A deep learning-based method for calculating discretized geodesics on riemannian manifolds

Abstract

The application discloses a kind of on Riemann manifold based on deep learning's discretization geodesic calculation method, it is related to geodesic measurement technical field.The present application, by unsupervised neural network, the data sample to be measured under a high-dimensional space is mapped to a low-dimensional Riemann manifold completely unsupervised, and guarantee the equivalence of data distribution structure in mapping process and the data structure in low-dimensional Riemann manifold space, also add the monte carlo integration error constraint for the discretization geodesic solving process in model, as the regularization term in model training process, so that the discretization solving process of geodesic is more accurate.The method of the present application can unsupervisedly reduce the dimension of any high-dimensional data, and ensure that high-dimensional data and low-dimensional manifold after dimension reduction meet the Riemann feature constraint of differential isomorphism and equidistant mapping, as far as possible guarantee that macroscopic topological structure between data samples and microscopic local distance measurement relationship is not changed in dimension reduction process.

Description

Technical Field

[0001] This invention relates to the field of geodesic measurement technology, and more specifically to a method for calculating discretized geodesics on Riemannian manifolds based on deep learning. Background Technology

[0002] With the explosive growth of information in the era of big data, the analysis and classification of high-dimensional data has become crucial. High-dimensional data often contains information redundancy, and we need to identify and utilize the special hidden structures existing in the original data while reducing its dimensionality.

[0003] Existing research has shown that many high-dimensional datasets are actually attached to lower-dimensional Riemannian manifolds. Therefore, how to reduce the dimensionality of data to a lower spatial dimension while ensuring that the hidden information in the data is preserved as little as possible during the dimensionality reduction process is a key problem facing manifold learning. Using the "latent variables" encoding of the dimensionality-reduced manifold space to describe the original high-dimensional data allows machine learning to perform faster and more efficient optimization calculations.

[0004] Distance metrics are one of the most important and core concepts in geometry, forming the foundation for all computational methods derived from geometry. In machine learning, because the statistical distribution of target sample data is often unclear, we emphasize the distance between samples, using distance metrics to determine the similarity between sample points for classification and representation learning. However, in Riemannian space, the shortest distance between points is no longer the "shortest line between two points" as in Euclidean space, but rather requires the use of geodesics to describe the local shortest distance between two points. Therefore, finding geodesic distances and paths that conform to the geometric characteristics of the original samples in an invisible low-dimensional Riemannian manifold is currently a key problem in machine learning. Summary of the Invention

[0005] To overcome the shortcomings and deficiencies of the existing technologies, this invention provides a deep learning-based method for discretized geodesic calculation on Riemannian manifolds. The purpose of this invention is to solve the problem of finding relatively correct geodesic paths when data samples are unevenly distributed and may contain noise. This deep learning-based method for discretized geodesic calculation on Riemannian manifolds proposes a neural network architecture. This neural network maps a high-dimensional data sample to a low-dimensional Riemannian manifold in a completely unsupervised manner, ensuring the equivalence of the data distribution structure during the mapping process with the data structure in the low-dimensional Riemannian manifold space. Furthermore, a Monte Carlo integral error constraint for the discretized geodesic solution process is added to the model as a regularization term during model training, making the discretized geodesic solution process more accurate. This method can perform unsupervised dimensionality reduction on arbitrary high-dimensional data, ensuring that the high-dimensional data and the reduced low-dimensional manifold conform to the Riemannian feature constraints of differential homogeneity and isometric mapping. It also ensures that the macroscopic topological structure and microscopic local distance metric relationships between data samples are not altered during the dimensionality reduction process. Based on the dimension-reduced manifold structure, a discretization computation method can be used to quickly calculate the geodesic path and distance between any two sample points in the Riemannian manifold space.

[0006] To address the problems existing in the prior art, the present invention is achieved through the following technical solution.

[0007] This invention provides a method for calculating discretized geodesics on Riemannian manifolds based on deep learning, the method comprising the following steps:

[0008] S1. Obtain the known data of the target object to be tested, and summarize the known data of the target object to be tested into a test data sample in a high-dimensional space;

[0009] S2. Use the test data samples obtained in step S1 as the input to the model and input them into an unsupervised neural network model for unsupervised learning.

[0010] The unsupervised neural network model includes a differential blank network and an isometric mapping network. During the unsupervised learning process, the differential blank network and the isometric mapping network alternately perform gradient descent optimization. Specifically, the model parameters calculated in the differential blank network are first fixed, and the model parameters of the isometric mapping network are trained with these model parameters as the optimization target. The model parameters in the isometric mapping network are then iteratively optimized.

[0011] S3. When the unsupervised neural network has been iterated and optimized a set number of times in step S2, or the loss function after iterative optimization meets a set threshold, the trained differential plagiarism network and the isometric mapping network are obtained.

[0012] S4. In the original Euclidean space of the data sample to be tested in step S1, select the starting point and ending point of the geodesic to be calculated. Use the trained differential monolithic network obtained in step S3 to map the starting point and ending point to the Riemann space. Select the region containing the starting point and ending point in the Riemann space as the discretization region. Use the equidistant mapping network trained in step S3 to obtain the Riemann metric of any position in the Riemann space. According to the movement cost calculation formula of discretization pathfinding, obtain the movement cost matrix of all points in the discretization region.

[0013] S5. Based on the movement cost matrix calculated in step S4, use the pathfinding algorithm to find the shortest path from the starting point to the ending point in the Riemann space. This shortest path is the geodesic path.

[0014] S6. Input the points on the geodesic path obtained in step S5 into the trained differential polynomial network obtained in step S3, and reconstruct the points on the geodesic path into Euclidean space for display.

[0015] More preferably, in step S2, the differential same-format network is a differential same-format autoencoder, which includes an encoder and a decoder. Receive high-dimensional space R m The sample data to be tested is mapped to E. n In space; decoder Receive E n Data in space, and mapping it back to R m Space; where E n Let represent an n-dimensional differential manifold, and assign a Riemannian metric to each point on the n-dimensional differential manifold, such that the n-dimensional differential manifold forms a Riemannian manifold; and They are all continuous functions.

[0016] More preferably, the Riemann metric is calculated from an extrinsic perspective, and its calculation formula is as follows:

[0017]

[0018]

[0019]

[0020] In the formula, dv represents the tangent vector in Riemann space; dv represents the Riemann manifold v-axis differential. Denote the v-axis basis vectors of Riemannian space; y represents the x-axis basis vector in Euclidean space; dy represents the y-axis differential in Euclidean space. Represent the y-axis basis vector in Euclidean space; dx represents the z-axis basis vector of the Riemannian manifold; du represents the u-axis differential on the Riemannian manifold; dx represents the x-axis differential in Euclidean space; dz represents the z-axis differential in Euclidean space. The u-axis basis vectors of the Riemannian manifold are represented by ; the Riemannian metric represents the Riemannian metric.

[0021] When decoder Once trained, the decoder can be used. The Jacobian matrix is ​​used to calculate the Riemannian metric of any coordinate point on an n-dimensional differential manifold.

[0022] A further preferred approach is to mesh the continuous Riemannian manifold into a discrete Riemannian manifold space; specifically,

[0023] Specialization of discrete sampling:

[0024]

[0025] In the formula, τ represents the directional spacing; μ represents the modulus spacing; θ net (x c ,τ) represents the differential x at any point on the Riemannian flow. c It can point to any angle within its tangent space; D net (μ) represents x at any point on the Riemannian stream. c The length of the module at the location;

[0026] The discretized grid points can move in 8 directions. The start and end points of the geodesics do not necessarily fall on the discretized grid. They are mapped to the nearest grid point using the following mapping function:

[0027] δ(n)={node|min node [Manhdist(n,node)]};

[0028] In the formula, δ(n) represents the mapping function that calculates and returns the nearest grid point node to the sample point n in Manhattan distance;

[0029] The discretized geodesic calculation equation is as follows:

[0030]

[0031] And there is,

[0032]

[0033]

[0034]

[0035] In the formula, δ(a) represents the mapping function of point a in the Riemannian manifold, that is, mapping point a to the nearest grid point; δ(b) represents the mapping function of point b in the Riemannian manifold, that is, mapping point b to the nearest grid point; μ represents the modulus interval; τ is the directional interval; The "°" in the figure represents a continuous mapping between two functions; The step size after discretization is represented by x; c This represents the position information at point c; θ net (x c ,τ) represents the discretized direction; g(c) represents the second-order tensor of the Riemannian metric.

[0036] Furthermore, the error analysis and reduction of the discretized geodesic calculation equations are performed using Monte Carlo integration. Based on the error analysis results, a regularization term that satisfies the error reduction constraint is designed.

[0037] Further preferably, in step S2, the differential monolithic network optimizes its parameterized network using the gradient descent method, with the loss function being:

[0038]

[0039] In the formula, This represents the reconstruction accuracy constraint of the differential monolithic autoencoder in the differential monolithic network; This means using the Spearman correlation coefficient to penalize the sample mapping to any two dimensions in a low-dimensional manifold. The correlation between them; This indicates that the Monte Carlo integration error in the discretized geodesic process is reduced by reducing the variance; α and β are adjustable penalty coefficients for the loss term;

[0040] In the above formula, n represents the total number of samples in the dataset; k represents the number of the k-th sample. This represents the reconstructed sample data; x k N represents the original data of the k-th sample; dim , denotes the set of dimensions of a Riemannian manifold; D denotes variance; G E This represents the distance calculation within the popular tangent space; Represents the inverse mapper of the differential homeomorphism; This represents a differential homeomorphism mapper.

[0041] Furthermore, the loss function of the isometric mapping network is designed as follows:

[0042]

[0043] In the formula, γ is the adjustable penalty coefficient of the loss term; G E This indicates the calculation of Euclidean distance within a local area; Represents an isometric mapper; This represents the sampling function.

[0044] In a further preferred embodiment, in step S2, the model parameters calculated in the differential monolithic network are first fixed. Using these model parameters as the optimization target, the model parameters of the isometric mapping network are trained. The model parameters in the isometric mapping network are then iteratively optimized. Specifically, this means...

[0045] The result calculated from the differential isotope Fixed, using this as the optimization objective, train the isometric mapping network. Thus, for isometric mapping networks The parameters in the data are iteratively optimized;

[0046] The overall loss function is:

[0047] Totalloss=Isometricloss+Diffeomorphismloss.

[0048] More preferably, in step S4, the formula for calculating the movement cost of discretized pathfinding represents the movement cost when moving from one grid point to the next grid point node_θ in the direction θ∈{0,τ,…,360-τ}:

[0049]

[0050] make

[0051]

[0052]

[0053] In the above formula, inf represents positive infinity; Represents any sample n; dataset represents a dataset; Rieman d (node, θ, μ) represents the distance μ at node node in the direction θ; Rieman d (node θ ,180-θ,μ) represents the value at node θ The distance of a step size μ in the direction of 180-θ; This indicates the existence of n;

[0054] When no sample n in any dataset can be mapped to a grid point via δ(n), treat this grid point as a wall and set all paths connecting it to infinity; otherwise, set the cost of moving a grid point in the θ direction to the Riemann metric unit inner product distance of the node grid point in the θ direction in the tangent space plus the average of the Riemann metric unit inner product distance of the next grid point node_θ in this direction when it moves to node in the tangent space.

[0055] Furthermore, preferably, when searching for the geodesics of any two samples in the field of view apparatus, if:

[0056]

[0057] st,

[0058]

[0059] Then gradually increase μ until the geodesic distance is not equal to inf.

[0060] Further preferred, the pathfinding algorithm in step S5 is either the Dijkstra algorithm or the Floyd-Warshall pathfinding algorithm.

[0061] In a further preferred embodiment, in step S1, the coordinates of the aircraft surface, the flight state, and the pressure coefficient at the corresponding point are obtained. The coordinates of the aircraft surface include three-dimensional spatial coordinates of the x, y, and z axes, and the flight state includes Mach number, angle of attack, and Reynolds number. These data are then aggregated to form a 7-dimensional test data sample.

[0062] Compared with the prior art, the beneficial technical effects of the present invention are as follows:

[0063] 1. Traditional nonlinear geodesic learning algorithms, such as Isomap and its variant L-Isomap, attempt to preserve pairwise geodesic distances between all points during the learning process, thereby minimizing data distortion during dimensionality reduction. This is achieved by applying the KNN algorithm to each sample point in a high-dimensional Euclidean space to find and calculate the Euclidean distances between its k closest sample points, forming an undirected graph metric matrix. Finally, Dijkstra's or Floyd-Warshall pathfinding algorithms are used to find the shortest path between any two sample nodes in the graph, approximating this path as a geodesic path between the two points in the manifold space. However, this method has significant drawbacks; when using Isomap and its derivatives, we need to assume that the distribution of all samples is uniform and dense. Because the KNN algorithm is used, each node in the metric matrix of the directed graph only records the Euclidean distances of its k closest nodes. Therefore, when there are sparse regions with uneven data distribution, the Isomap algorithm will choose to bypass the sparse regions and go towards the dense regions when calculating the geodesics, which may cause the geodesics to be distorted and elongated.

[0064] 2. Compared to existing technologies, the method of this invention first uses a differential homeomorphic autoencoder to perform unsupervised homeomorphic mapping on high-dimensional data, ensuring that the high-dimensional Euclidean space is equivalent to the mapped low-dimensional manifold in terms of macroscopic topological structure. Furthermore, the differential homeomorphic autoencoder neural network has a certain generalization learning effect, and can reconstruct the main feature distribution of the original data well even when faced with uneven data distribution or noise. In addition, we also use a discretization training method, performing discretization sampling in multiple directions around each data sample, which alleviates the problem of uneven data distribution to some extent. Finally, we use equidistant mapping to ensure that the distance metric between all samples and other sampling points in their local neighborhood is equivalent to the distance metric in the high-dimensional space. This makes the change in distance metric across the entire space after mapping to the low-dimensional manifold smoother and more continuous. Combining these features, our proposed method, compared to traditional Isomap-type geodesic learning methods, has better robustness to noisy or unevenly distributed high-dimensional data and can calculate geodesic paths more accurately. Attached Figure Description

[0065] Figure 1 This is a flowchart of the geodesic calculation method of the present invention;

[0066] Figure 2 This is a diagram of the unsupervised neural network model structure of the present invention;

[0067] Figure 3 This is a schematic diagram illustrating the principle of the differential co-fiber self-encoder of the present invention;

[0068] Figure 4This is a diagram of the discretized geodesic calculations of this invention;

[0069] Figure 5 This is a schematic diagram illustrating the process of solving the discretized geodesics of this invention;

[0070] Figure 6 This is a top view of the sample data distribution in the robustness test of the method of the present invention and the Isomap method under noise and uneven data distribution conditions.

[0071] Figure 7 This is a sample data distribution diagram (test diagram) in the robustness test of the method of the present invention and the Isomap method under noise and uneven data distribution conditions.

[0072] Figure 8 This is a top view showing the comparison of the robustness of the method of this invention and the Isomap method in the first round of robustness tests under noise and uneven data distribution conditions.

[0073] Figure 9 This is a side view showing the first round of robustness tests (comparison of the method of this invention and the Isomap method under noise and uneven data distribution conditions).

[0074] Figure 10 This is a top view showing the comparison of the robustness of the method of this invention and the Isomap method in the second round of robustness tests under noise and uneven data distribution conditions.

[0075] Figure 11 This is a side view showing a comparison of the second round of robustness tests between the method of this invention and the Isomap method under conditions of noise and uneven data distribution.

[0076] Figure 12 This is a top view showing the comparison of the robustness of the method of this invention and the Isomap method in the third round of robustness tests under noise and uneven data distribution conditions.

[0077] Figure 13 This is a side view showing the comparison of the robustness of the method of this invention and the Isomap method in the third round of robustness testing under noise and uneven data distribution conditions.

[0078] Figure 14 This is a modeling diagram of the wing of the DLR-F11 aircraft in an example of the method of the present invention;

[0079] Figure 15 This is a heat map of the pressure coefficient obtained by unsupervised dimensionality reduction of the 7-dimensional data of the DLR-F11 aircraft to 2-dimensional data using the method of this invention;

[0080] Figure 16This is a heatmap of the angle of attack number obtained by unsupervised dimensionality reduction of the 7-dimensional data of the DLR-F11 aircraft to 2-dimensional using the method of this invention. Detailed Implementation

[0081] The following are exemplary embodiments of the invention as defined by the claims and their equivalents, taken in conjunction with the accompanying drawings, to aid in a comprehensive understanding. The specific details described herein are to be considered exemplary only and not to limit the scope of the invention. Therefore, those skilled in the art can make various changes and modifications to the embodiments without departing from the scope and spirit of the invention.

[0082] Example 1

[0083] As a preferred embodiment of the present invention, please refer to the appendix to the specification. Figure 1 As shown in the figure, this embodiment discloses a method for calculating discretized geodesics on Riemannian manifolds based on deep learning. The method includes the following steps:

[0084] S1. Obtain the known data of the target object to be tested, and summarize the known data of the target object to be tested into a test data sample in a high-dimensional space;

[0085] S2. Use the test data samples obtained in step S1 as the input to the model and input them into an unsupervised neural network model for unsupervised learning.

[0086] The unsupervised neural network model includes a differential blank network and an isometric mapping network. During the unsupervised learning process, the differential blank network and the isometric mapping network alternately perform gradient descent optimization. Specifically, the model parameters calculated in the differential blank network are first fixed, and the model parameters of the isometric mapping network are trained with these model parameters as the optimization target. The model parameters in the isometric mapping network are then iteratively optimized.

[0087] S3. When the unsupervised neural network has been iterated and optimized a set number of times in step S2, or the loss function after iterative optimization meets a set threshold, the trained differential plagiarism network and the isometric mapping network are obtained.

[0088] S4. In the original Euclidean space of the data sample to be tested in step S1, select the starting point and ending point of the geodesic to be calculated. Use the trained differential monolithic network obtained in step S3 to map the starting point and ending point to the Riemann space. Select the region containing the starting point and ending point in the Riemann space as the discretization region. Use the equidistant mapping network trained in step S3 to obtain the Riemann metric of any position in the Riemann space. According to the movement cost calculation formula of discretization pathfinding, obtain the movement cost matrix of all points in the discretization region.

[0089] S5. Based on the movement cost matrix calculated in step S4, use the pathfinding algorithm to find the shortest path from the starting point to the ending point in the Riemann space. This shortest path is the geodesic path.

[0090] S6. Input the points on the geodesic path obtained in step S5 into the trained differential polynomial network obtained in step S3, and reconstruct the points on the geodesic path into Euclidean space for display.

[0091] Example 2

[0092] As another preferred embodiment of the present invention, this embodiment is a further detailed supplement and explanation of the technical solution of the present invention based on the above-described embodiment 1. (Refer to the appendix of the specification.) Figure 3 As shown, the differential blank network is a differential blank autoencoder, which includes an encoder and a decoder. Receive high-dimensional space R m The sample data to be tested is mapped to E. n In space; decoder Receive E n Data in space, and mapping it back to R m Space; where E n Let represent an n-dimensional differential manifold, and assign a Riemannian metric to each point on the n-dimensional differential manifold, such that the n-dimensional differential manifold forms a Riemannian manifold; and They are all continuous functions.

[0093] Specifically, this embodiment provides an unsupervised neural network model that meets the requirements of Embodiment 1 above. The specific structure of this unsupervised neural network model is shown in the appendix to the specification. Figure 2 As shown in the figure, the Sample(x,μ,τ) module performs discretization grid sampling around the sample point x according to equation (10). For any dimension d of x, v ∈N dim , calculate its interpolation in each dimension and And by permuting and combining the interpolations with those of all other dimensions, we obtain the interpolation set Sample for x. x Differential platinum mapper And the inverse mapper f of the differential isomorphism θ ` Corresponding to the encoder and decoder of the differential blank autoencoder, respectively, the differential blank mapper Differential inverse mapper and isometric mapping learner It is a parameterized trainable module for neural networks.

[0094] The Riemannian metric is calculated from an extrinsic perspective, and its formula is as follows:

[0095]

[0096]

[0097]

[0098] dv represents the tangent vector in Riemann space; dv represents the Riemann manifold v-axis differential. Denote the v-axis basis vectors of Riemannian space; y represents the x-axis basis vector in Euclidean space; dy represents the y-axis differential in Euclidean space. Represent the y-axis basis vector in Euclidean space; Let denote the z-axis basis vector of the Riemannian manifold; du denotes the u-axis differential on the Riemannian manifold; dx denotes the x-axis differential in Euclidean space; dx denotes the z-axis differential in Euclidean space. The u-axis basis vectors of the Riemannian manifold are represented; the Riemannian metric represents the Riemannian metric when decoded. Once trained, the decoder can be used. The Jacobian matrix is ​​used to calculate the Riemannian metric of any coordinate point on an n-dimensional differential manifold.

[0099] The continuous Riemannian manifold is meshed into a discrete Riemannian manifold space; specifically,

[0100] Specialization of discrete sampling:

[0101]

[0102] In the formula, τ represents the directional spacing; μ represents the modulus spacing; θ net (x c ,τ) represents the differential x at any point on the Riemannian flow. c It can point to any angle within its tangent space; Let x represent any point on the Riemannian stream. c The length of the module at the location;

[0103] The discretized grid points can move in 8 directions. The start and end points of the geodesics do not necessarily fall on the discretized grid. They are mapped to the nearest grid point using the following mapping function:

[0104] δ(n)={node|min node [Manhdist(n,node)]};

[0105] In the formula, δ(n) represents the mapping function that calculates and returns the nearest grid point node to the sample point n in Manhattan distance;

[0106] The discretized geodesic calculation equation is as follows:

[0107]

[0108] And there is,

[0109]

[0110]

[0111]

[0112] In the formula, δ(a) represents the mapping function of point a in the Riemannian manifold, that is, mapping point a to the nearest grid point; δ(b) represents the mapping function of point b in the Riemannian manifold, that is, mapping point b to the nearest grid point; μ represents the modulus interval; τ is the directional interval; The "°" in the figure represents a continuous mapping between two functions; The step size after discretization is represented by x; c This represents the position information at point c; θ net (x c ,τ) represents the discretized direction; g(c) represents the second-order tensor of the Riemannian metric.

[0113] The Monte Carlo integral of the discretized geodesic calculation equation is subjected to error analysis and reduction. Based on the error analysis results, a regularization term that satisfies the error reduction constraint is designed.

[0114] As one implementation of this embodiment, in step S2, the differential monolithic network optimizes its parameterized network using the gradient descent method, and the loss function used is:

[0115]

[0116] In the formula, This represents the reconstruction accuracy constraint of the differential monolithic autoencoder in the differential monolithic network; This means using the Spearman correlation coefficient to penalize the sample mapping to any two dimensions in a low-dimensional manifold. The correlation between them; This indicates that the Monte Carlo integration error in the discretized geodesic process is reduced by reducing the variance; α and β are adjustable penalty coefficients for the loss term;

[0117] n represents the total number of samples in the dataset; k represents the number of the k-th sample. This represents the reconstructed sample data; x k N represents the original data of the k-th sample; dim , denotes the set of dimensions of a Riemannian manifold; D denotes variance; GE This represents the distance calculation within the popular tangent space; Represents the inverse mapper of the differential homeomorphism; This represents a differential homeomorphism mapper.

[0118] The loss function of the isometric mapping network is designed as follows:

[0119]

[0120] In the formula, γ is the adjustable penalty coefficient of the loss term; G E This indicates the calculation of Euclidean distance within a local area; Represents an isometric mapper; This represents the sampling function.

[0121] In step S2, the model parameters calculated in the differential monolithic network are first fixed. Using these model parameters as the optimization objective, the model parameters of the isometric mapping network are trained. The model parameters in the isometric mapping network are then iteratively optimized. Specifically, this means...

[0122] The result calculated from the differential isotope Fixed, using this as the optimization objective, train the isometric mapping network. Thus, for isometric mapping networks The parameters in the data are iteratively optimized;

[0123] The overall loss function is:

[0124] Totalloss=Isometricloss+Diffeomorphismloss.

[0125] Example 3

[0126] As another preferred embodiment of the present invention, this embodiment further supplements and elaborates on the technical solution of the present invention based on the above-described Embodiment 1 or Embodiment 2. In this embodiment, the movement cost calculation formula for discretized pathfinding indicates that the movement cost when moving from one grid point to the next grid point node_θ in the direction θ∈{0,τ,…,360-τ} is:

[0127]

[0128] make

[0129]

[0130] inf represents positive infinity; Represents any sample n; dataset represents a dataset; Rieman d(node, θ, μ) represents the distance μ at node node in the direction θ; Rieman d (node θ ,180-θ,μ) represents the value at node θ The distance of a step size μ in the direction of 180-θ; This indicates the existence of n;

[0131] When no sample n in any dataset can be mapped to a grid point via δ(n), treat this grid point as a wall and set all paths connecting it to infinity; otherwise, set the cost of moving a grid point in the θ direction to the Riemann metric unit inner product distance of the node grid point in the θ direction in the tangent space plus the average of the Riemann metric unit inner product distance of the next grid point node_θ in this direction when it moves to node in the tangent space.

[0132] When searching for the geodesics of any two samples in a field of view machine, if:

[0133]

[0134] st,

[0135]

[0136] Then gradually increase μ until the geodesic distance is not equal to inf.

[0137] As an example of this embodiment, the pathfinding algorithm in step S5 is either the Dijkstra algorithm or the Floyd-Warshall pathfinding algorithm.

[0138] Example 4

[0139] In a preferred embodiment of the present invention, finding the Riemann geodesic distance for high-dimensional data that cannot be visualized is a crucial issue, because the distance metric between samples in machine learning can affect the performance of downstream classification or prediction tasks. The more accurately the distance metric between samples reflects the true distribution, the better the generalization effect will be in downstream tasks.

[0140] Refer to the instruction manual appendix Figure 14 As shown, in this embodiment, the coordinates (three-dimensional spatial coordinates of x, y, z axes), flight state (Mach number, angle of attack and Reynolds number) of the surface of the DLR-F11 aircraft and the pressure coefficient (Cp) of the corresponding point are used as input data for the unsupervised neural network model in this invention.

[0141] The model training process and the offline geodesic calculation process for any two samples are as follows:

[0142] 1. Input the seven-dimensional data (x, y, z, angle of attack, Mach, Reynolds, pressure coefficient) of the DLR-F11 aircraft into an unsupervised neural network model for unsupervised learning to reduce it to 2-dimensionality;

[0143] 2. When the unsupervised neural network has been iterated and optimized a set number of times, or when the loss function after iterative optimization meets a set threshold, the trained differential encoder is obtained. Differential inverse encoder and isometric mapper

[0144] 3. Select the start and end points of the geodesic you want to calculate in the original Euclidean space of the sample, and use the trained encoder. Map the start and end points to Riemannian space, and select the region containing the two points as the discretization region. Utilize an isometric mapper. Then, the Riemann metric g(c) of any Riemann space coordinate position c can be obtained by using the formula for calculating the Riemann metric under the extrinsic perspective. Thus, the movement cost matrix of all points in the discretized region can be obtained by using the movement cost calculation formula for discretized pathfinding.

[0145] 4. Based on this movement cost matrix, use Dijkstra's algorithm or other pathfinding algorithms to find the shortest path from the starting point to the ending point in Riemann space, which is the geodesic path.

[0146] After performing unsupervised dimensionality reduction to 2 dimensions on the above 7-dimensional data, the following results can be obtained: Figure 15 and attached Figure 16 The image shown, with attached Figure 15 A thermogram of the pressure coefficient is attached. Figure 16 The figure shows a heatmap of the angle of attack. As can be seen from the figure, the model of this invention can find Riemannian geodesics between data sample points across aerodynamic parameters, and these geodesics are not straight paths in Euclidean space. This indirectly illustrates that the aerodynamic parameters of the wing are actually distributed as a low-dimensional Riemannian manifold embedded in a high-dimensional (7-dimensional) space.

[0147] In downstream tasks, the adjacency matrix constructed using geodesic distance (GNN) has higher prediction accuracy than the adjacency matrix constructed using Euclidean distance. This also demonstrates that geodesic distance in Riemannian manifolds contains a more realistic and accurate distance metric between data samples compared to Euclidean distance.

[0148] Example 5

[0149] As another preferred embodiment of the present invention, this embodiment provides a method for calculating discretized geodesics on Riemannian manifolds based on deep learning.

[0150] In Euclidean space, describing the shortest distance between two points given their coordinates is straightforward. However, the space we inhabit is not simply Euclidean. Einstein's theory of relativity provides evidence that the universe operates in 4-dimensional Riemannian space. In Riemannian space, only distance calculations within local coordinates can use the inner product of Euclidean metric as the unit of distance description. On a macroscopic scale, any two points outside their neighborhood require contact geodesics to describe their shortest distance. With the rapid development of deep learning, research has shown that high-dimensional data also resides in spaces other than Euclidean. They are highly likely embedded within a low-dimensional Riemannian manifold structure. However, because high-dimensional data is difficult to visualize, and the distribution of unknown datasets cannot be uniformly described by a single probability distribution formula, accurately describing the distance relationships between these high-dimensional data points remains a challenge for deep learning and data mining research.

[0151] This embodiment provides a novel neural network structure with universal applicability, as detailed in the appendix to the specification. Figure 2 As shown, it can homomorphically map high-dimensional data to a low-dimensional manifold space under unsupervised conditions, while ensuring the metric isometry property of any sample in the local area during the mapping process as much as possible. The isometry and homeomorphism properties guarantee the equivalence of the data distribution structure in the high-dimensional Euclidean space and the data structure in the low-dimensional Riemannian manifold space. Furthermore, to more accurately and quickly find the geodesic path between any two points after model training, we also added a Monte Carlo integral error constraint for the discretization solution of the geodesic process as a regularization term during model training. This makes the discretization solution of the geodesic more accurate.

[0152] The entire network architecture can perform unsupervised dimensionality reduction on arbitrarily high-dimensional data, ensuring that both the high-dimensional data and the reduced low-dimensional manifold conform to the Riemannian characteristic constraints of differential homeomorphism and isometric mapping. It strives to ensure that the macroscopic topological structure and microscopic local distance metrics between data samples remain unchanged during the dimensionality reduction process. Based on the reduced manifold structure, a discretization computation method can be used to quickly calculate the geodesic path and distance between any two sample points in the Riemannian manifold space.

[0153] 1. Differential Monolithic Autoencoder: First, an autoencoder consists of two parts: an encoder and a decoder. We will write them as: and According to the definition of an autoencoder:

[0154]

[0155] The encoder receives an R m Spatial data, mapping it to En In space, and Decoder receives E n Data in space and mapping it back to R m space.

[0156] Assumption

[0157] This demonstrates that the autoencoder can losslessly encode and decode to reconstruct the original data x. i Then there is Furthermore, based on the fact that neural network training requires backpropagation, They are all continuous functions.

[0158] Therefore, based on the mapping function European-style space R m With hidden space E n Belongs to topological homogeneous billet Proved E n →R m It is a same-bill mapping, and if in E n A set of open covering {O} can be found on the table. A}, it can be known must Make It's also the opening of the market.

[0159] In E n Find any pair of open neighborhoods O A O b ;satisfy

[0160] According to the above formula (2), for have Therefore E is certified n It is an n-dimensional differential manifold. Therefore, under the assumption that "the autoencoder can encode and decode the original data almost losslessly," this autoencoder is a differential isomorphic mapper (such as...). Figure 3 (As shown).

[0161] A Riemannian manifold is a special type of differential manifold. A smooth second-order covariant tensor field g is defined locally at every point M on the manifold. (M, g) is called an n-dimensional Riemannian manifold, and g is the fundamental tensor or Riemannian metric of the manifold. Therefore, we need to assign a Riemannian metric to each point on the differential manifold to make it a Riemannian manifold.

[0162] The formula for calculating the Riemann metric from an extrinsic perspective is as follows:

[0163]

[0164]

[0165]

[0166] Equation (3) illustrates that when we have a trained decoder At that time, it can be used The Jacobian matrix is ​​used to calculate the Riemannian metric at any coordinate point on the differential manifold.

[0167] As the above derivation shows, assuming that the autoencoder can encode and decode the original data almost losslessly, then a homeomorphic differential manifold structure of the original data exists in the latent space of the encoder output. Furthermore, we can use the Jacobian matrix of the decoder to calculate the Riemannian metric at any position on the differential manifold. Knowing this, the next key problem is how to find the geodesic between any two points on the invisible Riemannian manifold structure.

[0168] 2. Discretization and Error Analysis

[0169] 2.1 Discretization of the geodesic formula on a manifold

[0170] The formula for calculating the geodesic distance between two points a and b on a manifold is as follows:

[0171] in It is a curve, δ is a curve This represents the movement parameter on the curve.

[0172] According to equation (4), the calculation of the geodesic is a definite integral problem. Although we know the Riemannian metric for any point on the manifold, the shortest distance between any two points on the Riemannian manifold is no longer a straight line. Therefore, there are infinitely many possible paths between the two points, meaning we need to solve infinitely many definite integrals to find the one with the smallest numerical solution as the geodesic distance.

[0173]

[0174] in Let S represent the set of all paths that start at point a and end at point b. R || represents the number of paths in the entire path set. This minimization problem has no analytical solution when dealing with unknown data distributions.

[0175] Formula (5) can be written in the following form:

[0176]

[0177] Among them, dx cIt is a differential component of a vector. A vector is a metric with magnitude and direction. The calculation of the integral in this formula mainly requires knowing the direction of motion along the geodesic at position c in the current manifold space. g(c) is a second-order tensor of the Riemannian metric, which controls the metric matrix at the current position c.

[0178] To facilitate discretization, we decouple the differential components of the vector in this formula and represent them separately as direction and magnitude:

[0179] in θ represents the magnitude of x, and θ represents the direction of x.

[0180] then,

[0181]

[0182] Where, θ * (x c ) = arccosθ; j and k represent any two vectors in the tangent space starting at point c.

[0183] θ on a 2D Riemannian manifold * (x) means that dx at any point can point to any angle in its tangent space. The magnitude of dx always tends towards 0.

[0184] However, at present This is still a problem without an analytical solution. We need to further discretize it and find an approximate solution. Therefore, we let:

[0185]

[0186] Where: θ ,-t (x,τ)∈{0,τ,…,360-τ} τ is the directional spacing and μ is the modulus spacing. With these two controllable variables, we can mesh a continuous Riemannian manifold into a discrete Riemannian manifold space.

[0187] In this example, we specialize in discrete sampling:

[0188]

[0189] From this special case formula, we can see that after discretization, the grid points can move in eight directions: up, down, left, and right. Figure 4 As shown in the diagram. Furthermore, since the start and end points of a geodesic do not necessarily fall on the discretized grid, we need to specially handle the start and end points, mapping them to the nearest grid point:

[0190] Let the mapping function be:

[0191] δ(n)={node|min node [Manhdist(n,node)]} (11)

[0192] This function calculates and returns the grid point node that is closest to sample point n in Manhattan distance.

[0193] Therefore, the discretized geodesic calculation equation is:

[0194]

[0195] And there are:

[0196]

[0197] With the discrete expression for geodesics (12), we can use a pathfinding algorithm to find the shortest path on the discrete grid. The problem now is how to analyze and control the error caused by the integral discretization.

[0198] 2.2 Error Analysis and Reduction Based on Monte Carlo Integrals

[0199] According to Khinchin's law of large numbers, any integral function can be written in the following form:

[0200]

[0201] Based on the above formula, the discretized geodesic integral in Section 2.1 can be derived as follows:

[0202]

[0203] Let among them In the above formula, pdf(c) is the random probability density function for selecting points on the geodesic path, and it must satisfy the following property:

[0204] 1. Positive definiteness: pdf(c) > 0;

[0205] 2. Normalization:

[0206] Any pdf(c) that satisfies the above requirements can be customized, commonly using uniform or Gaussian distributions. However, in the actual calculation of geodesic distances, each interpolation point on the path will only have its distance calculated once. Therefore, to fit the geodesic distance calculation scenario, we choose a uniform probability density as pdf(c). This ensures that each point on the geodesic path has a uniform probability X ~ U(a,b) of being selected for calculating the inner product metric, thus:

[0207] According to the central limit theorem for independent and identically distributed entities (De Moivre-Laplace theorem):

[0208] because The arithmetic mean is If it is an unbiased estimate, then it reflects, to a certain extent, the... The actual situation, therefore the absolute error It should meet the following description:

[0209] For any error size k, there exists a probability Make:

[0210] Here Represents the confidence level. It is usually possible to take represent It is a highly probable event.

[0211] We hereby order: So:

[0212] When n is sufficiently large, this distribution follows a Gaussian distribution: Therefore there is

[0213]

[0214] Confidence value in the above formula We define it ourselves, from which we can solve for ∈ under the corresponding confidence condition, which is the Monte Carlo integral error.

[0215] Based on the above derivation, the complete description of Monte Carlo integration error is as follows:

[0216] At confidence level In this case, the absolute error of the Monte Carlo integral is:

[0217]

[0218] From the above formula, we can see that there are two ways to reduce the estimation error:

[0219] 1. Increase the number of trials (n) as much as possible. If you want to reduce the error by half, you need to increase the number of trials by four times.

[0220] 2. Reduce sampling variance in:

[0221] From the error analysis above, we can see that in order to reduce the error, we need to design regularization terms that satisfy the following two requirements when training the neural network:

[0222] 1. pdf(c) should satisfy a uniform distribution as much as possible; sampling points on any path should be distributed as evenly as possible.

[0223] 2. In In the case of a uniform distribution, it is necessary to reduce the variance D[f(c)] of f(c) to make the distribution of f(c) as flat as possible.

[0224] Here, f(c) represents the length of the unit inner product under the Riemannian metric at point c. Figuratively speaking, we need the Riemannian metric to change more smoothly and uniformly in Riemannian space, so that the distance metric calculation results at each location in the manifold space do not vary significantly.

[0225] 3. Neural Network Model Structure

[0226] To prove that the manifolds in Riemannian space and the surfaces in the original Euclidean space output by the encoder are isometric mappings, assuming that the decoder is a differentiable homeomorphic mapping, we assume:

[0227] The Riemannian manifold space output by the encoder and the Euclidean space reconstructed by the decoder are two manifold structures (E, G). R ) and (S,G E Therefore, let the decoder be an inverse transform mapping of a differential homeomorphism:

[0228] f`:E→S(19);

[0229] If for and All vectors X and Y in the tangent space at point p on the manifold E satisfy: G R (X,Y)=G E [f`(X),f`(Y)](20);

[0230] Then f' is said to be from (E, G) R ) to (S,G E isometric embedding of ).

[0231] We designed a network structure that meets the above requirements to guarantee the isometric mapping and differential homeomorphism properties between the two manifolds, such as... Figure 2 As shown, Figure 2 In the sample (x,μ,τ) module, the sample point x is discretized into a grid according to equation (10). For any dimension d of x, the sampling is performed. v ∈N dim , calculate its interpolation in each dimension and And by permuting and combining the interpolations with those of all other dimensions, we obtain the interpolation set Sample for x. xThe modules labeled θ in the diagram. and as well as It is a parameterized trainable module for neural networks.

[0232] Let x∈E, and the set of interpolations with respect to x belong to the sample in the tangent space at x. x ∈T x E. At the same time, in order to... and Since it is a mapper that satisfies differential homeomorphism, we need to ensure that the reconstruction loss is as small as possible.

[0233] If x k Let represent a training sample as input, and let in, These are training samples reconstructed by a differential autoencoder.

[0234] We designed the following two parts of the loss function as the optimization objective of the model, so as to facilitate the optimization of the parameterized network above using the gradient descent method:

[0235]

[0236] The design principle of the loss function (21) is as follows: The first term in the above equation is the reconstruction accuracy constraint of the autoencoder to satisfy the assumptions of equation (2) as much as possible. The second term, in order to reduce the correlation between the dimensions of the target manifold mapped to the homeomorphism, uses the Spearman correlation coefficient to penalize the sample mapping to any two dimensions in the low-dimensional manifold. The correlation between them. The third term is obtained by reducing the variance according to equation (17) in section 2.2. The method reduces the Monte Carlo integration error in the discretized solution of the geodesic process. α and β are adjustable penalty coefficients for the loss term.

[0237] The loss function for the equidistant mapping part is designed as follows:

[0238]

[0239] Design principle of loss function (22): The above equation is designed to satisfy the isometric mapping constraint based on mean squared error (MseLoss) of equation (20). γ is the adjustable penalty coefficient of the loss term.

[0240] In summary, the entire network comprises a differential homeomorphic network and an isometric embedding network. During training, these two networks are alternately optimized using gradient descent: first, the values ​​calculated in the differential homeomorphic network are... Fix it, and use this as the optimization target to train isometric embeddings. Thus, for isometric mapping networks The parameters are iteratively optimized.

[0241] Overall loss function:

[0242] Total loss=Isometric loss+Diffeomorphism loss(23).

[0243] In actual optimization, the training times for the differential homeomorphism and isometric mapping parts of the network can be allocated in any x:y ratio. Because the calculation process of isometric mapping is more complicated, requiring the calculation of backpropagation based on the Jacobian matrix, the total number of parameters in the isometric mapping network should be more than that in the differential homeomorphism network.

[0244] 4. Discretization of the geodesic process

[0245] Through the unsupervised mapping above, we have a manifold structure in low-dimensional space. We can treat the discretized 2D Riemannian manifold as an undirected graph structure and use heuristic pathfinding algorithms (such as Dijkstra's algorithm) to find the shortest distance between any two points. To use the pathfinding algorithm, we need to assign a movement cost to the path between each grid point and its connected surrounding grid points.

[0246] Discretized pathfinding movement cost design method:

[0247] The cost of moving from one grid point to the next grid point node_θ in the direction θ∈{0,τ,…,360-τ} is:

[0248]

[0249] make:

[0250]

[0251] When no sample n in any dataset can be mapped to a grid point through δ(n), we treat this grid point as a "wall" and set all paths connected to it to infinity (inf).

[0252] Otherwise, the cost of moving a grid point in the θ direction is set to the Riemann metric unit inner product distance of the node grid point in the θ direction in the tangent space plus the average of the Riemann metric unit inner product distance of the next grid point node_θ in this direction when it moves towards node.

[0253] However, some problems still exist in the application: when When the μ value is too small, most of the discrete grid nodes in the space will become "walls", causing the geodesics between some sample points to be completely blocked.

[0254] Therefore, a mechanism for the gradual amplification of μ was proposed to address these two issues:

[0255] When searching for the geodesics of any two samples in a field of view machine, if:

[0256]

[0257] st;

[0258]

[0259] Then gradually increase μ until the geodesic distance is not equal to inf.

[0260] Refer to the instruction manual appendix Figure 5 The diagram illustrates a complete discretized geodesic solution process:

[0261] 1. Input the data into the new network architecture designed in Chapter 3 and perform unsupervised learning according to the loss function (23).

[0262] 2. When the model's loss is small enough or after a sufficient number of iterations, a well-trained differential homeomorphic encoder is obtained. and decoder and isometric mapper

[0263] 3. Select the starting and ending points of the geodesic you want to calculate in the original Euclidean space of the sample, and use an encoder. Map the start and end points to Riemannian space, and select the region containing the two points as the discretization region. Utilize an isometric mapper. Then, the Riemann metric g(c) of any Riemann space coordinate position c can be obtained through equation (3), and the movement cost matrix of all nodes in the discretized region can be obtained using equation (24).

[0264] 4. Based on this movement cost matrix, use Dijkstra's algorithm or other pathfinding algorithms to find the shortest path from the starting point to the ending point in Riemann space, which is the geodesic path.

[0265] 5. Input the points on this path into the decoder. Reconstruct and display it in Euclidean space.

[0266] Example 6

[0267] As another preferred embodiment of the present invention, this embodiment is a comparative experiment on the robustness of the discretized geodesic calculation method of the present invention and the Isomap method under noise and uneven data distribution conditions. The distribution map of the sample data is shown in the figure. Figure 6 and Figure 7 As shown.

[0268] The above sample data is obtained through the function z 2 =x 2 +y 2 The data is obtained by uniformly and randomly sampling 1000 samples within the range x∈[-3,3], y∈[-3,3]. In the figure, blue represents the sampled data samples, and red represents the geodesic sampling instances along the data manifold between the two points (3,0,9) and (-3,0,9) learned by Isomap.

[0269] We performed the following sparsity and noise reduction processing on these data samples:

[0270] Sparsity was applied within a circle 1 unit away from (0,±3), and 80% of the data samples within this range were randomly deleted.

[0271] And 100 oversampled sample points are uniformly added to the function surface from (0,3) to (0,-3).

[0272] In the function z 2 =x 2 +y 2 300 noise samples are randomly added within a 1-unit vertical distance of the curved surface.

[0273] After generating multiple random training samples, a geodesic learning comparison experiment was conducted. In the experiment, two points on the function, (3,0,9) and (-3,0,9), were selected as the start and end points of the geodesic. The red points represent samples along the learned geodesic path, the green points represent the actual geodesic path, and the blue points represent equidistant sampling reference points along the x-axis from (3,0,9) to (3,0,9) and from (0,3,9) to (0,-3,9) on the function surface. (See attached manual.) Figure 8-13 As shown in the comparative experiments above, under the same noise-contaminated data distribution conditions, our proposed method can learn a more continuous, smoother, and more accurate geodesic path than Isomap, indicating that our proposed method has better robustness when the data is uneven and contaminated by noise.

[0274] Although the inventive concept has been specifically shown and described with reference to exemplary embodiments thereof, those skilled in the art should understand that various changes in form and detail may be made therein without departing from the spirit and scope of the invention as defined by the claims.

Claims

1. A method for calculating discretized geodesics on Riemannian manifolds based on deep learning, characterized in that, The method includes the following steps: S1. Obtain known data of the target object to be tested. The target object to be tested is an aircraft. Obtain the coordinates of the aircraft surface, the flight state, and the pressure coefficient of the corresponding point. The coordinates of the aircraft surface include three-dimensional spatial coordinates of the x, y, and z axes. The flight state includes Mach number, angle of attack number, and Reynolds number. The coordinates of the aircraft surface, the flight state, and the pressure coefficient of the corresponding point are summarized to form a 7-dimensional space test data sample. S2. Use the test data samples obtained in step S1 as the input to the model and input them into an unsupervised neural network model for unsupervised learning. The unsupervised neural network model includes a differential blank network and an isometric mapping network. During the unsupervised learning process, the differential blank network and the isometric mapping network alternately perform gradient descent optimization. Specifically, the model parameters calculated in the differential blank network are first fixed, and the model parameters of the isometric mapping network are trained with these model parameters as the optimization target. The model parameters in the isometric mapping network are then iteratively optimized. S3. When the unsupervised neural network has been iterated and optimized a set number of times in step S2, or the loss function after iterative optimization meets a set threshold, the trained differential plagiarism network and the isometric mapping network are obtained. S4. In the original Euclidean space of the data sample to be tested in step S1, select the starting point and ending point of the geodesic to be calculated. Use the trained differential monolithic network obtained in step S3 to map the starting point and ending point to the Riemann space. Select the region containing the starting point and ending point in the Riemann space as the discretization region. Use the equidistant mapping network trained in step S3 to obtain the Riemann metric of any position in the Riemann space. According to the movement cost calculation formula of discretization pathfinding, obtain the movement cost matrix of all points in the discretization region. S5. Based on the movement cost matrix calculated in step S4, use the pathfinding algorithm to find the shortest path from the starting point to the ending point in the Riemann space. This shortest path is the geodesic path. S6. Input the points on the geodesic path obtained in step S5 into the trained differential polynomial network obtained in step S3, and reconstruct the points on the geodesic path into Euclidean space for display.

2. The method for calculating discretized geodesics on a Riemannian manifold based on deep learning as described in claim 1, characterized in that: In step S2, the differential blank network is a differential blank autoencoder, which includes an encoder and a decoder. Receiving high-dimensional space The sample data to be tested is mapped to... In space; decoder take over Data in space, and mapping it back Space; in which Let represent an n-dimensional differential manifold, and assign a Riemannian metric to each point on the n-dimensional differential manifold, such that the n-dimensional differential manifold forms a Riemannian manifold; and They are all continuous functions.

3. The method for calculating discretized geodesics on a Riemannian manifold based on deep learning as described in claim 2, characterized in that: The Riemannian metric is calculated from an extrinsic perspective, and its formula is as follows: ； ； ； In the formula, Represents the tangent vector in Riemann space; Indicating the Riemannian trend Axial differential; Representing Riemannian space Axial basis vectors; Indicating European-style space Axial basis vectors; Indicating European-style space Axial differential; Indicating European-style space Axial basis vectors; Representing Riemannian manifolds Axial basis vectors; It indicates that Riemannian fashion is popular. Axial differential; Indicating European-style space Axial differential; Representing European-style space Axial differential; Representing Riemannian manifolds Axis basis vectors; Riemannian metric represents the Riemannian metric when decoded. Once trained, the decoder can be used. The Jacobian matrix is ​​used to calculate the Riemannian metric of any coordinate point on an n-dimensional differential manifold.

4. The method for calculating discretized geodesics on a Riemannian manifold based on deep learning as described in claim 3, characterized in that: The continuous Riemannian manifold is meshed into a discrete Riemannian manifold space; specifically, Specialization of discrete sampling: ， ; In the formula, Indicates directional intervals; Indicates the modulus interval; Denotes the differential at any point on the Riemannian flow. It can point to any angle within its tangent space; Represents any point on the Riemannian flow The length of the module at the location; The discretized grid points can move in 8 directions. The start and end points of the geodesics do not necessarily fall on the discretized grid. They are mapped to the nearest grid point using the following mapping function: ； In the formula, This function calculates and returns the mapping of sample point n to the nearest grid point node in Manhattan distance; The discretized geodesic calculation equation is as follows: ； And there is, ； ； ； In the formula, In Riemannian manifolds The mapping function of points, that is The point is mapped to the nearest grid point; The mapping function for point b in a Riemannian manifold is used to map point b to the nearest grid point. Indicates the modulus interval; For directional spacing; " in " indicates a continuous mapping between two functions; This represents the step size after discretization; Indicates in Location information at the point; Indicates the direction after discretization; This represents the second-order tensor of the Riemannian metric.

5. The method for calculating discretized geodesics on a Riemannian manifold based on deep learning as described in claim 4, characterized in that: The Monte Carlo integral of the discretized geodesic calculation equation is subjected to error analysis and reduction. Based on the error analysis results, a regularization term that satisfies the error reduction constraint is designed.

6. A method for calculating discretized geodesics on a Riemannian manifold based on deep learning, as described in any one of claims 1-5, characterized in that: In step S2, the differential monolithic network optimizes its parameterized network using gradient descent, with the loss function being: ； In the formula, This represents the reconstruction accuracy constraint of the differential monolithic autoencoder in the differential monolithic network; This means using the Spearman correlation coefficient to penalize the sample mapping to any two dimensions in a low-dimensional manifold. The correlation between them; This indicates that the Monte Carlo integration error in the discretized geodesic process is reduced by decreasing the variance. , This is the adjustable penalty coefficient for the loss term; In the above formula, n represents the total number of samples in the dataset; k represents the number of the k-th sample. This represents the reconstructed sample data; This represents the original data of the k-th sample; The set of dimensions representing a Riemannian manifold; Indicates variance; This represents the distance calculation within the popular tangent space; Represents the inverse mapper of the differential homeomorphism; This represents a differential homeomorphism mapper.

7. The method for calculating discretized geodesics on a Riemannian manifold based on deep learning as described in claim 6, characterized in that: The loss function of the isometric mapping network is designed as follows: ； In the formula, This is the adjustable penalty coefficient for the loss term; This indicates the calculation of Euclidean distance within a local area; Represents an isometric mapper; This represents the sampling function.

8. The method for calculating discretized geodesics on a Riemannian manifold based on deep learning as described in claim 7, characterized in that: In step S2, the model parameters calculated in the differential monolithic network are first fixed. Using these model parameters as the optimization objective, the model parameters of the isometric mapping network are trained. The model parameters in the isometric mapping network are then iteratively optimized. Specifically, this means... The result calculated from the differential blank is now... Fixed, using this as the optimization objective, train the isometric mapping network. Thus, for the isometric mapping network The parameters in the data are iteratively optimized; The overall loss function is: .

9. A method for calculating discretized geodesics on a Riemannian manifold based on deep learning, as described in any one of claims 1-5, characterized in that: In step S4, the formula for calculating the movement cost of discretized pathfinding indicates that when moving from a grid point to... Move direction to the next grid point The cost of moving at that time is: ； make ; In the above formula, inf represents positive infinity; Represents any sample n; dataset represents a dataset; This indicates the direction at node node. Directional step size The distance; Indicates at node Orientation Directional step size The distance; This indicates that there exists an n; When no sample n can be found in any dataset, can it be achieved through... When mapping to a grid point, treat that grid point as a wall and set all paths connecting it to infinity; otherwise, map a grid point to... The movement cost in the direction is set to the node grid point direction. The Riemannian metric unit inner product distance in tangent space plus the next grid point in this direction. The average of the Riemannian unit inner product distance in the tangent space when moving toward the node.

10. The method for calculating discretized geodesics on a Riemannian manifold based on deep learning as described in claim 9, characterized in that: When searching for the geodesics of any two samples in a field of view machine, if: ， ， ； Then gradually enlarge until the geodesic distance is not equal to inf.

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