A method for reconstructing the gravitational field of an asteroid based on neural density fields embedded with mathematical knowledge

Abstract

The present invention relates to a method for reconstructing an asteroid gravitational field based on a neural density field with embedded mathematical knowledge, comprising: completing data point sampling based on an asteroid particle group model, constructing a function solver for converting spatial position to density; constructing a spatial sampling point function to obtain spatial density of spatial points at different positions; for spatial points, using an asteroid visual three-dimensional model to obtain celestial shell information, and screening spatial points inside and outside the celestial shell as prior information; screening the results of the density function solver based on the prior information; further using a partial differential equation solver to calculate the Laplace equation of the data point and using numerical integration to calculate the gravitational acceleration equation of the spatial point outside the celestial body; constructing a total loss function; performing error back propagation and optimizing network parameters at the same time; recalculating the density of spatial points inside the celestial body to obtain the internal density distribution of the asteroid; and further calculating the value of the gravitational acceleration at each spatial position to obtain the external gravitational field distribution of the asteroid.

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 essential celestial bodies in the solar system, crucial for our understanding of its formation and evolution. An asteroid's gravitational field is not only closely related to its mass and shape, but also profoundly influences key parameters such as orbital evolution and collision probability. High-precision modeling of an asteroid's gravitational field will enable more reliable planning of probe, rover, and lander missions. This includes the design and stability analysis of orbits, hovering orbits, transfer orbits, controlled attachment, and ballistic attachment orbits, as well as robust planning of surface patrol paths.

[0003] Traditional gravitational field reconstruction techniques are based on physics, primarily including spherical harmonics, polyhedron models, and particle group models. The spherical harmonics method forms a completely orthogonal function basis, allowing any continuous function defined on a sphere to be expressed as a linear combination of spherical harmonics. However, due to the limitations of Brillouin surfaces, it lacks global convergence and is therefore unsuitable for calculating asteroid gravitational fields. The polyhedron model method transforms the calculation of a small body's gravitational field into a calculation of the geometric relationship between a point in space and each edge and triangular face of a polyhedron. However, this method assumes uniform mass and is inconsistent with actual conditions. The particle group model treats asteroids as composed of numerous point masses of varying masses, calculating the overall gravitational field through numerical integration, fully accounting for the non-uniformity of the asteroid's mass distribution. However, this type of calculation is time-consuming and computationally intensive.

[0004] With the rapid development of deep learning technology, data-driven neural network methods have been applied to gravitational field calculations. This method can directly fit the gravitational potential function and offers advantages such as high flexibility, fast computational speed, and low memory usage. Despite significant progress in gravitational field reconstruction, many challenges remain in reconstructing the gravitational fields of asteroids. The irregular shapes, uneven mass distribution, and scarcity of available observational data complicate gravitational field reconstruction for most asteroids. Therefore, further improvement and optimization of gravitational field reconstruction techniques remain of great 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 an 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 difficulties 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 asteroid gravitational field based on an embedded mathematical knowledge neural density field. On the basis of limited gravitational field data, it combines physical knowledge, asteroid visual three-dimensional models and deep learning models to improve the training accuracy of the model and achieve high-precision reconstruction of the asteroid gravitational field.

[0007] The technical solution of the present invention is a method for reconstructing the asteroid gravitational field based on an embedded mathematical knowledge neural density field, comprising the following steps:

[0008] Step (1) Sampling data points based on the asteroid particle swarm model. The particle swarm model uses a finite number of particles or homogeneous spheres to fill the interior space of a small celestial body. The gravitational potential outside the asteroid is represented as the superposition of the gravitational potentials generated by all particles. These data points are assigned a basic density value, and the sum of the assigned basic density values ​​is 1. A density function solver based on the MLP network is constructed.

[0009] Step (2) Generate spatial target points by sampling in the [-1,1]³ hypercube space based on radius sampling. These target points will be used for subsequent gravitational acceleration calculations.

[0010] Step (3) introduces the celestial shell information based on the asteroid visual three-dimensional model, divides the hypercube space into a uniform grid structure according to the preset resolution, and uses the 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 judgment 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;

[0011] Step (4) Use the MLP-based density function solver to solve the density of the spatial 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;

[0012] Step (5) for the target point gravitational potential calculated in step (4), use a partial differential equation solver based on the automatic differentiation module to calculate the physical partial differential equations of the target point, including the Laplace equation and the gravitational acceleration equation;

[0013] Step (6) Based on 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 backpropagation and optimization of network parameters;

[0014] Step (7) evaluates the prediction results of the asteroid mass swarm model optimized in step (6) using sampling points at different heights from the asteroid surface, and displays the absolute and relative errors of the prediction results at different heights.

[0015] The advantages of the present invention compared with the prior art are:

[0016] (1) The present invention adopts a multi-sensor fusion method, using the three-dimensional reconstruction model obtained from the asteroid image for contour constraint 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.

[0017] (2) The present invention adopts a neural network strategy with embedded physical knowledge, which strengthens the constraints of the model in conforming to the actual physical equations during the training process and improves the reconstruction accuracy of the asteroid gravitational field.

[0018] (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.

[0019] 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

[0020] Figure 1 This is a flow chart of the asteroid gravitational field reconstruction method based on the neural density field with embedded mathematical knowledge. DETAILED DESCRIPTION

[0021] In order to make the objectives, technical solutions, and advantages of the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and examples. It should be understood that the specific embodiments described herein are intended only to illustrate the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other. To achieve the above-mentioned objectives, the present invention adopts the following technical solutions.

[0022] like Figure 1 As shown in FIG, the present invention is a method for reconstructing the asteroid gravitational field based on an embedded mathematical knowledge neural density field, and the specific implementation steps are as follows:

[0023] 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;

[0024] Step (2) Generate spatial target points by sampling in the [-1,1]³ hypercube space based on radius sampling. These target points will be used for subsequent gravitational acceleration calculations.

[0025] Step (3) introduces the celestial shell information based on the asteroid visual three-dimensional model, divides the hypercube space into a uniform grid structure according to the preset resolution, and uses the 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 judgment 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;

[0026] Step (4) Use the MLP-based density function solver to solve the density of the spatial 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;

[0027] Step (5) for the target point gravitational potential calculated in step (4), use a partial differential equation solver based on the automatic differentiation module to calculate the physical partial differential equations of the target point, including the Laplace equation and the gravitational acceleration equation;

[0028] Step (6) Based on 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 backpropagation and optimization of network parameters;

[0029] Step (7) evaluates the prediction results of the asteroid mass swarm model optimized in step (6) using sampling points at different heights from the asteroid surface, and displays the absolute and relative errors of the prediction results at different heights.

[0030] Furthermore, step (1) is specifically as follows: based on the asteroid mass group model, the gravitational potential function and gravitational acceleration of the data point are calculated using the mass point universal gravitation formula and numerical integration:

[0031] ,

[0032] ,

[0033] in, is the gravitational potential function at the data point, is the gravitational acceleration at the data point, represents the gravitational constant, Indicates the 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 Length of the module;

[0034] A multi-layer, multi-node MLP network is constructed as a density function solver. An MLP, or multi-layer perceptron, is a feedforward neural network that accepts a certain number of input features, extracts features from a fully connected hidden layer of multiple neurons, and ultimately outputs the desired result. The MLP network used in this method enables the asteroid point swarm model to implement a function fitting from spatial position to spatial point density.

[0035] Furthermore, step (2) is as follows: first, the entire sampling space is Normalized in three dimensions to Hypercube space:

[0036] ,

[0037] in, is the minimum value in the data set, is the maximum value in the data set, is the normalized spatial point coordinate, and the same operation is performed on the y-axis and z-axis:

[0038] ,

[0039] ,

[0040] 、 is the minimum value in the data set, 、 is the maximum value in the data set, 、 is the normalized spatial point coordinate;

[0041] 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, for the phi angle, use the average random generation The value of , and then calculate the angle by the arc cosine function to ensure that the target points are evenly distributed on the expected sphere; then calculate the radius r for the angle of the data point that has been randomly generated in the previous step. The value of the radius is generated according to the given upper and lower bounds, and then calculate a proportional factor minimal_radius_scale. 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:

[0042] ,

[0043] 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;

[0044] Based on the above operations, the spherical coordinate data of the spatial target point is obtained, and then they are converted into Cartesian coordinates:

[0045] ,

[0046] 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.

[0047] Furthermore, step (3) is specifically as follows: first, 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 the polyhedron;

[0048] Then, based on the defined grid resolution, the hypercube space is evenly divided into the desired grid size. To obtain information about 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's shell based on 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 using linear algebra methods.

[0049] In this method, grid points are defined as the starting points of rays, and the directions of the rays are all along the z-axis. 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 even, it is outside, otherwise it is inside. The result is saved as a mask in Boolean form. In this mask, the density of external points will be set to 0, and the density of 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.

[0050] Furthermore, step (4) is specifically as follows: the density value of the grid points in the hypercube space is calculated through the MLP network, where the density of the grid points in the outer space of the star 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 gravitational formula:

[0051] ,

[0052] 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 it 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.

[0053] Furthermore, step (5) is specifically 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 computational 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:

[0054] ,

[0055] 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 of the target point is then calculated by the partial differential equation solver:

[0056] ,

[0057] in, is the three-dimensional position coordinate of the target point, The second-order gradient of the gravitational potential function with respect to spatial position;

[0058] Furthermore, step (6) is specifically as follows: constructing a main loss function based on the target point gravitational acceleration calculated in step (4), and the main loss function adopts the MAE loss function with a mass normalization factor:

[0059] ,

[0060] 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.

[0061] Based on the two physical partial differential equations of the gravitational field calculated in step (5), the auxiliary loss function is constructed, and the gravitational potential function loss and Laplace equation loss are introduced into the data point part:

[0062] ,

[0063] ,

[0064] in, is the gravitational potential function loss and Laplace equation loss, are network parameters, is the total number of target points, For the The spatial position coordinates of the target point 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;

[0065] The total loss function is formed by combining the main loss function and the secondary loss function:

[0066] ,

[0067] in, is the weight of each loss item, is the total loss function.

[0068] Furthermore, step (7) is specifically as follows: after optimizing the asteroid point 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 performed based on the outer shell of the celestial 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 the absolute error and relative error are obtained by comparing with the true value of the gravitational acceleration.

[0069] It can be seen that the present invention can improve the accuracy of asteroid gravitational field reconstruction by using asteroid visual three-dimensional model, embedded physical knowledge neural network, and neural density field strategy.

[0070] Any matters not described in detail in the present specification are known to those skilled in the art. Although the above description of the present invention is based on illustrative embodiments to facilitate understanding of the present invention by those skilled in the art, it should be understood that the present invention is not limited to the scope of the specific embodiments. It should be apparent to those skilled in the art that various modifications within the spirit and scope of the present invention as defined and established by the appended claims are readily apparent, and all inventions utilizing the present invention are protected.

Claims

1. A method for reconstructing the asteroid gravitational field 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 swarm model uses a finite number of particles or homogeneous spheres to fill the interior space of a small celestial body. The gravitational potential outside the asteroid is represented as the superposition of the gravitational potentials generated by all particles. These data points are assigned a basic density value, and the sum of the assigned basic density values ​​is 1. A density function solver based on the MLP network is constructed. Step (2) Generate spatial target points by sampling in the [-1,1]³ hypercube space 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 asteroid visual three-dimensional model, divides the hypercube space into a uniform grid structure according to the preset resolution, and uses the 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 judgment 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 spatial 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 target point gravitational potential calculated in step (4), use a partial differential equation solver based on the 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) Based on 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 backpropagation and optimization of network parameters; Step (7) evaluates the prediction results of the asteroid mass swarm model optimized in step (6) using sampling points at different heights from the asteroid surface, and displays the absolute and relative errors of the prediction results at different heights.

2. The asteroid gravitational field reconstruction method based on the neural density field embedded with mathematical knowledge according to claim 1 is characterized by: The specific implementation method of step (1) is as follows: Based on the asteroid point mass model, the gravitational potential function and gravitational acceleration of the data points are calculated using the universal gravitational formula of the point mass and numerical integration: ， ， in, is the gravitational potential function at the data point, is the gravitational acceleration at the data point, represents the gravitational constant, Indicates the 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 Length of the module; 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 ultimately outputs the required results. The density function solver enables the asteroid particle swarm model to achieve function fitting from spatial position to spatial point density.

3. The asteroid gravitational field reconstruction method based on the neural density field embedded with mathematical knowledge according to claim 2 is characterized by: 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, is the normalized spatial point coordinate, 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, for the phi angle, use the average random generation The value of , and then calculate the angle by the arc cosine function to ensure that the target points are evenly distributed on the expected sphere; then calculate the radius r for the angle of the data point that has been randomly generated in the previous step. The value of the radius is generated according to the given upper and lower bounds, and then calculate a proportional factor minimal_radius_scale. 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 is 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 asteroid gravitational field reconstruction method 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: First, we use the asteroid visual 3D model and introduce it as the celestial boundary to obtain the surface, edge, and fixed point data of the celestial shell that belongs 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 based on the density of the grid points. The Moller-Trumbore algorithm uses linear algebra methods to determine whether the ray intersects with the triangle; the grid points are defined as the starting points of the rays, and the ray directions 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 even, it is outside, otherwise it is inside if it is odd. 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 asteroid gravitational field reconstruction method based on the neural density field embedded with mathematical knowledge according to claim 4 is characterized by: The specific implementation method of step (4) is as follows: The density function solver based on the MLP network is used to calculate the density value of the grid points in the hypercube space, where the density of the grid points in the outer space of the star 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 gravitational 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 it 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 asteroid gravitational field reconstruction method based on the neural density field embedded with mathematical knowledge according to claim 5 is characterized by: The specific implementation method of step (5) is as follows: First, we construct 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 of the target point is then calculated by 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 spatial position.

7. The method for reconstructing the asteroid gravitational field based on a neural density field embedded with 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 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 subsequent error backpropagation to focus on learning the deviation of uniform filling rather than just focusing 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 auxiliary 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 Laplace equation loss, are network parameters, is the total number of target points, For the The spatial position coordinates of the target point 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 item, is the total loss function.

8. The asteroid gravitational field reconstruction method based on the neural density field embedded with mathematical knowledge according to claim 7 is characterized by: The specific implementation method of step (7) is as follows: After optimizing the asteroid point 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 celestial 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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