A terrain profile continuous prediction method based on neural differential equation
By combining Long Short-Term Memory (LSTM) networks and neural differential equations, the problem of predicting irregularly sampled terrain profile data was solved, achieving continuous, stable, and high-precision prediction of terrain profiles.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- NANJING UNIV OF AERONAUTICS & ASTRONAUTICS
- Filing Date
- 2026-03-16
- Publication Date
- 2026-06-19
AI Technical Summary
Existing technologies struggle to effectively process irregularly sampled terrain profile data, resulting in discontinuous prediction results and low accuracy. Traditional methods are prone to destroying temporal information when processing irregularly sampled time series, thus affecting prediction performance.
By combining a Long Short-Term Memory (LSTM) network with neural differential equations and using an Ordinary Differential Equation (ODE) solver to continuously model the hidden states, a method for continuous prediction of terrain profiles is constructed to avoid the loss of temporal information.
It improves the stability and accuracy of terrain profile prediction, and can achieve continuous and smooth prediction results under irregular sampling and long time spans, breaking through the limitations of traditional methods.
Smart Images
Figure CN122241077A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of continuous prediction of topographic profiles, and specifically to a method for continuous prediction of topographic profiles based on neural differential equations. Background Technology
[0002] Topographic profile data belongs to the generalized time series. A time series refers to a series of information data arranged chronologically with fixed sampling times. This type of data is usually obtained through observation or recording and has strong uncertainty; therefore, it is essentially a sequence of random variables. Topographic profile data also exhibits characteristics such as discreteness and uncertainty, thus it can be classified as a generalized time series and can be modeled and analyzed using time series processing and prediction estimation methods. This invention uses deep learning methods as a foundation to study the prediction problem of irregularly sampled topographic profiles. This invention uses deep learning as an algorithm to study the prediction problem of irregularly sampled topographic profiles. Because time series with non-uniform intervals exist in many applications, and are difficult to model using standard recurrent neural networks (RNNs).
[0003] Long Short-Term Memory (LSTM) networks are commonly used deep learning models for processing high-dimensional, sequential data, and are widely applied to typical time series tasks such as text and speech. However, when time series exhibit irregular sampling characteristics, directly using traditional methods often fails to guarantee predictive effectiveness. A common approach is to divide the time axis into several equal-length intervals and estimate or summarize the observations using methods such as averaging. However, this preprocessing process can destroy temporal information, especially the latent variable characteristics inherent in the sampling time itself, thus weakening the model's predictive ability.
[0004] Currently, there are two main approaches to processing irregularly sampled time series data: one is to use generative models to fill in missing data and restore continuity, and the other is to directly input the time series and its timestamp information as a whole into a neural network, avoiding the traditional missing data completion problem and directly performing time series prediction and inference. This invention adopts the second approach, combining a Long Short-Term Memory (LSTM) network with neural differential equations to effectively process long-span and irregularly sampled terrain data. Compared to traditional methods, our innovation lies in continuously modeling the hidden states using an Ordinary Differential Equation (ODE) solver, avoiding the loss of temporal information during data processing and improving the stability and accuracy of predictions. Existing research has shown that using a basic LSTM model to process missing sequential input data allows its output features to serve as a key basis for prediction, demonstrating good performance in multi-label classification tasks of clinical time series.
[0005] Compared to traditional Recurrent Neural Networks (RNNs), Long Short-Term Memory (LSTM) networks introduce forget gates, input gates, and output gates into their structure, effectively mitigating problems such as vanishing and exploding gradients, and possessing stronger long-term dependency modeling capabilities. Especially when dealing with irregularly sampled and long-span terrain profile data, LSTM networks can maintain long-term memory and dynamic updates of key information, ensuring the continuity and smoothness of prediction results, thereby significantly improving prediction accuracy and stability, and exhibiting stronger robustness and generalization ability. This characteristic makes LSTM networks more advantageous in this field, representing a technological advancement distinct from existing Recurrent Neural Networks (RNNs). Summary of the Invention
[0006] Objective of the Invention: The technical problem to be solved by this invention is to address the shortcomings of existing technologies by providing a method for continuous prediction of terrain profiles based on neural differential equations. This method includes the following steps:
[0007] Step 1: Acquire terrain data of the target area and preprocess the terrain data to obtain terrain profile data. Construct a training dataset. The terrain data consists of terrain elevation data sequences collected by airborne sensors such as terrain radar and radio altimeters along the aircraft path at a fixed sampling frequency during flight. The online-acquired terrain elevation data sequences are jittered point-by-point to increase noise, thus constructing a terrain elevation data sequence dataset. Based on the maximum and minimum elevation values of individual points in the elevation data sequence dataset, all data in the dataset are normalized so that the values of points on each terrain elevation data sequence are located in the interval [0, 1]. This yields the terrain data training set. The terrain elevation data sequences collected along the flight path are arranged in sampling order to form a type of time series data with irregular sampling intervals.
[0008] Step 2: Construct a Long Short-Term Memory (LSTM) network as the encoder. Its input is the terrain elevation sequence data. The encoder generates the mean and variance of latent variables through a gating mechanism, assuming that these latent variables conform to a multidimensional Gaussian distribution. The spatial sampling sequence of the terrain profile along the flight path is mapped to a temporal input with sampling interval information according to the acquisition order. This temporal input is then fed into the gated recursive encoder, which outputs the hidden states and the initial values of the potential states required for continuous time evolution.
[0009] Step 3: Construct a neural differential equation dynamic model parameterized by a neural network, and use the initial value of the potential state as the initial condition. Call the ordinary differential equation solver to solve the potential state in continuous time to obtain the continuous potential state trajectory within the observation interval.
[0010] Step 4: Extrapolate the potential state based on the continuous-time dynamics model within the prediction interval to obtain the extrapolated potential state corresponding to the target time. Then, fuse the initial value of the potential state obtained in Step 2 with the continuous potential state trajectory obtained in Step 3 to obtain a joint potential representation. Input the joint potential representation into the decoding network and output the topographic elevation prediction value corresponding to the target time. The output of the decoding network is the prediction value under the normalized scale. Perform inverse normalization processing on the prediction value according to the maximum and minimum values recorded in Step 1 to restore it to the original topographic elevation scale, thereby obtaining the continuous prediction result of the topographic profile.
[0011] Step 1 includes: perturbing the online acquired terrain elevation sequence point by point to form noise-enhanced samples; normalizing the training dataset according to the maximum and minimum values of the terrain elevation sequence so that the sampled value of each terrain elevation sequence is within a preset interval; and scaling the terrain elevation values to reduce the numerical differences between different samples, making the curves predicted by the model more flat and smooth. Based on the maximum and minimum values of the single-point elevation in the elevation sequence dataset, normalize all data in the elevation sequence dataset so that the values of points on each terrain elevation sequence are within the interval [0, 1], thus obtaining the terrain data training set:
[0012] in, Is this the original terrain elevation value? and Let these represent the maximum and minimum terrain elevation values in the training dataset, respectively. This is to output normalized predicted values.
[0013] In step 2, the encoder's gating mechanism is configured as follows: the forget gate controls the retention ratio of historical elevation information in the terrain elevation sequence to avoid the accumulation of invalid information; the input gate combines the current terrain input and the previous hidden state with the candidate state to dynamically update the mean and variance of the hidden variables; the output gate adjusts the hidden state output according to the current terrain features to ensure that the encoded features are adapted to the continuous transformation requirements of the Ordinary Differential Equation (ODE) solver.
[0014] By combining time series data with timestamp information, and describing it using a Long Short Time Memory (LSTM) network, predictions can be made directly based on modeling the time series of missing data.
[0015] Step 2 also includes: mapping the spatial sampling point number along the track to a pseudo-time index, and constructing the spatial interval or timestamp difference between adjacent sampling points as sampling interval information as part of the input of the neural differential equation dynamics model, so that the interval change caused by irregular spatial sampling can be explicitly modeled by the continuous time dynamics model.
[0016] Step 3 includes the following steps:
[0017] Step 3.1: Let the corresponding times of two adjacent sampling points be respectively and Sampling interval The encoder infers the distribution of the initial latent variables based on the observation sequence, and samples them to obtain... The latent state is then extrapolated continuously in time using an ordinary differential equation ODE solver:
[0018] ,
[0019] in This represents a continuous-time dynamic function parameterized by a neural network. For the first along the flight path The pseudo-time index corresponding to each terrain sampling point This represents the pseudo-time interval between adjacent sampling points. For the first The status of each terrain sampling point Indicates from Continuous evolution to The state obtained afterwards This represents the operator for solving ordinary differential equations;
[0020] Step 3.2: Construct terrain feature vectors :
[0021] ,
[0022] in For the first Topographic elevation of each topographic sampling point Indicates the first The sampling time for each terrain sampling point; if higher-order features such as slope and curvature need to be introduced, the code dimension and data construction dimension need to be expanded simultaneously.
[0023] Calculate gate control variables and state updates:
[0024] ,
[0025] ,
[0026] ,
[0027] ,
[0028] ,
[0029] ,
[0030] in , , These represent fully connected layers with activation functions for the forget gate, input gate, and output gate, respectively. Indicates the first The state after gating update of each terrain sampling point Indicates candidate memory states, Indicates the first The hidden state of each terrain sampling point This indicates the hidden state of the previous terrain sampling point. , , Let represent the gate vectors for the forget gate, input gate, and output gate, respectively. Indicates the candidate state. , , , Represents the weight matrix. , , , Let represent the bias vectors updated after the forget gate, input gate, output gate, and gated operation, respectively, and tanh represent the activation function. This represents the element-by-element multiplication operation between corresponding elements of a vector.
[0031] In step 4, the extrapolated potential state includes uncertain characterization parameters, which are generated according to the following steps:
[0032] Step 4.1: From statistical mapping network Output the mean vector and scale vector based on the hidden state:
[0033] ,
[0034] Among them, statistical mapping network Input hidden state, output distribution parameters. For the first The vector of latent variable means at each location For the first A scale vector of latent variables at each location; generating the extrapolated latent state corresponding to the target time:
[0035] ,
[0036] in This represents the potential representation corresponding to the target time or target spatial location. It is a multidimensional standard normal distribution with a mean of 0 and a covariance of identity matrix I. Indicates the distribution from the standard normal distribution The random noise vector obtained by sampling.
[0037] In step 4, the predicted observations are obtained through the following steps:
[0038] Step 4.2: Decoding Prediction: Decoding the Latent Representation Input Decoder The predicted observations were obtained:
[0039] ,
[0040] in This represents the predicted elevation value at time t. This represents the trajectory of latent variables obtained from the evolution of neural differential equations.
[0041] Step 4 also includes:
[0042] Step 4.3: Represent the joint potential as:
[0043] ,
[0044] in Represents the prediction residual of the i-th sampling point, and... Input multilayer connection decoding network output:
[0045] ,
[0046] in This represents the model's predicted value for the i-th sample. For fusion function, This is the decoder network. The resulting joint latent representation... It simultaneously incorporates terrain context information represented by historical hidden states and future trend information represented by extrapolated latent variables, enabling the decoding network to output continuous and consistent terrain profile prediction results at any target time or target spatial location, and improving the continuity and stability of the predicted profile.
[0047] Step 4.4: After the decoding network outputs the normalized predicted values, the original terrain elevation scale is restored using the following inverse normalization formula:
[0048] ,
[0049] The above inverse normalization process yields the terrain prediction results corresponding to the actual terrain elevation.
[0050] Step 5 includes: Model training is accomplished by maximizing the likelihood function of the observed data, with the optimization objective expressed as:
[0051] ,
[0052] in To infer the distribution for the encoder, As a prior distribution, This represents the model training objective function. Represents topographic profile data. Represents the initial potential state. This represents the expectation of the distribution of latent variables. Indicates the probability of generating the observed data. Let KL divergence be the denoted KL divergence. In the above formula, the first term represents the prediction error, and the second term represents the latent variable regularization. To ensure good continuity and stability of the latent variable space, a variational inference-based objective function is introduced during model training. By minimizing the KL divergence between the reconstruction error and the latent distribution, the model parameters are jointly optimized.
[0053] The present invention also provides an electronic device, including a processor and a memory, the memory storing program code that, when executed by the processor, causes the processor to perform the steps of the method.
[0054] The present invention also provides a storage medium storing a computer program or instructions that, when the computer program or instructions are run on a computer, execute the steps of the method described.
[0055] NODE is an abbreviation for Neural Ordinary Differential Equation. The idea behind neural differential equations is to continuously transform the residual mapping (R) in a neural network into differential equations, thereby extending the originally discrete iterative process between layers into a continuous-time dynamic system model.
[0056] LSTM stands for Long Short-Term Memory, an improved recurrent neural network architecture used for modeling time series data, particularly suitable for tasks involving long-term dependencies. The basic idea of LSTM is to introduce cell states and gating mechanisms such as forget gates, input gates, and output gates into the network structure. This controls the retention and forgetting of information, ensuring that the current hidden state depends not only on the input data and the hidden state of the previous time step, but also retains or updates key information over a longer time span. Through this structure, LSTM effectively solves the gradient vanishing and gradient exploding problems that traditional recurrent neural networks easily encounter during long-sequence training, demonstrating stronger long-term dependency modeling capabilities.
[0057] The present invention has the following beneficial effects:
[0058] 1. By introducing neural differential equations, the originally discrete neural network training process is transformed into continuous dynamic modeling, making the prediction results smoother and more accurate in the time dimension.
[0059] 2. By combining the Long Short-Term Memory (LSTM) network structure to model the long-term dependencies of the input data, the problem of insufficient memory and gradient vanishing that is prone to occur in traditional methods is avoided, and the ability to characterize high-order sequence data of complex terrain is effectively improved.
[0060] 3. By using the ordinary differential equation ODE solver to extrapolate and solve random latent variables, continuous prediction of any future time point is achieved, breaking through the limitation of traditional methods that can only output results at fixed time points.
[0061] 4. The model can adapt to irregular sampling and long-term data input, with small prediction error, high accuracy, and strong generalization ability.
[0062] Therefore, this invention has significant advantages in continuous topographic profile prediction tasks, including high accuracy, good stability, and strong adaptability, and has good engineering application value and promotion prospects. Attached Figure Description
[0063] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments, and the advantages of the present invention in the above and / or other aspects will become clearer.
[0064] Figure 1 This is a schematic diagram of the fitting process structure (Long Short Time Memory Network (LSTM) encoder, differential equation solver, and Multilayer Perceptron Decoder (MLP Deconder).
[0065] Figure 2 This is a schematic diagram comparing the predicted values generated in an embodiment of the present invention with the actual data. Detailed Implementation
[0066] like Figure 1 As shown, this embodiment of the invention provides a method for continuous prediction of terrain profiles based on neural differential equations, including the following steps:
[0067] Step 1: Acquire terrain data for the target area and preprocess the terrain data to obtain terrain profile data. Construct a training dataset. The terrain data consists of terrain elevation data sequences acquired during flight by airborne sensors such as terrain radar and radio altimeters along the aircraft's flight path at a fixed sampling frequency. The online-acquired terrain elevation data sequences are jittered point-by-point to increase noise, thus constructing a terrain elevation data sequence dataset. Based on the maximum and minimum elevation values of individual points in the elevation data sequence dataset, all data in the dataset are normalized so that the values of points on each terrain elevation data sequence fall within the interval [0, 1], thereby obtaining the terrain data training set.
[0068] Step 2: Construct a Long Short-Term Memory (LSTM) network as the encoder. Its input is the terrain elevation sequence data. The encoder generates the mean and variance of latent variables through a gating mechanism, assuming that these latent variables conform to a multidimensional Gaussian distribution. The spatial sampling sequence of the terrain profile along the flight path is mapped to a temporal input with sampling interval information according to the acquisition order. This temporal input is then fed into the gated recursive encoder, which outputs the hidden states and the initial values of the potential states required for continuous time evolution.
[0069] Step 3: Construct a neural differential equation dynamic model parameterized by a neural network, and use the initial value of the potential state as the initial condition. Call the ordinary differential equation solver to solve the potential state in continuous time to obtain the continuous potential state trajectory within the observation interval.
[0070] Step 4: Extrapolate the latent state based on the continuous-time dynamics model within the prediction interval to obtain the extrapolated latent state corresponding to the target time. Then, fuse the latent state features obtained in Step 2 through the Long Short Time Memory (LSTM) network and the Ordinary Differential Equation (ODE) solver with the latent variable features generated by the ODE extrapolation network in Step 3 to obtain a joint latent representation. Input the joint latent representation into the decoding network and output the topographic elevation prediction value corresponding to the target time. The output of the decoding network is the prediction value at a normalized scale. Further, perform inverse normalization processing on the prediction value according to the maximum and minimum values recorded in Step 1 to restore it to the original topographic elevation scale, thereby obtaining the continuous prediction result of the topographic profile.
[0071] Step 1 includes:
[0072] The online-acquired terrain elevation sequence is perturbed point-by-point to generate noise-enhanced samples; the training dataset is normalized based on the maximum and minimum values of the terrain elevation sequence, ensuring that the sampled values of each terrain elevation sequence are within a preset interval; and the terrain elevation values are scaled to reduce the numerical differences between different samples, making the curves predicted by the model more flat and smooth. Based on the maximum and minimum values of the single-point elevation in the elevation sequence dataset, all data in the elevation sequence dataset are normalized so that the values of points on each terrain elevation sequence are within the interval [0, 1], thus obtaining the terrain data training set: .
[0073] Step 2 includes:
[0074] The spatial sampling point numbers along the flight path are mapped to pseudo-time indices, and the spatial intervals or timestamp differences between adjacent sampling points are constructed as sampling interval information and used as part of the input of the neural differential equation dynamics model, so that the interval changes caused by irregular spatial sampling can be explicitly modeled by the continuous time dynamics model.
[0075] Step 3 includes:
[0076] Step 3.1: Let the corresponding time of adjacent sampling points be... and The sampling interval is The encoder's Long Short-Term Memory (LSTM) network infers the distribution of the initial latent variables based on the observed sequences, and samples them to obtain... Then, the latent state is extrapolated continuously in time using the constant differential equation ODE:
[0077] ;
[0078] Step 3.2: Construct terrain feature vectors :
[0079] ,
[0080] If higher-order features such as slope and curvature need to be introduced, both the code dimension and the data construction dimension need to be expanded simultaneously:
[0081] Based on this, the gate control input and state update are calculated:
[0082] ,
[0083] ,
[0084] ,
[0085] ,
[0086] , ;
[0087] Step 4 includes:
[0088] Step 4.1: From statistical mapping network Based on hidden state Output the mean vector and scale vector:
[0089] ,
[0090] And generate the latent variables corresponding to the target time:
[0091] ,
[0092] Step 4.2 includes decoding and prediction: extrapolating the neural network constant differential equation (ODE) to obtain the latent sequence. Input Decoder The predicted observations were obtained:
[0093] ,
[0094] Step 4.3 includes: representing the joint potential as:
[0095] ,
[0096] And Input multilayer connection decoding network output:
[0097] ,
[0098] The resulting joint potential representation It simultaneously incorporates terrain context information represented by historical hidden states and future trend information represented by extrapolated latent variables, enabling the decoding network to output continuous and consistent terrain profile prediction results at any target time or target spatial location, and improving the continuity and stability of the predicted profile.
[0099] Step 4.4: After the decoding network outputs the normalized predicted values, the original terrain elevation scale is restored using the following inverse normalization formula:
[0100] ,
[0101] The above inverse normalization process yields the terrain prediction results corresponding to the actual terrain elevation.
[0102] Step 5 includes: Model training is accomplished by maximizing the likelihood function of the observed data, and its optimization objective can be expressed as:
[0103] ,
[0104] In the above formula, the first term represents the prediction error, and the second term represents the latent variable regularization. To ensure good continuity and stability of the latent variable space, a variational inference-based objective function is introduced during model training. By minimizing the KL divergence between the reconstruction error and the latent distribution, the model parameters are jointly optimized.
[0105] This embodiment selects 100 terrain data points, sets a random removal rate of 30%, and transforms the terrain into an irregular / unequal time series. The first 80 points are used as a single training sample, and perturbations are added to form a training sample set. Prediction is performed up to the 82nd terrain data point, and the results are obtained according to steps 1 to 4. Figure 2 The topographic profile shown is continuous, and the prediction realizes data input prediction over a long period of time, with good prediction accuracy and small error.
[0106] This invention provides a method for continuous prediction of terrain profiles based on neural differential equations. Many methods and approaches exist for implementing this technical solution; the above description is merely a preferred embodiment of the invention. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of this invention, and these improvements and modifications should also be considered within the scope of protection of this invention. All components not explicitly stated in this embodiment can be implemented using existing technologies.
Claims
1. A method for continuous prediction of terrain profiles based on neural differential equations, characterized in that, Includes the following steps: Step 1: Obtain terrain data of the target area, preprocess the terrain data to obtain terrain profile data, and construct a training dataset; The terrain data is a series of terrain height data collected by airborne sensors along the aircraft's flight path at a fixed sampling frequency during flight. The terrain height data acquired online is jittered point by point to increase noise and construct a terrain height data set. Based on the maximum and minimum values of the single-point elevation in the high-order sequence dataset, all data in the high-order sequence dataset are normalized so that the values of points on each terrain high-order sequence are in the interval [0, 1]. This yields the terrain data training set. The terrain high-order sequences collected along the flight path are arranged in the sampling order to form a type of time series data with irregular sampling intervals. Step 2: Construct a Long Short Time Memory (LSTM) network as an encoder. The input is the terrain height sequence data. The encoder generates the mean and variance of the latent variables through a gating mechanism. The spatial sampling sequence of the terrain profile along the flight path is mapped into a temporal input with sampling interval information according to the acquisition order. The described temporal input is fed into the gated recursive encoder, and the output is the hidden state and the initial value of the potential state required for continuous time evolution. Step 3: Construct a neural differential equation dynamic model parameterized by a neural network, and use the initial value of the potential state as the initial condition. Call the ordinary differential equation solver to solve the potential state in continuous time to obtain the continuous potential state trajectory within the observation interval. Step 4: Extrapolate the potential state based on the continuous-time dynamics model within the prediction interval to obtain the extrapolated potential state corresponding to the target time. Then, fuse the initial value of the potential state obtained in Step 2 with the continuous potential state trajectory obtained in Step 3 to obtain a joint potential representation. Input the joint potential representation into the decoding network and output the topographic elevation prediction value corresponding to the target time. The output of the decoding network is the prediction value under the normalized scale. Perform inverse normalization processing on the prediction value according to the maximum and minimum values recorded in Step 1 to restore it to the original topographic elevation scale, thereby obtaining the continuous prediction result of the topographic profile.
2. The method according to claim 1, characterized in that, Step 1 includes: perturbing the terrain height sequence obtained online point by point to form a noise enhanced sample; normalizing the training data set according to the maximum and minimum values of the terrain height sequence, so that the sampling values of each terrain height sequence are within a preset interval; and scaling the terrain height values to reduce the numerical differences between different samples; according to the maximum and minimum values of the single-point height in the height sequence data set, normalizing all data in the height sequence data set, so that the values of the points on each terrain height sequence are located in the interval [0, 1], thereby obtaining a terrain data training set: in, Is this the original terrain elevation value? and Let these represent the maximum and minimum terrain elevation values in the training dataset, respectively. This is to output normalized predicted values.
3. The method according to claim 2, characterized in that, In step 2, the encoder's gating mechanism is configured as follows: the forget gate controls the retention ratio of historical elevation information in the terrain elevation sequence; the input gate combines the current terrain input and the previous hidden state with the candidate state to dynamically update the mean and variance of the hidden variables; the output gate adjusts the hidden state output according to the current terrain features. By combining time series data with timestamp information, and describing it using a Long Short Time Memory (LSTM) network, predictions can be made directly based on modeling the time series of missing data.
4. The method according to claim 3, characterized in that, Step 2 also includes: mapping the spatial sampling point number along the track to a pseudo-time index, and constructing the spatial interval or timestamp difference between adjacent sampling points as sampling interval information as part of the input of the neural differential equation dynamics model, so that the interval change caused by irregular spatial sampling can be explicitly modeled by the continuous time dynamics model.
5. The method according to claim 4, characterized in that, Step 3 includes the following steps: Step 3.1: Let the corresponding times of two adjacent sampling points be respectively and Sampling interval The encoder infers the distribution of the initial latent variables based on the observation sequence, and samples them to obtain... The latent state is then extrapolated continuously in time using an ordinary differential equation ODE solver: ， in This represents a continuous-time dynamic function parameterized by a neural network. For the first along the flight path The pseudo-time index corresponding to each terrain sampling point This represents the pseudo-time interval between adjacent sampling points. For the first The status of each terrain sampling point Indicates from Continuous evolution to The state obtained afterwards This represents the operator for solving ordinary differential equations; Step 3.2: Construct terrain feature vectors : ， in For the first Topographic elevation of each topographic sampling point Indicates the first Sampling time for each terrain sampling point; Calculate gate control variables and state updates: ， ， ， ， ， ， in , , These represent fully connected layers with activation functions for the forget gate, input gate, and output gate, respectively. Indicates the first The state after gating update of each terrain sampling point Indicates candidate memory states, Indicates the first The hidden state of each terrain sampling point This indicates the hidden state of the previous terrain sampling point. , , Let represent the gate vectors for the forget gate, input gate, and output gate, respectively. Indicates the candidate state. , , , Represents the weight matrix. , , , Let represent the bias vectors updated after the forget gate, input gate, output gate, and gated operation, respectively, and tanh represent the activation function. This represents the element-by-element multiplication operation between corresponding elements of a vector.
6. The method according to claim 5, characterized in that, In step 4, the extrapolated potential state includes uncertain characterization parameters, which are generated according to the following steps: Step 4.1: From statistical mapping network Output the mean vector and scale vector based on the hidden state: ， Among them, statistical mapping network Input hidden state, output distribution parameters. For the first The vector of latent variable means at each location For the first A scale vector of latent variables at each location; generating the extrapolated latent state corresponding to the target time: ， in This represents the potential representation corresponding to the target time or target spatial location. It is a multidimensional standard normal distribution with a mean of 0 and a covariance of identity matrix I. Indicates the distribution from the standard normal distribution The random noise vector obtained by sampling.
7. The method according to claim 6, characterized in that, In step 4, the predicted observations are obtained through the following steps: Step 4.2: Decoding Prediction: Decoding the Latent Representation Input Decoder The predicted observations were obtained: ， in This represents the predicted elevation value at time t. This represents the trajectory of latent variables obtained from the evolution of neural differential equations.
8. The method according to claim 7, characterized in that, Step 4 also includes: Step 4.3: Represent the joint potential as: ， in Represents the prediction residual of the i-th sampling point, and... Input multilayer connection decoding network output: ， in This represents the model's predicted value for the i-th sample. For fusion function, For decoder networks; Step 4.4: After the decoding network outputs the normalized predicted values, the original terrain elevation scale is restored using the following inverse normalization formula: ； Step 5 includes: Model training is accomplished by maximizing the likelihood function of the observed data, with the optimization objective expressed as: ， in To infer the distribution for the encoder, For the prior distribution, This represents the model training objective function. Represents topographic profile data. Represents the initial potential state. This represents the expectation of the distribution of latent variables. Indicates the probability of generating the observed data. This represents the KL divergence.
9. An electronic device, characterized in that, It includes a processor and a memory, the memory storing program code that, when executed by the processor, causes the processor to perform the steps of the method as described in any one of claims 1 to 8.
10. A storage medium, characterized in that, It stores a computer program or instructions that, when run on a computer, perform the steps of the method as described in any one of claims 1 to 8.