GPR algorithm disclosed by the invention can be applied to the field of deep space exploration, provides technical support and reference for establishing the surrounding dynamic environment of the small celestial body in the small celestial body exploration task, and solves the related engineering problems.
Drawings
Fig. 1 is a flowchart of a gravitational field modeling method based on a gaussian process regression GPR algorithm disclosed in this embodiment.
FIG. 2 is a 433Eros gravitational acceleration calculation result in an example;
FIG. 3 shows the error distribution for 10000 checkpoints at 20km for 433 Eros.
Detailed Description
For a better understanding of the objects and advantages of the present invention, reference should be made to the following detailed description taken in conjunction with the accompanying drawings and examples.
This example calculated the gravitational acceleration g of a small celestial body 433Eros at 10000 checkpoints within 20km of the centroid, the 433Eros density being 2.67 × 1012kg/km3The gravitational constant of small celestial body is 0.4401X 10-3km3/s2. In order to prove the applicability of the method, the check points are randomly obtained, and the modeling result is compared with the result and the time calculated by the polyhedron method.
As shown in fig. 1, the method for modeling a small celestial body gravitational field based on gaussian process regression GPR disclosed in this embodiment is specifically implemented as follows:
step 1: and acquiring a training set of the gravitational field data by a polyhedral method.
In the training process of the training set, firstly, randomly taking 800 field points within a range of 20km from the centroid of the small celestial body to generate the training set, and then, acquiring the spherical coordinate position data lambda of the field points,
Figure BDA0001666605650000061
r is used as an input vector of a training set, the output of the training set is gravity acceleration g at a field point, and the gravity acceleration g is calculated by a polyhedral method.
The specific implementation method of the step 1 is as follows:
in the polyhedral method, the coordinate of an arbitrary point P is P (x, y, z), and the coordinate of P in a spherical coordinate system:
Figure BDA0001666605650000071
the training set will be trained using the ball coordinate position data, the conversion of the ball coordinates to cartesian coordinates being:
Figure BDA0001666605650000072
modeling of the small celestial body gravity field is essentially a process of finding the gravitational acceleration function g (x, y, z) at the gravity field midpoint P. The gravitational acceleration g is obtained from gravitational potential energy V (x, y, z), and the relationship between the gravitational acceleration g and the gravitational potential energy V (x, y, z) is as follows:
Figure BDA0001666605650000073
in the process of solving the gravitational acceleration g in the training set, the small celestial body is divided into a plurality of volume infinitesimals, S is a volume infinitesimal with mass dm in the small celestial body, and r is the distance from S to the check point P. The gravitational potential at the checkpoint is defined by a triple integral:
Figure BDA0001666605650000074
wherein G is 6.672x10-11 N.m2/kg2Is a universal gravitation constant. And finally, applying a Gaussian formula and a Green formula to derive the gravitational acceleration g as:
Figure BDA0001666605650000075
wherein R is the position vector of the inspection point P under the asteroid fixed connection coordinate system, eedgeRepresents an edge, reIs a vector from any point on the edge e of the polyhedron to the inspection point, and
Figure BDA0001666605650000076
Figure BDA0001666605650000077
Figure BDA0001666605650000078
Figure BDA0001666605650000079
Figure BDA00016666056500000710
wherein EeIs a 3 x 3 matrix;
Figure BDA0001666605650000081
an outer normal direction vector of edge e within plane A;
Figure BDA0001666605650000082
is the outer normal direction vector of the face A; re1, re2 is the distance from the check point to the two end points of the edge; e12 is the length of edge e;
Figure BDA0001666605650000083
is the outer normal direction vector of the face f.
In addition, the inspection point can be directly judged not to be outside the celestial body, in the formula
Figure BDA0001666605650000087
And judging the position of the check point for the Laplacian operator of the gravitational field, wherein the criterion is as follows:
Figure BDA0001666605650000084
and (4) obtaining an output value of the training set, namely the gravitational acceleration g at the input point, by calculation of a polyhedron method and combination of input point-sphere coordinates in the training set.
The input and output of the training set are:
Figure BDA0001666605650000085
wherein, giIs the gravitational acceleration at the ith data point in the input vector.
Formula (12) is a training set of gravitational field data obtained by the polyhedral method.
Step 2: and (3) establishing a Gaussian process regression GPR model by using the training set obtained in the step 1.
Training set D { (X)i,Yi) I | (1, 2 …, n } ═ X, Y), input vector
Figure BDA0001666605650000086
I.e. the spherical coordinate data described in step 1. Output scalar YiIs gravitational acceleration g at the ith pointi. The regression task is to learn according to a given training set to obtain an output scalar YiAnd the input vector XiThe mapping relation between the test points is finally determined according to the test point X*Calculating the output value Y with the maximum possibility*。
When a regression GPR model of the Gaussian process is established by using the training set obtained in the step 1, observed values Y of n training functionsi∈R，i＝1,2…n。
Given the training set, the noise ε -N (0, σ) is due to the noisy observations2) Then, there is an observed value:
Y＝f(X)+ε (13)
wherein X is an input vector, Y is an observed value containing noise, and f (X) is a function value. (x) the gaussian process is given a priori, i.e.:
f(X)～GP(m(X),k(X,X')) (14)
for the sake of symbol simplicity, the mean function m (x) is defined as 0, and the covariance function k is chosen differently.
The establishment of a regression GPR model of the Gaussian process is completed, and for the problem of the regression GPR of the Gaussian process, the target is to a new detection point X*Can obtain a corresponding Y*。
And step 3: and (3) predicting the gravitational acceleration at 10000 random check points by using the Gaussian process regression GPR model in the step (2), namely, rapidly and accurately modeling and calculating the gravitational field near the small celestial body by using the Gaussian process regression GPR method, reducing the calculated amount and improving the modeling speed.
Due to test data (X)*,Y*) And training data (X, Y) are all from the same distribution, and training data (X, Y) and test data (X) are obtained*,Y*) The joint distribution of (A) is:
Figure BDA0001666605650000091
where m is the mean function and K is the covariance function, also known as the kernel function.
Figure BDA0001666605650000092
By using the kernel function, the input-output relation of the regression GPR prediction model of the Gaussian process can be obtained, and Y can be obtained by combining the formula (15)*The predicted value of (2).
After the training set shown in formula (12) is obtained in step 1, a square exponential form (EQ) is selected as a kernel function form, a zero mean value is used as a mean function, and the expression is as follows:
Figure BDA0001666605650000093
after adding noise, k (X, X') is:
Figure BDA0001666605650000094
Y*the condition distribution of (1):
Figure BDA0001666605650000095
selecting the mean value of the distribution as Y*I.e.:
Figure BDA0001666605650000096
thus, Y is obtained*The parameter θ referred to in the kernel function k (X, X') shown in equation (18) [ < l >, σ >, is [ < l >, σ >f,σn]And l is a variance measure,
Figure BDA0001666605650000101
as signal variance, σnIs the variance of the noise. The variance scale l, the signal variance
Figure BDA0001666605650000102
Variance parameter sigma of noisenReferred to as a hyper-parameter. The maximum likelihood method is used for solving the hyperparameter, firstly, a negative log-likelihood function of prior probability distribution is obtained, then, a partial derivative is solved for the likelihood function, and the minimum value of the likelihood function is solved through a conjugate gradient method, so that the optimal solution of the hyperparameter is obtained.
The likelihood function L (θ) is:
Figure BDA0001666605650000103
and (3) performing partial derivation on the likelihood function to obtain:
Figure BDA0001666605650000104
after the partial derivative of the likelihood function is obtained, the optimal hyper-parameter is obtained through a conjugate gradient method, the obtained hyper-parameter is substituted into the formula (18) and the formula (20) to obtain the predicted value of the gravitational acceleration g, namely, the gravitational field near the celestial body is quickly and accurately modeled and calculated by using a Gaussian process regression GPR method, the calculated amount can be reduced, and the modeling speed is increased.
Further comprising the application step 4: the method from the step 1 to the step 3 is applied to the field of deep space exploration, provides technical support and reference for establishment of the dynamic environment around the celestial body in the small celestial body exploration task, and solves the related engineering problem.
The simulation parameters are shown in table 1:
TABLE 1 simulation parameters
Number of learning samples 800
Number of test samples 10000
Small celestial body density (kg/km)3) 2.67×1012
Point range (Km) 20
Gravitational constant of small celestial body (km)3/s2) 0.4401×10-3
The modeling accuracy and time are shown in table 2:
TABLE 2 modeling accuracy and time
Mean error (%) Polyhedral method time(s) GPR method time(s)
1.22 611.81 0.71
As can be seen from table 2 and fig. 2 to 3, the small celestial body gravitational field modeling method based on the gaussian process regression GPR can realize rapid and accurate modeling of the gravitational field near the small celestial body, and particularly, compared with the conventional method, the operation time is greatly improved, so that the efficient modeling calculation method can greatly reduce the modeling time of the gravitational field, and is beneficial to shortening the design period of the small celestial body detection task.
The above detailed description is intended to illustrate the objects, aspects and advantages of the present invention, and it should be understood that the above detailed description is only exemplary of the present invention and is not intended to limit the scope of the present invention, and any modifications, equivalents, improvements and the like made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (5)

1. The small celestial body gravitational field modeling method based on Gaussian process regression GPR is characterized by comprising the following steps: the method comprises the following steps:
step 1: acquiring a training set of gravitational field data by a polyhedral method;
in the training process of the training set, firstly, the training is carried out within a preset range around the small celestial bodyRandomly taking points, and comparing the spherical coordinate position data lambda of the field points,
Figure FDA0001666605640000015
r is used as an input vector of a training set, the output of the training set is gravity acceleration g at a field point, and the gravity acceleration g is calculated by a polyhedral method;
step 2: establishing a regression GPR model of the Gaussian process by using the training set obtained in the step 1;
and step 3: and (3) predicting the gravitational acceleration at the inspection point by using the Gaussian process regression GPR model in the step (2), namely, rapidly and accurately performing modeling calculation on the gravitational field near the small celestial body by using the Gaussian process regression GPR method, reducing the calculation amount and improving the modeling speed.
2. The gaussian process regression GPR based small celestial body gravitational field modeling method of claim 1, wherein: the method also comprises an application step 4, wherein the method in the steps 1 to 3 is applied to the field of deep space exploration, technical support and reference are provided for the establishment of the surrounding dynamic environment of the small celestial body in the small celestial body exploration task, and the related engineering problems are solved.
3. The method for modeling a small celestial gravity field based on gaussian process regression GPR of claim 1 or 2, characterized in that: the specific implementation method of the step 1 is as follows:
in the polyhedral method, the coordinate of an arbitrary point P is P (x, y, z), and the coordinate of P in a spherical coordinate system:
Figure FDA0001666605640000011
the training set will be trained using the ball coordinate position data, the conversion of the ball coordinates to cartesian coordinates being:
Figure FDA0001666605640000012
modeling the gravity field of the small celestial body is substantially a process of solving a gravity acceleration function g (x, y, z) at the midpoint P of the gravity field; the gravitational acceleration g is obtained from gravitational potential energy V (x, y, z), and the relationship between the gravitational acceleration g and the gravitational potential energy V (x, y, z) is as follows:
Figure FDA0001666605640000013
in the process of solving the gravitational acceleration g in the training set, the small celestial body is divided into a plurality of volume infinitesimals, S is a volume infinitesimal with mass dm in the small celestial body, and r is the distance from S to a check point P; the gravitational potential at the checkpoint is defined by a triple integral:
Figure FDA0001666605640000014
and finally, applying a Gaussian formula and a Green formula to derive the gravitational acceleration g as:
Figure FDA0001666605640000021
wherein R is the position vector of the inspection point P under the asteroid fixed connection coordinate system, eedgeRepresents an edge, reIs a vector from any point on the edge e of the polyhedron to the inspection point, and
Figure FDA0001666605640000022
Figure FDA0001666605640000023
Figure FDA0001666605640000024
Figure FDA0001666605640000025
Figure FDA0001666605640000026
wherein EeIs a 3 x 3 matrix;
Figure FDA0001666605640000027
an outer normal direction vector of edge e within plane A;
Figure FDA0001666605640000028
is the outer normal direction vector of the face A; r ise1,re2The distances from the check point to two end points of the edge; e.g. of the type12Is the length of edge e;
Figure FDA0001666605640000029
is the outer normal direction vector of plane f;
in addition, the inspection point can be directly judged not to be outside the celestial body, in the formula
Figure FDA00016666056400000212
And judging the position of the check point for the Laplacian operator of the gravitational field, wherein the criterion is as follows:
Figure FDA00016666056400000210
calculating by a polyhedral method, and combining input point-ball coordinates in a training set to obtain an output value of the training set, namely gravitational acceleration g at an input point;
the input and output of the training set are:
Figure FDA00016666056400000211
wherein, giThe gravity acceleration g at the ith data point in the input vector is obtained;
formula (12) is a training set of gravitational field data obtained by the polyhedral method.
4. The gaussian process regression GPR based small celestial body gravitational field modeling method of claim 3, wherein: the step 2 is realized by the following specific method:
training set D { (X)i,Yi) I | (1, 2 …, n } ═ X, Y), input vector
Figure FDA00016666056400000213
Namely the spherical coordinate data in the step 1; output scalar YiIs the gravitational acceleration g at the ith point; the regression task is to learn according to a given training set to obtain an output scalar YiAnd the input vector XiThe mapping relation between the test points is finally determined according to the test point X*Calculating the output value Y with the maximum possibility*；
When a regression GPR model of the Gaussian process is established by using the training set obtained in the step 1, observed values Y of n training functionsi∈R，i＝1,2…n；
Given the training set, the noise ε -N (0, σ) is due to the noisy observations2) Then, there is an observed value:
Y＝f(X)+ε (13)
wherein X is an input vector, Y is an observed value containing noise, and f (X) is a function value; (x) the gaussian process is given a priori, i.e.:
f(X)～GP(m(X),k(X,X')) (14)
wherein, for the sake of symbolic simplicity, the mean function m (x) is defined as 0, and the covariance function k has different choices;
the establishment of a regression GPR model of the Gaussian process is completed, and for the problem of the regression GPR of the Gaussian process, the target is to a new detection point X*Can obtain a corresponding Y*。
5. The Gaussian process regression GPR-based small celestial body gravitational field modeling method as defined in claim 4, wherein: the concrete implementation method of the step 3 is as follows:
due to test data (X)*,Y*) And training data (X, Y) are all from the same distribution, and training data (X, Y) and test data (X) are obtained*,Y*) The joint distribution of (A) is:
Figure FDA0001666605640000031
wherein m is a mean function, K is a covariance function, also called a kernel function;
Figure FDA0001666605640000032
by using the kernel function, the input-output relation of the regression GPR prediction model of the Gaussian process can be obtained, and Y can be obtained by combining the formula (15)*The predicted value of (2);
after the training set shown in formula (12) is obtained in step 1, a square exponential form (EQ) is selected as a kernel function form, a zero mean value is used as a mean function, and the expression is as follows:
Figure FDA0001666605640000033
after adding noise, k (X, X') is:
Figure FDA0001666605640000041
Y*the condition distribution of (1):
Figure FDA0001666605640000047
selecting the mean value of the distribution as Y*I.e.:
Figure FDA0001666605640000042
thus, Y is obtained*The parameter θ referred to in the kernel function k (X, X') shown in equation (18) [ < l >, σ >, is [ < l >, σ >f,σn]And l is a variance measure,
Figure FDA0001666605640000043
as signal variance, σnIs the variance of the noise; the variance scale l, the signal variance
Figure FDA0001666605640000044
Variance parameter sigma of noisenReferred to as hyper-parameters; the maximum likelihood method is used for solving the hyperparameters, firstly, a negative log-likelihood function of prior probability distribution is obtained, then, a partial derivative is solved for the likelihood function, and the minimum value of the likelihood function is solved through a conjugate gradient method, so that the optimal solution of the hyperparameters is obtained;
the likelihood function L (θ) is:
Figure FDA0001666605640000045
and (3) performing partial derivation on the likelihood function to obtain:
Figure FDA0001666605640000046
after the partial derivative of the likelihood function is obtained, the optimal hyper-parameter is obtained through a conjugate gradient method, and the obtained hyper-parameter is substituted into the formula (18) and the formula (20) to obtain the predicted value of the gravitational acceleration g, so that the gravitational field near the celestial body can be rapidly and accurately modeled and calculated through a Gaussian process regression GPR method, the calculated amount can be reduced, and the modeling speed can be improved.
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