An asteroid gravity field reconstruction method based on physical information neural network

Abstract

This invention relates to a method for reconstructing the gravitational field of an asteroid based on a physical information neural network, comprising the following steps: sampling data points and generating auxiliary points based on an asteroid mass swarm model, and constructing a gravitational potential function solver; preprocessing the data points and generated auxiliary points, and calculating the gravitational potential of each sampling point; calculating the Laplace equation and gravitational curl equation of the auxiliary points and the gravitational acceleration equation of the data points using a partial differential equation solver for the gravitational potential of the sampling points; constructing a total loss function based on the equation calculation results and the mean square error of the gravitational potential; performing backpropagation of the error for the loss function, and simultaneously optimizing the network parameters and the weights of each loss; evaluating the model prediction results for the optimized model, focusing on sampling in key areas with large prediction errors, and finally adding the sampling results to the auxiliary point dataset for further training.

Description

Technical Field

[0001] This invention relates to the field of asteroid gravitational field reconstruction, and more specifically to a method for asteroid gravitational field reconstruction based on a physical information neural network. Background Technology

[0002] Asteroids are among the most important celestial bodies in the solar system, playing a crucial role in 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 important parameters such as orbital evolution and collision probability. High-precision modeling of the gravitational fields of asteroids allows for more reliable design of probes, rovers, and landers, including the design and stability analysis of orbits such as orbital, hovering, transfer, controlled, and ballistic attachment orbits, as well as robust planning of surface exploration trajectories.

[0003] Traditional gravitational field reconstruction techniques are physics-driven and mainly include the spherical harmonic function method, the polyhedral model method, and the particle swarm model method. The spherical harmonic function method uses spherical harmonics to expand the gravitational potential function of an asteroid, providing an analytical solution. However, it is limited by the Brillouin surface and cannot be applied to asteroid gravitational field calculations. The polyhedral model method treats the celestial body as composed of many polyhedra and uses mathematical methods to calculate the overall gravitational field, fully considering the irregular three-dimensional shape of the asteroid. However, it is based on the assumption of uniform mass, which does not reflect reality. The particle swarm model method treats the celestial body as composed of a large number of different particles and uses numerical integration to calculate the overall gravitational field, fully considering the uneven mass distribution of the asteroid. However, the numerical integration computation is extremely intensive and time-consuming.

[0004] With the development of deep learning technology, data-driven neural network methods have emerged for calculating gravitational fields. These methods directly fit the gravitational potential function using neural networks, offering flexibility, speed, and low memory consumption. Despite significant progress in gravitational field reconstruction techniques, challenges remain in reconstructing asteroid gravitational fields. The irregular shapes, uneven mass distribution, and limited observational data of most asteroids contribute to the difficulty of gravitational field reconstruction. Therefore, improving and optimizing gravitational field reconstruction techniques remains crucial.

[0005] The shortcomings of existing technologies for gravitational field reconstruction methods for asteroids are mainly reflected in the following aspects: (1) Asteroid gravitational field data is sparse, making it difficult for the network to learn an accurate representation of the gravitational field; (2) The mass distribution of asteroids is uneven, and the network predicts the gravitational potential function inaccurately in areas with large mass variations; (3) Existing deep learning methods use gradient descent to optimize the model, but gradient descent has the problem of difficulty in multi-objective optimization. Summary of the Invention

[0006] The technical problem solved by this invention is to overcome the shortcomings of existing technologies and provide a method for reconstructing the gravitational field based on asteroids. Based on limited gravitational field data, this method combines physical knowledge and deep learning models to improve the training speed and inference efficiency of the model, thereby achieving high-precision reconstruction of the gravitational field of asteroids.

[0007] The technical solution of this invention is a method for reconstructing the gravitational field of an asteroid based on a physical information neural network, comprising the following steps:

[0008] Step (1) Based on the asteroid mass swarm model, data points are sampled and auxiliary points are generated. The data points and auxiliary points together constitute the sampling points, and a gravitational potential function solver based on the transformer-MLP network is constructed.

[0009] Step (2) For the data points sampled in step (1) and the generated auxiliary points, perform data normalization and 4D position encoding, and calculate the gravitational potential of each sampling point through the transformer-MLP network;

[0010] Step (3) For the gravitational potential of each sampling point calculated in step (2), use the partial differential equation solver based on the automatic differential module to calculate the physical partial differential equations of the auxiliary points, including the Laplace equation and the gravitational acceleration curl equation, as well as the physical partial differential equations of the data points, including the gravitational acceleration equation.

[0011] Step (4) Based on the calculation results of the three physical partial differential equations in step (3), the loss of the three physical partial differential equations is combined with the mean square error of the gravitational potential to construct the total loss function.

[0012] Step (5) Perform backpropagation of the error for the loss function constructed in step (4), and optimize the network parameters and each loss weight at the same time;

[0013] Step (6) evaluates the model prediction results using gravitational field data near the asteroid after optimization in (5), finds key areas with large prediction errors, samples key areas, and finally adds the sampling results to the auxiliary point dataset for continued training.

[0014] Furthermore, the specific implementation method of step (1) is as follows:

[0015] Based on the asteroid swarm model, the gravitational potential function and gravitational acceleration of the data points are calculated using the gravitational formula of the particles and numerical integration:

[0016]

[0017]

[0018] Where U is the gravitational potential function at the data point, a is the gravitational acceleration at the data point, G represents the gravitational constant, and m j This represents the mass of the j-th particle. This represents the position vector of a data point relative to the origin of the coordinate system. r represents the position vector of a data point relative to the j-th mass point. i and r ij Position vector and The modulus length;

[0019] Data points are sampled using the gravitational potential function formula and the gravitational acceleration formula. Then, auxiliary points are randomly and uniformly generated in the sampling space. A transformer-MLP network is built as the gravitational potential function solver, and combined with the partial differential equation solver, the PINNs model can calculate the gravitational acceleration at the sampling points.

[0020] Furthermore, the specific implementation method of step (2) is as follows:

[0021] First, the auxiliary points and data points obtained in step (2) are normalized using a data normalization method that considers physical relationships:

[0022]

[0023]

[0024]

[0025] Where x,U are the three-dimensional position coordinates and gravitational potential of the sampling point before normalization, x * U * ,a * These are the normalized position coordinates, gravitational potential, and gravitational acceleration. x0,U0 are the minimum values ​​of the position coordinate components and gravitational potential at the sampling point. s U s It represents the range of position coordinates and gravitational potentials across all sampling points. It is the first-order partial derivative of the gravitational potential function with respect to spatial position; this normalization method normalizes the sampled data to the [0,1] interval; after data normalization, the normalized position coordinates are then 4D position encoded:

[0026]

[0027]

[0028]

[0029]

[0030] Where x, y, z are the three-dimensional position coordinates before encoding, and r, s, t, u are the four-dimensional position coordinates after encoding; the encoded four-dimensional position coordinates are fed into transformer-MLP to regress the gravitational potential at the sampling point.

[0031] Furthermore, the specific implementation method of step (3) is as follows:

[0032] First, a partial differential equation solver is constructed using the automatic differentiation module. Then, the gravitational acceleration equation for the data points is calculated using the partial differential equation solver.

[0033]

[0034] Where R is the three-dimensional position coordinate of the data point. The first-order gradient of the gravitational potential function with respect to the spatial location is given by 'a', where 'a' is the gravitational acceleration at the data point. Then, the Laplace equation and the curl equation for the auxiliary point are calculated using a partial differential equation solver.

[0035]

[0036]

[0037] Where R is the three-dimensional position coordinate of the data point. It is the second-order gradient of the gravitational potential function with respect to spatial position. It is the curl of gravitational acceleration, a x ,a y ,a z It is the component of gravitational acceleration in three directions.

[0038] Furthermore, the specific implementation method of step (4) is as follows:

[0039] Based on the three physical partial differential equations of the gravitational field calculated in step (3), a loss function is constructed, incorporating the gravitational potential function loss and the gravitational acceleration equation loss for the data points:

[0040]

[0041]

[0042] Among them, L U L a R' represents the gravitational potential function loss and gravitational acceleration loss, Θ represents the network parameters, N represents the total number of data points, and R' represents the total number of data points. i Let i be the spatial coordinates of the i-th data point relative to the origin. Let U be the gravitational potential predicted by the network, and U be the actual gravitational potential obtained from sampling. Let be the first-order gradient of the gravitational potential predicted by the network with respect to the location of the data point, and let a be the gravitational acceleration of the sampled data point.

[0043] In the auxiliary point section, we introduce losses from the Laplace equation for the gravitational field and the curl equation for gravitational acceleration:

[0044]

[0045]

[0046] Among them, L PDE L a The losses are the Laplace equation loss and the gravitational acceleration curl equation loss, where Θ is the network parameter, N is the total number of auxiliary points, and R is the total number of auxiliary points. i Let be the spatial coordinates of the i-th auxiliary point relative to the origin. Let be the second-order gradient of the gravitational potential predicted by the network with respect to the location of the data point. The gravitational curl predicted by the network;

[0047] The total loss function is constructed by combining the data point loss term and the auxiliary point loss term:

[0048] L(Θ)=λ U L U (Θ)+λ a L a (Θ)+λ PDE L PDE (Θ)+λ curl L curl (Θ)

[0049] Where, λ U ,λ a ,λ PDE ,λ curl The weights of the four loss terms.

[0050] Furthermore, the specific implementation method of step (5) is as follows:

[0051] Based on the total loss function constructed in step (4), the sum of all weights in the total loss function is set to 1, and an adaptive weighting scheme is introduced to reconstruct the total loss function:

[0052]

[0053] Among them, L U L a L represents the gravitational potential function loss and gravitational acceleration loss for the data points. PDE L curlThe model is then subjected to Laplace equation loss and gravitational acceleration curl loss for auxiliary points. Subsequently, the model backpropagates based on the reconstructed total loss function, while simultaneously optimizing the network parameters and the weights of each loss term.

[0054] Furthermore, the specific implementation method of step (6) is as follows:

[0055] After optimizing the model for a certain number of rounds based on step (5), an adaptive sampling operation will be performed. 50 test areas will be randomly sampled within the gravitational field, and each area will consist of 20 Gaussian-distributed sampling points. Then, the average acceleration residuals in the 50 test areas will be used to find the area with the largest acceleration residual as the fault area. Finally, 200 additional sampling points will be sampled in the fault area according to the Gaussian distribution and added to the auxiliary point dataset to participate in the subsequent training of the network. Through adaptive sampling based on residuals, the model will focus on sampling in areas with low prediction accuracy during the training process.

[0056] The advantages of this invention compared to the prior art are:

[0057] (1) The present invention adopts a data sampling method that combines data points and auxiliary points. By introducing auxiliary points to calculate the loss of physical equations, additional physical constraints are introduced into the neural network, which reduces the model's demand for data and improves the model's accuracy.

[0058] (2) The present invention adopts an adaptive loss weight strategy, which realizes the dynamic adjustment of the weight of each loss term during the training process, thus alleviating the problem of multi-objective optimization in gradient descent.

[0059] (3) The present invention adopts an adaptive sampling strategy, which enables the model to be guided by the training residual during the training process, and to perform additional sampling in areas with large gravitational changes, thereby achieving key sampling of fault areas and improving the prediction accuracy of fault areas.

[0060] In summary, the method employed in this invention is simple in principle and can achieve the goal of rapidly and accurately reconstructing the gravitational field of an asteroid. Attached Figure Description

[0061] Figure 1 This is a flowchart of an asteroid gravitational field reconstruction method based on a physical information neural network according to the present invention. Detailed Implementation

[0062] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the protection scope of the present invention.

[0063] like Figure 1 As shown, the specific implementation steps of the asteroid gravitational field reconstruction method based on physical information neural network of the present invention are as follows:

[0064] Step 1: Based on the asteroid swarm model, calculate the gravitational potential function and gravitational acceleration of the data points using the gravitational formula for particles and numerical integration:

[0065]

[0066]

[0067] Where U is the gravitational potential function at the data point, a is the gravitational acceleration at the data point, G represents the gravitational constant, and m j This represents the mass of the j-th particle. This represents the position vector of a data point relative to the origin of the coordinate system. r represents the position vector of a data point relative to the j-th mass point. i and r ij Position vector and The length of the module.

[0068] Data points are sampled using the gravitational potential function formula and the gravitational acceleration formula. Then, auxiliary points are randomly and uniformly generated in the sampling space. These auxiliary points are used to calculate the physical equation loss of the gravitational field in subsequent calculations. No real labels are required, thus reducing the model's data requirements. Finally, a transformer-MLP network is built as the gravitational potential function solver, and combined with a partial differential equation (PDE) solver, the PINNs model can calculate the gravitational acceleration at the sampling points.

[0069] Step 2: First, normalize the data using a data normalization method that considers physical relationships:

[0070]

[0071]

[0072]

[0073] Where x,U are the three-dimensional position coordinates and gravitational potential of the sampling point before normalization, x * U * ,a * These are the normalized position coordinates, gravitational potential, and gravitational acceleration. x0,U0 are the minimum values ​​of the position coordinate components and gravitational potential at the sampling point. s U sIt represents the range of position coordinates and gravitational potentials across all sampling points. This is the first-order partial derivative of the gravitational potential function with respect to spatial location; this normalization method normalizes the sampled data to the [0,1] interval, facilitating neural network training. After data normalization, the normalized position coordinates are then coded in 4D:

[0074]

[0075]

[0076]

[0077]

[0078] Where x, y, z are the three-dimensional position coordinates before encoding, and r, s, t, u are the four-dimensional position coordinates after encoding; finally, the encoded four-dimensional position coordinates are fed into transformer-MLP to regress the gravitational potential at the sampling point.

[0079] Step 3: First, construct a partial differential equation solver using the automatic differentiation module, and then use the partial differential equation solver to calculate the gravitational acceleration equation for the data points:

[0080]

[0081] Where R is the three-dimensional position coordinate of the data point. The first-order gradient of the gravitational potential function with respect to spatial location is given, where 'a' is the gravitational acceleration at the data point. Then, the Laplace equation and the curl equation for the auxiliary point are calculated using a partial differential equation solver.

[0082]

[0083]

[0084] Where R is the three-dimensional position coordinate of the data point. It is the second-order gradient of the gravitational potential function with respect to spatial position. It is the curl of gravitational acceleration, a x ,a y ,a z It is the component of gravitational acceleration in three directions.

[0085] Step 4: Construct the loss function, introducing gravitational potential function loss and gravitational acceleration equation loss for the data points:

[0086]

[0087]

[0088] Among them, L U L a R' represents the gravitational potential function loss and gravitational acceleration loss, Θ represents the network parameters, N represents the total number of data points, and R' represents the total number of data points. i Let i be the spatial coordinates of the i-th data point relative to the origin. Let U be the gravitational potential predicted by the network, and U be the actual gravitational potential obtained from sampling. Let be the first-order gradient of the gravitational potential predicted by the network with respect to the location of the data point, and let a be the gravitational acceleration of the sampled data point.

[0089] In the auxiliary point section, we introduce losses from the Laplace equation for the gravitational field and the curl equation for gravitational acceleration:

[0090]

[0091]

[0092] Among them, L PDE L a The losses are the Laplace equation loss and the gravitational acceleration curl equation loss, where Θ is the network parameter, N is the total number of auxiliary points, and R is the total number of auxiliary points. i Let be the spatial coordinates of the i-th auxiliary point relative to the origin. Let be the second-order gradient of the gravitational potential predicted by the network with respect to the location of the data point. The gravitational curl predicted by the network;

[0093] The total loss function is constructed by combining the data point loss term and the auxiliary point loss term:

[0094] L(Θ)=λ U L U (Θ)+λ a L a (Θ)+λ PDE L PDE (Θ)+λ curl L curl (Θ)

[0095] Where, λ U ,λ a ,λ PDE ,λ curl The weights of the four loss terms.

[0096] Step 5: Reconstruct the total loss function, setting the sum of all weights in the total loss function to 1, and introduce an adaptive weighting scheme to reconstruct the total loss function:

[0097]

[0098] Among them, L U L aL represents the gravitational potential function loss and gravitational acceleration loss for the data points. PDE L curl The model employs Laplace's equation loss and gravitational curl loss for auxiliary points. Then, it backpropagates based on the reconstructed total loss function, simultaneously optimizing network parameters and the weights of each loss term. This adaptive loss weight strategy alleviates the multi-objective optimization difficulties of gradient descent and improves the model's training convergence speed.

[0099] Step 6: After a certain number of model optimization rounds, adaptive sampling will be performed. Fifty test regions will be randomly sampled within the gravitational field, each consisting of 20 Gaussian-distributed sampling points. The average acceleration residuals within these 50 regions will then be used to identify the region with the largest acceleration residual as the fault region. Finally, an additional 200 sampling points will be sampled from the fault region according to a Gaussian distribution and added to the auxiliary point dataset for subsequent network training. This residual-based adaptive sampling allows the model to focus on sampling regions with low prediction accuracy during training, mitigating the decrease in prediction accuracy caused by the uneven mass distribution of the asteroid.

[0100] Therefore, this invention can improve the training speed and prediction accuracy of physical information neural networks when training data is limited by introducing auxiliary points, adaptive weights, and adaptive sampling strategies for asteroid gravitational fields.

[0101] Contents not described in detail in this specification are common knowledge to those skilled in the art. Although illustrative specific embodiments of the invention have been described above to facilitate understanding by those skilled in the art, it should be understood that the invention is not limited to the scope of the specific embodiments. Various modifications will be readily apparent to those skilled in the art as long as they fall within the spirit and scope of the invention as defined and determined by the appended claims, and all inventions utilizing the inventive concept are protected.

Claims

1. A method for reconstructing the gravitational field of an asteroid based on a physical information neural network, characterized in that, Includes the following steps: Step (1) Sampling of data points and generation of auxiliary points based on the asteroid mass swarm model. The data points and auxiliary points together constitute the sampling points, and a gravitational potential function solver based on the transformer-MLP network is constructed. Step (2) For the data points sampled in step (1) and the generated auxiliary points, perform data normalization and 4D position encoding, and calculate the gravitational potential of each sampling point through the transformer-MLP network; Step (3) For the gravitational potential of each sampling point calculated in step (2), use the partial differential equation solver based on the automatic differential module to calculate the physical partial differential equations of the auxiliary points, including the Laplace equation and the gravitational acceleration curl equation, as well as the physical partial differential equations of the data points, including the gravitational acceleration equation. Step (4) Based on the calculation results of the three physical partial differential equations in step (3), the loss of the three physical partial differential equations is combined with the mean square error of the gravitational potential to construct the total loss function. Step (5) Perform backpropagation of the error for the loss function constructed in step (4), and optimize the network parameters and each loss weight at the same time; Step (6) For the model optimized in (5), use the gravitational field data near the asteroid to evaluate the model prediction results, find the key areas with large prediction errors, sample the key areas, and finally add the sampling results to the auxiliary point dataset to continue training. The specific implementation method of step (3) is as follows: First, a partial differential equation solver is constructed using the automatic differentiation module. Then, the gravitational acceleration equation for the data points is calculated using the partial differential equation solver. in, These are the three-dimensional position coordinates of the data points. It is the first-order gradient of the gravitational potential function with respect to spatial location. It is the gravitational acceleration at the data point location; then, the Laplace equation and the acceleration curl equation at the auxiliary point are calculated using a partial differential equation solver: in, These are the three-dimensional position coordinates of the data points. It is the second-order gradient of the gravitational potential function with respect to spatial position. It is the curl of gravitational acceleration. It is the component of gravitational acceleration in three directions.

2. The asteroid gravitational field reconstruction method based on physical information neural network according to claim 1, characterized in that: The specific implementation method of step (1) is as follows: Based on the asteroid swarm model, the gravitational potential function and gravitational acceleration of the data points are calculated using the gravitational formula of the particles and numerical integration: in, It is the gravitational potential function at the data point. It is the gravitational acceleration at the data point. Represents the gravitational constant. Indicates the first The mass of a point mass This represents the position vector of a data point relative to the origin of the coordinate system. Indicates the data point relative to the first The position vector of a particle. and Position vector and The modulus length; Data points are sampled using the gravitational potential function formula and the gravitational acceleration formula. Then, auxiliary points are randomly and uniformly generated in the sampling space. A transformer-MLP network is built as the gravitational potential function solver, and combined with the partial differential equation solver, the PINNs model can calculate the gravitational acceleration at the sampling points.

3. The asteroid gravitational field reconstruction method based on physical information neural network according to claim 1, characterized in that: The specific implementation method of step (2) is as follows: First, the auxiliary points and data points obtained in step (2) are normalized using a data normalization method that considers physical relationships: in, These are the three-dimensional position coordinates and gravitational potential of the sampling points before normalization. These are the normalized position coordinates, gravitational potential, and gravitational acceleration. It is the value that minimizes the position coordinate components and gravitational potential at the sampling point. It represents the range of position coordinates and gravitational potentials across all sampling points. It is the first-order partial derivative of the gravitational potential function with respect to spatial location; this normalization method normalizes the sampled data to... Interval; after data normalization, 4D position encoding is then performed on the normalized position coordinates: in, These are the three-dimensional position coordinates before encoding. These are the encoded four-dimensional position coordinates; the encoded four-dimensional position coordinates are fed into the transformer-MLP to regress the gravitational potential at the sampling point.

4. The asteroid gravitational field reconstruction method based on physical information neural network according to claim 1, characterized in that: The specific implementation method of step (4) is as follows: Based on the three physical partial differential equations of the gravitational field calculated in step (3), a loss function is constructed, and the data point part introduces the gravitational potential function loss and the gravitational acceleration equation loss: in, For the loss of gravitational potential function and the loss of gravitational acceleration, For network parameters, The total number of data points. For the first The spatial coordinates of each data point relative to the origin. For the gravitational potential predicted by the network, To obtain the true gravitational potential from the sampling, This represents the first-order gradient of the gravitational potential predicted by the network with respect to the location of the data points. The data points obtained from sampling represent the gravitational acceleration. In the auxiliary point section, we introduce losses from the Laplace equation for the gravitational field and the curl equation for gravitational acceleration: in, Losses are due to the Laplace equation and the gravitational curl equation. For network parameters, The total number of auxiliary points. For the first The spatial coordinates of each auxiliary point relative to the origin. Let be the second-order gradient of the gravitational potential predicted by the network with respect to the location of the data point. The gravitational curl predicted by the network; The total loss function is constructed by combining the data point loss term and the auxiliary point loss term: in, The weights of the four loss terms.

5. The asteroid gravitational field reconstruction method based on physical information neural network according to claim 1, characterized in that: The specific implementation method of step (5) is as follows: Based on the total loss function constructed in step (4), let the sum of all weights in the total loss function be 1, and introduce an adaptive weight scheme to reconstruct the total loss function: in, The gravitational potential function loss and gravitational acceleration loss are for the data points. The model is then subjected to Laplace equation loss and gravitational acceleration curl loss for auxiliary points. Subsequently, the model backpropagates based on the reconstructed total loss function, while simultaneously optimizing the network parameters and the weights of each loss term.

6. The asteroid gravitational field reconstruction method based on physical information neural network according to claim 1, characterized in that: The specific implementation method of step (6) is as follows: After optimizing the model for a certain number of rounds based on step (5), an adaptive sampling operation will be performed. 50 test areas will be randomly sampled within the gravitational field, and each area will consist of 20 Gaussian-distributed sampling points. Then, the average acceleration residuals in the 50 test areas will be used to find the area with the largest acceleration residual as the fault area. Finally, 200 additional sampling points will be sampled in the fault area according to the Gaussian distribution and added to the auxiliary point dataset to participate in the subsequent training of the network. Through adaptive sampling based on residuals, the model will focus on sampling in areas with low prediction accuracy during the training process.

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