Porous medium seepage field rapid prediction method based on random neural operator efficient training

Abstract

The invention discloses a porous medium seepage field rapid prediction method based on principal component analysis and a random neural network. Firstly, principal component analysis dimensionality reduction is performed on a permeability field and a pressure field, and an encoder and a decoder are constructed; then, a random neural network is constructed, the hidden layer weight and bias of the random neural network are fixed after being initialized, and the output layer weight is trained only by solving the linear least square problem so as to learn the potential space mapping relation; preferably, the prediction stability and precision can be improved by integrating a plurality of networks; furthermore, a prediction interval with statistical guarantee can be provided for the integrated model by using a conformal prediction method, and the prediction uncertainty is quantified. According to the method, the training time is shortened to the second level, the calculation cost is remarkably reduced while high prediction precision is kept, high robustness is achieved, and an efficient tool is provided for real-time simulation and rapid decision making of oil and gas exploitation, underground water simulation and other scenes.

Description

Technical Field

[0001] This invention relates to the field of computational fluid dynamics and industrial numerical intelligent simulation technology, and in particular to a method for rapid prediction of seepage fields in porous media based on efficient training of random neural operators, which can be used to achieve efficient and high-precision rapid prediction of seepage fields in porous media. Background Technology

[0002] The seepage problem in porous media, typically described by Darcy's law or its derivatives, is a core physical process in many industrial fields, including oil and gas extraction, groundwater hydrology, environmental engineering, and chemical separation. High-precision, high-efficiency numerical simulations of this type of problem are crucial for scientific prediction and engineering optimization. Currently, mainstream numerical methods for solving seepage fields in porous media include the finite difference method, the finite element method, and the finite volume method. These methods directly discretize and solve physical equations, possessing a solid mathematical foundation and high versatility. However, when facing strongly nonlinear problems or "multi-query" scenarios requiring extensive parameterization (such as for different permeability fields), the computational cost of these methods becomes extremely high. Each parameter change necessitates a complete re-meshing, matrix assembly, and equation solving, with single simulations taking anywhere from hours to days, failing to meet the urgent needs of real-time industrial prediction and rapid optimization. Especially in scenarios such as optimizing oil and gas field development plans and conducting emergency assessments of groundwater pollutant migration, it is often necessary to rapidly simulate dozens or even hundreds of different geological models (represented by changes in permeability fields) within a few hours or even less to assess development risks or the extent of pollution. Faced with such engineering requirements of "multiple options and rapid decision-making," the computational efficiency of traditional numerical methods has become a key bottleneck restricting their application.

[0003] To overcome the bottlenecks of traditional methods, deep learning-based operator learning methods have emerged in recent years, aiming to directly learn the mapping relationship from the input parameter field to the output solution field. Representative methods include deep operator networks and Fourier neural operators. Once trained, such models can quickly predict new inputs. However, these methods generally rely on deep neural networks for end-to-end gradient optimization training, which has two inherent drawbacks: First, the training process involves non-convex optimization of millions or even billions of parameters, resulting in huge computational overhead, typically requiring hours to days on high-performance computing devices; second, model performance is extremely sensitive to the choice of optimization algorithm, hyperparameter settings, and initialization, leading to instability in the training process, easy getting trapped in local optima, and fluctuations in the model's generalization ability.

[0004] In addition, some studies have attempted to combine dimensionality reduction methods such as principal component analysis with shallow stochastic neural networks (such as extreme learning machines) to improve efficiency. However, such methods typically employ single-hidden-layer networks, which have limited nonlinear expressive power and are difficult to accurately characterize the complex multi-scale features in seepage fields. Furthermore, the random initialization of network parameters lacks guidance adapted to specific data distributions, leading to unstable model performance and significant room for improvement in prediction accuracy for complex problems.

[0005] Therefore, how to further improve the prediction accuracy and stability of data-driven methods for complex nonlinear seepage problems while ensuring extremely high computational efficiency remains a technical challenge that urgently needs to be solved in this field. Summary of the Invention

[0006] Traditional numerical simulation methods for porous media seepage fields in industrial scenarios such as oil and gas reservoir simulation and groundwater environmental assessment are computationally intensive and cannot meet the requirements for real-time analysis and rapid comparison of multiple schemes. Existing data-driven surrogate models suffer from high training costs and insufficient predictive stability under complex heterogeneous conditions. This invention aims to provide a rapid prediction method for porous media seepage fields that is extremely fast to train, highly accurate, and robust. This method can quickly construct a surrogate model after obtaining a limited number of high-fidelity simulation samples, enabling millisecond-level prediction of pressure distribution in new permeability fields (such as new geological models), thereby significantly improving the efficiency of engineering design and decision-making.

[0007] To achieve the above-mentioned objectives, the technical solution adopted by this invention is as follows: A rapid prediction method for seepage fields in porous media based on efficient training of stochastic neural operators is proposed. Its core lies in constructing an efficient hybrid framework of "data dimensionality reduction—stochastic feature extraction—closed-loop linear solution." In this invention, "stochastic neural operator" is synonymous with "stochastic neural network," specifically referring to a feedforward network model where the hidden layer weights and biases are randomly initialized and then fixed, and the output layer weights are trained solely by solving a linear least squares problem. Its function is to directly approximate the mapping operator from the input parameter field to the output solution field. The method specifically includes the following steps: Step 1: Obtain multiple sets of known porous media permeability fields and their corresponding pressure field data to form a training sample set.

[0008] Step 2 involves performing principal component analysis on the permeability field and pressure field sample sets respectively, extracting their respective principal component bases. Based on these bases, an encoder and decoder are constructed. Specifically, the encoder projects the input physical field (permeability field or pressure field) onto the corresponding principal component base, and the resulting projection coefficients constitute a low-dimensional latent encoding vector. The decoder uses these latent encoding vectors as weighting coefficients to linearly combine the corresponding principal component bases to reconstruct a high-dimensional physical field. Thus, the encoder maps the high-dimensional permeability field and pressure field to a low-dimensional latent space, obtaining the corresponding latent encoding vectors, thereby transforming the task of learning an infinite-dimensional permeation operator into learning a finite-dimensional latent space mapping relationship. For example, in this step, the number of principal components can be selected by setting a cumulative variance contribution rate threshold; generally, this threshold is greater than or equal to 90%.

[0009] Step 3: Construct a random neural network with at least one hidden layer to learn the latent spatial mapping relationship defined in Step 2. The hidden layers of the random neural network employ non-linear activation functions. The weights and bias parameters of each hidden layer are randomly initialized using an energy-matching strategy before training and then fixed thereafter. In the random neural network, when the number of hidden layers is greater than one, a dense skip connection architecture is adopted, meaning that each hidden layer in the network, except for the last hidden layer, has a direct skip connection to the output layer. The output layer weights and combines the inputs of all skip paths and the output of the last hidden layer to produce the final output. This allows the features extracted by all non-final hidden layers to directly participate in the composition of the final output while avoiding information dissipation in the deep network, and to fuse with the deep features extracted by the last hidden layer, thereby achieving full utilization of multi-scale features. It can be understood that when the random neural network contains only one hidden layer, the above dense skip connection architecture degenerates into a standard single hidden layer structure without skip connections. Regardless of the number of hidden layers, the initialization of parameters for each layer employs a data energy-based matching strategy, and the output layer weights are globally optimized in a single operation by solving a linear least squares problem. Therefore, the single-hidden-layer architecture, as a special form of this invention, is also within the scope of the efficient training framework described in this invention. For example, the hidden layers of the stochastic neural network in this step use a nonlinear activation function, selected from one of the following: hyperbolic tangent, sigmoid, Gaussian, trigonometric sine, trigonometric cosine, polynomial, ReLU, LeakyReLU, or ELU.

[0010] In one embodiment, the parameters of the hidden layer are initialized using an energy matching strategy; when the number of hidden layers is greater than 1, a hierarchical energy matching strategy is adopted, specifically: for the first hidden layer, the initialization range parameter of its weights is calculated based on the total sample energy of the input latent code and the output latent code; for each subsequent hidden layer, the initialization range parameter of the weights of that layer is calculated based on the total sample energy of the output of the previous hidden layer and the total sample energy of the output latent code.

[0011] In one embodiment, the hierarchical energy matching strategy is implemented in the following way: Let the total sample energy of the input latent code be . The total sample energy of the output latent encoding is For the first... Each hidden layer has a weight initialization range parameter. Determined by the following formula: in, In the formula, For the first The total sample energy of the output of each hidden layer is defined. ; For the first Width of the hidden layer; and These are preset hyperparameters used to constrain the initialization range; This is the first The energy matching coefficient set by the layer is used to adjust the target output energy. Distribute proportionally to each layer, and satisfy the following conditions: ,in This represents the total number of hidden layers. The overall sample energy is defined as: for a set of sample vectors Data matrix The total sample energy is ,in Representing vectors Norm.

[0012] Step 4: Train the stochastic neural network using the latent encoding vectors of the training samples obtained in Step 2. That is, input the latent encoding vectors of the training samples into the stochastic neural network, and perform forward propagation to obtain the output feature matrices of all hidden layers; by solving a linear least squares problem with the feature matrix as the design matrix and the target output latent encoding as the observation value, calculate the global optimal solution of all weight matrices connecting the hidden layers to the output layer in one go, and complete the network training.

[0013] Step 4-1 (preferred step): To improve prediction stability, an ensemble learning enhancement strategy is adopted: Steps 3 and 4 are executed independently and repeatedly to construct multiple stochastic neural network models with different random initialization parameters; in the subsequent step 5, when making predictions, the low-dimensional latent vector to be predicted is input into the multiple stochastic neural network models respectively, and the output latent vectors of all stochastic neural network models are arithmetically averaged to obtain the final predicted output latent vector in an ensemble manner.

[0014] In one embodiment, step 4 involves inputting the latent encoding vector of the training samples obtained in step 2 into the random neural network, and forward propagating to obtain the output feature matrices of all hidden layers. By solving a linear least squares problem with the feature matrix as the design matrix and the target output latent encoding as the observation, the global optimal solution of all weight matrices connecting the hidden layers to the output layer is calculated in one go, thus completing the network training. The linear least squares problem is solved by a numerical method, which is selected from one of the following methods: calculating the Moore-Penrose generalized inverse, QR decomposition, singular value decomposition, Cholesky decomposition, or solving the normal equation.

[0015] Step 5: For the new permeability field to be predicted, firstly, the encoder in step 2 is used to map it into a low-dimensional latent vector; then, the low-dimensional latent vector is input into the stochastic neural network model trained in step 4 (or the ensemble model obtained in step 4-1) to obtain the predicted output latent vector; finally, the decoder in step 2 is used to reconstruct the output latent vector into the complete predicted pressure field.

[0016] Step 6 (optional): To further enhance the engineering reliability of the prediction results, this method also includes an uncertainty quantification step based on conformal prediction. Specifically, using the multiple stochastic neural network models (ensemble model) and an independent calibration dataset, the mean and standard deviation of the predicted pressure field are calculated at discrete grid points in the solution domain. Based on the true and predicted values ​​of the calibration dataset, a calibration score set is calculated, and its quantiles are determined according to a preset confidence level. Finally, for any permeability field to be predicted, a statistically guaranteed prediction interval can be generated at each grid point. This interval covers the true pressure field with a definite probability, thereby quantifying the uncertainty of the model prediction and providing a quantitative basis for risk assessment and engineering decision-making.

[0017] In one embodiment, this step can be specifically described as follows: Step 6-1: Using the calibration dataset, predict and reconstruct the pressure field of each calibration sample through the multiple random neural network models, and calculate its predicted mean and standard deviation at discrete grid points in the solution domain; Step 6-2: Based on the predicted mean, standard deviation, and corresponding real pressure field, construct a calibration score set for the calibration dataset; Step 6-3: Determine the quantiles of the calibration score set according to the preset confidence level. ; Step 6-4: For the permeability field to be predicted, calculate the mean value of the predicted pressure field at the discrete grid points in the solution domain. with standard deviation The standard deviation is then numerically stabilized to obtain... ,in A preset positive threshold is set; then a prediction interval is generated. .

[0018] Step 6-2, which involves constructing the calibration score set, includes: for each sample in the calibration dataset, calculating the calibration score point-by-point on the discrete grid points in the solution domain. ,in For the first Each sample at the grid point The true stress value, and These are the predicted mean and standard deviation, respectively. To The value after numerical stabilization. Set a preset positive threshold; then use all grid points of all calibration samples. Values ​​are collected as a set of calibration scores. ,in To calibrate the sample size, This represents the number of grid points.

[0019] The present invention also provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the program, it implements the steps of the rapid prediction method for porous media seepage field based on efficient training of random neural operators described in the present invention.

[0020] The present invention also provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of the method for rapid prediction of porous media seepage field based on efficient training of random neural operators as described in the present invention.

[0021] Compared with the prior art, the technical solution provided by the present invention has the following significant advantages: 1. Significantly improved training efficiency: By fixing the hidden layer parameters and transforming network training into a closed linear least squares problem, the time-consuming backpropagation and non-convex optimization process of traditional deep neural networks are completely avoided, greatly reducing computational costs and time overhead.

[0022] 2. High prediction accuracy: Compared with the traditional single-hidden-layer random neural network architecture, the structure of this invention, which includes at least one hidden layer, and the skip connection design used when the number of hidden layers is greater than 1, significantly enhances the model's ability to capture multi-scale features and nonlinear relationships, thereby enabling it to more accurately approximate complex seepage physical mappings in the dimensionality-reduced latent space.

[0023] 3. Robust and easy to deploy in engineering: The proposed energy-matching initialization strategy reduces sensitivity to random initialization, while the ensemble learning strategy further smooths prediction fluctuations. The entire method is well-defined, training is stable, and results are reliable, making it highly suitable for integration into industrial software or simulation platforms, meeting the high stability requirements of practical engineering.

[0024] 4. Excellent Framework Versatility: This invention is a fully data-driven, non-intrusive solution that does not rely on the specific discretization scheme or solver knowledge of partial differential equations. Therefore, the framework is easily transferable and applied to the rapid forward prediction and inverse problem solving of other physical fields such as heat conduction, elasticity, and wave propagation, exhibiting good scalability.

[0025] 5. Provides a reliable measure of uncertainty: By combining ensemble learning with conformal prediction, it provides a statistically guaranteed prediction interval for the prediction results, quantifies the uncertainty of the model, enhances the credibility and engineering applicability of the method, and adds almost no additional computational cost.

[0026] In summary, this invention, through its innovative network architecture and training mechanism, successfully resolves the long-standing core contradiction between "computational accuracy" and "solution efficiency" in the prediction of seepage fields in porous media. It provides an efficient and reliable technical tool for real-time simulation, parameter optimization, and rapid decision-making in fields such as oil and gas resource development, groundwater hydrological simulation, and environmental engineering assessment. Attached Figure Description

[0027] Figure 1 This is an overall flowchart of the method for rapid prediction of seepage field in porous media according to an embodiment of the present invention.

[0028] Figure 2 This is a schematic diagram of a standard stochastic neural network structure: solid lines represent fixed weights and biases after initialization, and dashed lines represent adjustable output layer weights during training.

[0029] Figure 3 This is a schematic diagram of a two-hidden-layer random neural network structure: solid lines represent fixed parameters, dashed lines represent trainable parameters, and the jump connections from the first hidden layer to the output layer are shown.

[0030] Figure 4 This is an integrated PCA according to a preferred embodiment of the present invention. A schematic diagram of a random neural network architecture.

[0031] Figure 5 The following is a comparison of the prediction results of the embodiments not integrated in this invention: (a) Input permeability field; (b) Reference pressure field (numerical solution); (c) Predicted pressure field of this invention; (d) Absolute error distribution.

[0032] Figure 6 This invention integrates ( Comparison of prediction results of the embodiments: (a) Input permeability field; (b) Reference pressure field (numerical solution); (c) Predicted pressure field of the present invention; (d) Absolute error distribution.

[0033] Figure 7 The method of this invention is relatively effective on 200 test samples. Error box plot (including comparison of results between the non-integrated and integrated groups).

[0034] Figure 8 Here is a visualization of the uncertainty quantification results in this embodiment of the invention: (a) the predicted mean of the pressure field; (b) the reference pressure field (numerical solution); (c) the width of the prediction interval; and (d) the coverage indicator of the actual pressure field (where green areas indicate coverage and red areas indicate no coverage). Detailed Implementation

[0035] To make the technical problems, technical solutions, and beneficial effects of this invention clearer and more explicit, the invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the embodiments described herein are only for explaining the technical principles of this invention and are not intended to limit the scope of protection of this invention.

[0036] This invention presents a rapid prediction method for seepage fields in porous media based on principal component analysis and random neural networks. The method can be implemented on an electronic device with a processor and memory: the memory stores a computer program, and the processor executes the program to complete the steps described in the following embodiments. This embodiment uses Darcy flow in a two-dimensional steady-state heterogeneous porous medium as an application object to demonstrate the specific process of the prediction method and verify its technical effectiveness. The governing equations are as follows: And at the boundary Apply Dirichlet boundary conditions .in, It is a heterogeneous permeability field. For the pressure field to be sought, Given a known source term. The objective of this invention is to learn from the permeability field. to pressure field Mapping operators .

[0037] The following embodiments use the random neural network containing two hidden layers as an example to illustrate the method of the present invention in detail.

[0038] like Figure 1 As shown, the overall process of this method includes the following steps: Step 1: Collect or generate spatial distribution data of permeability of porous media and its corresponding pressure field data.

[0039] Multiple sets of known porous media permeability fields and their corresponding pressure fields are acquired to form a training sample set. The data can be sourced from: 1. High-fidelity numerical simulation: such as multiple representative geological models constructed based on geological modeling software, and pressure field results obtained using finite element or finite volume method solvers.

[0040] 2. Physical experiments: such as permeability and pressure difference data obtained from core displacement experiments.

[0041] 3. Field data: Under specific conditions, the permeability distribution is obtained by combining downhole pressure gauge data with inversion data.

[0042] To improve the model's generalization ability, the constructed training set should cover as much as possible the range of permeability distribution that may occur in the work area, including high-permeability channels, low-permeability interlayers, and random heterogeneous patterns.

[0043] To verify the effectiveness of this invention, data generated by numerical simulation is used in this embodiment for verification. The specific data generation and discretization process is as follows: 1. In a uniform grid Define computational domain Total number of discrete points And apply zero Dirichlet boundary conditions at the boundary of the computational domain.

[0044] 2. Generate Gaussian random fields (Pick The data is then binarized with a threshold of 0 to obtain a discrete heterogeneous permeability field. 3. For each permeability field on the above grid, take the source term. The reference pressure field is obtained by solving the discrete equations using the finite difference method. .

[0045] 4. A total of 1200 samples were generated and randomly divided into training sets. With test set .

[0046] Step 2: Use Principal Component Analysis (PCA) for dimensionality reduction.

[0047] For the input field (permeability) and output field (pressure field) of the training set, construct sample matrices and calculate the covariance matrix respectively, and perform eigenvalue decomposition to obtain the principal component basis. and Before performing principal component analysis, the training set samples need to be mean-reduced to obtain centered samples. Number of principal components. Based on the set cumulative variance contribution rate threshold Select (e.g.) ≥90%, this embodiment takes =90%), that is, choose the smallest one. Make in It is a sequence of eigenvalues.

[0048] In this embodiment, considering the balance between computational efficiency and prediction accuracy, the dimensionality reduction method was ultimately selected. At this point, the cumulative variance contribution rate of the input field is 92.4%, and that of the output field is 99.9%. Dimensionality reduction through principal component analysis effectively filters out high-frequency geometric noise and extracts dominant flow features. This setting, while ensuring that core features are not lost, effectively avoids interference from binarized high-frequency noise, thereby improving the overall computational efficiency and generalization ability of the model.

[0049] In this step, encoding and decoding operations are defined.

[0050] Define an encoder that maps a high-dimensional physical field to a low-dimensional feature vector. Specifically, input the encoder. Arbitrary permeability field It is mapped to its projection coefficient vector on the input principal component basis: in, The mean of the training samples in the penetration field. This represents the inner product operation. Similarly, the output encoder... pressure field Mapped to the corresponding low-dimensional feature vector: .

[0051] Define a decoder that reconstructs a high-dimensional physical field from a low-dimensional feature vector. Specifically, output the decoder. any low-dimensional vector Reconstructed into a pressure field: in, This is the mean of the training samples for the pressure field. This reconstruction process is the inverse of the encoding principle shown in formula (2), that is, using... For coefficients on the output principal component basis Perform a linear combination and add a mean field. .

[0052] Through the principal component analysis dimensionality reduction and reconstruction mechanism described above, infinite-dimensional operators are learned from the original problem. The task is transformed into learning mappings in a finite-dimensional latent space. That is, satisfying This mapping relationship will be fitted by a subsequent random neural network.

[0053] Step 3: Construct a random neural network. In this embodiment, the random neural network contains two hidden layers.

[0054] This embodiment uses a stochastic neural network with two hidden layers and skip connections to learn the mapping. Its structure is as follows Figure 3 As shown. This architecture is designed for traditional single-hidden-layer stochastic neural networks (its structural diagram is shown in...). Figure 2 This is a significant improvement over existing technologies, aiming to simultaneously capture low-frequency global features and high-frequency local details in a potential spatial map.

[0055] Specifically, the network of this invention comprises two parallel paths: (1) Main path: Input layer → First hidden layer → Second hidden layer → Output layer.

[0056] (2) Jump path: Input layer → First hidden layer → Output layer.

[0057] In network architecture design, the introduction of skip connections is of great significance. Skip connections can directly and effectively transmit the macroscopic background trend of the pressure field (usually related to the low-frequency component of the permeability field) to the output layer, thereby avoiding information dissipation during the transmission process in deep networks and further improving prediction accuracy.

[0058] Let the input latent vector be Then the outputs of the first hidden layer and the second hidden layer are respectively: Among them, the weight matrix , and bias vector , These are the parameters that need to be randomly initialized and fixed in this invention. This differs from traditional single-layer networks which have only one set of fixed parameters. In this embodiment, both hidden layers use the hyperbolic tangent activation function, i.e. .

[0059] To ensure the effectiveness of initialization, a hierarchical energy matching strategy is used to initialize the hidden layer parameters of the stochastic neural network. This embodiment takes a case with two hidden layers as an example to specifically illustrate the hierarchical energy matching strategy: based on the input latent encoding, the output of the first hidden layer, and the total sample energy of the output latent encoding, the weight initialization range parameters of the two hidden layers are calculated respectively. and This ensures that the initial state of the two-layer network matches the data distribution characteristics.

[0060] First, calculate the range parameters of the first hidden layer. The weights of the first hidden layer Each element is from a uniform distribution Mid-sampling, of which Calculated using the following formula: Then, using the initialized first hidden layer weights and bias Calculate the output matrix of the first hidden layer and calculate Overall sample energy Next, the range parameters of the second hidden layer are calculated. The weights of the second hidden layer Each element is from a uniform distribution Mid-sampling, of which Calculated using the following formula: In the above formula, and These are the latent encodings of the training set input. and output latent encoding The overall sample energy is defined as the average of the second moments of all sample vectors. Specifically, for the input latent code, we have... Similarly, For the first Hidden layer output Its total sample energy is defined as . and These are the widths of the first hidden layer and the second hidden layer, respectively. and It is a preset energy matching coefficient that satisfies This is used to allocate the contribution ratio of the two network layers to the output energy (in this embodiment, we take...). , ). , , , The boundary values ​​set for experience are calculated using the following formula: in, and This is an empirical hyperparameter used to control the boundaries of the initial weights, avoiding activation function saturation or overly weak features. Empirically, it is usually set in a positive range near 1. In this embodiment, it is set to... .

[0061] The bias term uses a conventional small-range random initialization, i.e. ,in This represents the dimension of the input layer, i.e., the number of neurons in the previous layer. The hidden layer parameters, after being initialized according to the above distribution, remain fixed throughout the training process (e.g., ...). Figure 3 (As shown by the solid line).

[0062] The above initialization strategy for the two hidden layers embodies the core idea of ​​hierarchical energy matching: that is, the initialization range parameters of each layer are based on its input features (for the first layer, it is...). For subsequent layers, it is the energy output of the previous layer. ) and the energy of the final output target To calculate, and through matching coefficients This is used to allocate the expected contribution of each layer to the output energy. It can be understood that this strategy can be applied to networks with different numbers of hidden layers. For networks containing... The network with the nth hidden layer, the nth ( Initialization range parameters of the layer It can be determined by the following general formula: in, ; For the first Hidden layer width, For the first The energy matching coefficient of the layer, and satisfying , This represents the total number of hidden layers.

[0063] When the network has only one hidden layer ( When this occurs, the strategy degenerates into single-layer energy matching, at which point the matching coefficient... This embodiment uses... The following is a detailed explanation and verification using the example of the case where the matching coefficient was used. , This is only one preferred allocation method.

[0064] Step 4: Train the network using linear least squares.

[0065] set up and Let represent the learnable weight matrices from the first hidden layer to the output layer and from the second hidden layer to the output layer, respectively. Then the network... The final output is contributed by both the jump path and the main path: The only parameters that need to be trained are the weight matrices connecting the two hidden layers to the output layer. and (like Figure 3 (As shown by the dashed line). The output vector of the hidden layer. Stacked into a matrix , target vector Stacked into a matrix Therefore, training a stochastic neural network with two hidden layers is transformed into solving a standard linear least squares problem: This problem has a globally optimal solution, which can be obtained through numerical methods, such as calculating the augmented output feature matrix. Moore-Penrose generalized inverse matrix : This process avoids time-consuming backpropagation iterations and non-convex optimization processes.

[0066] Step 4-1: Ensemble learning to enhance robustness (preferred step).

[0067] To reduce prediction fluctuations caused by random initialization and further improve the model's stability and generalization ability, this embodiment may optionally employ an ensemble learning strategy. For example... Figure 4 As shown, step 3 is repeated while keeping the network structure unchanged. Each time, the hidden layer parameters are initialized using independent random seeds, thus obtaining... PCA with the same structure but different internal random feature maps Stochastic Neural Network Submodel In the prediction phase, for the same input The outputs of each sub-model are statistically combined to obtain the final predicted value. In this preferred embodiment, the combination method uses an arithmetic mean: Those skilled in the art will know that, The value of can range from 1 to any positive integer; to balance computational cost and accuracy gain, it is usually 1. Assuming the range is 2–30, meaning the number of random neural networks constructed is 2 to 30, this embodiment preferably uses the range 2–30. .when In this case, this step degenerates into single-model prediction, which still achieves the basic function of the present invention. Therefore, the above integration method is only used to illustrate the preferred implementation and does not constitute a limitation on the scope of protection of the present invention.

[0068] Step 5: Prediction Process.

[0069] For a new geological model or a new permeability field to be predicted, the steps for predicting its pressure field are as follows: 1. Encoding: .

[0070] 2. Latent spatial mapping: or .

[0071] 3. Decoding and Reconstruction: .

[0072] The prediction process can be completed in milliseconds, enabling real-time forecasting. It can also be seamlessly integrated into existing reservoir numerical simulation workflows or groundwater simulation platforms as a rapid simulation module or real-time early warning module, providing engineers with immediate feedback.

[0073] Step 6: Uncertainty quantification (optional step).

[0074] To provide a measure of the reliability of the prediction results, this embodiment optionally employs an uncertainty quantification method based on conformal prediction. This method uses a calibration set to calibrate the predictions of the ensemble model to construct a prediction interval with statistical guarantees, thereby providing a reliable basis for engineering decisions. The specific implementation steps are as follows: 1. Prepare the calibration dataset: In the total data samples generated in step 1, a portion is independently allocated as the calibration set (100 samples in this example) to ensure that the calibration set, training set and test set are independent and identically distributed.

[0075] 2. Calculate the calibration score: using the score obtained from training in step 4-1. A random neural network model (in this embodiment) The prediction is performed on the calibration set samples. For each sample in the calibration set, the mean prediction of the ensemble model is calculated at discrete grid points in the solution domain. and the predicted standard deviation To avoid numerical instability caused by excessively small standard deviations, a standard deviation stabilization process is performed: in A preset positive threshold number (taken in this embodiment) This operation ensures that the standard deviation is always not lower than the preset minimum effective value. Then, the calibration score is calculated point by point: in, For the first One calibration sample at grid point The true pressure value. The scores of all grid points in all calibration samples are collected into a calibration score set. ,in To calibrate the sample size, This represents the number of grid points.

[0076] 3. Calculate calibration quantiles: given confidence level (In this embodiment) (i.e., at a 95% confidence level), calculate the quantiles of the calibration score set. Specifically, the set of calibration scores Sort in ascending order, and take the [number]th [item]. Each fraction as ,in To calibrate the total score.

[0077] 4. Constructing the prediction interval: For a new permeability field, obtain the following from step 4-1: After reconstructing the model's predictions in step 5, the mean prediction is calculated at the discrete grid points in the solution domain. and the predicted standard deviation Similarly, numerical stabilization is applied to the prediction standard deviation: ,in The same preset positive threshold as in step 2 is used. Therefore, at each grid point... The prediction interval is: In this embodiment, the calibration score calculation (Formula (13)) and the prediction interval construction (Formula (14)) adopt the same numerical stabilization processing method (i.e. This ensured the quantiles. The statistical guarantee is maintained in actual forecasts. This approach avoids numerical instability caused by excessively small standard deviations, and does not unnecessarily affect reasonable standard deviation estimates, thus achieving robustness of the forecast interval.

[0078] To verify the effectiveness and optimal performance of the method of this invention, the proposed non-integrated stochastic neural network scheme with two hidden layers was tested on 200 independent test samples. ) and integration solutions ( The network hidden layer width is set to [value]. , The range parameters are calculated according to formulas (6) and (7). , This was used for model initialization. All experiments were conducted on a computer platform configured with an Intel i7 processor and an NVIDIA RTX 2060 graphics card.

[0079] Figure 5 The following are the prediction results of the non-integrated scheme for a typical high-contrast permeability sample: (a) input permeability field; (b) finite difference reference solution; (c) the predicted pressure field of the present invention; (d) absolute error distribution. Figure 6 Given the same sample ensemble ( The corresponding results of the proposed solution are shown below. It can be seen that the pressure field predicted by this invention is visually highly consistent with the numerical reference solution, the absolute error field amplitude is extremely low, and the error amplitude is further reduced after integration, with a more uniform spatial distribution.

[0080] Figure 7 This study summarizes the relative performance of non-integrated and integrated solutions across 200 test samples. Box plot of error distribution. The median prediction error of the non-integrated scheme is... The median error of the integrated scheme is This represents a decrease of approximately 11.8% compared to the non-integrated solution. Meanwhile, the error distribution of the integrated solution is more concentrated, with its interquartile range decreasing from... Reduced to The maximum error is due to Down to This fully demonstrates the effectiveness of ensemble learning strategies in improving prediction stability and robustness.

[0081] To further verify the reliability of this invention in industrial safety assessment, uncertainty quantification was performed on the same 200 test samples. Figure 8 The results of a typical test sample are presented: (a) the predicted mean pressure field; (b) the reference pressure field; (c) the predicted interval width; and (d) the actual pressure field coverage indicator (where green areas represent coverage and red areas represent uncovered areas). It can be seen that the predicted mean pressure field is visually highly consistent with the reference pressure field; the predicted interval width exhibits a non-uniform spatial distribution, reflecting the model's uncertainty in predicting different regions; the coverage indicator shows that the reference pressure field falls within the predicted interval for most spatial points. Statistical results show that the actual coverage rate of this method reaches 95.8%, close to the set 95% confidence level; the average interval width is... This indicates that the model has a high confidence level. This result verifies the effectiveness of the uncertainty quantification method based on conformal prediction within the framework of this invention, providing reliable decision support for engineering applications.

[0082] In terms of efficiency, this invention demonstrates a significant advantage. Training the non-ensemble model takes approximately 1.7 seconds, while the ensemble approach takes approximately 7.2 seconds. Compared to traditional gradient descent-based deep learning methods, this method utilizes PCA dimensionality reduction to decrease the computational scale and combines a stochastic neural network to directly solve for the least-squares analytical solution, avoiding the lengthy weight iteration process and achieving 2... Three orders of magnitude efficiency improvement. The single-sample inference time is 4 milliseconds for the non-ensemble model and 90 milliseconds for the ensemble model, both in the millisecond range, fully meeting the needs of industrial real-time simulation. Furthermore, uncertainty quantization can be completed in approximately 0.44 seconds for calibration and interval construction. Compared to traditional Bayesian sampling-based uncertainty quantization methods, this method does not incur additional training costs, requiring only simple post-processing of the existing ensemble model, truly achieving low-cost, high-guarantee uncertainty quantification.

[0083] In summary, this specific embodiment confirms the PCA provided by the present invention. The stochastic neural network method can achieve high-precision seepage field prediction with a training cost in the order of seconds; its optimized ensemble mode further improves prediction accuracy and stability without significantly increasing computational overhead. This invention effectively solves the contradiction between "computational speed" and "prediction accuracy" in traditional methods, providing a practical solution for real-time simulation and rapid analysis in related industrial fields.

Claims

1. A method for rapid prediction of seepage field in porous media based on efficient training of random neural operators, characterized in that, Includes the following steps: Step 1: Obtain multiple sets of known porous media permeability fields and their corresponding pressure field data to form a training sample set; Step 2: Perform principal component analysis on the permeability field sample set and the pressure field sample set respectively, and extract the principal component basis for each. Based on the principal component basis, an encoder and a decoder are constructed. The encoder is configured to project the input physical field onto the corresponding principal component basis to obtain projection coefficients as latent encoding vectors. The decoder is configured to use the latent encoding vectors as coefficients to linearly combine the principal component basis to reconstruct the physical field. Thus, the encoder maps the high-dimensional permeability field and pressure field to a low-dimensional latent space to obtain the corresponding latent encoding vectors, thereby transforming the learning of an infinite-dimensional permeation operator into the learning of a finite-dimensional latent space mapping. Step 3: Construct a stochastic neural network with hidden layers to learn the latent space mapping defined in Step 2; wherein, all weights and bias parameters of the hidden layers are randomly initialized before training and then fixed; in the stochastic neural network, when the number of hidden layers is greater than 1, the output of each hidden layer except the last one is directly connected to the output layer through an independent skip path; the output layer performs a weighted combination of the inputs of all skip paths and the output of the last hidden layer to produce the final output; Step 4: Train the random neural network using the latent encoding vectors of the training samples obtained in Step 2; Step 5: For the new permeability field to be predicted, first use the encoder in step 2 to map it into a low-dimensional potential vector; then input it into the random neural network trained in step 4 to obtain the predicted output potential vector; finally use the decoder in step 2 to reconstruct it into a complete predicted pressure field.

2. The method for rapid prediction of porous media seepage field based on efficient training of random neural operators according to claim 1, characterized in that, In step 2, the number of principal components is selected by setting a cumulative variance contribution rate threshold, wherein the threshold is greater than or equal to 90%.

3. The method for rapid prediction of porous media seepage field based on efficient training of random neural operators according to claim 1, characterized in that, In step 3, the hidden layer of the random neural network uses a nonlinear activation function, which is selected from one of the hyperbolic tangent function, sigmoid function, Gaussian function, trigonometric sine function, trigonometric cosine function, polynomial function, ReLU function, LeakyReLU function, or ELU function.

4. The method for rapid prediction of porous media seepage field based on efficient training of random neural operators according to claim 1, characterized in that, In step 3, the parameters of the hidden layer are initialized using an energy matching strategy. When the number of hidden layers is greater than 1, a hierarchical energy matching strategy is adopted. Specifically, for the first hidden layer, the initialization range parameter of its weights is calculated based on the total sample energy of the input latent code and the output latent code. For each subsequent hidden layer, the initialization range parameter of the weights of that layer is calculated based on the total sample energy of the output of the previous hidden layer and the total sample energy of the output latent code.

5. The method for rapid prediction of porous media seepage field based on efficient training of random neural operators according to claim 4, characterized in that, The hierarchical energy matching strategy is implemented in the following way: Let the total sample energy of the input latent code be . The total sample energy of the output latent encoding is For the first... Each hidden layer has a weight initialization range parameter. Determined by the following formula: in, In the formula, For the first The total sample energy output by each hidden layer is defined. ; For the first Width of the hidden layer; and These are preset hyperparameters used to constrain the initialization range; This is the first The energy matching coefficient set by the layer is used to direct the target output energy. Distribute proportionally to each layer, and satisfy the following conditions: ,in This represents the total number of hidden layers. The overall sample energy is defined as: for a set of sample vectors Data matrix The total sample energy is ,in Representing vectors Norm.

6. The method for rapid prediction of seepage field in porous media based on efficient training of random neural operators according to claim 1, characterized in that, After step 4 and before step 5, step 4-1 is also included: ensemble learning enhancement; The ensemble learning enhancement specifically involves: independently repeating steps 3 and 4 multiple times to construct multiple stochastic neural networks with different random initialization parameters; during prediction in step 5, the low-dimensional latent vector to be predicted is input into multiple stochastic neural networks respectively, and the output latent vectors of all stochastic neural networks are arithmetically averaged to obtain the final predicted output latent vector in an ensemble manner.

7. The method according to claim 1, characterized in that, In step 4, the latent encoding vectors of the training samples obtained in step 2 are input into the random neural network, and forward propagation is used to obtain the output feature matrices of all hidden layers. By solving a linear least squares problem with the feature matrix as the design matrix and the target output latent encoding as the observation, the global optimal solution of all weight matrices connecting the hidden layers to the output layer is calculated at once, thus completing the network training. The linear least squares problem is solved by a numerical method, which is selected from one of the following methods: calculating the Moore-Penrose generalized inverse, QR decomposition, singular value decomposition, Cholesky decomposition, or solving the normal equation.

8. The method for rapid prediction of porous media seepage field based on efficient training of random neural operators according to claim 6, characterized in that, Following step 4-1, step 6 is also included: uncertainty quantification; the uncertainty quantification utilizes the multiple stochastic neural network models to provide a statistically guaranteed prediction interval for the predicted pressure field based on the conformal prediction method, specifically including: Step 6-1: Using the calibration dataset, predict and reconstruct the pressure field of each calibration sample through the multiple random neural network models, and calculate its predicted mean and standard deviation at discrete grid points in the solution domain; Step 6-2: Based on the predicted mean, standard deviation, and corresponding real pressure field, construct a calibration score set for the calibration dataset; Step 6-3: Determine the quantiles of the calibration score set according to the preset confidence level. ; Step 6-4: For the permeability field to be predicted, calculate the mean value of the predicted pressure field at the discrete grid points in the solution domain. with standard deviation The standard deviation is then numerically stabilized to obtain... ,in A preset positive threshold is set; then a prediction interval is generated. .

9. The method for rapid prediction of seepage field in porous media based on efficient training of random neural operators according to claim 9, characterized in that, Step 6-2, which involves constructing the calibration score set, includes: for each sample in the calibration dataset, calculating the calibration score point-by-point on the discrete grid points in the solution domain. ,in For the first Each sample at the grid point The true stress value, and These are the predicted mean and standard deviation, respectively. To The value after numerical stabilization. Set a preset positive threshold; then use all grid points of all calibration samples. Values ​​are collected as a set of calibration scores. ,in To calibrate the sample size, This represents the number of grid points.

10. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the program, it implements the steps of the method for rapid prediction of porous media seepage field based on efficient training of random neural operators as described in any one of claims 1 to 9.

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