A green function-based deep operator network method for ocean current velocity prediction

Abstract

This invention discloses a deep operator network-based ocean current velocity prediction method based on the Green's function, belonging to the field of ocean current prediction technology. The method includes: constructing a deep operator network model based on the Green's function; training the deep operator network model by transforming the fitting process of the incompressible two-dimensional Navier-Stokes equation solution operator into an integral kernel fitting process to obtain a pre-trained model, and learning the Green's function and its convergence parameters; collecting temporal ocean current velocity data of the sea surface as training data; initializing the pre-trained model using the Green's function and its convergence parameters, and performing transfer learning on the initialized pre-trained model using the training data to obtain an ocean current velocity prediction model, thereby predicting the ocean current velocity distribution in the target sea area at future times. This method combines mathematical theory with deep learning, effectively enhancing the modeling ability of nonlocal dynamics using the Green's function, improving computational efficiency and generalization performance, and providing an efficient and stable computational method for ocean current velocity prediction.

Description

Technical Field

[0001] This invention belongs to the field of ocean current prediction technology, specifically relating to an ocean current velocity prediction method based on a depth operator network using Green's function. Background Technology

[0002] Ocean current velocity refers to the speed at which water moves in the ocean over time. It is influenced by multiple factors, including wind, temperature, salinity, tides, ocean currents, and the Earth's rotation, exhibiting complex spatiotemporal variations. Ocean currents are one of the main manifestations of ocean current velocity and can be classified into different types based on scale, such as large-scale ocean currents, boundary currents, and tidal currents. They have a profound impact on the global climate system, marine ecological environment, and human activities. Accurate prediction of ocean current velocity is of great significance for shipping safety, climate monitoring, fisheries management, and extreme weather warnings. Ocean observation data typically includes information such as sea surface temperature, sea surface height, and sea surface humidity, providing a solid foundation for studying ocean flow characteristics and velocity trends. 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 continuously improved, further deepening our understanding of ocean dynamic processes and providing more accurate and reliable scientific support for shipping, fisheries, meteorology, and climate research. Against this backdrop, how to efficiently and accurately predict ocean current velocity and reveal the laws governing ocean current evolution has become an important research direction in the fields of marine science and engineering applications.

[0003] In ocean current velocity prediction, the incompressible two-dimensional Navier-Stokes equations serve as the fundamental equations describing ocean hydrodynamics, accurately characterizing the mass and momentum conservation relationships of ocean fluids. These equations consist of a continuity equation and a momentum equation, where the continuity equation ensures the fluid is incompressible, meaning the fluid volume remains constant during flow.

[0004]

[0005] The momentum equation describes the motion of a 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, and usually contains two components u. x and u yρ represents the velocity along the x and y directions, respectively; p represents the pressure at the fluid's current location, determining the fluid's acceleration and direction; ρ represents the fluid density; v represents the kinematic viscosity, reflecting the fluid's viscosity; and f represents the external force term, such as wind stress or tidal force. In marine environments, 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 current velocity.

[0008] Since the Navier-Stokes equations are often difficult to solve analytically, numerical methods are required in practical applications. Common methods include the finite difference method (FDM), the finite element method (FEM), and the finite volume method (FVM). These methods discretize the equations, divide the ocean region into a grid, and iteratively solve for the evolution of flow velocity over time at each node. Among them, the finite difference method (FDM) is suitable for structured grids but struggles to accurately describe complex topographic boundaries; the finite element method (FEM) can flexibly handle irregular boundaries but has a relatively higher computational cost, making it unsuitable for real-time prediction; the finite volume method (FVM) has good conservation properties and is suitable for complex grids and large-scale simulations, but its numerical accuracy depends on the discretization scheme and turbulence model used.

[0009] Although the above methods have been widely used in ocean numerical simulation, their high accuracy and controllability have provided important support for many studies, in high-resolution, long-scale ocean current velocity prediction tasks, they often face problems such as long computation time, high resource consumption, and slow response speed, making it difficult to meet the needs of real-time and large-scale prediction.

[0010] Deep Operator Networks (DeepONets) are a class of deep models used to learn function-to-function mappings, directly establishing the mapping relationship between the input and solution of partial differential equations. By inputting initial conditions, boundary conditions, and other parameters into the network as functions, DeepONets can quickly output the solution function for the entire region, thereby achieving efficient prediction of ocean current fields. They exhibit advantages in high computational speed and inference efficiency, making them particularly suitable for real-time prediction and data-driven modeling scenarios.

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

[0012] Although deep operator networks have improved the computational efficiency of ocean current velocity prediction to some extent, they still have problems such as limited expressive power, insufficient training efficiency, and difficulty in accurately capturing long-distance interactions when dealing with nonlocal ocean dynamics. Summary of the Invention

[0013] To address the aforementioned shortcomings in existing technologies, this invention provides a deep operator network-based ocean current velocity prediction method that solves the problems of limited expressive power, low training efficiency, and unstable prediction accuracy of existing deep operator networks in nonlocal ocean dynamic modeling, thereby achieving more efficient and accurate prediction of ocean current velocity distribution.

[0014] To achieve the aforementioned objectives, the present invention employs the following technical solution: a method for predicting ocean current velocity using a depth operator network based on the Green's function, comprising the following steps:

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

[0016] The deep operator network model is trained to transform the fitting process of the incompressible two-dimensional Navier-Stokes equation solver into the fitting process of the integral kernel, thus obtaining a pre-trained model. During the training process, the Green function and its convergence parameters used to characterize the incompressible two-dimensional Navier-Stokes equation solver are learned.

[0017] Collect temporal ocean current velocity data at the sea surface as training data;

[0018] The pre-trained model is initialized using the learned Green's function and its convergence parameters, and the pre-trained model is then transferred to the training data to obtain an ocean current velocity prediction model.

[0019] Using ocean current velocity prediction models, we can predict the ocean current velocity distribution in a target sea area at future times.

[0020] The beneficial effects of this invention are as follows:

[0021] (1) This invention constructs a depth operator network model for ocean current velocity 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. The Green function is efficiently fitted in the frequency domain by combining the Fourier network structure, thereby improving the model's ability to express nonlocal ocean dynamic processes.

[0022] (2) Compared with the traditional deep operator network method, the training efficiency of the method of the present invention is significantly improved, and through the combination of pre-training and transfer learning, more accurate prediction of future current velocity distribution is achieved in complex marine environments.

[0023] Furthermore, the deep operator network model based on the Green function transforms the solution process of partial differential equations into an integral process by using the Green function as the integral kernel function.

[0024] Furthermore, the deep 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 any position within the integration region Ω, called the source point, the integration kernel function G(x,x′) is the Green's function, representing the contribution of a 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 intensity of the source term applied at x′; wherein, the integration kernel function G(x,x′) satisfies:

[0026] LG(x,x′)=δ(x―x′)

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

[0028] The beneficial effects of the above-mentioned further solutions are as follows:

[0029] In the above scheme, by transforming the solution process of partial differential equations into an integral expression based on Green's functions, the model can model the solver operators with a clearer structure, thereby improving its ability to characterize physical processes. This method can explicitly represent the global influence of source terms on the target region. Especially when the operator possesses translation invariance, the integral can be simplified to a convolution operation, further reducing computational complexity and improving computational efficiency and model stability. Simultaneously, this structure provides a solid mathematical foundation for the learning and generalization of Green's functions, enhancing the model's interpretability.

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

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

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

[0033] The output layer is used to perform 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, the branch network embeds a Fourier network module with a Green function, comprising a first fully connected layer, multiple Fourier layers, and a second fully connected layer connected in sequence.

[0035] Each of the Fourier layers includes a Fast Fourier Transform (FFT) unit, a Fourier Domain Linear Transform (FFT) unit, and an Inverse Fast Fourier Transform (IFFT) unit connected in sequence.

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

[0037] The first fully connected layer performs feature mapping on the function values ​​of the input function at multiple sampling points to obtain a fixed-dimensional representation vector;

[0038] By extracting features from a fixed-dimensional representation vector through multiple Fourier layers, a spatial domain representation is obtained.

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

[0040] In each Fourier layer:

[0041] The Fast Fourier Transform (FFT) unit performs a Fast Fourier Transform on a fixed-dimensional representation vector, converting it from spatial domain data into a spectral domain signal.

[0042] The Fourier linear transform unit performs element-wise multiplication of the spectral domain signal with a learnable parameter matrix, which is used to fit the Green's function representation in the frequency domain.

[0043] The spatial domain representation is obtained by restoring the results of the Fourier linear transform unit to the spatial domain through the inverse fast Fourier transform.

[0044] The beneficial effects of the above-mentioned further solutions are as follows:

[0045] In the above scheme, by introducing a Fourier transform mechanism into the branch network, the feature vector after mapping the input source term function is transformed from the spatial domain to the frequency domain signal. A linear transform layer is then used to efficiently approximate the Green's function integral kernel, and its spatial domain representation is recovered through inverse Fourier transform, thus completing the modeling of the integral operation. Compared to directly fitting the integral kernel in the spatial domain, this method reduces learning complexity and improves training efficiency and fitting accuracy. Furthermore, the frequency domain-based representation more easily captures global features, helping the model accurately express nonlocal dynamic processes and improving overall prediction performance.

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

[0047]

[0048] In the formula, x represents the location coordinates of the solution to be predicted, N represents the number of training samples, and f i Let represent the value of the input source term function for the i-th sample at the source term point. This represents the operator pairs learned by the deep operator network model from the input function f. i The output, i.e., the predicted value of the solution function at point x, 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-mentioned further solutions are as follows:

[0050] In the above scheme, training via the supervised learning mechanism enables the deep operator network to automatically learn the implicit mapping relationship between source terms and solutions without explicitly introducing physical constraints into partial differential equations, thus effectively modeling complex ocean hydrodynamic characteristics. Compared to traditional methods, this scheme exhibits better training stability and generalization ability.

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

[0052] The location coordinates from the training data are input into the backbone network of the deep operator network model, and the corresponding time-series ocean current velocity data are input into the branch network of the deep operator network model.

[0053] Forward propagation is performed in a deep operator network model based on the input data;

[0054] In several iterations, 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 function integral kernel is dynamically updated step by step to obtain the ocean current velocity prediction model.

[0055] The beneficial effects of the above-mentioned further solutions are as follows:

[0056] In the above approach, the model is initialized using a pre-trained Green's function and its convergence parameters during transfer learning, and then dynamically updated by incorporating the actual latitude and longitude of the target sea area and time-series ocean current data. This allows the model to better adapt to the hydrodynamic characteristics of the new region. This approach not only improves the model's prediction accuracy in the target scenario but also enhances its ability to model nonlocal dynamic processes, contributing to more accurate and stable predictions of ocean current evolution trends. Attached Figure Description

[0057] Figure 1 The flowchart of the ocean current velocity prediction method based on Green's function depth operator network provided by this invention is shown.

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

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

[0060] The specific embodiments of the present invention are described below to enable those skilled in the art to understand the present invention. However, it should be understood that the present invention is not limited to the scope of the specific embodiments. For those skilled in the art, various changes are obvious as long as they are within the spirit and scope of the present invention as defined and determined by the appended claims. All inventions utilizing the concept of the present invention are protected.

[0061] This invention provides a method for predicting ocean current velocity using a deep operator network based on the Green's function. This method achieves prediction of future ocean current velocities by performing transfer learning on a pre-trained model built on the incompressible two-dimensional Navier-Stokes equations. The specific implementation method is as follows: Figure 1 As shown, it includes the following steps:

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

[0063] The deep operator network model is trained to transform the fitting process of the incompressible two-dimensional Navier-Stokes equation solver into the fitting process of the integral kernel, thus obtaining a pre-trained model. During the training process, the Green function and its convergence parameters used to characterize the incompressible two-dimensional Navier-Stokes equation solver are learned.

[0064] Collect temporal ocean current velocity data at the sea surface as training data;

[0065] The pre-trained model is initialized using the learned Green's function and its convergence parameters, and the pre-trained model is then transferred to the training data to obtain an ocean current velocity prediction model.

[0066] Using ocean current velocity prediction models, we can predict the ocean current velocity distribution in a target sea area at future times.

[0067] In this embodiment of the invention, the deep operator network model based on the Green function transforms the solution process of partial differential equations into an integration process by using the Green function as the integration kernel function. That is, the solution function u(x) of the partial differential equation at spatial location x is expressed as the integral form of the source term function f(x′) defined in the physical space range Ω and the integration 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 location to be solved), x′ is any position within the integration region Ω, called the source point, the integration kernel function G(x,x′) is the Green's function, representing the contribution of a 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 intensity of the source term applied at x′; where, for a given partial differential equation, its corresponding linear operator L and Dirac delta function source term δ(xx′), the integration kernel function G(x,x′) satisfies:

[0069] LG(x,x′)=δ(x―x′)

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

[0071] When the linear operator L is translation invariant (i.e., the operator's properties do not change with spatial position), the Green's function G(x,x′) can be simplified in one step to the displacement difference form G(x―x′), thus reducing the integral to 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 function includes a backbone network, branch networks, and an output layer.

[0074] The core network (Trunk Net) extracts the feature representation of the solution to be predicted based on its location coordinates (x,t). The branch network (Branch Net) extracts the global feature representation of the input function (e.g., source term function, initial conditions, etc.) at multiple sampling points. The output layer performs an inner product calculation on the features output by the core network and the branch network to obtain the predicted value u(x,t) at the prediction region (x,t), thus realizing the mapping from the input function space to the target solution space.

[0075] Specifically, in Figure 2 In the deep operator network model shown, f(x) represents the source term function of the branch network input, and (x,t) represents the spatial and temporal coordinates of the backbone network input; b1 to b p For the branch network output, t1 to t p The main network output is used to obtain the predicted output u(x,t) through inner product operation; FFT and IFFT represent Fast Fourier Transform and Inverse Fast Fourier Transform, respectively, and LT represents Fourier Linear Transform.

[0076] In this embodiment of the invention, the Fourier network module with Green's function embedded in the branch network includes 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 branch network structure, the processing of the input function of a partial differential equation by a Fourier network module embedded with a Green's function includes:

[0078] The first fully connected layer performs feature mapping on the function values ​​of the input function at multiple sampling points to obtain a fixed-dimensional representation vector I(x);

[0079] The spatial domain representation is obtained by extracting features from the fixed-dimensional representation vector I(x) through multiple Fourier layers.

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

[0081] In each Fourier layer:

[0082] The fixed-dimensional representation vector I(x) is transformed into a spectral domain signal by performing a Fast Fourier Transform (FFT) unit, so as to achieve efficient computation in the spectral domain.

[0083] The Fourier linear transform unit performs element-wise multiplication of the spectral domain signal with a learnable parameter matrix, which is used to fit the Green's function representation in the frequency domain.

[0084] The result of the Fourier linear transform unit is restored to the spatial domain by inverse fast Fourier transform, resulting in the spatial domain representation O(x).

[0085] This embodiment provides the aforementioned deep operator network model based on the Green's function, the purpose of which is to transform the learning process of partial differential equation solvers into a fitting problem of integral kernels. In the context of this invention, the purpose of this model is to perform transfer learning on a pre-trained model based on incompressible two-dimensional Navier-Stokes equations, thereby enabling the prediction of future current velocities in ocean regions. Therefore, this invention proposes the aforementioned deep operator network ocean current velocity prediction method, which avoids the problems of long computation time, high computational resource consumption, and difficulty in meeting real-time and large-scale prediction requirements of traditional numerical solution methods. Furthermore, by combining pre-training and transfer learning, it further improves the convergence speed of deep learning during training and the accuracy of solving partial differential equations, enabling more accurate prediction of future current velocity distributions in complex ocean environments.

[0086] In this embodiment of the invention, a data-driven supervised learning approach is used to train the deep operator network model. The training loss function is the mean square error between the predicted solution operator output and the true solution of the incompressible two-dimensional Navier-Stokes equation, expressed as:

[0087]

[0088] In the formula, x represents the location coordinates of the solution to be predicted, N represents the number of training samples, and f i Let represent the value of the input source term function for the i-th sample at the source term point. This represents the operator pairs learned by the deep operator network model from the input function f. i The output, i.e., the predicted value of the 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 supervised learning loss function achieves high-precision fitting of the partial differential equation solver by minimizing the difference between the model's prediction and the actual observation data. This method automatically captures the implicit mapping relationship between the source function and the solution function without explicitly introducing physical constraint terms, thus improving training stability and generalization ability, and providing an efficient and accurate foundation 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 equations. These equations, as a fundamental physical model describing ocean hydrodynamics, comprehensively reflect the laws of mass and momentum conservation in fluids and can effectively capture the dynamic evolution of ocean velocity fields.

[0090] In this embodiment, during the training of the deep operator network model based on the Green's function, the branch network inputs the source term function f(x), reflecting the initial conditions, boundary conditions, or other driving factors affecting fluid motion; the backbone network inputs the query point coordinates (x,t) to determine the target location and time to be solved. Since the Navier-Stokes equations describe the physical mapping relationship between the source term function and the solution function, training enables the network output to approximate the integral expression defined by the equations. 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, efficiently capturing global nonlocal information and simplifying complex convolution operations into pointwise multiplication, thereby significantly reducing computational complexity. Simultaneously, the Fourier transform can achieve noise reduction and data compression, highlighting key low-frequency physical information, thus improving the robustness and stability of the model. Through this method, the model can approximate the integral kernel with a more explicit physical structure and provide a clear structured expression for the subsequent fusion of the inner product of the backbone and branch networks.

[0091] In one example of this embodiment, when collecting training data, taking the OSCAR dataset as an example, surface current velocity data observed in the Mediterranean Sea from 2020 to 2024 were selected, with a spatial resolution of 0.25° longitude × 0.25° latitude grid cells and a temporal resolution of 1 day. Using this dataset firstly ensures the authenticity and reliability of the data, as it is collected from actual observations and can truly reflect the dynamic changes of surface currents in the sea area; secondly, the dataset has high spatiotemporal resolution, allowing the data to meticulously describe local and global current characteristics, which is beneficial for the model to capture subtle changes; simultaneously, to facilitate model training and temporal modeling, these time-series data are divided into 10-day windows, with the first 8 days used as training data and the last 2 days as test data. This data organization method reflects the evolution trend of currents in the short term and facilitates the model learning time dependencies, thereby improving prediction accuracy. In summary, this dataset not only provides the model with high-quality, continuous training samples but also significantly improves the model's prediction accuracy and generalization ability for future current trends in complex marine environments.

[0092] In embodiments of the present invention, such as Figure 3 As shown, the process of performing transfer learning on the initialized pre-trained model using training data includes:

[0093] The location coordinates from the training data are input into the backbone network of the deep operator network model, and the corresponding time-series ocean current velocity data are input into the branch network of the deep operator network model.

[0094] Forward propagation is performed in a deep operator network model based on the input data;

[0095] In several iterations, 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 function integral kernel is dynamically updated step by step, so that the model can quickly adapt to the actual dynamic characteristics of the target sea area after several iterations, thus obtaining the ocean current velocity prediction model.

[0096] In each forward propagation, the branch network receives the discretized source term function of the target sea area observation data, and outputs an approximate convolution result after multiple FFT, linear transformation LT, and IFFT operations. The backbone network takes the grid query point coordinates and temporal features as input and outputs the corresponding spatial-temporal feature vector. The two are fused at the output layer through an inner product operation to generate a prediction of the ocean current velocity distribution. The model dynamically updates the Green's function integral kernel, so that the predicted values ​​gradually converge with the actual observation data, thus ensuring good convergence and prediction accuracy for multi-scale, significantly nonlinear ocean hydrodynamic systems.

[0097] After transfer learning, the deep operator network model predicts data for the remaining test periods and compares the results with actual observations to evaluate the model's generalization ability and accuracy in new scenarios. Experiments demonstrate that by combining pre-training and transfer learning, the model not only inherits the physical priors of the incompressible two-dimensional Navier-Stokes equations but also achieves rapid adaptation to complex ocean current dynamics guided by real observation data from the Mediterranean Sea, providing a solid guarantee for accurate prediction of future ocean current trends.

[0098] In summary, compared with traditional numerical solution methods, this invention fully integrates pre-training and transfer learning strategies, enabling the Green's function-based deep operator network to more accurately capture the nonlocal dynamic characteristics of ocean fluids. During the Fourier transform and integral kernel approximation of the source term function, the model achieves efficient solution to the incompressible two-dimensional Navier-Stokes equations through data-driven supervised learning by dynamically updating the Green's function parameters. For ocean current velocity data without explicit analytical expressions, the model requires no complex preprocessing and directly uses discrete data as network input. Supported by high-quality time-series observation data, the model inherits prior physical information and flexibly adapts to the dynamic characteristics of different sea areas, thus significantly improving prediction accuracy and stability. This improvement provides an efficient, stable, and highly generalizable solution for real-time ocean current velocity prediction, strongly supporting the accurate prediction of future current velocity distribution in complex ocean environments.

[0099] Specific embodiments have been used to illustrate the principles and implementation methods of this invention. The descriptions of the embodiments above are only for the purpose of helping to understand the method and core ideas of this invention. At the same time, for those skilled in the art, there will be changes in the specific implementation methods and application scope based on the ideas of this invention. Therefore, the content of this specification should not be construed as a limitation of this invention.

[0100] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the principles of the invention, and should be understood that the scope of protection of the invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of this invention.

Claims

1. A Green function based deep operator network ocean current velocity prediction method, characterized in that, Includes the following steps: Construct a deep operator network model based on the Green's function; The deep operator network model based on the Green's function transforms the solution process of partial differential equations into an integral process by using the Green's function as the integral kernel function, which is expressed as: In the formula, Let x be the solution function of the partial differential equation at the query point x. For the integration region Any position within the range is called the source point. For the integration region, the integration kernel function The Green function represents the source point. After applying a unit intensity source term to the query point The contribution of the solution function. The source term function represents the function acting on. Source term strength at; where, integral kernel function satisfy: wherein is a unit impulse source at the source point , denotes a linear operator; The deep operator network model is trained to transform the fitting process of the incompressible two-dimensional Navier-Stokes equation solver into the fitting process of the integral kernel, thus obtaining a pre-trained model. During the training process, the Green function and its convergence parameters used to characterize the incompressible two-dimensional Navier-Stokes equation solver are learned. Collect temporal ocean current velocity data at the sea surface as training data; The pre-trained model is initialized using the learned Green's function and its convergence parameters, and the pre-trained model is then transferred to the training data to obtain an ocean current velocity prediction model. Using ocean current velocity prediction models, we can predict the ocean current velocity distribution in a target sea area at future times.

2. The method of claim 1, wherein, The deep operator network model based on the Green function includes a backbone network, branch networks, and an output layer; The backbone network is used to extract the corresponding feature representation based on the location coordinates of the solution to be predicted; The branch network is used to extract a global feature representation of the input function based on the function values ​​of the input function at multiple sampling points in the partial differential equation; The output layer is used to perform 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.

3. The method of claim 2, wherein, The branch network contains a Fourier network module with an embedded Green function, which includes 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 (FFT) unit, a Fourier Domain Linear Transform (FFT) unit, and an Inverse Fast Fourier Transform (IFFT) unit connected in sequence.

4. The method of claim 3, wherein, The Fourier network module's processing of the input function of the partial differential equation includes: The first fully connected layer performs feature mapping on the function values ​​of the input function at multiple sampling points to obtain a fixed-dimensional representation vector; By extracting features from a fixed-dimensional representation vector through multiple Fourier layers, a spatial domain representation is obtained. The extracted spatial domain representation is mapped to a P-dimensional vector through the second fully connected layer; In each Fourier layer: The Fast Fourier Transform (FFT) unit performs a Fast Fourier Transform on a fixed-dimensional representation vector, converting it from spatial domain data into a spectral domain signal. The Fourier linear transform unit performs element-wise multiplication of the spectral domain signal with a learnable parameter matrix, which is used to fit the Green's function representation in the frequency domain. The spatial domain representation is obtained by restoring the results of the Fourier linear transform unit to the spatial domain through the inverse fast Fourier transform.

5. The method of claim 1, wherein, The deep operator network model is trained using a data-driven supervised learning approach. The training loss function is the mean square error between the predicted solution operator output and the true solution of the incompressible two-dimensional Navier-Stokes equation, expressed as: In the formula, x represents the position coordinates of the solution to be predicted. Indicates the number of training samples. Indicates the first The value of the input source term function for each sample at the source term point. This represents the operator pairs learned by the deep operator network model from the input function. The output, i.e., the predicted value of the solution function at point x, Indicates the first The value of the true solution function at x corresponding to each input.

6. The method of claim 1, wherein, The process of transferring learning from the initialized pre-trained model using training data includes: The position coordinates in the training data are input into a backbone network of the deep operator network model, and the corresponding time-series ocean current velocity data are input into a branch network of the deep operator network model; Based on the input data, forward propagation is performed in the deep operator network model; In a plurality of iteration processes, based on the forward propagation result, the error between the output of the pre-training model and the observation value in the training data is minimized, and the Green function integral kernel is dynamically updated step by step to obtain the ocean current velocity prediction model.

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