Asteroid gravitational field reconstruction method based on embedded mathematical knowledge neural density field

Abstract

The invention relates to an asteroid gravitational field reconstruction method based on an embedded mathematical knowledge neural density field, and the method comprises the steps: completing the sampling of data points based on an asteroid mass point group model, and constructing a function solver from a spatial position to density; constructing a spatial sampling point function to obtain spatial densities of spatial points at different positions; aiming at the space points, utilizing an asteroid visual three-dimensional model to obtain star shell information, and screening the space points as prior information inside and outside a star shell; screening results of the density function solver according to prior information; further calculating a Laplace equation of the data points by using a partial differential equation solver, and calculating a gravitational acceleration equation of external space points of the star by using numerical integration; constructing a total loss function; error back propagation is carried out, and network parameters are optimized; recalculating the internal space point density of the star to obtain the internal density distribution of the asteroid; and further calculating the value of the gravitational acceleration of each position in the space to obtain asteroid external gravitational field distribution.

Description

Technical Field

[0001] The present invention relates to the field of asteroid gravitational field reconstruction, and in particular to an asteroid gravitational field reconstruction method based on an embedded mathematical knowledge neural density field. Background Art

[0002] Asteroids are indispensable celestial bodies in the solar system and are of great significance to our understanding of the formation and evolution of the solar system. The gravitational field of an asteroid is not only closely related to its mass and shape, but also has a profound impact on key parameters such as orbital evolution and collision probability. If the gravitational field of an asteroid can be modeled with high precision, it will be possible to more reliably plan the missions of probes, rovers, and landers, including the design and stability analysis of orbits, hovering orbits, transfer orbits, controlled attachment, and ballistic attachment orbits, as well as the robust planning of surface patrol paths.

[0003] Traditional gravitational field reconstruction techniques are based on physics, and mainly include spherical harmonics, polyhedron model, and particle group model. The spherical harmonics method constitutes a set of completely orthogonal function bases, and continuous functions defined on the sphere can be written as linear combinations of spherical harmonics, but there are limitations on the Brillouin surface, and it does not have global convergence, so it cannot be applied to the calculation of asteroid gravitational fields; the polyhedron model method transforms the calculation of the gravitational field of a small celestial body into the calculation of the geometric relationship between the spatial point and each edge and triangular face of the polyhedron, but it is based on the uniform mass assumption and does not conform to the actual situation; the particle group model rule regards asteroids as composed of many particles of different masses, and calculates the overall gravitational field through numerical integration, fully considering the non-uniformity of the mass distribution of asteroids, but this type of calculation has a high time cost and a huge amount of calculation.

[0004] With the rapid development of deep learning technology, data-driven neural network methods have been applied to the calculation of gravitational fields. This method can directly fit the gravitational potential function and has the advantages of high flexibility, fast calculation speed and low memory usage. Although significant progress has been made in gravitational field reconstruction technology, there are still many challenges in the reconstruction of asteroid gravitational fields. The reconstruction of gravitational fields becomes complicated due to the irregular shapes, uneven mass distribution and scarce available observational data of most asteroids. Therefore, further improvement and optimization of gravitational field reconstruction technology still has important research value.

[0005] Regarding the research on gravitational field reconstruction methods for asteroids, the shortcomings of existing technologies are mainly reflected in the following aspects: (1) Asteroid gravitational field data are scarce and uncertain, and it is difficult for the network to learn accurate gravitational field representation; (2) Many reconstruction methods make simplifying assumptions in the initial model construction process, which affects the reconstruction accuracy; (3) Existing methods have difficulty in dealing with the complex numerical calculations of asteroid gravitational field reconstruction. Summary of the invention

[0006] The technical problem solved by the present invention is: to overcome the shortcomings of the existing technology and provide a method for reconstructing the gravitational field of an asteroid based on a neural density field with embedded mathematical knowledge. On the basis of limited gravitational field data, it combines physical knowledge, asteroid visual three-dimensional model and deep learning model to improve the training accuracy of the model and realize high-precision reconstruction of the asteroid gravitational field.

[0007] The technical solution of the present invention is a method for reconstructing the gravitational field of an asteroid based on a neural density field with embedded mathematical knowledge, comprising the following steps: Step (1) Sampling data points based on the asteroid particle group model. The particle group model uses a finite number of particles or homogeneous spheres to fill the internal space of a small celestial body, and the gravitational potential outside the asteroid is expressed as the superposition of the gravitational potential generated by all particles. These data points will be assigned a basic density value, and the sum of the assigned basic density values ​​is 1, and a density function solver based on the MLP network is constructed; Step (2) Generate spatial target points by sampling in the hypercube space of [-1,1]³ based on radius sampling. These target points will be used for subsequent gravitational acceleration calculations. Step (3) introduces the celestial shell information based on the visual three-dimensional model of the asteroid, divides the hypercube space into a uniform grid structure according to the preset resolution, and uses a ray casting algorithm to determine whether each grid point is inside or outside the celestial body, where the density of the external grid points will be set to 0; at the same time, for the target points generated in step (2), the same determination is made as to whether they are inside or outside the celestial body, where the internal target points will be eliminated for subsequent gravitational acceleration calculations; Step (4) Use the MLP-based density function solver to solve the density of the spatial grid points, and then use the numerical integration method in the entire hypercube space to solve the gravitational acceleration and gravitational potential of the target point; Step (5) for the gravitational potential of the target point calculated in step (4), using a partial differential equation solver based on an automatic differentiation module to calculate the physical partial differential equations of the target point, including the Laplace equation and the gravitational acceleration equation; Step (6) for the partial differential equation calculation results of step (5), the two physical partial differential equation losses are combined with the mean absolute error of gravitational acceleration to construct a total loss function; the loss function is used for error back propagation and optimization of network parameters; Step (7) evaluates the prediction results of the asteroid particle swarm model optimized in step (6) using sampling points at different heights from the surface of the asteroid body, and displays the absolute error and relative error of the prediction results at different heights.

[0008] The advantages of the present invention compared with the prior art are: (1) The present invention adopts a multi-sensor fusion method, using the three-dimensional reconstruction model obtained from the asteroid image to perform contour constraints and the gravitational acceleration obtained by the acceleration sensor to reconstruct the gravitational field, thereby improving the reconstruction accuracy of the asteroid's gravitational field.

[0009] (2) The present invention adopts a neural network strategy with embedded physical knowledge, which strengthens the constraints of the model in the training process to fit the actual physical equations and improves the reconstruction accuracy of the asteroid gravitational field.

[0010] (3) The present invention adopts a neural density field strategy to realize the mathematical mapping from spatial position to density, which can express the suddenly disappearing density part and discontinuous density distribution structure, and improve the accuracy of asteroid gravitational field reconstruction.

[0011] In summary, the method adopted by the present invention is simple in principle and can achieve the purpose of reconstructing the asteroid gravitational field and corresponding mass distribution with high precision. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] Figure 1 This is a flow chart of a method for reconstructing the gravitational field of an asteroid based on a neural density field with embedded mathematical knowledge. DETAILED DESCRIPTION

[0013] In order to make the purpose, technical scheme and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention. In addition, the technical features involved in each embodiment of the present invention described below can be combined with each other as long as they do not conflict with each other. To achieve the above-mentioned purpose, the present invention adopts the following technical scheme.

[0014] like Figure 1 As shown, the present invention is a method for reconstructing the asteroid gravitational field based on the neural density field with embedded mathematical knowledge, and the specific implementation steps are as follows: Step (1) Sampling data points based on the asteroid particle group model; the particle group model uses a finite number of particles or homogeneous spheres to fill the internal space of the asteroid, and the gravitational potential outside the asteroid is expressed as the superposition of the gravitational potentials generated by all particles; these data points will be assigned a basic density value, and the sum of the assigned basic density values ​​is 1, and a density function solver based on the MLP network is constructed; Step (2) Generate spatial target points by sampling in the hypercube space of [-1,1]³ based on radius sampling. These target points will be used for subsequent gravitational acceleration calculations. Step (3) introduces the celestial shell information based on the visual three-dimensional model of the asteroid, divides the hypercube space into a uniform grid structure according to the preset resolution, and uses a ray casting algorithm to determine whether each grid point is inside or outside the celestial body, where the density of the external grid points will be set to 0; at the same time, for the target points generated in step (2), the same determination is made as to whether they are inside or outside the celestial body, where the internal target points will be eliminated for subsequent gravitational acceleration calculations; Step (4) Use the MLP-based density function solver to solve the density of the spatial grid points, and then use the numerical integration method in the entire hypercube space to solve the gravitational acceleration and gravitational potential of the target point; Step (5) for the gravitational potential of the target point calculated in step (4), using a partial differential equation solver based on an automatic differentiation module to calculate the physical partial differential equations of the target point, including the Laplace equation and the gravitational acceleration equation; Step (6) for the partial differential equation calculation results of step (5), the two physical partial differential equation losses are combined with the mean absolute error of gravitational acceleration to construct a total loss function; the loss function is used for error back propagation and optimization of network parameters; Step (7) evaluates the prediction results of the asteroid particle swarm model optimized in step (6) using sampling points at different heights from the surface of the asteroid body, and displays the absolute error and relative error of the prediction results at different heights.

[0015] Furthermore, step (1) is specifically as follows: based on the asteroid particle group model, the gravitational potential function and gravitational acceleration of the data point are calculated using the particle's universal gravitation formula and numerical integration: , , in, is the gravitational potential function at the data point, is the gravitational acceleration at the data point, is the gravitational constant, Indicates The mass of a particle, Represents the position vector of the data point relative to the coordinate origin, Indicates the data point relative to The position vector of a particle, and Position vector and The module length; A multi-layer, multi-node MLP network is built as a density function solver. MLP, or multi-layer perceptron, is a feedforward neural network that accepts a certain number of input features, relies on a fully connected multi-neuron hidden layer to extract features and finally outputs the desired results. The MLP network used in this method enables the asteroid mass group model to achieve function fitting from spatial position to spatial point density.

[0016] Further, step (2) is as follows: first, the entire sampling space is in Normalized in three dimensions to Hypercube space: , in, is the minimum value in the data set, is the maximum value in the data set. are the normalized spatial point coordinates, and the same operation is performed on the y-axis and z-axis: , , , is the minimum value in the data set, , is the maximum value in the data set. , is the normalized spatial point coordinate; After the sampling space is normalized, the normalized position coordinates are spherically sampled. The spherical sampling process first generates two random angles theta and phi. Theta is the angle around the z-axis and the range is , phi is the angle from the z-axis to the point, and the phi angle is generated using an average random The value of the arc cosine function is used to calculate the angle to ensure that the target points are evenly distributed on the expected sphere; then the radius r is calculated for the angle of the data points randomly generated in the previous step. The value of the radius is generated according to the given upper and lower bounds, and then a proportional factor minimal_radius_scale is calculated. The value of the proportional factor is in , used to indicate the ratio of the minimum radius to the maximum radius, and then generate a random number uni between [minimal_radius_scale, 1] with the same number of target points for subsequent radius calculation. The final radius value formula is as follows: , in, is the upper bound of the spherical sampling radius, is the radius of the final target point. By performing a cube root transformation on uni, the generated radius can be more evenly distributed in three-dimensional space. This is because in three-dimensional space, the volume is proportional to the cube of the radius; Based on the above operations, the spherical coordinate data of the spatial target point are obtained, and then they are converted into Cartesian coordinates: , in, is the angle around the z-axis, is the angle from the z-axis to the point, is the radius of the target point, are the Cartesian coordinates of the sampling points.

[0017] Furthermore, step (3) is specifically as follows: firstly, using the existing asteroid visual 3D model file, introducing it as the celestial body boundary, and obtaining the surface, edge, and fixed point data of the celestial body shell belonging to a polyhedron; Then, based on the defined grid resolution, the hypercube space is evenly divided into the desired grid size. In order to obtain the information of any point in the hypercube space, the Moller-Trumbore algorithm is first used to calculate whether these grid points are inside or outside the star shell according to the density of the grid points. The Moller-Trumbore algorithm is an efficient algorithm for calculating the intersection of triangles and rays. Its basic idea is to determine whether a ray intersects a triangle through linear algebra methods; In this method, the grid points are defined as the starting points of the rays, and the directions of the rays are all in the z-axis direction. The number of intersections between the rays emitted by each grid point and the outer shell of the star is calculated to determine whether the grid point is inside or outside the star. If the number of intersections is an even number, it is outside, otherwise it is inside. The result is saved in Boolean form as a mask. In this mask, the density of the external points will be set to 0, and the density of the internal points will be calculated normally through the MLP network, but the internal points will not be used in the subsequent gravitational acceleration calculation process.

[0018] Furthermore, step (4) is specifically as follows: the density value of the grid points in the hypercube space is calculated by the MLP network, wherein the density of the grid points in the outer space of the star body based on step (3) is set to 0, and then the target point obtained based on step (2) is traversed, the position of the target point is recorded using the index, the distance between the target point and all data points is calculated, and the gravitational acceleration is calculated according to the universal gravitation formula: , in, is the density of the corresponding data point, is the vector from the target point to the data point, is the gravitational acceleration. Then, using the numerical integration method, Integrate in three directions, use the trapezoidal rule, calculate the average value of adjacent points and multiply by half of the grid spacing to get the integral value in three directions, and finally get the gravitational acceleration value of the target point.

[0019] Furthermore, step (5) is as follows: first, a partial differential equation solver is constructed through the automatic differentiation module. The automatic differentiation module in this method comes from the autograd function in pytorch, which uses a dynamic calculation graph to automatically calculate the gradient of the tensor and uses the partial differential equation solver to calculate the gravitational acceleration equation of the target point: , in, is the three-dimensional position coordinate of the target point, is the first-order gradient of the gravitational potential function with respect to the spatial position, is the gravitational acceleration at the target point; the Laplace equation at the target point is then calculated using the partial differential equation solver: , in, is the three-dimensional position coordinate of the target point, The second-order gradient of the gravitational potential function with respect to the spatial position; Furthermore, step (6) is specifically as follows: construct a main loss function based on the gravitational acceleration of the target point calculated in step (4), and the main loss function adopts the MAE loss function with a mass normalization factor: , in, is the MAE loss of gravitational acceleration, is the total number of target points, is the predicted value of gravitational acceleration, is the true value of gravitational acceleration, is the mass normalization factor. The mass normalization factor provides a free parameter for the main loss function, allowing the subsequent error back propagation to focus on learning the deviation of uniform filling instead of focusing only on the absolute value of the mass density.

[0020] The sub-loss function is constructed based on the two physical partial differential equations of the gravitational field calculated in step (5), and the gravitational potential function loss and Laplace equation loss are introduced in the data point part: , , in, is the gravitational potential function loss and the Laplace equation loss, are network parameters, is the total number of target points, For the The spatial position coordinates of the target points relative to the coordinate origin, is the gravitational potential calculated for the network, is the real gravitational potential, The second-order gradient of the gravitational potential predicted by the network with respect to the data point location; The total loss function is formed by combining the main loss function and the secondary loss function: , in, is the weight of each loss term, is the total loss function.

[0021] Furthermore, step (7) is specifically as follows: after optimizing the asteroid particle group model for a certain number of times based on step (6), height sampling is performed at three different heights from the surface of the asteroid. The height sampling is performed based on the outer shell of the stellar body obtained by the three-dimensional reconstruction result, and the space points at a certain height from the outer shell are calculated. The gravitational acceleration of the space points at different heights is predicted, and the absolute error and relative error are obtained by comparing with the true value of the gravitational acceleration.

[0022] It can be seen that the present invention can improve the reconstruction accuracy of the asteroid gravitational field by using the asteroid visual three-dimensional model, the embedded physical knowledge neural network, and the neural density field strategy.

[0023] Contents not described in detail in the specification of the present invention belong to the common knowledge of professionals in the field. Although the above illustrative specific embodiments of the present invention are described to facilitate the understanding of the present invention by those skilled in the art, it should be clear that the present invention is not limited to the scope of the specific embodiments. For those of ordinary skill in the art, as long as various changes are within the spirit and scope of the present invention defined and determined by the attached claims, these changes are obvious, and all inventions and creations using the concept of the present invention are protected.

Claims

1. A method for reconstructing the gravitational field of an asteroid based on a neural density field with embedded mathematical knowledge, characterized in that: The following steps are involved: Step (1) sampling data points based on the asteroid mass group model; The particle group model uses a finite number of particles or homogeneous spheres to fill the internal space of a small celestial body, and the gravitational potential outside the asteroid is represented as the superposition of the gravitational potential generated by all particles; these data points will be assigned a basic density value, and the sum of the assigned basic density values ​​is 1, and a density function solver based on the MLP network is constructed; Step (2) Generate spatial target points by sampling in the hypercube space of [-1,1]³ based on radius sampling. These target points will be used for subsequent gravitational acceleration calculations. Step (3) introduces the celestial shell information based on the visual three-dimensional model of the asteroid, divides the hypercube space into a uniform grid structure according to the preset resolution, and uses a ray casting algorithm to determine whether each grid point is inside or outside the celestial body, where the density of the external grid points will be set to 0; at the same time, for the target points generated in step (2), the same determination is made as to whether they are inside or outside the celestial body, where the internal target points will be eliminated for subsequent gravitational acceleration calculations; Step (4) Use the density function solver based on the MLP network to solve the density of the space grid, and then use the numerical integration method in the entire hypercube space to solve the gravitational acceleration and gravitational potential of the target point; Step (5) for the gravitational potential of the target point calculated in step (4), using a partial differential equation solver based on an automatic differentiation module to calculate the physical partial differential equations of the target point, including the Laplace equation and the gravitational acceleration equation; Step (6) for the partial differential equation calculation results of step (5), the two physical partial differential equation losses are combined with the mean absolute error of gravitational acceleration to construct a total loss function; the loss function is used for error back propagation and optimization of network parameters; Step (7) evaluates the prediction results of the asteroid particle swarm model optimized in step (6) using sampling points at different heights from the surface of the asteroid body, and displays the absolute error and relative error of the prediction results at different heights.

2. The method for reconstructing the asteroid gravitational field based on the neural density field with embedded mathematical knowledge according to claim 1, characterized in that: The specific implementation method of step (1) is as follows: Based on the asteroid particle group model, the gravitational potential function and gravitational acceleration of the data points are calculated using the particle's universal gravitation formula and numerical integration: ， ， in, is the gravitational potential function at the data point, is the gravitational acceleration at the data point, is the gravitational constant, Indicates The mass of a particle, Represents the position vector of the data point relative to the coordinate origin, Indicates the data point relative to The position vector of a particle, and Position vector and The module length; A multi-layer and multi-node MLP network is built as a density function solver. MLP, or multi-layer perceptron, is a feedforward neural network that accepts a certain number of input features, relies on a fully connected multi-neuron hidden layer to extract features and finally outputs the required results. The density function solver enables the asteroid particle group model to realize function fitting from spatial position to spatial point density.

3. The method for reconstructing the asteroid gravitational field based on the neural density field with embedded mathematical knowledge according to claim 2, characterized in that: The specific implementation method of step (2) is as follows: First, the entire sampling space is Normalized in three dimensions to Hypercube space: ， in, is the minimum value in the data set, is the maximum value in the data set. are the normalized spatial point coordinates, and the same operation is performed on the y-axis and z-axis: ， ， , is the minimum value in the data set, , is the maximum value in the data set. , is the normalized spatial point coordinate; After the sampling space is normalized, the normalized position coordinates are spherically sampled. The spherical sampling process first generates two random angles theta and phi. Theta is the angle around the z-axis and the range is , phi is the angle from the z-axis to the point, and the phi angle is generated using an average random The value of the arc cosine function is used to calculate the angle to ensure that the target points are evenly distributed on the expected sphere; then the radius r is calculated for the angle of the data points randomly generated in the previous step. The value of the radius is generated according to the given upper and lower bounds, and then a proportional factor minimal_radius_scale is calculated. The value of the proportional factor is in , used to indicate the ratio of the minimum radius to the maximum radius, and then generate a random number uni between [minimal_radius_scale, 1] with the same number of target points for subsequent radius calculation. The final radius value formula is as follows: ， in, is the upper bound of the spherical sampling radius, is the radius of the final target point; by performing a cube root transformation on uni, the generated radius is distributed more evenly in the three-dimensional space; Based on the above operations, the spherical coordinate data of the spatial target point are obtained, and then they are converted into Cartesian coordinates: ， in, is the angle around the z-axis, is the angle from the z-axis to the point, is the radius of the target point, are the Cartesian coordinates of the target point.

4. The method for reconstructing the asteroid gravitational field based on the neural density field with embedded mathematical knowledge according to claim 3 is characterized by: The specific implementation method of step (3) is as follows: Firstly, the visual 3D model of the asteroid is used to introduce it as the boundary of the celestial body, and the surface, edge and fixed point data of the celestial body shell belonging to the polyhedron are obtained; Then, based on the defined grid resolution, the hypercube space is evenly divided into the desired grid size. In order to obtain the information of any point in the hypercube space, the Moller-Trumbore algorithm is first used to calculate whether these grid points are inside or outside the star shell according to the density of the grid points. The Moller-Trumbore algorithm uses linear algebra to determine whether the ray intersects with the triangle; the grid points are defined as the starting points of the rays, and the directions of the rays are all in the z-axis direction. The number of intersections between the rays emitted by each grid point and the star shell is calculated to determine whether the grid point is inside or outside the star. If the number of intersections is an even number, it is outside, otherwise it is odd, it is inside. The result is saved in Boolean form as a mask. In this mask, the density of the external points will be set to 0, and the density of the internal points will be calculated normally through the density function solver based on the MLP network, but the internal points will not be used in the subsequent gravitational acceleration calculation process.

5. The method for reconstructing the asteroid gravitational field based on the neural density field with embedded mathematical knowledge according to claim 4 is characterized by: The specific implementation method of step (4) is as follows: The density value of the grid points in the hypercube space is calculated by the density function solver based on the MLP network, where the density of the grid points outside the star space based on step (3) is set to 0. Then, the target point obtained based on step (2) is traversed, the position of the target point is recorded using the index, the distance between the target point and all data points is calculated, and the gravitational acceleration is calculated according to the universal gravitation formula: ， in is the density of the corresponding data point, is the vector from the target point to the data point, is the gravitational acceleration, and then using the numerical integration method, in Integrate in three directions, use the trapezoidal rule, calculate the average value of adjacent points and multiply by half of the grid spacing to get the integral value in three directions, and finally get the gravitational acceleration value of the target point.

6. The method for reconstructing the asteroid gravitational field based on the neural density field with embedded mathematical knowledge according to claim 5, characterized in that: The specific implementation method of step (5) is as follows: First, build a partial differential equation solver through the automatic differentiation module, and use the partial differential equation solver to calculate the gravitational acceleration equation of the target point: ， in, is the three-dimensional position coordinate of the target point, is the first-order gradient of the gravitational potential function with respect to the spatial position, is the gravitational acceleration at the target point; the Laplace equation at the target point is then calculated using the partial differential equation solver: ， in, is the three-dimensional position coordinate of the target point, The second-order gradient of the gravitational potential function with respect to the spatial position.

7. The method for reconstructing the asteroid gravitational field based on the neural density field with embedded mathematical knowledge according to claim 6, characterized in that: The specific implementation method of step (6) is as follows: The main loss function is constructed based on the gravitational acceleration of the target point calculated in step (4). The main loss function uses the MAE loss function with a mass normalization factor: ， in, is the MAE loss of gravitational acceleration, is the total number of target points, is the predicted value of gravitational acceleration, is the true value of gravitational acceleration, is the mass normalization factor; the mass normalization factor provides a free parameter for the main loss function, allowing subsequent error back propagation to focus on learning the deviation of uniform filling instead of focusing only on the absolute value of the particle density; Based on the two physical partial differential equations of the gravitational field calculated in step (5), the secondary loss function is constructed, and the gravitational potential function loss and Laplace equation loss are introduced into the data point part: ， ， in, is the gravitational potential function loss and the Laplace equation loss, are network parameters, is the total number of target points, For the The spatial position coordinates of the target points relative to the coordinate origin, is the gravitational potential calculated for the network, is the real gravitational potential, The second-order gradient of the gravitational potential predicted by the network with respect to the data point location; The total loss function is formed by combining the main loss function and the secondary loss function: ， in, is the weight of each loss term, is the total loss function.

8. The method for reconstructing the asteroid gravitational field based on the neural density field with embedded mathematical knowledge according to claim 7 is characterized by: The specific implementation method of step (7) is as follows: After optimizing the asteroid particle group model for a certain number of rounds based on step (6), height sampling is performed at three different heights from the asteroid surface. The height sampling is based on the outer shell of the stellar body obtained by the three-dimensional reconstruction result, and the spatial points at a certain height from the outer shell are calculated. The gravitational acceleration of the spatial points at different heights is predicted and compared with the true value of the gravitational acceleration to obtain the absolute error and relative error.

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