A coupled neural operator model fusing fourier space-time coding and multi-field attention

Abstract

The application belongs to the technical field of three-dimensional reservoir percolation, and particularly relates to a coupling neural operator fusing Fourier space-time coding and multi-field attention. The neural operator realizes joint prediction and efficient simulation of key physical quantities such as pressure, oil phase saturation and water phase saturation. Aiming at the problems of calculation bottleneck and weak generalization ability of traditional methods in processing high-dimensional space input, time evolution and physical coupling field modeling, the coupling neural operator of the application introduces a coupling correlation mechanism between physical fields and space-time through multi-representation structure and attention enhancement, effectively improves the collaborative efficiency between tasks, the space-time information utilization rate and the expression ability of the model to the complex flow field interaction behavior, and has the characteristics of fast overall reasoning, low persistent cost, fast and stable convergence in the optimization process and the like. The fusion of Fourier high-dimensional space-time feature mapping enables the model to have stronger space-time change perception ability, and improves the modeling stability of the time evolution process.

Description

Technical Field

[0001] The present invention belongs to the technical field of three-dimensional oil reservoir seepage, and in particular relates to a coupled neural operator integrating Fourier spatiotemporal coding and multi-field attention. Background Art

[0002] Currently, the three-dimensional reservoir flow equation is a fundamental mathematical model describing the migration of multiphase fluids, such as oil, gas, and water, through porous media. It is a core research topic in reservoir engineering, geological engineering, and energy development. While traditional numerical methods (such as finite difference, finite volume, and finite element methods) offer high physical interpretability and a mature theoretical framework for complex fluid systems, they suffer from significant computational overhead in high-dimensional nonlinear systems, long-term dynamic simulations, and complex scenarios, making them inadequate for practical applications. As a typical multi-physics coupled system, the dynamic evolution of spatiotemporal variables, such as pressure and saturation fields, in three-dimensional reservoir flow systems is not only constrained by physical mechanisms such as Darcy's law and conservation equations, but also by complex coupling due to heterogeneous formation permeability and nonlinear effects of fluid rheology. These strong nonlinear interactions across physical domains make it difficult for traditional methods to strike a balance between computational efficiency and accuracy, leading to a dilemma in tackling large-scale, complex models.

[0003] With the rapid iteration of artificial intelligence and intelligent computing, data-driven methods are becoming an emerging means to solve complex fluid simulations such as reservoir seepage. Deep learning, with its powerful feature extraction and nonlinear mapping capabilities, has opened up a new perspective for complex physical field modeling. Given that neural networks can effectively express complex physical fields constrained by partial differential equations and extract high-dimensional nonlinear dependencies, researchers have widely applied neural networks to complex physical field modeling. Unlike deep neural network models that rely on complex architectures, data-driven neural operators provide higher computational efficiency and a lighter learning process. Neural operators establish a mapping relationship between input function solutions, effectively control basic physical operations, and can handle different physical scenarios, avoiding the computational bottlenecks of traditional numerical methods.

[0004] The three-dimensional reservoir percolation process has geological structure differences, initial state variability, complex fluid properties, and multiple well locations and well control strategies, etc., which lead to the highly nonlinear and multiphase coupling characteristics of reservoir flow behavior. In view of this background, various advanced methods based on neural operators are used to cope with the challenges brought by the diversity of reservoir simulation parameters. Although neural operators and deep neural network models (such as convolutional networks and time series networks) are widely used in the prior art to model physical fields, there are still many challenges in dealing with the strong coupling system of multiple physical quantities such as pressure-saturation in three-dimensional reservoir percolation. The current method generally uses variable decoupling modeling method, ignoring the complex space-time interaction between pressure and saturation and other physical fields, which is difficult to effectively capture the global coupling behavior, resulting in insufficient prediction accuracy, large model overhead, and limited generalization ability. SUMMARY

[0005] The present application proposes a coupled neural operator that fuses Fourier space-time encoding and multi-field attention, which solves the problems of insufficient multi-field prediction accuracy, high model space-time cost, and weak generalization ability in three-dimensional reservoir percolation.

[0006] The technical solution of the present application is as follows:

[0007] A coupled neural operator that fuses Fourier space-time encoding and multi-field attention includes five parts:

[0008] Preprocessing, normalization processing of pressure field data, space-time index encoding and Fourier high-dimensional space-time encoding, through scale processing of pressure field distribution, space-time index information and high-dimensional features generated by Fourier encoding in three-dimensional reservoir, avoiding the gradient imbalance problem caused by the dimension difference of different variables, and enhancing the modeling ability of the model to the global structure;

[0009] Operator parameters, consisting of three independent parts of pressure parameters, mixed space-time parameters and saturation parameters, ensure that pressure parameters, saturation parameters and mixed space-time information parameters are reasonably combined, so that the operator can effectively capture the reservoir dynamics evolution characteristics in the downstream representation process;

[0010] Implicit representation, including pressure representation, saturation representation and space-time representation, mapping input data to representation vectors through implicit adaptive learning; through adaptive neural network structure, the original input data (including pressure, saturation and space-time coordinates) are mapped to continuous and differentiable representation vectors to capture the potential distribution law and nonlinear interaction between multiple variables, thereby providing a unified feature basis for the downstream attention mechanism;

[0011] Coupling attention, combining pressure representation , mixed space-time representation and saturation representation Coupling through a feature attention mechanism to extract representative feature information;

[0012] Operator error, including pressure field error and saturation error, is used to evaluate the accuracy of the neural operator model in predicting the pressure field and saturation field. By comparing the error between the model output and the high-precision reference solution, the pressure error and the saturation error are calculated respectively, so as to quantify the fitting ability, the generalization performance and the modeling effect of the multi-physical coupling relationship of the model.

[0013] Through the above technical solutions, the coupling neural operator is designed through the synergistic design of preprocessing, operator parameters, implicit representation, coupling attention and operator error feedback mechanism, effectively solving the problems of insufficient prediction accuracy of multi-physical field, high time and space calculation cost of operator model and weak generalization ability in three-dimensional reservoir percolation modeling. Preprocessing normalizes and Fourier encodes pressure, saturation and space-time variables to improve data expression stability and multi-scale perception ability. Implicit representation maps input to continuous latent vectors to capture nonlinear coupling relationships between variables. Operator parameters are optimized through multi-task loss to enhance modeling accuracy. Coupling attention mechanism dynamically models the interaction between physical fields and highlights key regional features. Operator error feedback guides the continuous convergence of the operator model to improve generalization performance. The modules work together to build an efficient end-to-end coupling modeling system.

[0014] Optionally, in the preprocessing, a max-min normalization strategy is used to normalize the pressure field data:

[0015] ;

[0016] wherein, is the original pressure data, , are the minimum and maximum values of the pressure in the data set, respectively, is the normalized pressure data, whose range is limited to [0, 1].

[0017] Optionally, in the preprocessing, the space-time index encoding is normalized as follows:

[0018] ;

[0019] wherein, , , , denotes the three-dimensional reservoir space grid scale, is the time scale, , is the horizontal coordinate, is the vertical coordinate, t is the time identifier, , is the horizontal discrete grid index, is the vertical discrete grid index, is the discrete grid index in time, is the maximum time scale of the reservoir fluid evolution process.

[0020] Optionally, during preprocessing, when Fourier high-dimensional space-time coding is normalized, the space-time index is first encoded using relative normalization, and then the fixed-dimensional features are mapped using the space-time frequency matrix, and the space-time index is encoded using relative normalization:

[0021] ;

[0022] in, 、 、 Represents the Fourier space coordinates, indicating the position of a physical point in three-dimensional space, represents the Fourier time coordinate, 、 、 represent 、 、 The scale of direction, represents the time scale;

[0023] The Fourier high-dimensional space-time coding is normalized as follows:

[0024] ;

[0025] in, Used to represent the normalized relative space-time coordinates, is the Fourier single frequency matrix, is the frequency number, is the number of high-dimensional single-frequency Fourier features; ; 、 Represent the high-dimensional features of sine and cosine frequencies respectively.

[0026] Optionally, in the operator parameters, the pressure parameter is defined as:

[0027] ;

[0028] The saturation parameter is defined as:

[0029] ;

[0030] Dimensions of two sets of parameters and the number of 3D reservoir grid cells Consistent, that is , Indicates the number of saturation field grids;

[0031] The mixed space-time parameters include normalized space-time parameters and Fourier high-dimensional space-time parameters:

[0032] ;

[0033] in, Indicates the spatial and temporal coordinates of the identification source, Represents Fourier coding, and obtains Fourier high-dimensional spatiotemporal features, represents hybrid Fourier-coded spatiotemporal coding;

[0034] ;

[0035] in, is the number of Fourier high-dimensional space-time features.

[0036] Optionally, in implicit representation, the initial pressure data is passed through the representation mapping function Perform implicit characterization:

[0037] ;

[0038] in, represents the pressure characterization function, For the Pressure grid values, Indicates that the mathematical length is The real number space of , which is the characterization vector of pressure;

[0039] Through neural network nonlinear mapping learning, the serialized initial pressure field Compress or project to a pressure representation vector:

[0040] ;

[0041] in, is the pressure representation dimension, Indicates the The pressure value of the dimension;

[0042] The spatiotemporal information includes normalized coordinates and Fourier encoding. The representation mapping process is as follows:

[0043] ;

[0044] in, Indicates that the mathematical length is The real number space of , that is, the representation vector of time;

[0045] Get the mixed space-time representation:

[0046] ;

[0047] in, represents the normalized spatiotemporal index coordinate, According to the Fourier high-dimensional space-time coding obtained from the normalized coordinates, It is a mixed spatiotemporal representation dimension;

[0048] Initial saturation field data, i.e. serialization ,pass Perform implicit characterization:

[0049] ;

[0050] in, represents the saturation characterization function, It represents the saturation representation space;

[0051] Through nonlinear mapping, the saturation sequence data The projection to saturation characterization length is vector:

[0052] .

[0053] Optionally, in coupled attention, the coupling operator combined with the attention mechanism is described as follows:

[0054] ;

[0055] in, represents the coupling operator function, 、 They are the initial pressure and saturation inputs, which represent the initial pressure at each spatial position (such as a grid point) in the three-dimensional reservoir area. The pressure and saturation distribution, It is a hybrid Fourier coding space-time coding;

[0056] and The core of the enhanced representation process is to use weighted attention , and Two-layer attention mechanism setting 、 、 The three sets of weighted attention are as follows:

[0057] = = ;

[0058] in, To represent the attention score for stress, is the spatiotemporal representation attention score, Characterize the attention score for saturation and the attention score for 、 and Calculated by linear adaptive learning module:

[0059] ;

[0060] in, represents the upper layer representation vector, , The three representation vectors are set to have equal dimensions to comply with the algebraic operation rules. Used to distinguish pressure, time and space, saturation to represent attention scores, The attention weight and deviation is the adaptive learning optimization parameter, To represent The attention weight transpose, is the Sigmoid activation function, represents the saturation representation vector, and the feature attention score can be specifically described as:

[0061] ;

[0062] and The calculation process is:

[0063] .

[0064] Optionally, in the operator error, the initial saturation field is set to the oil saturation field, and the oil phase saturation prediction value is obtained by coupling the operator calculation ,Right now:

[0065] ;

[0066] in, middle represents the spatial vector (x, y, z) coordinates, t represents time, Refers to the predicted saturation value of (x, y, z) at time t;

[0067] The oil saturation field error is calculated as follows:

[0068] ;

[0069] in, is a combination parameter, including the oil phase saturation attention parameter attention , initial saturation characterization parameters , shared spatiotemporal representation hybrid attention parameters ,Right now , Refers to the spatial position vector of the i-th in three-dimensional space, for Department The actual saturation value at the moment;

[0070] The saturation field involving the oil and water phases satisfies the conservation constraints, and the water phase saturation field can be indirectly calculated:

[0071] ;

[0072] Then, the water phase saturation field error is:

[0073] ;

[0074] The oil-water two-phase error is shown in the above formula: , the saturation error of the actual solution equation is ;

[0075] The pressure field error is calculated as follows:

[0076] ;

[0077] same All are combined parameters, and the attention parameter is pressure attention , the initial characterization parameters are , both share parameters ,Right now , for Department Real pressure value at all times;

[0078] The operator error is the sum of the pressure and oil phase saturation errors, that is:

[0079]

[0080] in, .

[0081] Optionally, in the operator error, the neural operator adaptive learning process is described as follows:

[0082] ;

[0083] in, represents the learned joint parameters of the coupling operators;

[0084] This adaptive learning process aims to minimize ,Right now ,satisfy:

[0085] ;

[0086] in , is the learning rate, is the combined prediction error gradient of pressure and saturation, is the accuracy threshold, which controls whether the prediction error meets the acceptable range. is the set of all parameters of the coupling operator, is the true pressure value, is the actual saturation value.

[0087] After adopting the above technical solution, the beneficial effects of the present invention are:

[0088] This paper proposes a coupled neural operator model that integrates Fourier spatiotemporal coding with a multi-field attention mechanism. This neural operator model extracts global multi-scale features of reservoir flow processes through Fourier spatiotemporal coding. It then leverages a multi-task architecture to share high-dimensional representations and dynamically models the interactions between different physical variables using a multi-field attention mechanism, thereby improving the accuracy and efficiency of three-dimensional reservoir flow modeling.

[0089] This paper proposes a coupled neural operator that integrates Fourier spatiotemporal coding and multi-field attention. Fourier spatiotemporal coding uses Fourier transforms to extract global features, enhancing the neural operator's ability to perceive spatiotemporal dependencies and information at different scales. By sharing high-dimensional spatiotemporal features through a multi-task architecture, the implicit relationships between different physical fields are learned. The multi-field attention mechanism is then used to dynamically model the interactions between different physical fields and the shared spacetime, thereby improving the model's generalization and prediction accuracy. Compared to standalone neural operators, this model improves training, inference, and storage efficiency while maintaining accuracy. Experiments demonstrate that the proposed neural operator can more effectively solve the physical system for modeling and simulating the flow of fluids (such as crude oil and water) in porous media within underground reservoirs in three-dimensional space, and achieves coordinated optimization across multiple tasks (such as pressure distribution prediction and saturation evolution). This is intended to effectively improve the modeling efficiency and prediction accuracy of pressure and saturation changes under different underground reservoir conditions. From the perspective of engineering applications, the present invention aims to improve the accuracy and real-time performance of pressure and saturation distribution prediction in reservoir simulation to meet the needs of rapid decision-making in complex geological scenarios.

[0090] The present invention designs a hybrid coupled operator network that integrates Fourier spatiotemporal coding and multi-field attention. The proposed coupled operator has the ability to share the spatiotemporal representation process and theoretically has time and space cost advantages for solving coupled multi-field equations. The introduction of Fourier coding aims to enhance the spatiotemporal data representation capability and improve the operator's high-dimensional ability to cope with spatiotemporal changes, which is conducive to the operator using high-dimensional spatiotemporal features to more effectively fit the spatiotemporal state field and the target spatiotemporal physical field, allowing the operator to solve equations in deeper spatiotemporal layers. The adaptive weighted attention mechanism aims to enable the coupled operator to mine key representation information, which is conducive to the operator efficiently extracting the differences between the initial state field and the spatiotemporal state field changes, so that the operator has the ability to mine potential information and improve the operator's ability to fit the spatiotemporal physical field.

[0091] The present invention focuses on the difficulties in multi-field coupling modeling in three-dimensional reservoir seepage problems, and proposes a coupled neural operator that integrates Fourier space-time coding and multi-field attention, realizing the joint prediction and efficient simulation of key physical quantities such as pressure, oil phase saturation and water phase saturation. It addresses the problems of computational bottlenecks and weak generalization capabilities of traditional methods in processing high-dimensional spatial input, temporal evolution and physical coupling field modeling. The present invention designs a coupled neural operator that integrates Fourier high-dimensional space-time and attention mechanisms. Through multi-representation structure and attention enhancement, it introduces the coupling correlation mechanism between physical field and space-time, effectively improving the collaborative efficiency between operator tasks, the utilization of space-time information and the model's ability to express complex flow field interaction behaviors. Compared with multi-field independent operators, the coupled neural operator of the present invention has the characteristics of fast overall reasoning, low persistence cost and fast and stable convergence of the optimization process. The integration of Fourier high-dimensional space-time feature mapping enables the neural operator to have a stronger ability to perceive space-time changes and improves the stability of modeling of time evolution processes. BRIEF DESCRIPTION OF THE DRAWINGS

[0092] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0093] Figure 1 It is a network architecture of coupled neural operators;

[0094] Figure 2 This is a schematic diagram of the pressure distribution prediction error of the reservoir seepage simulation at time steps of 40, 60, and 80;

[0095] Figure 3This is a schematic diagram of the oil saturation prediction error of reservoir seepage simulation at time steps of 40, 60, and 80;

[0096] Figure 4 This is a schematic diagram of the water saturation prediction error in reservoir seepage simulation at time steps of 40, 60, and 80. DETAILED DESCRIPTION

[0097] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0098] The embodiment of the present application discloses a coupled neural operator that integrates Fourier spatiotemporal coding and multi-field attention.

[0099] In an embodiment, a network architecture of coupled neural operators integrating Fourier spatiotemporal coding and multi-field attention is shown in FIG. Figure 1 As shown in Figure 1, the network architecture consists of five parts: preprocessing, operator parameters, implicit representation, coupled attention, and operator error. The details are as follows:

[0100] 1. Preprocessing

[0101] The preprocessing phase primarily processes the initial physical and spatiotemporal states. The physical states include the initial pressure field and the initial saturation field. In depth operators, data of varying dimensions significantly impacts the adaptive learning process, causing gradients to explode or vanish. To reduce data distribution bias and ensure that different features have consistent statistical properties, the present invention normalizes both the initial physical field data and the spatiotemporal state data.

[0102] Since the saturation value falls within the real number range, normalization is not required. However, the magnitude of the pressure field data is too large, usually varying in the KPa or even MPa level. If it is directly calculated using the operator, the numerical range will be too large, affecting the convergence of the characterization function fitting in the operator. Therefore, the present invention adopts the maximum-minimum normalization strategy to normalize the pressure field data:

[0103] ;

[0104] in, is the original pressure data, 、 are the minimum and maximum pressure values ​​in the data set, is the normalized pressure data, and its range is limited to [0,1].

[0105] Spatiotemporal data processing includes spatiotemporal index coding and Fourier high-dimensional spatiotemporal coding. Global normalization coding can uniformly scale all data, ensuring that different physical quantities are mapped to similar numerical ranges, which is more conducive to the operator learning the scale differences of data of different dimensions. In spatiotemporal index coding, the present invention adopts global spatiotemporal index coding. The global normalized spatiotemporal index is normalized based on the maximum index of the entire spatiotemporal computational domain, and the spatiotemporal index coding is normalized as follows:

[0106] ;

[0107] in, , 、 、 represents the spatial grid scale of the three-dimensional reservoir, is the time scale, 、 、 Both represent spatial coordinates, 、 is the horizontal coordinate, is the vertical coordinate, i.e. the horizontal and vertical position of a physical grid in three-dimensional space, and t is the time identifier. 、 is the horizontal discrete grid index, is the vertical discrete grid index, is the discrete grid index in time. is the maximum time scale of the reservoir fluid evolution process.

[0108] Fourier coding uses random Fourier transform to map normalized coordinates to high-dimensional space, so that the code contains information in different frequency spaces. The core of Fourier transform is to use multiple sets of sine and cosine functions. Due to the characteristics of sine and cosine functions, all frequency data are periodic, and the greater the numerical distance between different frequencies, the more it can reflect the high-dimensional information of the frequency. Global normalized spatiotemporal index, whether it is spatial coordinates or time coordinates, the normalized numerical range is the same, reducing the scale difference between different physical quantities. Relative normalization is to normalize each physical quantity separately, and there are incremental differences in the numerical range. When the spatiotemporal scale difference is large, the codes are independent of each other.

[0109] In order to make the Fourier features better represent the data scale, we first use relative normalization to encode the spatiotemporal index, and then use the spatiotemporal frequency matrix to map the fixed dimension features. Relative normalization encoding spatiotemporal index:

[0110] ;

[0111] in, 、 、 Represents the Fourier space coordinates, indicating the position of a physical point in three-dimensional space, represents the Fourier time coordinate, 、 、 Represents the directions of space ( 、 、 direction) scale, Represents the time scale.

[0112] The Fourier high-dimensional space-time coding is normalized and the Fourier coding function is defined as follows:

[0113] ;

[0114] in, Used to represent the normalized relative space-time coordinates, is the Fourier single frequency matrix, is the frequency number, is the number of high-dimensional single-frequency Fourier features; ; 、 Represent the high-dimensional features of sine and cosine frequencies respectively.

[0115] The coupling operator uses Fourier encoding to obtain a high-dimensional embedding vector after Fourier feature mapping of the input space-time coordinates, which is used to enhance the model's ability to model multi-scale spatial structures and long-term dependencies.

[0116] 2. Operator Parameters

[0117] The operator parameters are composed of three independent parts: pressure parameters, saturation parameters, and mixed space-time parameters. The core goal is to unify the normalized data format of the operator parameters to facilitate the subsequent calculation of implicit representations.

[0118] 1. Physical field parameters

[0119] In order to avoid the loss of important information due to local sampling, the present invention uses the full sample of the initial state field as a parameter.

[0120] (1) The pressure parameter is defined as:

[0121] ;

[0122] in, Represents the number of pressure field grids.

[0123] (2) The saturation parameter is defined as:

[0124] ;

[0125] Dimensions of two sets of parameters and the number of 3D reservoir grid cells Consistent, that is , Indicates the number of saturation field grids.

[0126] 2. Mixed spatiotemporal parameters

[0127] The mixed space-time parameters include normalized space-time parameters and Fourier high-dimensional space-time parameters:

[0128] ;

[0129] in, Indicates the spatial and temporal coordinates of the identification source, Represents Fourier coding, and obtains Fourier high-dimensional spatiotemporal features, represents hybrid Fourier-coded spatiotemporal coding.

[0130] ;

[0131] in, is the number of Fourier high-dimensional space-time features.

[0132] This construction ensures the rational combination of pressure, saturation, and spatiotemporal information, enabling the operator to effectively capture reservoir dynamic evolution characteristics in the downstream characterization process.

[0133] 3. Implicit Representation

[0134] The implicit representation process mainly includes three parts: pressure representation, spatiotemporal representation, and saturation representation. The core goal is to map the input data into a representation vector through implicit adaptive learning in order to extract features for subsequent calculations.

[0135] 1. Initial pressure data (normalized, serialized ) by characterizing the mapping function Perform implicit characterization:

[0136] ;

[0137] in, represents the pressure characterization function, For the Pressure grid values, Indicates that the mathematical length is The real number space of , that is, the characterization vector of pressure.

[0138] Through neural network nonlinear mapping learning, the serialized initial pressure field Compress or project to a pressure representation vector:

[0139] ;

[0140] in, is the pressure representation dimension, Indicates the The pressure value of the dimension.

[0141] 2. Spatiotemporal information includes normalized coordinates and Fourier encoding, The representation mapping process is as follows:

[0142] ;

[0143] in, Indicates that the mathematical length is The real number space of , that is, the representation vector of time.

[0144] Get the mixed space-time representation:

[0145] ;

[0146] in, represents the normalized space-time index coordinate, According to the Fourier high-dimensional space-time coding obtained from the normalized coordinates, It is a mixed space-time representation dimension.

[0147] 3. Initial saturation field data (serialized )pass Perform implicit characterization:

[0148] ;

[0149] in, represents the saturation characterization function, Indicates the saturation representation space.

[0150] Through nonlinear mapping, the saturation sequence data The projection to saturation characterization length is vector:

[0151] ;

[0152] These nonlinear representation vectors serve as reservoir and spatiotemporal features in the subsequent calculation process of the coupled neural operator to learn the nonlinear process of reservoir fluid flow.

[0153] 4. Coupled Attention

[0154] In this operator network framework, pressure representation , mixed spatiotemporal representation and saturation characterization Coupling is performed through the feature attention mechanism to extract representative feature information and improve the multi-field coupling modeling capabilities.

[0155] In the implicit representation process, the representation vector is mixed and input into the two attention mechanism layers. and , the coupling operator combined with the attention mechanism is described as follows:

[0156] ;

[0157] represents the coupling operator function, 、 They are the initial pressure and saturation inputs, which represent the initial pressure at each spatial position (such as a grid point) in the three-dimensional reservoir area. The pressure and saturation distribution, It is a hybrid Fourier coding and space-time coding.

[0158] Two attention mechanisms share a mixed high-dimensional Fourier encoding representation , to ensure that the influence of space-time characteristics on the two parts of physical quantities is consistent. Characterize the vector by pressure and space-time representation vector The interaction between the pressure and the flow is used to capture the pressure-dominated characteristics of the flow process. Characterizing Vectors by Saturation and space-time representation vector This design ensures the co-evolution of the pressure-saturation field while utilizing hybrid high-dimensional Fourier encoding to provide richer spatiotemporal feature information, thereby improving the learning ability and physical consistency of the neural operator.

[0159] and The core of the enhanced representation process is to use weighted attention , and Two-layer attention mechanism setting 、 、 The three sets of weighted attention are as follows:

[0160] = = ;

[0161] in, To represent the attention score for stress, is the spatiotemporal representation attention score, Characterize the attention score for saturation and the attention score for 、 and Calculated by linear adaptive learning module:

[0162] ;

[0163] Here, represents the upper layer representation vector, , The dimensions of the three representation vectors are set to be equal to comply with algebraic operation rules. Used to distinguish pressure, time and space, and saturation to represent attention scores. The attention weight and deviation is the adaptive learning optimization parameter, To represent The attention weights are transposed. Is the Sigmoid activation function, used to normalize the weights. represents the saturation characterization vector.

[0164] The feature attention score can be specifically described as:

[0165] ;

[0166] and The calculation process is:

[0167] ;

[0168] This feature attention module performs weighted filtering of pressure, spatiotemporal, and saturation representation vectors, enhancing key features of the upper-layer representation. This coupled approach of shared spatiotemporal attention enhances data-driven adaptive learning capabilities and helps better characterize the complex nonlinear relationships in reservoir flow.

[0169] 5. Operator Error

[0170] The operator error consists of two parts: pressure field error and saturation field error. The present invention sets the initial saturation field as the oil saturation field and calculates the oil phase saturation prediction value through the coupling operator. ,Right now:

[0171] ;

[0172] in, middle represents the spatial vector (x, y, z) coordinates, t represents time, Refers to the predicted saturation value of (x, y, z) at time t.

[0173] The oil saturation field error is calculated as follows:

[0174] ;

[0175] in, is a combination parameter, including the oil phase saturation attention parameter attention , initial saturation characterization parameters , shared spatiotemporal representation hybrid attention parameters ,Right now . Refers to the spatial position vector of the i-th in three-dimensional space, for Department The actual saturation value at the moment.

[0176] The saturation field involving the oil and water phases satisfies the conservation constraints, and the water phase saturation field can be indirectly calculated:

[0177] ;

[0178] Then the water phase saturation field error is:

[0179] ;

[0180] The oil-water two-phase error is shown in the above formula: , the saturation error of the actual solution equation is .

[0181] The pressure field error is calculated as follows:

[0182] ;

[0183] same All are combined parameters, and the attention parameter is pressure attention , the initial characterization parameters are , both share parameters ,Right now , for Department The actual pressure value at each moment.

[0184] Different from the actual solution, the operator solution involves fitting the pressure and oil phase saturation. The parameters are adapted to learn the pressure and saturation fields. Therefore, the operator error is the sum of the pressure and oil phase saturation errors, that is:

[0185] ;

[0186] in, .

[0187] Based on a network architecture coupled with neural operators, an adaptive learning mechanism for pressure distribution prediction and saturation evolution modeling was designed for the multi-physical variable prediction task of oil-water mixed fluid seepage in three-dimensional reservoirs. The neural operator adaptive learning process is described as follows:

[0188] ;

[0189] in, Represents the learned joint parameters of the coupling operators.

[0190] This adaptive learning process aims to minimize ,Right now ,satisfy:

[0191] ;

[0192] in, is the learning rate, is the combined prediction error gradient of pressure and saturation, is the accuracy threshold, which controls whether the prediction error meets the acceptable range. represents the initial saturation characterization parameter, represents the initial characterization parameters, Indicates the oil phase saturation attention parameter attention, represents the shared spatiotemporal representation hybrid attention parameter, is the set of all parameters of the coupling operator, is the true pressure value, is the actual saturation value.

[0193] The coupled neural operator utilizes an adaptive learning mechanism to jointly optimize a multi-field physical representation network, a Fourier spatiotemporal encoding module, and an inter-field attention coupling structure. This allows it to automatically learn complex spatiotemporal evolution relationships from the input initial state and spatiotemporal parameters without relying on grid iteration, enabling joint pressure-saturation prediction of oil-water fluids at different spatial locations and time scales. The hybrid Fourier spatiotemporal encoding technique extracts global frequency-domain features of the input spatiotemporal structure, effectively enhancing the model's ability to perceive multi-scale, remotely dependent spatiotemporal structures. The shared spatiotemporal representation mechanism establishes a unified high-dimensional representation space between different physical variables through a multi-task structure, enabling information sharing and coupled modeling between physical variables, greatly optimizing the spatiotemporal efficiency of pressure distribution prediction and saturation evolution computation. The multi-field attention mechanism dynamically models the interactions between various physical fields, highlighting key regions and coupled dependencies, thereby improving the accuracy of pressure distribution prediction and saturation evolution. This design addresses key issues in three-dimensional reservoir seepage problems, including insufficient multi-physics field prediction accuracy, high spatiotemporal computational costs, and poor cross-scenario generalization.

[0194] 6. Evaluation

[0195] For the three-dimensional oil-water mixture seepage experiment, the three-dimensional grid , the same grid size as SPE1. The reservoir geological model is set to heterogeneous geology, which is generated by uniform random distribution: dimensionless porosity ; Dimensional permeability Well parameters are fixed, with injection wells at a constant flow rate and outflow wells at a constant pressure. Regarding reservoir fluids, the oil-water density is fixed. The three-dimensional reservoir geology and fluid parameters are shown in Table 1 below:

[0196] Table 1. Ranges of reservoir parameters used in the experiments.

[0197]

[0198] As shown in Table 1, the experimental parameter settings cover the common fluid characteristics and reservoir physical parameters in actual reservoirs, which not only ensures physical rationality but also expands the range of training data space, which is conducive to the adaptability and generalization ability of the reaction operator model to complex reservoir conditions.

[0199] The experiment systematically evaluated the effectiveness and robustness of the coupled neural operator proposed in this invention in dealing with key issues such as insufficient multi-physics field prediction accuracy, high model spatiotemporal computational cost, and weak cross-scenario generalization ability in dealing with three-dimensional oil reservoir seepage problems from the aspects of coding dimension design, multi-task structure construction, and input feature modeling strategy.

[0200] To evaluate the impact of Fourier spatiotemporal encoding dimensions on operator model performance, we conducted comparative experiments using different Fourier encoding dimensions (32, 64, and 128). These experiments were conducted on networks with three and four layers, each containing 256 neurons. All models employed a fusion encoding scheme, combining Fourier encoding with normalized coordinates as spatiotemporal representation parameters. The results are shown in Table 2.

[0201] Table 2 shows the comparison of the average L2 error of the prediction of the pressure field and saturation field for the operator model integrating global grid coding and relative Fourier coding, which has 3 layers and 4 layers of 256 neurons respectively.

[0202]

[0203] As shown in Table 2, a Fourier encoding dimension of 64 achieves optimal performance, achieving the lowest average L2 error for both pressure and saturation fields in both model structures (3 and 4 layers). In the 3-layer model, the pressure error is 0.001775 and the saturation error is 0.001550. In the 4-layer model, the pressure error is further reduced to 0.001492 and the saturation error is 0.000970, improving pressure prediction by 15.9% and saturation prediction by 37.4%. Compared with the 128-dimensional encoding model, the 64-dimensional model reduces pressure prediction error by 51.8% (3-layer) and 31.1% (4-layer), respectively, and saturation error by 11.9% and 8.1%, respectively. Similarly, compared with the 32-dimensional model, the 64-dimensional model reduces pressure prediction error by 31.2% (3-layer) and 9.0% (4-layer), and saturation error by 14.3% and 3.6%, respectively. While the prediction errors of the low-dimensional model (32-dimensional Fourier encoding) were of the same order of magnitude, they were slightly lacking in fine-scale representation. The error of the high-dimensional model (128-dimensional Fourier encoding) increased significantly, particularly in the three-layer network, where the pressure prediction error more than doubled. This suggests that excessively high encoding dimensions may introduce redundant information, exacerbate overfitting, and reduce model stability. The four-layer model outperformed the three-layer model in both 64- and 128-dimensional encoding, demonstrating its stronger feature modeling capabilities. However, in the 32-dimensional encoding, the errors of the two models were similar, indicating that when the input feature dimensionality is insufficient, increasing network depth has limited performance improvements and may even lead to training instability.

[0204] 2. To explore the role of joint learning mechanisms in multi-physics field modeling, this experiment compared the performance of three neural operator architectures for pressure and saturation field prediction: independent prediction, multi-task prediction, and coupled prediction using Fourier encoding and attention. All models were based on the same fused spatiotemporal encoding and trained uniformly in 3-layer and 4-layer 256-neuron architectures. The prediction results on the same test set are shown in Table 2.

[0205] Table 3 compares the prediction errors of the 3-layer and 4-layer 256-neuron structures in the pressure field and saturation field using the operator model that combines global grid coding and relative Fourier coding.

[0206]

[0207] As shown in Table 3, the prediction error of the joint prediction neural operator in both architectures is higher than that of the corresponding independent prediction neural operators. The pressure field error increases from 0.001775 to 0.002981, an increase of approximately 67.9%. The saturation field error increases from 0.001550 to 0.001926, an increase of approximately 24.3%. Similarly, the error in the four-layer architecture shows a similar upward trend. This result indicates that directly adopting a multi-task learning mechanism with parameter sharing fails to improve accuracy in the current modeling context and may even cause a "negative transfer" effect due to optimization conflicts between tasks or feature interference. In contrast, the coupled neural operator performs best across all evaluation metrics, with significant improvements. In the four-layer architecture model, the pressure field error decreases from 0.001492 for a single task to 0.001213, a decrease of approximately 18.7%. The saturation error decreases from 0.000970 to 0.000666, a decrease of approximately 31.3%. The three-layer structure also exhibits a consistent trend, with pressure and saturation prediction errors decreasing by 29.5% and 56.5%, respectively. This result demonstrates that the task attention mechanism can effectively alleviate task interference, enable explicit modeling of inter-task dependencies and dynamic feature allocation, and enhance the model's ability to characterize complex coupled physical fields. A comprehensive comparison of the three-layer and four-layer models reveals that the error changes are relatively small under the two structures, indicating that network depth has a relatively limited impact on prediction performance at this experimental scale. The performance improvement primarily stems from improvements in task modeling, particularly the introduction of the attention mechanism, which effectively focuses on task-relevant features and suppresses irrelevant disturbances.

[0208] This experiment used an operator model with a unified network structure (4 layers, 256 neurons) to compare the time and space costs of the prediction phase between a traditional independent task neural operator and a coupled neural operator model that introduced Fourier encoding and attention mechanisms. This approach evaluated the computational efficiency of different task learning strategies in practical deployments. The results of the inference time and space costs are shown in Table 4.

[0209] Table 4 Comparison of the spatiotemporal costs of the independent prediction neural operator and the coupled neural operator model with a fixed 4-layer 256-neuron structure in the prediction of pressure field and saturation field.

[0210]

[0211] As can be seen from Table 4, the coupled neural operator model of hybrid Fourier coding and attention mechanism significantly reduces the inference time (reduced by about 70.9%). Compared with the traditional independent prediction neural operator model that needs to process two branches of pressure and saturation respectively, the coupled neural operator compresses feature redundancy by sharing the spatiotemporal representation encoder, introducing the attention mechanism, and adopting a more compact joint prediction structure, which is closer to the end-to-end joint reasoning method, helping to improve the inference parallelism and cache utilization. In addition, the coupled neural operator model saves about 22% space in storage. The fusion of the moment representation encoder, attention module and shared feature space of this model makes the model parameters more compact as a whole. Although it may contain more complex feature interaction structures in the representation fusion stage, the overall parameter redundancy is significantly reduced, which is suitable for model compression and edge deployment scenarios. In terms of theoretical time complexity, the calculation of the coupled neural operator is mainly composed of The decision has better schedulability; the traditional independent prediction neural operator needs to process two physical field tasks independently, which is more decentralized in reasoning scheduling. In terms of space complexity, although the coupled neural operator introduces an additional attention module , but through branch structure sharing and representation compression, it is still more efficient than the traditional structure overall.

[0212] 3. In three-dimensional reservoir seepage problems, the uncertainty of the initial pressure and saturation fields directly affects prediction accuracy and model generalization. To this end, this experiment systematically evaluates the adaptability of the proposed method to different initial condition perturbations by constructing three representative initial state samples. Sample A: Only the pressure field is randomly perturbed, and the saturation field remains fixed; Sample B: The pressure field is fixed, and the saturation field is randomly perturbed; Sample C: Both the pressure and saturation fields are perturbed. All samples are constructed based on Gaussian random fields to simulate the uncertainty of the reservoir's initial state. Gaussian random fields are data fields with continuity and spatial correlation. Compared with high-frequency jump perturbations generated by uniformly distributed random numbers, they are more suitable for describing the spatial continuity of the reservoir's physical state.

[0213] To analyze the impact of input feature modeling on model generalization performance, the present invention sets up three multi-task prediction models: a basic joint prediction operator that only uses globally normalized spatiotemporal variables as input; a Fourier-coded joint prediction operator that Fourier-codes the normalized spatiotemporal index variables to enhance the model's ability to express spatial frequency changes; and a coupled neural operator that introduces an attention mechanism based on the Fourier-coded joint prediction operator to improve the model's selective modeling capabilities for key dimensions and feature regions. The branch representation network structure remains consistent; the pressure field branch uses the maximum-minimum normalized initial pressure field as input, and the saturation branch uses the vectorized saturation field as input. The spatiotemporal variables in the operator network are uniformly input after global position normalization, while the Fourier-coded spatiotemporal variables are locally normalized. The average correlated L2 error and the variance of the correlated L2 error for the multi-task prediction evaluation in the three groups of samples are shown in Tables 5 and 6.

[0214] Table 5 Mean L2 error of the model prediction of coupled pressure and saturation in different multi-type Gaussian random initial state field samples A, B, and C of the reservoir

[0215]

[0216] As shown in Table 5, the Fourier-coded joint prediction operator reduces the average error by approximately 95% to 98% compared to the basic joint prediction operator, demonstrating that Fourier coding significantly enhances the model's expressive power under perturbations. The coupled neural operator further reduces the average error by approximately 50% compared to the Fourier-coded joint prediction operator, demonstrating that the attention mechanism effectively complements the local selectivity of the encoding. For the most complex perturbation, sample C, the pressure error decreases from 0.0967 for the basic joint prediction operator to 0.0009 for the coupled neural operator, a decrease of 99.1%. The saturation error decreases from 0.0773 to 0.0006, a decrease of 99.2%. This demonstrates that the coupled neural operator model exhibits extremely high robustness and predictive accuracy in complex perturbation fields.

[0217] Table 6 Standard deviation of the L2 error of the model prediction of coupled pressure and saturation for different multi-type Gaussian random initial state field samples A, B, and C in the reservoir

[0218]

[0219] As shown in Table 6, the standard deviation of the error for the basic joint prediction operator model ranged from 0.058 to 0.083 across all samples, indicating large prediction fluctuations and poor stability. In contrast, the Fourier-coded joint prediction operator compressed the standard deviation to approximately 0.0001 to 0.0007, reducing error fluctuation by approximately two orders of magnitude. The coupled neural operator further improved stability, with the standard deviation for saturation predictions for samples B and C dropping to 0.0004 and 0.0002, respectively. Overall, the coupled neural operator maintained minimal error fluctuations under various perturbations, demonstrating high robustness and repeatability.

[0220] In order to explore the predictive ability of the model in the process of time evolution, this experiment further carried out process prediction experimental analysis. The prediction error of the reservoir system in continuous time advancement was visualized and analyzed. Unlike static single-step prediction, process prediction emphasizes the dynamic evolution ability of the model's prediction output at the previous moment as the input at the subsequent moment, so it is closer to the step-by-step prediction and deduction scenario in the real simulation task. This part of the experiment uses the sample C with the strongest disturbance as the benchmark data set, and evaluates the error accumulation and generalization ability of the model under long time series conditions by setting different prediction step sizes. To test the model's adaptability to prediction tasks at different time scales, prediction step sizes of 40, 60, and 80 were set respectively, and the spatial errors of the three types of physical quantities were visualized, as shown below. Figure 2 、 Figure 3 、 Figure 4 The model's accuracy retention in pressure, oil saturation, and water saturation is evaluated from a multi-scale time span.

[0221] from Figure 2 It can be seen that at different time steps, the model's prediction of the pressure field can maintain a good global trend, and the predicted values ​​are highly consistent with the true values ​​in terms of spatial distribution. In the error plot, the absolute error is concentrated at the boundary and in the local gradient mutation area, and the overall error amplitude remains within the order of magnitude, indicating that the proposed model still has good stability and accuracy retention capabilities in predictions with longer time steps. Step size 40: The prediction error distribution is relatively uniform, and the error is mainly concentrated in the edge grid, indicating that the model has strong spatial generalization capabilities within the initial step size. Step size 60: The error increases slightly, and some areas show error diffusion, indicating that the prediction difficulty increases with the increase in step size. Step size 80: The error control ability is still acceptable, and the local deviation at the boundary is slightly enhanced, but the global structure has not undergone significant degradation.

[0222] As a variable with strong nonlinear evolution, the prediction results of oil phase saturation require higher modeling capabilities. Figure 3As can be seen from the figure, the accurate value matches the predicted value well, the structural outline is clear, the predicted map can capture the main saturation front and spatial migration trend, the error map shows a certain degree of random noise and local offset phenomenon, but the error amplitude in the dominant area is still maintained at around , especially when the step length is 80, the model can still maintain the ability to capture the nonlinear evolution trend, indicating that the model has certain robustness in modeling long time series.

[0223] like Figure 4 As shown, the evolution of water phase saturation shows a complementary relationship with the oil phase, and the overall error trend is similar to Figure 3 The predicted saturation front is essentially consistent with the actual saturation front, demonstrating an accurate description of the migration interface. Errors are concentrated in the transition region and boundary, primarily due to minor translation errors or errors in shock wave amplitude prediction. The model maintains relatively stable prediction performance at larger step sizes, demonstrating excellent timescale migration capabilities.

[0224] The coupled neural operator proposed in this paper, which integrates Fourier spatiotemporal coding and multi-field attention, can effectively address technical challenges in three-dimensional reservoir flow modeling, such as the difficulty in coupling multiple physical variables, high computational cost, and weak generalization capabilities. In actual reservoir development, flow processes are often influenced by complex geological structures, uncertain initial conditions, and nonlinear coupled governing equations. Traditional methods typically model the pressure and saturation fields separately, failing to capture the dynamic interactions between them. This results in large prediction errors and low solution efficiency, particularly under varying reservoir conditions (such as permeability heterogeneity and different injection and production methods).

[0225] This method uses Fourier space-time coding to map spatial position and time information into high-dimensional frequency domain features, enhancing the model's perception of long time spans and global spatial structure, effectively avoiding the error accumulation caused by time-step iteration. Furthermore, by employing multi-field joint learning and an attention mechanism, it achieves dynamic coupling modeling between physical fields such as pressure and saturation, improving the model's generalization performance under different initial conditions.

[0226] In the task of three-dimensional reservoir seepage prediction, this neural operator can be widely used in reservoir numerical simulation alternative modeling, rapid parameter sensitivity analysis and real-time decision support systems.

[0227] The above are only preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the technical solutions of the present invention should be included in the protection scope of the present invention.

Claims

1. A coupled neural operator model integrating Fourier spatiotemporal coding and multi-field attention, characterized by: The network architecture of the coupled neural operator consists of five parts, as follows: Preprocessing: normalizing the pressure field data, spatiotemporal index coding, and Fourier high-dimensional spatiotemporal coding in the three-dimensional reservoir; Operator parameters, including pressure parameters, mixed time and space parameters, and saturation parameters, ensure that pressure parameters, saturation parameters, and mixed time and space information parameters are reasonably combined so that the operator can effectively capture the reservoir dynamic evolution characteristics in the downstream characterization process; Implicit representation, including pressure representation, saturation representation, and spatiotemporal representation, maps the original input data into a continuously differentiable representation vector through an adaptive neural network structure to capture its potential distribution law and nonlinear interactions between multiple variables; Coupling attention to represent stress , mixed spatiotemporal representation and saturation characterization , coupled through the feature attention mechanism to extract representative feature information; Operator error, including pressure field error and saturation error, is used to evaluate the accuracy of the neural operator model in pressure field and saturation field prediction; In implicit representation, the initial pressure data is represented by the mapping function Perform implicit characterization: ; in, represents the pressure characterization function, For the Pressure grid values, Indicates that the mathematical length is The real number space of , which is the characterization vector of pressure; Through neural network nonlinear mapping learning, Compress or project the serialized initial pressure field into a pressure representation vector: ; in, is the pressure representation dimension, Indicates the The pressure value of the dimension; The spatiotemporal information includes normalized coordinates and Fourier encoding. The representation mapping process is as follows: ; in, Indicates that the mathematical length is The real number space of , that is, the representation vector of time; Get the mixed space-time representation: ; in, represents the normalized spatiotemporal index coordinate, According to the Fourier high-dimensional space-time coding obtained from the normalized coordinates, It is a mixed spatiotemporal representation dimension; Initial saturation field data, i.e. serialization ,pass Perform implicit characterization: ; in, represents the saturation characterization function, It represents the saturation representation space; Through nonlinear mapping, the saturation sequence data The projection to saturation characterization length is vector: 。 2. The coupled neural operator model integrating Fourier spatiotemporal coding and multi-field attention according to claim 1, characterized in that: In preprocessing, the maximum-minimum normalization strategy is used to normalize the pressure field data: ； in, is the original pressure data, 、 are the minimum and maximum pressure values ​​in the data set, is the normalized pressure data, and its range is limited to [0,1].

3. The coupled neural operator model integrating Fourier spatiotemporal coding and multi-field attention according to claim 1, characterized in that: During preprocessing, the spatiotemporal index coding is normalized as follows: = ； in, , 、 、 represents the spatial grid scale of the three-dimensional reservoir, is the time scale, 、 is the horizontal coordinate, is the vertical coordinate, t is the time mark, 、 is the horizontal discrete grid index, is the vertical discrete grid index, is the discrete grid index in time, is the maximum time scale of the reservoir fluid evolution process.

4. The coupled neural operator model integrating Fourier spatiotemporal coding and multi-field attention according to claim 3, characterized in that: In preprocessing, when Fourier high-dimensional space-time coding is normalized, the space-time index is first encoded using relative normalization, and then the fixed-dimensional features are mapped using the space-time frequency matrix, and the space-time index is relatively normalized: ； in, 、 、 represents the Fourier space coordinates, represents the Fourier time coordinate, 、 、 represent 、 、 The scale of direction, represents the time scale; The Fourier high-dimensional space-time coding is normalized as follows: ； in, represents the normalized relative space-time coordinates, is the Fourier single frequency matrix, is the frequency number, is the number of high-dimensional single-frequency Fourier features; ; 、 Represent the high-dimensional features of sine and cosine frequencies respectively.

5. The coupled neural operator model integrating Fourier spatiotemporal coding and multi-field attention according to claim 1, characterized in that: In the operator parameters, the pressure parameter is defined as: ; The saturation parameter is defined as: ; Dimensions of two sets of parameters and the number of 3D reservoir grid cells Consistent, that is , Indicates the number of saturation field grids; The mixed space-time parameters include normalized space-time parameters and Fourier high-dimensional space-time parameters: ； in, Indicates the spatial and temporal coordinates of the identification source, Represents Fourier coding, and obtains Fourier high-dimensional spatiotemporal features, represents hybrid Fourier-coded spatiotemporal coding; ； in, is the number of Fourier high-dimensional space-time features.

6. The coupled neural operator model integrating Fourier spatiotemporal coding and multi-field attention according to claim 1, characterized in that: In coupled attention, the coupling operator combined with the attention mechanism is described as follows: ; in, represents the coupling operator function, 、 They are the initial pressure and saturation inputs, indicating that each spatial position in the three-dimensional reservoir area is at the initial moment The pressure and saturation distribution, It is a hybrid Fourier coding space-time coding; and The core of the enhanced representation process is to use weighted attention 、 and Two-layer attention mechanism setting 、 、 The three sets of weighted attention are as follows: = = ; in, To represent the attention score for stress, is the spatiotemporal representation attention score, Characterize the attention score for saturation and the attention score for 、 and Calculated by linear adaptive learning module: ; in, represents the upper layer representation vector, , The three representation vectors are set to have equal dimensions to comply with algebraic operation rules. Used to distinguish pressure, time and space, saturation to represent attention scores, The attention weight and deviation is the adaptive learning optimization parameter, To represent The attention weight transpose, is the Sigmoid activation function, represents the saturation representation vector, and the feature attention score can be specifically described as: ； and The calculation process is: 。 7. The coupled neural operator model integrating Fourier spatiotemporal coding and multi-field attention according to claim 1, characterized in that: In the operator error, the initial saturation field is set as the oil saturation field, and the oil phase saturation prediction value is obtained through the coupling operator calculation ,Right now: ; in, middle represents the spatial vector (x, y, z) coordinates, t represents time, Refers to the predicted saturation value of (x, y, z) at time t; The oil saturation field error is calculated as follows: ; in, is a combination parameter, including the oil phase saturation attention parameter attention , initial saturation characterization parameters , shared spatiotemporal representation hybrid attention parameters ,Right now , Refers to the spatial position vector of the i-th in three-dimensional space, for Department The actual saturation value at the moment; The saturation field involving the oil and water phases satisfies the conservation constraints, and the water phase saturation field can be indirectly calculated: ; Then, the water phase saturation field error is: ； The oil-water two-phase error is shown in the above formula: , the saturation error of the actual solution equation is ; The pressure field error is calculated as follows: ; same All are combined parameters, and the attention parameter is pressure attention , the initial characterization parameters are , both share parameters ,Right now , for Department Real pressure value at all times; The operator error is the sum of the pressure and oil phase saturation errors, that is: ; in, .

8. The coupled neural operator model integrating Fourier spatiotemporal coding and multi-field attention according to claim 7, characterized in that: In the operator error, the neural operator adaptive learning process is described as follows: ; in, represents the learned joint parameters of the coupling operators; The adaptive learning process aims to minimize ,Right now ,satisfy: ; in, is the learning rate, is the combined prediction error gradient of pressure and saturation, is the accuracy threshold, which controls whether the prediction error meets the acceptable range. is the set of all parameters of the coupling operator, is the true pressure value, is the actual saturation value.

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