Deep operator network ocean flow velocity prediction method based on Green function

Abstract

The invention discloses a depth operator network ocean flow velocity prediction method based on a Green function, and belongs to the technical field of ocean current prediction, and the method comprises the steps: constructing a depth operator network model based on the Green function; the depth operator network model is trained, the fitting process of an incompressible two-dimensional Navier-Stokes equation solution operator is converted into the fitting process of an integral kernel, a pre-training model is obtained, and a Green function and convergence parameters thereof are learned; collecting sea area surface time sequence ocean flow velocity data as training data; and initializing the pre-training model by using the Green function and the convergence parameter thereof, and carrying out transfer learning on the initialized pre-training model by using the training data to obtain an ocean flow velocity prediction model, thereby predicting the ocean flow velocity distribution of the target sea area at the future moment. According to the method, the mathematical theory and deep learning are combined, the modeling ability for non-local dynamics is effectively enhanced by using the Green function, the calculation efficiency and generalization performance are improved, and an efficient and stable calculation method is provided for ocean flow velocity prediction.

Description

Technical Field

[0001] The present invention belongs to the technical field of ocean current prediction, and particularly relates to a method for predicting ocean current velocity based on a depth operator network of Green's function. Background Technique

[0002] Ocean current velocity refers to the movement speed of water bodies in the ocean over time. It is affected by multiple factors such as wind, temperature, salinity, tides, ocean currents, and the Earth's rotation, showing complex spatio-temporal variation characteristics. Ocean currents are one of the main manifestations of ocean current velocity and can be classified into different types such as large-scale ocean currents, boundary currents, and tidal currents according to scale, having a profound impact on the global climate system, marine ecological environment, and human activities. Accurately predicting ocean current velocity is of great significance for shipping safety, climate monitoring, fishery management, extreme weather warning, etc. Ocean observation data usually contains multiple information such as sea surface temperature, sea surface height, and sea surface humidity, providing a solid foundation for studying the characteristics of ocean flow and the changing trend of current velocity. In recent years, with the rapid development of big data and artificial intelligence technologies, the accuracy and real-time performance of ocean current velocity prediction have been continuously improved, further deepening people's understanding of ocean dynamic processes and providing more accurate and reliable scientific support for fields such as shipping, fisheries, meteorology, and climate research. In this context, how to efficiently and accurately predict ocean current velocity and reveal the evolution law of ocean currents has become an important research direction in the current field of ocean science and engineering applications.

[0003] In ocean current velocity prediction, the incompressible two-dimensional Navier-Stokes equation, as the basic equation describing ocean fluid dynamics, can accurately describe the mass conservation and momentum conservation relationships of ocean fluids. This equation consists of a continuity equation and a momentum equation, where the continuity equation ensures that the fluid is incompressible, that is, the fluid volume remains constant during the flow process:

[0004]

[0005] The momentum equation describes the motion law of the fluid under the action of external forces:

[0006]

[0007] where u = u(x, y, t) represents the velocity vector of the fluid at position (x, y) at time t, usually including two components u x and u y, respectively representing the velocities along the x and y directions; p represents the pressure at the current position of the fluid, which determines the acceleration and flow direction of the fluid; ρ represents the fluid density; v represents the kinematic viscosity, reflecting the viscous degree of the fluid; f represents the external force term, such as wind stress or tidal force. In the ocean environment, this equation can be used to simulate the flow characteristics of ocean currents, including complex dynamic processes such as circulation structures, boundary currents, and tidal currents, and is an important mathematical model for studying the evolution of ocean flow velocities.

[0008] Since it is usually difficult to obtain an analytical solution to the Navier-Stokes equation, numerical methods need to be relied on for solution in practical applications. Common methods include the finite difference method (FDM), the finite element method (FEM), and the finite volume method (FVM). They discretize the equation, divide the ocean area into grids, and iteratively solve the evolution of the flow velocity over time at each node. Among them, the finite difference method (FDM) is suitable for structured grids but difficult to accurately describe complex terrain boundaries; the finite element method (FEM) can flexibly handle irregular boundaries but has a relatively higher computational cost and is not conducive to real-time prediction; the finite volume method (FVM) has good conservation properties, is suitable for complex grids and large-scale simulations, and its numerical accuracy depends on the discretization format and turbulence model adopted.

[0009] Although the above methods have been widely used in ocean numerical simulations and their high accuracy and controllability provide important support for many studies, in high-resolution and long-time-scale ocean flow velocity prediction tasks, they often face problems such as long calculation time, large resource consumption, and slow response speed, and are difficult to meet the requirements of real-time and large-scale prediction.

[0010] The Deep Operator Network (DeepONet) is a class of deep models for learning function-to-function mappings, which can directly establish the mapping relationship between the input and solution of partial differential equations. By inputting initial conditions, boundary conditions, etc. into the network in the form of functions, DeepONet can quickly output the solution function of the entire region, thus realizing the efficient prediction of the ocean flow velocity field, showing the advantages of high computational speed and inference efficiency, and is especially suitable for real-time prediction and data-driven modeling scenarios.

[0011] To enhance the physical consistency of the model, the Physics-informed DeepONet introduces the partial differential equation residual as a loss term during the training process, enabling the model to actively satisfy physical constraints when relying on limited observational data, thereby improving its applicability and generalization ability in data-scarce scenarios.

[0012] Although the deep operator network has improved the computational efficiency of ocean current prediction to a certain extent, there are still problems such as limited expressive power, insufficient training efficiency, and difficulty in accurately capturing long-distance interactions in dealing with non-local ocean dynamic processes. Summary of the Invention

[0013] Aiming at the above deficiencies in the prior art, a method for predicting ocean current based on the Green function of the deep operator network provided by the present invention solves the problems of limited expressive power, low training efficiency, and unstable prediction accuracy in the non-local ocean dynamic modeling of the existing deep operator network, thus realizing a more efficient and accurate prediction of the ocean current distribution.

[0014] To achieve the above invention purpose, the technical solution adopted by the present invention is: a method for predicting ocean current based on the Green function of the deep operator network, including the following steps:

[0015] Construct a deep operator network model based on the Green function;

[0016] Train the deep operator network model, transform the fitting process of the incompressible two-dimensional Navier-Stokes equation solution operator into the fitting process of the integral kernel, obtain the pre-trained model, and learn the Green function and its convergence parameters for characterizing the incompressible two-dimensional Navier-Stokes equation solution operator during the training process;

[0017] Collect the time-series ocean current data on the sea surface as training data;

[0018] Initialize the pre-trained model with the learned Green function and its convergence parameters, and perform transfer learning on the initialized pre-trained model using the training data to obtain the ocean current prediction model;

[0019] Use the ocean current prediction model to predict the ocean current distribution at future times in the target sea area.

[0020] The beneficial effects of the present invention are:

[0021] (1) The present invention constructs a deep operator network model for realizing ocean current prediction. By introducing the Green function integral kernel, the learning process of the partial differential equation solution operator is transformed into the fitting problem of the integral kernel, and the Green function is efficiently fitted in the frequency domain by combining the Fourier network structure, thereby improving the expressive power of the model for non-local ocean dynamic processes.

[0022] (2) Compared with the traditional deep operator network method, the method of the present invention significantly improves the training efficiency, and through the combination of pre-training and transfer learning, a more accurate prediction of the future current distribution is realized in a complex ocean environment.

[0023] Furthermore, in the depth operator network model based on the Green's function, the solution process of the partial differential equation is transformed into an integral process by using the Green's function as the integral kernel function.

[0024] Furthermore, the depth operator network model based on the Green's function is expressed as: u(x) = ∫ Ω G(x, x′)f(x′)dx′

[0025] In the formula, u(x) is the solution function of the partial differential equation at the query point x, x′ is an arbitrary position within the integration region Ω, called the source point, the integral kernel function G(x, x′) is the Green's function, representing the contribution of applying a unit-intensity source term at the source point x′ to the solution function at the query point x, and f(x′) is the source term function, representing the source term intensity acting at x′; among them, the integral kernel function G(x, x′) satisfies:

[0026] LG(x, x′) = δ(x ― x′)

[0027] In the formula, δ(x ― x′) is the unit impulse source at the source point x′, and L represents the linear operator.

[0028] The beneficial effects of the above further solution are as follows:

[0029] In the above solution, by transforming the solution process of the partial differential equation into an integral expression based on the Green's function, the model can model the solution operator with a clearer structure, thereby enhancing the ability to depict physical processes. This method can explicitly represent the global influence of the source term on the target region. Especially when the operator has translational invariance, the integral can be simplified to a convolution operation, further reducing the computational complexity, improving the computational efficiency and the model stability. At the same time, this structure provides a solid mathematical foundation for the learning and generalization of the Green's function, enhancing the interpretability of the model.

[0030] Furthermore, the depth operator network model based on the Green's function includes a backbone network, a branch network, and an output layer;

[0031] The backbone network is used to extract the corresponding feature representation according to the position coordinates of the solution to be predicted;

[0032] The branch network is used to extract the global feature representation of the input function according to the function values of the input function in the partial differential equation at multiple sampling points;

[0033] The output layer is used to perform an inner product calculation on the feature vectors output by the backbone network and the branch network to obtain the predicted value at the solution to be predicted.

[0034] Furthermore, a Fourier network module of the Green's function is embedded in the branch network, which includes a first fully connected layer, a plurality of Fourier layers, and a second fully connected layer connected in sequence;

[0035] Each of the Fourier layers includes a fast Fourier transform unit, a Fourier domain linear transformation unit, and an inverse fast Fourier transform unit connected in sequence.

[0036] Furthermore, the processing process of the input function of the partial differential equation by the Fourier network module includes:

[0037] Feature map the function values of the input function at multiple sampling points through the first fully connected layer to obtain a representation vector of a fixed dimension;

[0038] Extract features from the representation vector of the fixed dimension through a plurality of Fourier layers to obtain a spatial domain representation;

[0039] Map the extracted spatial domain representation to a P-dimensional vector through the second fully connected layer;

[0040] Wherein, in each Fourier layer:

[0041] Perform a fast Fourier transform on the representation vector of the fixed dimension through the fast Fourier transform unit to convert it from spatial domain data to a frequency spectrum domain signal;

[0042] Perform an element-wise multiplication operation on the frequency spectrum domain signal and a learnable parameter matrix through the Fourier linear transformation unit, and the parameter matrix is used to fit the representation of the Green's function in the frequency domain;

[0043] Restore the operation result of the Fourier linear transformation unit to the spatial domain through the inverse fast Fourier transform to obtain a spatial domain representation.

[0044] The beneficial effects of the above further solution are:

[0045] In the above solution, by introducing a Fourier transform mechanism in the branch network, after mapping the feature vector after the input source term function from the spatial domain to the frequency domain signal, the Green's function integral kernel is efficiently approximated using a linear transformation layer, and its spatial domain representation is restored through the inverse Fourier transform to complete the modeling of the integral operation. Compared with directly fitting the integral kernel in the spatial domain, this method can reduce the learning complexity, improve the training efficiency and fitting accuracy. At the same time, the frequency domain-based representation is more likely to capture global features, which helps the model accurately express non-local dynamic processes and improve the overall prediction performance.

[0046] Furthermore, the deep operator network model is trained by a data-driven supervised learning method. The training loss function is the mean square error between the output of the prediction solution operator of the incompressible two-dimensional Navier-Stokes equation and the true solution, expressed as:

[0047]

[0048] where x represents the position coordinates of the solution to be predicted, N represents the number of training samples, f i represents the value of the input source term function of the i-th sample at the source term point, represents the output of the operator learned by the deep operator network model for the input function f i , that is, the value of the predicted solution function at point x, and u i (x) represents the value of the true solution function corresponding to the i-th input at x.

[0049] The beneficial effects of the above further solution are as follows:

[0050] In the above solution, through the above supervised learning mechanism for training, the deep operator network can automatically learn the implicit mapping relationship between the source term and the solution without explicitly introducing the physical constraint terms of the partial differential equation, so as to effectively model the complex hydrodynamic characteristics of the ocean. Compared with the traditional method, this solution has better training stability and generalization ability.

[0051] Furthermore, the process of using the training data for transfer learning of the initialized pre-trained model includes:

[0052] Input the position coordinates in the training data into the backbone network of the deep operator network model, and input the corresponding time-series ocean flow velocity data into the branch network of the deep operator network model;

[0053] Based on the input data, perform forward propagation in the deep operator network model;

[0054] In several iterative processes, based on the forward propagation results, minimize the error between the output of the pre-trained model and the observed values in the training data, and gradually and dynamically update the Green's function integral kernel to obtain an ocean flow velocity prediction model.

[0055] The beneficial effects of the above further solution are as follows:

[0056] In the above solution, by initializing the model using the pre-trained Green's function and its convergence parameters during the transfer learning process, and dynamically updating it in combination with the actual longitude, latitude, and time-series ocean current data of the target sea area, the model can better adapt to the hydrodynamic characteristics of the new area. This solution not only improves the prediction accuracy of the model in the target scenario, but also enhances the modeling ability for non-local dynamic processes, contributing to a more accurate and stable prediction of the ocean current evolution trend. Description of the Drawings

[0057] Figure 1 It is a flowchart of the method for predicting ocean current velocity based on the deep operator network using Green's function provided by the present invention.

[0058] Figure 2 It is a schematic diagram of the model structure of the deep operator network based on Green's function provided by the present invention.

[0059] Figure 3 It is a schematic diagram of the transfer learning process of the pre-trained model provided by the present invention. Detailed Embodiments

[0060] The following describes the detailed embodiments of the present invention to facilitate those skilled in the art of the present technology to understand the present invention. However, it should be clear that the present invention is not limited to the scope of the detailed embodiments. For those of ordinary skill in the art of the present technology, as long as various changes are within the spirit and scope of the present invention defined and determined by the appended claims, these changes are obvious, and all inventions created using the concept of the present invention are within the scope of protection.

[0061] The embodiment of the present invention provides a method for predicting ocean current velocity based on the deep operator network using Green's function. This method realizes the prediction of the future ocean current velocity in the ocean area by performing transfer learning on the pre-trained model constructed based on the incompressible two-dimensional Navier-Stokes equation. The specific implementation method is as Figure 1 shown, and includes the following steps:

[0062] Construct a deep operator network model based on Green's function;

[0063] Train the deep operator network model, convert the fitting process of the solution operator of the incompressible two-dimensional Navier-Stokes equation into the fitting process of the integral kernel, obtain the pre-trained model, and learn the Green's function and its convergence parameters for characterizing the solution operator of the incompressible two-dimensional Navier-Stokes equation during the training process;

[0064] Collect the time-series ocean current velocity data on the sea surface as training data;

[0065] Initialize the pre-trained model using the learned Green's function and its convergence parameters, and perform transfer learning on the initialized pre-trained model using the training data to obtain an ocean current velocity prediction model;

[0066] Use the ocean current velocity prediction model to predict the ocean current velocity distribution in the target sea area at future times.

[0067] In the embodiment of the present invention, the deep operator network model based on the Green's function transforms the solution process of the partial differential equation into an integral process by using the Green's function as the integral kernel function, that is, the solution function u(x) of the partial differential equation at the spatial position x is expressed as the integral form of the source term function f(x′) defined within the physical space range Ω and the integral kernel function G(x,x′), which is expressed as: u(x) = ∫ Ω G(x,x′)f(x′)dx′

[0068] In the formula, u(x) is the solution function of the partial differential equation at the query point x (i.e., the position to be solved), x′ is an arbitrary position within the integral region Ω, called the source point, the integral kernel function G(x,x′) is the Green's function, representing the contribution of the unit intensity source term applied at the source point x′ to the solution function at the query point x, and f(x′) is the source term function, representing the source term intensity acting at x′; among them, for a given partial differential equation, its corresponding linear operator L and the Dirac δ function source term δ(x - x′), the integral kernel function G(x,x′) satisfies:

[0069] LG(x,x′) = δ(x ― x′)

[0070] In the formula, δ(x ― x′) is the unit impulse source at the point source x′, and L represents the linear operator.

[0071] When the linear operator L has translational invariance (i.e., the operator property does not change with the spatial position), the Green's function G(x,x′) can be simplified to the displacement difference form G(x ― x′) in one step, thereby degrading the integral into a convolution operation:

[0072] u(x) = ∫ Ω G(x ― x′)f(x′)dx′

[0073] In this embodiment, the deep operator network model based on the Green's function includes a backbone network, a branch network, and an output layer.

[0074] Among them, the Trunk Net is used to extract the corresponding feature representation according to the position coordinates (x, t) of the solution to be predicted; the Branch Net is used to extract the global feature representation of the input function (such as the source term function, initial condition, etc.) in the partial differential equation based on the function values at multiple sampling points; the output layer is used to perform an inner product calculation on the features output by the Trunk Net and the Branch Net to obtain the predicted value u(x, t) at the prediction region (x, t), realizing the mapping from the input function space to the target solution space.

[0075] Specifically, in Figure 2 the shown deep operator network model, f(x) represents the source term function input to the Branch Net, and (x, t) represents the spatial and temporal coordinates input to the Trunk Net; b1 to b p are the outputs of the Branch Net, and t1 to t p are the outputs of the Trunk Net. Finally, the predicted output u(x, t) is obtained through an inner product operation; FFT and IFFT respectively represent the fast Fourier transform and the inverse fast Fourier transform, and LT represents the Fourier linear transform.

[0076] In the embodiment of the present invention, a Fourier network module of the Green's function is embedded in the Branch Net, including a first fully connected layer, multiple Fourier layers, and a second fully connected layer connected in sequence; each Fourier layer includes a fast Fourier transform (FFT) unit, a Fourier domain linear transform (LT) unit, and an inverse fast Fourier transform (IFFT) unit connected in sequence.

[0077] Based on the structure of the Branch Net, the processing process of the Fourier network module of the Green's function for the input function of the partial differential equation includes:

[0078] Performing feature mapping on the function values of the input function at multiple sampling points through the first fully connected layer to obtain a representation vector I(x) with a fixed dimension;

[0079] Performing feature extraction on the representation vector I(x) with a fixed dimension through multiple Fourier layers to obtain a spatial domain representation;

[0080] Mapping the extracted spatial domain representation to a P-dimensional vector through the second fully connected layer;

[0081] Among them, in each Fourier layer:

[0082] Performing a fast Fourier transform on the representation vector I(x) with a fixed dimension through the fast Fourier transform unit to convert it from spatial domain data to a frequency spectrum domain signal for efficient calculation in the frequency spectrum domain;

[0083] Perform an element-wise multiplication operation on the spectral-domain signal and a learnable parameter matrix through a Fourier linear transformation unit, where the parameter matrix is used to fit the representation of the Green's function in the frequency domain;

[0084] Restore the operation result of the Fourier linear transformation unit to the spatial domain through an inverse fast Fourier transform to obtain the spatial-domain representation O(x).

[0085] This embodiment provides the above-mentioned deep operator network model based on the Green's function, and its purpose is to transform the learning process of the partial differential equation solution operator into an integral kernel fitting problem; in the scenario of the present invention, the purpose of this model is to perform transfer learning on a pre-trained model based on the incompressible two-dimensional Navier-Stokes equation, so as to be able to predict the future flow velocity in the ocean area. Therefore, the present invention proposes the above-mentioned deep operator network ocean flow velocity prediction method, which avoids the problems of long calculation time, large consumption of computing resources, and difficulty in meeting real-time and large-scale prediction of traditional numerical solution methods. At the same time, through the combination of pre-training and transfer learning, it further improves the convergence speed of deep learning during the training process and the solution accuracy of partial differential equations, and can more accurately predict the future flow velocity distribution in a complex ocean environment.

[0086] In the embodiment of the present invention, the deep operator network model is trained by using a data-driven supervised learning method, and the training loss function is the mean square error between the predicted solution operator output of the incompressible two-dimensional Navier-Stokes equation and the true solution, which is expressed as:

[0087]

[0088] In the formula, x represents the position coordinate of the solution to be predicted, N represents the number of training samples, f i represents the value of the input source term function of the i-th sample at the source term point, represents the output of the operator learned by the deep operator network model for the input function f i That is, the value of the predicted solution function at point x, u i (x) represents the value of the true solution function corresponding to the i-th input at x.

[0089] In this embodiment, the above-mentioned supervised learning loss function realizes high-precision fitting of the solution operator of the partial differential equation by minimizing the difference between the model prediction result and the true observation data. This method can automatically capture the implicit mapping relationship between the source term function and the solution function without explicitly introducing physical constraint terms, improving the training stability and generalization ability, and providing an efficient and accurate basis for prediction and transfer learning in different sea areas. In this embodiment, a deep operator network model based on the Green's function is used to solve the incompressible two-dimensional Navier-Stokes equation. As a basic physical model describing ocean hydrodynamics, this equation comprehensively reflects the laws of mass conservation and momentum conservation of fluids and can effectively capture the dynamic evolution law of the ocean velocity field.

[0090] In this embodiment, during the training process of the deep operator network model based on the Green's function, the branch network inputs the source term function f(x), which reflects the initial conditions, boundary conditions, or other driving factors affecting fluid motion; the main network inputs the query point coordinates (x,t) to determine the target position and time to be solved. Since the Navier-Stokes equation describes the physical mapping relationship between the source term function and the solution function, the network output is trained to approximate the integral expression defined by the equation. A series of Fourier transform operations in the branch network are used to transform the input signal from the spatial domain to the frequency domain to efficiently capture global non-local information and simplify complex convolution operations into pointwise multiplications, thus significantly reducing the computational complexity. At the same time, the Fourier transform can achieve noise reduction and data compression, highlighting key low-frequency physical information, and thus improving the robustness and stability of the model. Through this method, the model can approximate the integral kernel with a clearer physical structure and provide a clear structured expression for the subsequent inner product fusion of the main network and the branch network.

[0091] In an example of this embodiment, when collecting training data, taking the data from the OSCAR dataset as an example, the surface ocean current velocity data observed in the Mediterranean Sea from 2020 to 2024 is selected. The spatial resolution is a grid cell of 0.25° longitude × 0.25° latitude, and the time resolution is 1 day. Using this dataset first ensures the authenticity and reliability of the data because it is collected from actual observations and can truly reflect the dynamic changes of the surface ocean current in the sea area. Secondly, this dataset has a high spatio-temporal resolution, enabling the data to describe the local and global ocean current characteristics in detail, which is conducive to the model capturing minute changes. At the same time, for the convenience of model training and time series modeling, these time series data are divided into windows of 10 days, where the first 8 days are used as training data and the last 2 days are used as test data. This data organization method can not only reflect the evolution trend of the ocean current in the short term but also facilitate the model to learn time dependence, thereby improving the prediction accuracy. In summary, this dataset not only provides high-quality and continuous training samples for the model but also significantly improves the prediction accuracy and generalization ability of the model for future ocean current trends in complex ocean environments.

[0092] In the embodiment of the present invention, as Figure 3 shown, the process of performing transfer learning on the pre-trained model after initialization using the training data includes:

[0093] Input the position coordinates in the training data into the backbone network of the deep operator network model, and input the corresponding time series ocean current velocity data into the branch network of the deep operator network model;

[0094] Based on the input data, perform forward propagation in the deep operator network model;

[0095] In several iterative processes, based on the forward propagation results, minimize the error between the output of the pre-trained model and the observed values in the training data, and gradually and dynamically update the Green's function integral kernel, so that the model can quickly adapt to the actual dynamic characteristics of the target sea area after several rounds of iteration, and obtain an ocean current velocity prediction model.

[0096] Among them, in each forward propagation process, the branch network receives the discretized source term function of the observed data in the target sea area, and outputs an approximate convolution result after operations such as multiple FFTs, linear transforms LT, and IFFTs. The backbone network inputs the grid query point coordinates and time characteristics and outputs the corresponding spatio-temporal feature vectors. The two are fused through an inner product operation in the output layer to generate a prediction of the ocean current velocity distribution. The model ensures good convergence and prediction accuracy for the multi-scale and highly non-linear ocean hydrodynamic system by dynamically updating the Green's function integral kernel, making the predicted values gradually converge to the real observed data.

[0097] After the transfer learning is completed, the deep operator network model predicts the data for the remaining test period and compares the results with the actual observations to evaluate the generalization ability and accuracy of the model in the new scenario. Experiments have shown that by combining pre-training and transfer learning, the model not only inherits the physical prior of the incompressible two-dimensional Navier-Stokes equations, but also achieves a rapid adaptation to complex ocean current dynamics under the guidance of real observational data in the Mediterranean Sea, providing a solid guarantee for the accurate prediction of future ocean current trends.

[0098] In summary, compared with traditional numerical solution methods, the present invention fully integrates pre-training and transfer learning strategies, making the deep operator network based on the Green's function more accurate in capturing the non-local dynamic characteristics of ocean fluids. During the Fourier transform of the source term function and the approximation of the integral kernel, by dynamically updating the Green's function parameters, the model uses data-driven supervised learning to achieve an efficient solution of the incompressible two-dimensional Navier–Stokes equations. For ocean current velocity data without a clear analytical expression, the model directly uses the discrete data as the network input without complex preprocessing. With the support of high-quality time-series observational data, the model not only inherits the physical prior information but also can flexibly adapt to the dynamic characteristics of different sea areas, thus significantly improving the prediction accuracy and stability. This improvement provides an efficient, stable and well-generalized solution for the real-time prediction of ocean current velocities, strongly supporting the accurate prediction of future velocity distributions in complex ocean environments.

[0099] Specific embodiments are used in the present invention to elaborate on the principles and implementation manners of the present invention. The description of the above embodiments is only used to help understand the method and its core idea of the present invention; at the same time, for those of ordinary skill in the art, according to the idea of the present invention, there will be changes in the specific implementation manners and application scopes. In summary, the content of this specification should not be construed as a limitation to the present invention.

[0100] Those of ordinary skill in the art will realize that the embodiments described here are for helping readers understand the principles of the present invention, and it should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. Those of ordinary skill in the art can make various other specific deformations and combinations that do not depart from the essence of the present invention based on the technical revelations disclosed in the present invention, and these deformations and combinations are still within the protection scope of the present invention.

Claims

1. A method for predicting ocean current velocity based on a deep operator network of Green's function, characterized in that, It includes the following steps: Construct a deep operator network model based on the Green's function; Train the deep operator network model, transform the fitting process of the solution operator of the incompressible two-dimensional Navier-Stokes equation into the fitting process of the integral kernel, obtain the pre-trained model, and learn the Green's function and its convergence parameters for characterizing the solution operator of the incompressible two-dimensional Navier-Stokes equation during the training process; Collect the time-series ocean current velocity data on the sea surface as training data; Initialize the pre-trained model using the learned Green's function and its convergence parameters, and perform transfer learning on the initialized pre-trained model using the training data to obtain an ocean current velocity prediction model; Use the ocean current velocity prediction model to predict the ocean current velocity distribution at future times in the target sea area.

2. The method according to claim 1, characterized in that, The deep operator network model based on the Green's function transforms the solution process of the partial differential equation into an integral process by using the Green's function as the integral kernel function.

3. The method according to claim 1, characterized in that, The deep operator network model based on the Green's function is expressed as: u(x) = ∫ Ω G(x, x′)f(x′)dx′ In the formula, u(x) is the solution function of the partial differential equation at the query point x, x′ is an arbitrary position within the integration region Ω, called the source point, the integral kernel function G(x, x′) is the Green's function, representing the contribution of a unit-strength source term applied at the source point x′ to the solution function at the query point x, and f(x′) is the source term function, representing the source term strength acting at x′; among them, the integral kernel function G(x, x′) satisfies: LG(x, x′) = δ(x - x′) In the formula, δ(x - x′) is the unit impulse source at the source point x′, and L represents the linear operator.

4. The method according to claim 1, wherein The deep operator network model based on the Green's function includes a backbone network, a branch network, and an output layer; The backbone network is used to extract the corresponding feature representation according to the position coordinates of the solution to be predicted; The branch network is used to extract the global feature representation of the input function according to the function values of the input function in the partial differential equation at multiple sampling points; The output layer is used to perform an inner product calculation on the feature vectors output by the backbone network and the branch network to obtain the predicted value at the solution to be predicted.

5. The method according to claim 4, wherein A Fourier network module of the Green's function is embedded in the branch network, including a first fully connected layer, multiple Fourier layers, and a second fully connected layer connected in sequence; Each of the Fourier layers includes a fast Fourier transform unit, a Fourier domain linear transform unit, and an inverse fast Fourier transform unit connected in sequence.

6. The method according to claim 5, characterized in that, The processing process of the Fourier network module for the input function of the partial differential equation includes: Performing feature mapping on the function values of the input function at multiple sampling points through the first fully connected layer to obtain a representation vector of a fixed dimension; Performing feature extraction on the representation vector of the fixed dimension through multiple Fourier layers to obtain a spatial domain representation; Mapping the extracted spatial domain representation to a P-dimensional vector through the second fully connected layer; Among them, in each Fourier layer: Performing a fast Fourier transform on the representation vector of the fixed dimension through the fast Fourier transform unit to convert it from spatial domain data to a frequency domain signal; The spectral domain signal is subjected to element-wise multiplication with a learnable parameter matrix by a Fourier linear transformation unit, and the parameter matrix is used to fit the representation of the Green's function in the frequency domain; The operation result of the Fourier linear transformation unit is restored to the spatial domain by inverse fast Fourier transform to obtain a spatial domain representation.

7. The method according to claim 1, characterized in that, The deep operator network model is trained in a data-driven supervised learning manner, and the training loss function is the mean square error between the predicted solution operator output of the incompressible two-dimensional Navier-Stokes equation and the true solution, expressed as: where x represents the position coordinate of the solution to be predicted, N represents the number of training samples, and f i represents the value of the input source term function of the i-th sample at the source term point, represents the output of the operator learned by the deep operator network model for the input function f i , that is, the value of the predicted solution function at point x, and u i (x) represents the value of the true solution function corresponding to the i-th input at x.

8. The method according to claim 1, wherein The process of using the training data for transfer learning on the initialized pre-trained model includes: Inputting the position coordinates in the training data into the backbone network of the deep operator network model, and inputting the corresponding time-series ocean current velocity data into the branch network of the deep operator network model; Based on the input data, forward propagation is performed in the deep operator network model; In several iterative processes, based on the forward propagation results, the error between the output of the pre-trained model and the observed values in the training data is minimized, and the Green's function integral kernel is gradually and dynamically updated to obtain an ocean current velocity prediction model.

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