A liquid lead-bismuth medium oxygen concentration prediction method and an oxygen concentration field construction method

Abstract

The application discloses a liquid lead-bismuth oxygen concentration prediction method and an oxygen concentration field construction method, relates to the oxygen concentration field prediction technical field of a lead-bismuth experimental device, and the liquid lead-bismuth oxygen concentration field construction method comprises the following steps: taking a plurality of oxygen electromotive force sensor measuring points in a lead-bismuth experimental device as graph nodes, constructing a three-source fusion adjacency matrix, and extracting spatial coupling features between the measuring points by K-order adaptive diffusion graph convolution; long-term and short-term time sequence dependencies and key time step information in a historical window are captured by combining a bidirectional long short-term memory network and a time sequence attention mechanism; oxygen electromotive force prediction values and oxygen concentration prediction values of each measuring point under a plurality of future prediction steps are obtained; a prediction step adaptive physical constraint module is constructed to dynamically adjust a physical loss weight, and multi-branch consistency distillation is performed; finally, spatial interpolation reconstruction is performed in combination with spatial positions of the measuring points to obtain a continuous oxygen concentration field in the liquid lead-bismuth experimental device. High-precision and high-robustness oxygen concentration field space-time prediction is achieved.

Description

Technical Field

[0001] This invention relates to the field of spatiotemporal prediction technology of oxygen concentration field in lead-bismuth experimental apparatus, and in particular to a method for predicting oxygen concentration in liquid lead-bismuth and a method for constructing oxygen concentration field. Background Technology

[0002] In lead-based reactors and lead-bismuth experimental setups, oxygen concentration is a crucial parameter affecting the corrosion of structural materials, the stability of protective oxide films, and the safe operation of the circuit. The oxygen electromotive force (EMF) output by the oxygen sensor reflects changes in oxygen concentration and serves as a vital monitoring signal for the liquid lead-bismuth oxygen control system. However, oxygen sensors typically operate for extended periods in high-temperature, corrosive liquid lead-bismuth environments and complex thermal-hydraulic conditions. Their sensitive elements, reference electrodes, electrical connections, and packaging structures may experience performance degradation, resulting in a limited sensor lifespan and susceptibility to problems such as measurement drift, abnormal fluctuations, increased noise, intermittent failures, and partial data loss. Furthermore, the oxygen EMF sequence under actual operating conditions exhibits nonlinearity, non-stationarity, multi-point spatial coupling, and temperature dependence. Factors such as sensor aging, data loss, acquisition anomalies, and operational disturbances make it difficult to guarantee the continuity, accuracy, and reliability of the monitoring data over the long term.

[0003] In existing technologies, a common graph convolution model is used, which extracts time-domain and frequency-domain features through a frequency domain transformation module, a time-domain decomposition module, a convolution feature extraction module, and a time-series feature extraction module to perform multi-feature fusion prediction.

[0004] However, existing methods have the following main shortcomings: First, traditional time-series models such as LSTM, GRU, and BiLSTM primarily model time dependencies, making it difficult to fully utilize the geometric positions, flow directions, and spatial coupling relationships between oxygen sensors. Existing methods typically treat the data from each measurement point as independent time-series sequences and process them separately, ignoring the spatial correlations between multiple measurement points caused by loop flow paths and spatial topology.

[0005] Second, conventional graph convolution models rely on a fixed adjacency matrix, making it difficult to reflect the dynamic changes in sensor correlation strength under varying operating conditions. In liquid lead-bismuth circuits, the correlation strength between measuring points changes under different operating conditions (such as flow rate changes and temperature fluctuations), and fixed graph structures cannot adapt to this dynamic characteristic.

[0006] Third, pure data-driven models are prone to error accumulation, trend drift, and physically inconsistent predictions in long-term forecasting. Lacking physical constraints, data-driven models may deviate from the physically feasible range during multi-step extrapolation.

[0007] Fourth, common industrial environments, such as noise, random packet loss, and partial network outages, lead to incomplete input data, causing existing models to exhibit significantly reduced predictive performance under these extreme conditions. Current methods lack a systematic and robust training mechanism to address sensor noise and data gaps.

[0008] Fifth, fixed-strength PINN physical constraints are not applicable to all prediction time steps. In the short time domain, they may interfere with data-driven fitting, while in the medium to long time domain, they help suppress trend drift and physically unreasonable results. Existing physical constraint methods typically use fixed weights and cannot adaptively adjust according to the prediction time step.

[0009] Therefore, there is a need for a method that can simultaneously take into account spatial topology modeling, temporal dependency modeling, robustness against noise and packet loss, and adaptive physical constraints of prediction time step. Summary of the Invention

[0010] To overcome the shortcomings of the prior art, this invention proposes a method for predicting oxygen concentration in liquid lead-bismuth and a method for constructing an oxygen concentration field.

[0011] To achieve the above objectives, the present invention employs the following technical solution: a method for predicting oxygen concentration in liquid lead-bismuth, comprising: Step S1: Using several oxygen electromotive force sensor measurement points set in the lead-bismuth experimental device as graph structure nodes, collect the oxygen electromotive force time series data and temperature time series data of each measurement point, and fuse the static physical prior graph, dynamic state graph and adaptive learning graph to construct a three-source fusion adjacency matrix. Step S2: Based on the three-source fusion adjacency matrix, perform K-order diffusion graph convolution on the node input feature matrix composed of the node input features of all nodes at each time step within the historical input window, extract the spatial coupling features between each measurement point, and obtain the spatially enhanced node feature sequence composed of the spatially enhanced node features of all historical time steps under this window; the node input features include one or more of the following: oxygen electromotive force data, valid data labeling, measurement point location embedding, node identity embedding, temperature time series data mapping results, and node features after local time series enhancement; Step S3: Perform bidirectional temporal modeling and temporal attention weighted aggregation on the spatially enhanced node feature sequence to extract the temporal dependencies and key time step information within the historical input window, and obtain comprehensive temporal features; Step S4: Based on the comprehensive time series features, perform spatiotemporal fusion and multi-step prediction to obtain the oxygen electromotive force prediction results of each measuring point under the future P prediction time steps; Step S5: Based on the physical conversion relationship between oxygen electromotive force, temperature and oxygen concentration, convert the oxygen electromotive force prediction result into the oxygen concentration prediction result at each measuring point.

[0012] Preferably, in step S1, the three-source fusion adjacency matrix A is represented as: ; in, This is a static physical prior diagram; This is a dynamic state diagram; For adaptive learning graphs; , , These are the learnable fusion weights for the three types of adjacency matrices.

[0013] Preferably, the static physical prior map The construction method is as follows: A flow direction topology matrix is ​​constructed based on the spatial arrangement of the oxygen electromotive force sensor in the lead-bismuth experimental device loop, its location in the pipe section, upstream and downstream relationships, and flow direction; a historical correlation matrix between the measuring points is calculated based on the oxygen electromotive force time-series data and temperature time-series data of each measuring point in the training set; the flow direction topology matrix and the historical correlation matrix are fused element-by-element and normalized to obtain a static physical prior map. ; The dynamic state diagram The construction method is as follows: the temperature time series data in the current historical input window is mapped to hidden space state features, their statistical state variables are calculated and concatenated, and a dynamic adjacency matrix corresponding to the current sample is generated through a nonlinear mapping network, thereby obtaining the dynamic state diagram. ; The adaptive learning graph The construction method is as follows: A first learnable node embedding matrix and a second learnable node embedding matrix are assigned to each oxygen electromotive force measurement point, used to characterize the latent features of the receiving end and the latent features of the transmitting end of the node, respectively; the inner product of the two sets of learnable node embedding matrices is calculated to obtain the correlation score between measurement points, which is then activated by ReLU to obtain a non-negative correlation strength matrix; a mask sparsity filter is performed using the mean of each row in the non-negative correlation strength matrix as a threshold, and then normalization is performed row by row to obtain the adaptive learning graph formed by the adaptive learning graph adjacency matrix. .

[0014] Preferably, in step S2, the spatial augmentation node features of the t-th historical time step in the current historical input window. Represented as: ; Where σ(·) represents the nonlinear activation function; K represents the upper bound of the diffusion convolution order; and k represents the diffusion order number. A is the three-source fusion adjacency matrix; , where I is the identity matrix, representing the information of the node itself preserved in the 0th-order diffusion; This represents the node input feature matrix of all oxygen electromotive force measurement points at the t-th historical time step in the current historical input window; The learnable weight parameters represent the k-th order diffusing convolution branch; This represents the bias term corresponding to the k-th order diffusion characteristic; Step S3 includes: The spatial augmentation node features of each historical time step are aggregated by node dimension to obtain the temporal input feature sequence F; The temporal input feature sequence F is input into a bidirectional long short-term memory network, and temporal dependencies are extracted from the forward temporal order and the reverse temporal order, respectively, to obtain the temporal output feature sequence L; Attention weights for each historical time step are calculated based on the temporal output feature sequence L. The temporal output features of all historical time steps are then weighted and aggregated to obtain the global temporal features. ; The temporal output features of the last historical time step of the bidirectional long short-term memory network. With the aforementioned time-series global features By splicing the data, we obtain the comprehensive temporal features. .

[0015] Preferably, step S4 includes: The comprehensive time series features The spatiotemporal fusion prediction features, serving as the current historical input window, are input to the multi-step prediction output network. , to obtain the depth prediction term ; Simultaneously, the change ΔX of the oxygen electromotive force observation value at each measuring point within the current historical input window relative to the historical anchor point B is extracted and processed through an autoregressive residual mapping network. Obtain the residual prediction term ; The depth prediction term D, the residual prediction term R, and the historical anchor point B are added together, and then reconstructed to obtain the oxygen electromotive force prediction results for all N measurement points under the future P prediction steps: ; in, ; p represents the future prediction time step number; i represents the oxygen electromotive force measurement point number. This represents the predicted oxygen electromotive force at the i-th measurement point in the p-th prediction step. This indicates that the output result will be reconstructed into a prediction matrix form; the historical anchor point B is calculated based on the effective historical observation values ​​of the oxygen electromotive force of each measuring point within the current historical input window.

[0016] Preferably, step S5 includes: Obtain the predicted oxygen electromotive force value at the i-th measurement point in the p-th prediction step. and their corresponding temperatures Using physical conversion functions Convert the predicted oxygen electromotive force and corresponding temperature into predicted oxygen concentration characterization values: ; in, The oxygen electromotive force, temperature, and oxygen concentration are determined based on the Nernst equation or oxygen sensor calibration relationship; the predicted oxygen concentration is characterized by the oxygen concentration itself or the logarithm of the oxygen concentration; the corresponding temperature... The temperature sequence obtained is extrapolated from the historical input window, or provided by an external temperature prediction model, or obtained from a set operating curve or measured temperature data. Using the predicted oxygen concentration characterization value as the predicted oxygen concentration, the discrete oxygen concentration prediction results for all N measurement points under the next P prediction steps are obtained. : ; in, , This represents the predicted oxygen concentration at the i-th measurement point in the p-th prediction step.

[0017] Preferably, the oxygen concentration prediction model is constructed through the above steps S1-S5, and the training process includes: Step S6: Construct a prediction time step adaptive physical constraint module, dynamically adjust the physical loss weights according to the prediction time step P, apply physical consistency constraints to the oxygen electromotive force prediction results, and use the total training loss. Train the model: ; The adaptive physical loss weights are used to predict the time step P and the number of training rounds e. Data-driven loss prediction; Total physical loss ; Step S7: Construct a multi-branch consistent distillation robust training module, using the prediction result of the pure input branch as the teacher signal, apply consistent distillation constraints to the perturbed input branch and the missing input branch, and perform robust training of the model.

[0018] Preferably, in step S6, the total physical loss of the prediction time step adaptive physical constraint module is... The calculation method is as follows: ; in, Indicates the loss of the saturation boundary barrier. Indicates loss of safety anchor points. Indicates a loss due to a safety trend. Indicates smoothing loss; , , , These represent the weighting coefficients corresponding to boundary fence loss, safety anchor point loss, safety trend loss, and smoothing loss, respectively. The adaptive physical loss weight The predicted step size P is dynamically determined based on the current training epoch e: when the predicted step size P is less than or equal to the first preset threshold... =0; when the prediction step size P is greater than the first preset threshold. After each warm-up round, the weight is gradually increased to the preset upper limit of the corresponding prediction step size interval; Step S7 includes: using the undisturbed historical input window as the pure input branch to obtain the pure prediction result. Random noise perturbation is applied to the oxygen electromotive force time series data in the historical input window to obtain the perturbation input branch and perturbation prediction results. A missing data mask matrix is ​​constructed to mask some measurement point data in the historical input window, thus obtaining the missing input branch and the missing prediction result. Based on the pure prediction results As a teacher signal, the disturbance prediction result and the missing prediction results Apply a uniform distillation constraint; the uniform distillation loss is expressed as: ; in, The distillation consistency metric function can be represented by mean square error or other distance metrics. This indicates that the gradient operation is stopped.

[0019] Preferably, in step S6, the saturation boundary barrier loss for: ; Where N represents the number of oxygen electromotive force measurement points; P represents the prediction time step; i represents the oxygen electromotive force measurement point number; and p represents the future prediction time step number. This represents the predicted oxygen concentration at the i-th oxygen electromotive force measurement point in the p-th prediction time step. This represents the physical lower boundary of oxygen concentration at the i-th oxygen electromotive force measurement point in the future p-th predicted time step; This represents the upper boundary of oxygen concentration saturation at the i-th oxygen electromotive force measurement point in the p-th predicted time step in the future; Represents a linear activation function; Safety anchor point loss for: ; in, This represents the predicted oxygen concentration characterization value at the i-th oxygen electromotive force measurement point at the p-th prediction time step; The oxygen concentration characterization value at the historical physical anchor point of the i-th oxygen electromotive force measurement point at the p-th prediction time step; This represents the safety aiming point gating variable for the i-th oxygen electromotive force measurement point in the p-th predicted time step. Safety Trend Loss Defined as: ; in, The predicted oxygen concentration trend at the i-th oxygen electromotive force measurement point in the p-th prediction time step; This represents the safety trend gating variable for the i-th oxygen electromotive force measurement point at the p-th prediction time step in the future; This indicates the oxygen concentration trend at the corresponding historical anchor point.

[0020] A method for constructing an oxygen concentration field in liquid lead-bismuth involves acquiring oxygen electromotive force and temperature data in liquid lead-bismuth under test conditions, using the aforementioned oxygen concentration prediction method in liquid lead-bismuth to obtain oxygen concentration prediction results, and performing spatial interpolation reconstruction based on the spatial location of each measuring point and the oxygen concentration prediction results to obtain a continuous oxygen concentration field in the liquid lead-bismuth experimental device. The spatial interpolation employs any one of the following: inverse distance weighted interpolation, radial basis function interpolation, Kriging interpolation, spline interpolation, and path distance interpolation based on loop geometry topology; and the oxygen concentration field obtained by interpolation is subjected to physical feasibility projection, constraining it to a preset physical feasibility interval.

[0021] The advantages of this invention are: (1) This invention uses multiple oxygen electromotive force sensor measurement points in a lead-bismuth experimental device as graph nodes, integrates a static physical prior graph, a dynamic state graph, and an adaptive learning graph to construct a three-source fusion adjacency matrix, extracts spatial coupling features between measurement points through K-order adaptive diffusion graph convolution, captures long-short-term temporal dependencies and key time step information within the historical window by combining a bidirectional long short-term memory network and a temporal attention mechanism, uses a multi-step prediction output module to obtain the oxygen electromotive force prediction values ​​of each measurement point under multiple prediction step sizes in the future, and converts the oxygen electromotive force prediction values ​​into oxygen concentration prediction values ​​based on the Nernst equation, constructs a prediction step size adaptive physical constraint module to dynamically adjust the physical loss weights, and enhances the physical consistency of the prediction results, improves the stability of the model under sensor noise and random data missing conditions through a multi-branch consistent distillation robust training mechanism, and finally performs spatial interpolation reconstruction based on the spatial location of the measurement points to obtain the continuous oxygen concentration field in the liquid lead-bismuth experimental device. It fully explores the spatial coupling relationship and temporal dependency relationship of multiple measurement points to achieve high-precision and high-robust spatiotemporal prediction of oxygen concentration field.

[0022] (2) This invention uses multiple oxygen electromotive force measurement points as graph nodes to construct a three-source fusion adjacency matrix composed of a static physical prior graph, a dynamic state graph, and an adaptive learning graph. This matrix can simultaneously characterize the spatial arrangement of measurement points, loop flow relationships, dynamic correlations under changing operating conditions, and implicit coupling relationships that are difficult to express through artificial priors. Based on this, K-order adaptive diffusion graph convolution is performed, which can simultaneously capture the direct spatial coupling relationships between adjacent measurement points and the higher-order indirect spatial coupling relationships between non-adjacent measurement points, effectively making up for the shortcomings of traditional time series models in utilizing spatial topological information.

[0023] (3) This invention extracts the temporal dependencies within the historical input window from both the forward and reverse directions using BiLSTM, and combines this with a temporal attention mechanism to adaptively focus on key historical time steps that contribute significantly to future multi-step predictions. The integrated temporal features simultaneously contain local state information at the end of the historical window and global key time step information extracted by the attention mechanism, enabling a more comprehensive capture of long- and short-term dependencies and complex dynamic change patterns in the oxygen electromotive force sequence.

[0024] (4) This invention employs a multi-branch consensus distillation robust training mechanism, setting up a clean input branch, a perturbation input branch, and a missing input branch. The clean branch serves as the teacher signal to apply consensus distillation constraints to the perturbation and missing branches. This mechanism enables the model to learn prediction patterns under normal, noisy, and missing input conditions simultaneously during training, effectively suppressing prediction drift caused by abnormal inputs and maintaining stable prediction performance even in scenarios with random sensor loss, strong noise, and partial network outages.

[0025] (5) This invention uses a prediction step-adaptive Safe-PINN physical constraint module to dynamically adjust the physical loss weights according to the prediction step size: in short-term prediction, physical constraints are turned off or weakened to avoid interfering with data-driven fitting; in medium- and long-term prediction, physical constraints are gradually enabled to suppress trend drift and non-physical jitter. At the same time, based on a safety gating mechanism, boundary guardrails, safety anchors, safety trends, and smoothing physical losses are designed so that physical constraints only play a role when they are conducive to improving the rationality of prediction.

[0026] (6) This invention enables simultaneous prediction of multiple oxygen electromotive force measurement points under multiple prediction step sizes. Furthermore, it converts the oxygen electromotive force into discrete oxygen concentration through the physical conversion relationship between oxygen electromotive force, temperature, and oxygen concentration. Finally, it combines the spatial location of the measurement points with spatial interpolation, graph propagation, and physical constraint reconstruction to obtain a continuous oxygen concentration field. This continuous oxygen concentration field can be used for oxygen control regulation, abnormal trend early warning, structural material corrosion risk assessment, sensor status assessment, and intelligent operation and maintenance of liquid metal circuits. Attached Figure Description

[0027] Figure 1 This is a diagram of the model architecture of the present invention; Figure 2 This is a flowchart of the data preprocessing training and inference process of the present invention; Figure 3 This is a diagram illustrating the prediction effect of the multi-prediction time step method of the present invention. Figure 4 This is an example diagram illustrating noise and packet loss robustness in an embodiment of the present invention; Figure 5 This is a graph showing the convergence curve of the training loss of the model in this invention. Figure 6 This is a schematic diagram of the oxygen concentration field construction of the present invention. Detailed Implementation

[0028] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0029] Example 1 like Figures 1-5 As shown, a method for predicting oxygen concentration in liquid lead-bismuth is proposed, including: First, a model for predicting oxygen concentration in liquid lead-bismuth is constructed. This model includes an adaptive diffusion graph convolution module, a bidirectional temporal attention module, a multi-step prediction output module, an oxygen electromotive force-oxygen concentration conversion module, a prediction step size adaptive physical constraint module, and a multi-branch consistent distillation robust training module. Specifically, it includes: In a lead-bismuth experiment, several oxygen electromotive force (EMF) sensors are set up, with each sensor location considered a measurement point. Time-series data of the oxygen EMF and temperature are collected from all measurement points. Let the number of oxygen EMF sensors be N, then the number of measurement points is also N. The i-th oxygen EMF measurement point is denoted as . ,in The measuring point may have spatial coordinates, the location of the pipe section, the flow direction relationship, the historical oxygen electromotive force sequence, and the corresponding temperature or other external state characteristics.

[0030] To simultaneously describe the stable physical connections between measurement points, the dynamic state-related relationships under different input windows, and the implicit coupling relationships that are difficult to express directly through manual priors, this invention constructs a three-source fusion adjacency matrix A composed of a static physical prior graph, a dynamic state graph, and an adaptive learning graph. Specifically: Step S1: Construct the three-source fusion adjacency matrix A. Using all oxygen electromotive force measurement points as nodes in the graph structure, and the temperature time series as the external operating condition feature, the static physical prior graph, dynamic state graph, and adaptive learning graph are fused to obtain the three-source fusion adjacency matrix A, which can simultaneously characterize the spatial arrangement of measurement points, loop flow relationships, historical data associations, input state changes, and learnable implicit relationships. The corresponding three-source fusion graph is then obtained. ; in, This is a static physical prior diagram; This is a dynamic state diagram; For adaptive learning graphs; , , For the learnable fusion weights of the three adjacency matrices, a certain initial weight is first set. During model training, these weights are automatically optimized as the loss propagates backpropagated. The model will autonomously learn the optimal contribution ratio of the three graphs, which is then normalized by Softmax and satisfies the following: .

[0031] (1) Static physical prior diagram

[0032] It is used to describe the relatively stable physical connections and historical statistical correlations between oxygen electromotive force (EMF) measurement points. This diagram is mainly constructed from the spatial arrangement of the measurement points, the flow direction relationships, and the correlation of historical sequences of oxygen EMF at multiple measurement points. It consists of two parts: first, a flow topology matrix constructed based on the spatial arrangement of the oxygen EMF sensors in the lead-bismuth experimental apparatus loop, their location in the pipe section, upstream and downstream relationships, and the flow direction. The flow direction topology matrix is ​​used to represent the physical connections that may propagate the oxygen electromotive force signal or oxygen concentration disturbance along the flow path in the loop. If there is a direct flow direction association, adjacent pipe segment association, or preset physical connection relationship between the i-th measuring point and the j-th measuring point, then let Otherwise, =0. Meanwhile, to preserve the historical information of each measuring point, a self-connection can be added to the flow direction topology matrix, that is, let... >0.

[0033] Second, the correlation matrix between the measurement points is calculated based on the time-series data of oxygen potential and temperature from all oxygen potential measurement points in the training set. This historical correlation matrix can be obtained using Pearson correlation coefficient, maximum information coefficient, mutual information, distance correlation coefficient, or other methods that can characterize the correlation strength of historical sequences across multiple measurement points.

[0034] Then, the flow direction topology matrix and the correlation matrix are fused and normalized element by element to obtain a static physical prior map A that characterizes the position of each oxygen electromotive force sensor measuring point, the flow direction, and the relationship between the oxygen electromotive force at the corresponding measuring point. s .

[0035] The first step here uses historical oxygen electromotive force data to calculate the historical correlation between measurement points.

[0036] (2) Dynamic state diagram A d It is used to describe the dynamic changes in the correlation strength between multiple oxygen electromotive force measurement points under different input windows or different operating states across the entire dataset. Unlike static physical prior graphs, dynamic state graphs are not global fixed graphs, but rather sample-level adjacency matrices generated in real time based on external state features in the current historical input window. These external state features can include operating states such as flow rate, pump speed, and heating power.

[0037] In this embodiment, temperature time series is used as an external state feature to characterize loop state changes, and a dynamic state diagram is constructed to describe the dynamic changes in the correlation strength between oxygen electromotive force sensor measuring points. This diagram is generated based on the temperature time series: first, the temperature time series data within the historical window is mapped to high-dimensional hidden space temperature state features; then, the mean, variance, and other statistical state quantities of the temperature features within the historical window are calculated and concatenated; and finally, a dynamic adjacency matrix corresponding to the current sample is generated through a nonlinear mapping network (an adjacency matrix corresponding to the time step is generated through the nonlinear mapping network, and a dynamic state diagram A is generated for each sample). d Each sample participates in the construction of the three-source fusion using its own dynamic state graph. The dynamic graph is at the historical window level, not a globally fixed graph.

[0038] (3) Adaptive learning graph A l This adaptive learning graph is used to capture latent, hidden measurement point relationships that cannot be explicitly expressed by artificial topology, historical statistical correlations, or temperature conditions. The model learns only one globally fixed adaptive learning graph structure, which is shared by all samples after training. These implicit coupling relationships cannot be directly derived from the raw sensor data using fixed formulas. The model relies on the backpropagation mechanism of prediction errors to automatically discover effective measurement point connections during training, thereby supplementing the spatial correlation features not covered by static and dynamic graphs. The specific implementation method is as follows: First, two independent sets of learnable node embedding matrices are assigned to each oxygen electromotive force measurement point. These two sets of learnable node embedding matrices are used to characterize the latent features of the transmitting and receiving ends of the nodes, respectively, so that the learned adjacency relationships have directionality. The correlation score between the measurement points is calculated using the inner product of these two sets of node embedding matrices, and then a non-negative correlation strength matrix is ​​obtained through ReLU activation. ; The dimensions and structure of the two sets of learnable node embedding matrices are manually set, with initial values ​​obtained through random initialization; the specific values ​​are automatically optimized during training through backpropagation of the prediction loss. Each row of each matrix corresponds to an oxygen electromotive force measurement point, and each row is an m-dimensional vector, where m can be determined based on experimental needs and testing. Initially, these two matrices are obtained through random initialization. These two matrices are used to describe the mutual influence between the two measurement points. Since the influence of one measurement point on another is not necessarily equal to their opposite influence due to the directionality of liquid lead-bismuth flow between measurement points, two matrices are set up. This allows the model to learn asymmetric potential measurement point relationships. The other two learnable node embedding matrices are not directly calculated based on measurement point coordinates, temperature, or historical correlation coefficients, nor are they pre-assigned specific physical values. Their matrix dimensions are set to N during model design. M: N rows correspond to U1, U2, U3, and UN, respectively, representing N oxygen electromotive force measurement points. M columns represent the M-dimensional latent features of each measurement point. During model initialization, the elements of the two matrices are generated through a random normal distribution. In subsequent training, they serve as network parameters and are continuously updated based on the backpropagation results of the prediction loss, ultimately forming a node latent relationship representation adapted to the prediction task.

[0039] The reason for using two sets of embeddings instead of one set is to allow the measurement point relationship to be directional. The first set of embeddings can be understood as the receiving / converging role of the node, and the second set of embeddings can be understood as the output / contributing role of the node. Therefore, it is allowed that the "intensity of information converged from U1 to U2" and the "intensity of information converged from U2 to U1" are different.

[0040] To avoid an overly dense learning graph, a masking process is applied to the original association strength matrix. Sparse filtering is performed using the mean of each row as a threshold, resulting in a binary mask matrix (where 1 represents retaining strong connections and 0 represents removing weak connections). The association matrix and the mask matrix are multiplied element-wise, with the scores for weak connections set to zero and local minimum values ​​filled in the corresponding positions. Finally, Softmax normalization is performed row-by-row to obtain the final adaptive learning graph A, formed by the adjacency matrix of the adaptive learning graph. l .

[0041] During model training, the adaptive learning graph not only participates in the construction of the three-source fusion adjacency matrix but also further enters the diffusion graph convolution module. The model performs backpropagation based on the loss between the multi-step oxygen electromotive force prediction results and the true labels for each batch on the training set, updating the embedding matrices of two learnable nodes in each batch. This allows the adaptive learning graph to gradually learn implicit measurement point coupling relationships beneficial to the prediction task. If a potential measurement point connection can reduce prediction error, its corresponding association score will be enhanced during training; if a connection is not beneficial to prediction or interferes with it, its weight will be weakened or masked.

[0042] The two learnable node embedding matrices, like the weights and biases of ordinary network layers, are all trainable parameters of the model. The only difference is that the parameters of ordinary linear layers are used to implement feature transformation, while these two parameter matrices are specifically used to construct adaptive graph relationships between measurement points.

[0043] In each training batch, the model first generates an adaptive learning graph using two node embedding matrices, then fuses it with the static and dynamic graphs, and proceeds to the diffusion graph convolution, BiLSTM, attention, and prediction output modules to obtain multi-step prediction results. Then, based on the loss between the prediction results and the true labels, the gradient of this loss with respect to each element of the two node embedding matrices is calculated in reverse; finally, the AdamW optimizer updates its values ​​according to the gradients. After multiple training batches and rounds of training, the two node embedding matrices are gradually adjusted so that the constructed node relationship graph is more conducive to reducing prediction errors and improving prediction stability under noisy and missing conditions.

[0044] (4) Three-source graph fusion adjacency matrix A: After obtaining the static physical prior map Dynamic status diagram of each historical input window and adaptive learning graph Then, learnable fusion weights are set for the three types of graphs. , , The normalized fusion weights are obtained through the Softmax function: , Then, for each input sample, construct its corresponding three-source fusion adjacency matrix: Among them, static physical prior diagrams A globally shared graph, used to provide stable physical topology and historical correlation priors; a dynamic state graph. This is a sample-level graph used to adjust the correlation strength of measurement points based on external state features in the current historical input window; adaptive learning graph. This is a global implicit relation graph automatically learned during the training process, used to supplement the potential spatial coupling relationships that cannot be fully expressed by artificial topology, historical correlation and external state features.

[0045] The final three-source fusion adjacency matrix The data is then fed into a subsequent diffusion map convolution module to extract spatial features of the direct and higher-order indirect correlations between multiple oxygen electromotive force measurement points.

[0046] Obtain the three-source fusion adjacency matrix A corresponding to all time windows.

[0047] In this method, all relevant data within a historical input window is treated as a sample.

[0048] Step S2: Construct an adaptive diffusion graph convolution module.

[0049] After obtaining the three-source fusion adjacency matrix A corresponding to the current historical input window in step S1, an adaptive diffusion graph convolution module is constructed to extract features of the spatial coupling relationship between multiple oxygen electromotive force measurement points based on the three-source fusion adjacency matrix. This module uses the three-source fusion adjacency matrix A corresponding to the current historical input window as the basis for graph propagation, enabling the node features of each measurement point to undergo multi-order diffusion propagation along the graph structure, thereby extracting the spatial dependency features between directly related measurement points and higher-order indirectly related measurement points. Specifically: The node input feature matrix, composed of the node input features of all oxygen electromotive force measurement points at the t-th historical time step in the current historical input window, is denoted as... The node input features may include one or more of the following: oxygen electromotive force observations, valid data labels, measurement point location embeddings, node identity embeddings, external state feature mapping results, and node features enhanced by local time series.

[0050] Based on the three-source fusion adjacency matrix A obtained in step S1, the node input feature matrix is... Perform K-order diffusion propagation to obtain spatially enhanced node features; spatially enhanced node features at the t-th historical time step in the current historical input window. Represented as: ; In the formula, This represents the spatially enhanced node feature obtained at the t-th historical time step in the current historical input window; the subscript g represents graph, i.e., graph structure feature or graph diffusion feature, and has no other special physical meaning; σ(·) represents the nonlinear activation function; K represents the upper limit of the diffusion convolution order; k represents the diffusion order number. A is the three-source fusion adjacency matrix corresponding to the current historical input window obtained in step S1; denoted by the k-th power of the three-source fusion adjacency matrix A, it represents the spatial association relationship after propagation through the k-th order neighborhood. It is not another separately constructed adjacency matrix. , where I is the identity matrix, representing the information of the node itself preserved in the 0th-order diffusion; This represents the node input feature matrix of all oxygen electromotive force measurement points at the t-th historical time step in the current historical input window; This represents the linear mapping weight matrix corresponding to the k-th order diffusion feature, which is also the learnable weight parameters of the k-th order diffusion convolution branch. This represents the bias term corresponding to the k-th order diffusion characteristic.

[0051] Through the aforementioned diffusion graph convolution, the 0th-order diffusion is used to preserve the node features of each oxygen electromotive force measurement point, the 1st-order diffusion is used to aggregate information from one-hop neighboring measurement points, and the 2nd-order and higher-order diffusions are used to aggregate higher-order neighboring information formed through transmission from multiple measurement points. Thus, the model can simultaneously capture the direct spatial coupling relationship between adjacent measurement points and the higher-order indirect spatial coupling relationship between non-adjacent measurement points, based on the three-source fusion measurement point relationship constructed in step S1.

[0052] For a historical input window of length T, the above diffusion map convolution process is applied sequentially to the node input features of each historical time step within the window to obtain a spatially augmented node feature sequence composed of spatially augmented node features from all historical time steps within the window. : ; Where T is the length of the historical input window. The spatially enhanced node feature sequence serves as the input to the subsequent temporal feature extraction module, used to further extract the temporal variation features and long-short-term dependencies of the oxygen electromotive force. It should be noted that the three-source fusion adjacency matrix A corresponding to the current historical input window is obtained by fusing the static physical prior graph, dynamic state graph, and adaptive learning graph in step S1. For different historical input windows, the dynamic state graph can change with the external state features, so the fused three-source fusion adjacency matrix A can also change with the historical input window; within the same historical input window, the three-source fusion adjacency matrix A can be applied to the node input features of each historical time step within that window. If in other embodiments the dynamic state graph is generated according to time steps, the three-source fusion adjacency matrix of the corresponding time step can be used for the diffusion propagation of that time step. This invention does not limit the update granularity of the dynamic state graph.

[0053] Through the aforementioned adaptive diffusion graph convolution module, the model can transform the measurement point relationships represented by the three-source fusion adjacency matrix into spatially enhanced node features, realizing the transformation from measurement point relationship modeling to spatial coupling feature extraction. Since the adjacency matrix upon which diffusion propagation is based integrates static physical priors, dynamic state changes, and adaptively learned implicit relationships, this module can more fully exploit the spatial relationships between multiple oxygen electromotive force measurement points in the liquid lead-bismuth circuit, improving the accuracy and stability of subsequent multi-step oxygen electromotive force prediction.

[0054] Step S3: Construct a bidirectional temporal attention module. The spatially augmented node feature sequence under the current historical input window is obtained in step S2. Subsequently, a bidirectional temporal attention module is constructed to further extract the temporal dependencies of oxygen electromotive force changes within the historical input window, and adaptively focus on key historical time steps that contribute significantly to future multi-step predictions. Specifically: First, for each time step, the node features... Perform node-dimensional concatenation, pooling, or linear mapping to obtain temporal input features suitable for temporal network input. This forms a temporal input feature sequence. : ; in, This represents the temporal input feature at the t-th historical time step. The temporal input feature may include spatially enhanced node features obtained by diffusion graph convolution, or it may be further fused with external state features, local temporal convolution enhancement features, or other auxiliary features.

[0055] Then, the temporal input feature sequence F is input into a bidirectional long short-term memory network (BiLSTM). Temporal dependencies within the historical input window are extracted from both the forward and reverse temporal order to obtain the temporal output feature sequence of the BiLSTM. : ; in, This represents the temporal output feature at the t-th historical time step. Represented as: ); in, This represents the hidden state of the forward LSTM at the t-th historical time step. This represents the hidden state of the inverse LSTM at the t-th historical time step, and Concat(·) represents the concatenation operation along the feature dimension.

[0056] Finally, to highlight key time steps, inflection points, or abrupt change regions in the historical input window that contribute significantly to future predictions, a time-series attention mechanism is further constructed. This includes: First, the importance score for each historical time step is calculated based on the time-series output feature sequence L; the importance score for the t-th historical time step is: ; in, Let represent the unnormalized attention score at the t-th historical time step; Represents the attention feature transformation matrix; This represents the attention feature transformation bias term; Represents the attention score mapping vector; express Transpose of; This represents the hyperbolic tangent nonlinear activation function.

[0057] Then, the importance scores of each historical time step are normalized using the Softmax function to obtain the attention weights for each historical time step; the attention weight for the t-th historical time step is calculated as follows: ; in, Let represent the normalized attention weight at the t-th historical time step, satisfying: j represents the summation index of the time steps during the Softmax normalization process. Let represent the unnormalized attention score corresponding to the j-th historical time step.

[0058] Next, the temporal feature sequence output by BiLSTM is weighted and aggregated using the attention weights of all historical time steps under the historical input window to obtain the temporal global features corresponding to each historical input window. : ; in, This represents the temporal global features obtained after weighted aggregation through a temporal attention mechanism, used to characterize key time slice information within the historical input window that contributes significantly to future multi-step predictions.

[0059] Finally, the temporal output features of the last historical time step of BiLSTM are... As the end-state feature of the window, and aggregated with attention to obtain the temporal global feature. The data is then stitched together to form a comprehensive temporal feature for each historical input window. : ; in, This represents the comprehensive temporal features of the historical input window; Concat(·) indicates a concatenation operation along the feature dimension. This feature simultaneously contains local state information at the end of the historical window and global key time step information extracted by the attention mechanism, which can be used as input for subsequent spatiotemporal fusion prediction modules.

[0060] Through the aforementioned bidirectional temporal attention module, the model can further capture the forward and backward temporal dependencies of the oxygen electromotive force sequence within the historical input window, based on the spatial enhancement node features extracted in step S2, and adaptively highlight the historical time steps that are more important for future predictions, thereby improving the temporal expression ability and prediction stability of multi-step oxygen electromotive force prediction.

[0061] Step S4: Construct the spatiotemporal fusion and multi-step prediction output module. The comprehensive temporal features corresponding to the current historical input window are obtained in step S3. Subsequently, a spatiotemporal fusion and multi-step prediction output module is constructed to generate oxygen electromotive force prediction results for each oxygen electromotive force measurement point at multiple future prediction time steps. This is because it incorporates comprehensive temporal characteristics. The spatially enhanced node feature sequence obtained in step S2 and the bidirectional temporal attention mechanism in step S3 are combined to obtain the sequence. Therefore, it simultaneously contains spatial coupling information between oxygen electromotive force measurement points, temporal dependency information within the historical window, and attention focusing information for key historical time steps. Specifically, it includes: Integrating time series characteristics The spatiotemporal fusion prediction features, serving as the current historical input window, are input into the multi-step prediction output module. Let the prediction time step be P, then the model outputs the predicted oxygen electromotive force (EMF) results for all oxygen EMF measurement points in the next P prediction time steps. : ; in, This represents the multi-step prediction results corresponding to the historical input window; This represents the prediction results for all oxygen electromotive force measurement points at the p-th prediction time step in the future. .

[0062] For the p-th time step in the future, the predicted oxygen electromotive force is as follows. It can be represented as: ; Where i is the oxygen electromotive force measurement point number; This represents the predicted oxygen electromotive force (EMF) value at the i-th oxygen EMF measurement point at the p-th time step in the future. .

[0063] In one implementation, a fully connected multi-step output network is used to map the comprehensive temporal features to obtain the oxygen electromotive force prediction results for N measurement points at P future time steps. : ; in, This indicates a multi-step prediction output network. This indicates that the output will be reconstructed as The prediction matrix is ​​in the form of a multi-step prediction output network. This network can consist of one or more fully connected layers, normalization layers, nonlinear activation functions, and dropout layers.

[0064] Furthermore, to improve the ability of the prediction results to preserve historical end-states, an autoregressive residual branch can be introduced into the multi-step prediction output module. Based on the historical observation sequence of oxygen electromotive force at each oxygen electromotive force measurement point in the current historical input window, the change relative to the historical anchor point (i.e., the mean oxygen electromotive force under the historical window) is extracted, and the residual prediction term for the next P prediction time steps is obtained through an autoregressive mapping network. : ; in, This represents the change in the oxygen electromotive force observation sequence relative to the historical anchor point within the current historical input window. Let R represent the autoregressive residual mapping network, and let R represent the residual prediction term for each oxygen electromotive force measurement point at the next P prediction time steps. Simultaneously, based on spatiotemporal fusion features, a deep prediction term is obtained through the main prediction network: ; And by combining the historical anchor point B, the final multi-step prediction result is obtained: ; Where B represents the historical anchor point term calculated from the historical valid observations of each oxygen electromotive force measurement point in the current historical input window, used to provide the prediction benchmark; D represents the depth prediction term; and R represents the residual prediction term. If no autoregressive residual branch is set, it can be directly generated by the multi-step prediction output network. In the event of missing sensors or partially invalid observations, the historical anchor point B can be obtained by calculating the mean, extracting the end value, or weighting and aggregating the effective oxygen electromotive force observations within the historical window based on valid markers, in order to avoid invalid data interfering with the prediction benchmark.

[0065] Step S5: Construct the oxygen electromotive force to oxygen concentration conversion module The future is obtained through the aforementioned steps. After predicting the oxygen electromotive force (EMF) at each measurement point for each prediction time step, an oxygen EMF-oxygen concentration conversion module is constructed. This module converts the predicted oxygen EMF values ​​at discrete measurement points into corresponding predicted or characteristic oxygen concentration values, providing foundational data for the subsequent construction of a continuous oxygen concentration field. The multi-step oxygen EMF prediction results output by the model are as follows: ; in: ; Indicates the future number The first predicted time step Predicted oxygen electromotive force values ​​at each oxygen electromotive force measurement point.

[0066] In one implementation, the physical conversion relationship between oxygen electromotive force and oxygen concentration is constructed based on the Nernst equation or oxygen sensor calibration relationships. For an oxygen electromotive force sensor in liquid lead-bismuth, the oxygen electromotive force... Thermodynamic temperature With oxygen concentration The transformation relationship between them satisfies the following form: ; in, This represents the oxygen electromotive force. Represents thermodynamic temperature. Indicates oxygen concentration. , , , These are constants or configurable parameters determined by the oxygen sensor type, reference electrode, calibration parameters, or physical relationships of the liquid lead-bismuth system.

[0067] Based on the physical conversion relationship between oxygen electromotive force, temperature, and oxygen concentration, let the future... The first predicted time step The temperature corresponding to each oxygen electromotive force measurement point is: The predicted oxygen concentration characterization value is: ; in, This represents the conversion function determined by the physical relationship between oxygen electromotive force, temperature, and oxygen concentration. This represents the oxygen concentration characterization value obtained by converting the predicted oxygen electromotive force. The oxygen concentration characterization value can be the oxygen concentration itself, the logarithm of the oxygen concentration, or other equivalent characterization forms.

[0068] In one specific implementation, when the logarithm of oxygen concentration is used as the characterization, we have: ; If the physical transformation relationship in step S5 takes the following form: ; Then the future The first predicted time step The predicted logarithmic value of oxygen concentration at each measuring point is: ; The corresponding predicted oxygen concentration can be expressed as: ; in, Indicates the future number The first predicted time step Predicted oxygen concentration at each oxygen electromotive force measurement point.

[0069] From this, we can obtain the future. Prediction results of discrete oxygen concentration at all oxygen electromotive force measurement points under each prediction time step: ; in: ; It should be noted that the temperature corresponding to the future prediction time step... It can be obtained by extrapolating the temperature sequence from the historical input window, or by using an external temperature prediction model, a set operating curve, a control system setpoint, or measured temperature data. This invention does not limit the method of obtaining future temperature information, nor does it limit the specific inverse transformation form between the oxygen concentration characterization value and the actual oxygen concentration.

[0070] Through the oxygen electromotive force to oxygen concentration conversion module, the model output is further converted from the future multi-step oxygen electromotive force prediction result to the future multi-step discrete oxygen concentration prediction result, so that the prediction result has a clear physical meaning and provides a discrete measurement point data basis for the subsequent reconstruction of the oxygen concentration spatial distribution field.

[0071] Step S6: Construct Prediction Time Step Adaptive Physical Constraints (Safe-PINN). Construct a prediction time step adaptive Safe-PINN physical constraint module. This module establishes physical losses based on the physical conversion relationship between oxygen electromotive force, temperature, and oxygen concentration, and dynamically determines the physical loss weights according to the prediction time step. Physical losses are disabled or weakened in short-term predictions, while physical loss constraints are gradually enabled in medium- and long-term predictions. This improves the physical consistency and trend stability of multi-step prediction results and avoids negative transfer caused by fixed-strength physical constraints on short-term data-driven fitting.

[0072] In one test implementation, the Safe-PINN physical loss weight is adjusted in segments according to the prediction time step: when the prediction time step is <15, the physical loss weight is set to zero, making the 1-step, 3-step, and 15-step predictions primarily data-driven fitting; because my ablation experiments found that the 1-step, 3-step, and 15-step data-driven predictions have higher accuracy, when the step size is greater than 15, physical loss is gradually enabled after a preset warm-up round; when it is greater than 30, long-term physical constraints are gradually enabled after a preset warm-up round. Thus, in short-term predictions, physical constraints are avoided from interfering with data fitting, and in medium- to long-term predictions, the risks of trend drift, boundary overflows, and non-physical jitter are reduced.

[0073] The Safe-PINN physical constraint module is mainly used in the model training phase, where it participates in backpropagation by constructing physical loss. In the model prediction phase, no real labels are required, and the model can directly output the prediction results of future multi-step oxygen concentration based on the current historical input window.

[0074] Through the aforementioned spatiotemporal fusion and multi-step prediction output module, the model can uniformly map spatial coupling features, historical temporal dependence features, and key time step attention features into future multi-step oxygen electromotive force (EMF) prediction results, achieving synchronous prediction of multiple oxygen EMF measurement points at different prediction time steps in the future. This output result can be further used for subsequent physical constraint calculations, oxygen EMF-oxygen concentration conversion, and the construction of a continuous oxygen concentration field.

[0075] Let the multi-step oxygen electromotive force prediction result output by the model be... ,in, , This represents the predicted oxygen electromotive force value at the i-th oxygen electromotive force measurement point at the p-th prediction time step in the future. , During the training phase, the actual oxygen electromotive force label is: ,in, , This represents the actual oxygen electromotive force value at the i-th oxygen electromotive force measurement point at the p-th predicted time step in the future.

[0076] Physical conversion relationship between oxygen electromotive force, temperature, and oxygen concentration To construct the Safe-PINN physical loss, the oxygen electromotive force and corresponding temperature predicted by the model are converted into oxygen concentration characterization values.

[0077] Suppose the future... The first predicted time step The temperature corresponding to each oxygen electromotive force measurement point is: The predicted oxygen concentration characterization value and the actual oxygen concentration characterization value are expressed as follows: ; ; in, It represents the conversion function determined by the physical relationship between oxygen electromotive force, temperature, and oxygen concentration; Indicates the future number The first predicted time step The predicted oxygen concentration characterization value corresponding to each oxygen electromotive force measurement point; Indicates the future number The first predicted time step The actual oxygen concentration characterization value corresponding to each oxygen electromotive force measurement point. The oxygen concentration characterization value can be the oxygen concentration itself, the logarithm of the oxygen concentration, or other equivalent characterization forms.

[0078] From the above conversion relationship between oxygen electromotive force, temperature, and oxygen concentration, the logarithmic value of oxygen concentration can be obtained: ; Therefore, for predicting oxygen electromotive force The corresponding logarithmic representation of the predicted oxygen concentration can be written as: ; For the true oxygen electromotive force The corresponding logarithmic representation of the true oxygen concentration can be written as: ; In one specific implementation, when a specific oxygen sensor calibration relationship is used, the logarithmic representation of oxygen concentration can be expressed as: ; Correspondingly, the predicted logarithmic value of oxygen concentration and the actual logarithmic value of oxygen concentration can be expressed as follows: ; ; It should be noted that the specific parameters described above are only one implementation method. In practical applications, the parameters can be adjusted according to the type of oxygen electromotive force sensor, the reference electrode material, calibration experimental results, or the thermodynamic relationship of the liquid lead-bismuth system. The temperature corresponding to the future prediction time step. It can be obtained by extrapolating the temperature sequence from the historical input window, or by using an external temperature prediction model, a set operating curve, or measured temperature data. This invention does not limit the method of obtaining future temperature information.

[0079] (2) Adaptive physical loss weights for prediction time steps : Let the prediction time step be P, the current training round be e, and the Safe-PINN physical loss weights be... The adaptive physical loss weights for the prediction time step are used to dynamically adjust the contribution of the physical loss to the overall training objective based on the prediction time step and the training phase. Let the prediction time step segmentation threshold be: ; And set the upper limit of the physical constraint weights corresponding to different prediction time step intervals as follows: ; The adaptive physical loss weights for the prediction time step are defined as follows: When P≤ hour: =0; when <P≤ hour: After the warm-up phase, the constraint is gradually increased to the upper limit of the first level constraint. when <P≤ hour: After the warm-up phase, the constraint is gradually increased to the upper limit of the second level of constraints; ... when <P≤ hour: After the warm-up phase, the constraint limit is gradually increased to the (n-1)th level constraint limit. When P>n: After the warm-up phase, the upper bound of the long-term constraint is gradually increased to the nth level. That is: ; Where P represents the prediction time step, which is the number of future prediction time steps output by the model in one go; Indicates the current training round number; This represents the Safe-PINN physical loss weights when the prediction time step is P and the training epochs are e. This represents the preset threshold for segmentation of the prediction time step, and satisfies... ; The upper limit of the physical constraint weights corresponding to different prediction time step intervals; This represents the upper limit of the physical constraint weights corresponding to the m-th prediction time step interval, where r(e) represents the asymptotic boosting function that varies with the number of training epochs. The asymptotic boosting function r(e) is defined as: ; in, This represents the number of physical constraint warm-up rounds, a training scheduling hyperparameter used to control the training rounds in which the Safe-PINN physical loss begins to participate in training. This represents the number of asymptotic enhancement rounds of the physical constraints. It is a training scheduling hyperparameter used to control the number of training rounds required for the Safe-PINN physical loss to be enhanced from 0 to the upper limit of the weights for the corresponding prediction time step interval.

[0080] This scheduling method ensures that the step size is less than or equal to... The prediction retains the PINN module but excludes the physical loss from training to avoid physical constraints interfering with short-term data-driven fitting; when the prediction time step is greater than... In the step-by-step forecasting, Safe-PINN physical constraints are gradually enabled to enhance the trend stability and physical consistency of medium- and long-term forecasts.

[0081] (3) Saturation Boundary Barrier Loss: To avoid the predicted oxygen concentration characterization value exceeding the physically feasible range, a saturation boundary barrier loss is constructed. The penalty is significantly lower than the non-physical prediction of the thermodynamic saturation boundary; assuming the future... The first predicted time step The lower boundary for characterizing oxygen concentration at each measuring point is: The upper boundary of saturation is The specific values ​​of the upper and lower boundaries need to be determined based on the thermodynamic relationship of the liquid lead-bismuth system, the safe operating range of oxygen control, material protection requirements, the effective measurement range of the sensor, or experimental calibration results. (Saturation boundary barrier loss) Defined as: ; Where N represents the number of oxygen electromotive force measurement points; P represents the prediction time step, i.e., the number of future prediction time steps output by the model; i represents the oxygen electromotive force measurement point number, ( ); p represents the future prediction time step number, ( );N P represents the product of the number of oxygen electromotive force measurement points and the prediction time step, which is used to normalize the loss; This represents the predicted oxygen concentration at the i-th oxygen electromotive force measurement point in the p-th prediction time step. This represents the physical lower boundary of oxygen concentration at the i-th oxygen electromotive force measurement point in the future p-th predicted time step; This represents the upper boundary of oxygen concentration saturation at the i-th oxygen electromotive force measurement point in the p-th predicted time step in the future; This represents a linear activation function.

[0082] When predicting oxygen concentration characterization values This loss incurs a penalty when the value is below the physical lower boundary or above the saturation upper boundary; when predicting the oxygen concentration characterization value. When the loss is within the physically feasible range, it does not incur a penalty.

[0083] (4) Safety Anchor Loss: To utilize the physical states already available in the current historical input window to assist in future predictions and to avoid negative migration caused by unreliable anchors, a safety anchor gating mechanism is introduced. The historical physical anchors are jointly determined by the effective historical oxygen electromotive force observations, temperature states, and the physical conversion relationship between oxygen electromotive force, temperature, and oxygen concentration within the current historical input window.

[0084] Anchor constraints are activated only when the historical physical anchor point oxygen concentration characterization value is closer to the actual oxygen concentration characterization value than the current predicted oxygen concentration characterization value.

[0085] Let the historical physical anchor point oxygen concentration characterization value be obtained from the valid historical observations, temperature status, and physical transformation relationship in the current historical input window. It indicates the future number The first predicted time step Characteristic values ​​of oxygen concentration at each measuring point. Construct gated variables for safety anchor points: ; in, This represents the safety aiming point gating variable for the i-th oxygen electromotive force measurement point in the p-th predicted time step. This represents the true oxygen concentration characterization value at the i-th oxygen electromotive force measurement point at the p-th predicted time step. This represents the predicted oxygen concentration value at the i-th oxygen electromotive force measurement point at the p-th prediction time step. If the historical physical anchor point is closer to the actual oxygen concentration value than the current model prediction, then the anchor point constraint is enabled; otherwise, the anchor point constraint is not enabled.

[0086] Safety anchor point loss Defined as: ; in, Indicates the safety anchor point loss; N represents the number of oxygen electromotive force measurement points; P represents the prediction time step; i represents the oxygen electromotive force measurement point number; p represents the future prediction time step number; N P represents the product of the number of oxygen electromotive force measurement points and the prediction time step, used for loss normalization. This loss is only enabled when historical physical anchors are closer to the true labels than the current model predictions, thus avoiding unreliable anchors from forcibly pulling back the model predictions.

[0087] (5) Safety Trend Loss: To enhance trend consistency in medium- and long-term time-domain forecasts, a safety trend loss is introduced. Trend constraints are only activated when the historical anchor point oxygen concentration trend is closer to the actual oxygen concentration trend than the current predicted oxygen concentration trend. Let the predicted oxygen concentration trend, the actual oxygen concentration trend, and the historical anchor point oxygen concentration trend be respectively: ; ; ; in, This represents the predicted change trend of the i-th oxygen electromotive force measurement point in the future p-th prediction time step; This indicates the corresponding actual trend of change; This indicates the trend of change in the corresponding historical physical anchor points. The above trend item applies to... .

[0088] To prevent erroneous trend constraints from affecting model predictions, this embodiment introduces a safe trend gating variable. Its definition is: ; in, Let represent the safety trend gating variable for the i-th oxygen electromotive force measurement point at the p-th prediction time step in the future. This is considered when the historical physical trend is closer to the actual trend than the current model's predicted trend. Enable trend constraints; otherwise Trend constraints are not enabled.

[0089] Safety Trend Loss Defined as: ; in, This represents the safety trend loss. This loss is only activated when the historical physical trend is closer to the true trend than the current model predicts, thereby improving the trend consistency of predictions in the medium to long time domains.

[0090] (6) Safety Smoothing Loss: Only non-physical jitter where the predicted curvature significantly exceeds the true curvature is penalized. To suppress non-physical jitter in the predicted sequence, the second difference between the predicted logarithmic oxygen concentration sequence and the true logarithmic oxygen concentration sequence is defined as follows: ; ; in, This represents the predicted second-order difference at the i-th oxygen electromotive force measurement point at the p-th predicted time step in the future, used to characterize the curvature of the predicted sequence. This represents the corresponding true second-order difference, used to characterize the curvature of the true sequence. The above second-order difference term is applicable to... .

[0091] Let the safety tolerance threshold be Safe and smooth loss Defined as: ; in, Indicates safe and smooth loss; Indicates the safety tolerance threshold; denoted as linear rectified function; N(P-2) represents the number of measurement-prediction time steps involved in the second-order difference calculation, used for loss normalization.

[0092] This loss only penalizes non-physical jitter where the predicted curvature significantly exceeds the true curvature and surpasses the safety tolerance threshold, without over-smoothing normal trends. In such cases, this safe loss smoothing feature may not be enabled.

[0093] In summary, the physical losses of Safe-PINN consist of saturation boundary barrier losses, safety anchor point losses, safety trend losses, and smoothing losses, with the total physical losses being... Represented as: ; in, Indicates the loss of the saturation boundary barrier. Indicates loss of safety anchor points. Indicates a loss due to a safety trend. Indicates smoothing loss; , , , These represent the weighting coefficients corresponding to boundary guardrail loss, safety anchor point loss, safety trend loss, and smoothing loss, respectively.

[0094] Let the data-driven prediction loss of the model be... It can employ mean squared error, Huber loss, weighted mean squared error, or other loss functions suitable for multi-step time series prediction. After incorporating adaptive physical loss weights based on the prediction time step, the total training loss of the model without a robust training mechanism is reduced. for: ; in, This is used to dynamically adjust the contribution of Safe-PINN physical loss to the overall training objective based on the predicted time step and training epochs; This represents the total physical loss.

[0095] Through the aforementioned prediction time step adaptive Safe-PINN physical constraint module, this invention can maintain the fitting ability of the data-driven model in short-term prediction and gradually introduce physical consistency constraints in medium- and long-term prediction, thereby reducing the risks of trend drift, boundary crossing and non-physical jitter in the multi-step prediction process, and thus improving the reliability of oxygen electromotive force prediction and subsequent oxygen concentration field construction in liquid lead-bismuth.

[0096] Step S7: Construct a robust training mechanism for multi-branch consistent distillation. After constructing the prediction time step adaptive Safe-PINN physical constraint module in Step S6, a robust training module for multi-branch consistent distillation is further constructed. This module is used to improve the prediction stability and generalization ability of the model under conditions of sensor noise, random missing data, local network disconnection, or abnormal measurement points. The robust training module for multi-branch consistent distillation mainly functions during the model training phase. Its basic idea is: using the normal historical input window as the pure input branch, the historical input window with a certain proportion of noise perturbation as the perturbation input branch, and the historical input window with missing measurement points or local occlusion as the missing input branch; under the condition of shared model parameters, the multi-step prediction results of each branch are obtained respectively, and the prediction results of the pure input branch are used as the teacher signal to apply consistent distillation constraints to the perturbation input branch and the missing input branch, so that the prediction results under perturbation conditions or missing conditions are consistent with the prediction results under pure conditions, thereby improving the generalization ability and prediction stability of the model under abnormal input conditions.

[0097] Let the current history input window be: ; in, Indicates the first The input features for each historical time step include oxygen electromotive force observations from multiple oxygen electromotive force measurement points, valid data markers, temperature, or other external state features.

[0098] (1) Pure input branch A clean input branch uses an undisturbed history input window as input, represented as: ; Will Inputting the aforementioned spatiotemporal prediction model yields the multi-step oxygen electromotive force prediction results for the pure input branch: ; in, This represents a spatiotemporal prediction model composed of the aforementioned three-source fusion graph construction, adaptive diffusion graph convolution, bidirectional temporal attention, and multi-step prediction output module. Indicates model parameters.

[0099] The prediction loss for a clean input branch is defined as: ; in, This indicates a true multi-step oxygen electromotive force label. The prediction error loss function can be represented by mean squared error, Huber loss, weighted mean squared error, or other loss functions suitable for multi-step time series prediction.

[0100] (2) Disturbance input branch The perturbation input branch is used to simulate sensor measurement noise or field interference. A random noise perturbation is applied to the effective oxygen electromotive force observations in the current historical input window to obtain the perturbation input window: ; in, This represents a random perturbation term, which can follow a Gaussian distribution, a uniform distribution, or a bounded perturbation distribution set according to the sensor's measurement error range. For temperature, valid markings, or other features that are not suitable for perturbation, no perturbation may be applied or a separate perturbation strategy may be used.

[0101] By inputting the perturbation input window into the same spatiotemporal prediction model, multi-step prediction results of the perturbation input branch are obtained: ; The prediction loss of the perturbation input branch is defined as: ; This branch allows the model to actively engage with noisy inputs during the training phase, improving its robustness to sensor noise and measurement disturbances.

[0102] (3) Missing input branch The missing input branch is used to simulate random packet loss from the sensor, local network outages, measurement point occlusion, or local invalid observations. A missing input mask matrix is ​​constructed. ,in Indicates the first This historical step is the first The data from each oxygen electromotive force measurement point are valid. This indicates that the corresponding measurement point data is obscured or missing.

[0103] In one implementation, a missing input window can be represented as: ; in, This indicates element-wise multiplication. This indicates a placeholder value, interpolated value, or preset invalid flag indicating a missing location. The invalid flag can be used in conjunction with the valid data flags in the aforementioned node input features, enabling the model to distinguish between true observations and missing placeholder values.

[0104] Inputting the missing input window into the same spatiotemporal prediction model yields multi-step prediction results for the missing input branch: ; The prediction loss for missing input branches is defined as: ; Through this branch, the model can learn stable predictive capabilities under conditions of missing measurement points or incomplete local historical data.

[0105] (4) Consistent distillation constraint To ensure that perturbed and missing input branches maintain prediction behavior similar to normal input conditions under abnormal input conditions, this invention uses the prediction result of the clean input branch as the teacher signal to apply a uniform distillation constraint to the perturbed and missing input branches. The prediction results of the clean, perturbed, and missing input branches are as follows: , and Then the loss of uniform distillation Defined as: ; in, Indicates the loss of uniformity during distillation. The distillation consistency metric function can be represented by mean square error or other distance metrics. This indicates stopping gradient operations, used to prevent the consistency distillation loss from back-updating the teacher prediction results of the pure input branch, thus avoiding perturbation branches or missing branches from back-interfering with the main prediction path.

[0106] In other embodiments, either the perturbation input branch or the missing input branch can be selected, or it can be further extended to multiple noise intensities, multiple missing ratios, or multiple types of abnormal input branches; the uniform distillation constraint can be generalized in the same way.

[0107] (5) Total loss during robust training By combining the clean input branch, the perturbed input branch, the missing input branch, and the consistency distillation constraint, a robust training loss for multi-branch consistency distillation is obtained. : ; in, This represents the robust training loss for multi-branch consistent distillation; , , These represent the weighting coefficients corresponding to the perturbation input branch loss, the missing input branch loss, and the consistency distillation loss, respectively.

[0108] The total training loss of the model, ultimately used for backpropagation and parameter updates. The objective is represented as: ; in, The loss represents the total loss during model training, specifically the loss of the pure input branch. The total physical loss already includes the prediction time step adaptive Safe-PINN, meaning the Safe-PINN physical loss participates in the multi-branch consistency distillation robust training loss through the pure input branch. In this embodiment, the Safe-PINN physical loss is only applied to the prediction results of the pure input branch; the perturbed input branch and the missing input branch do not apply the Safe-PINN physical loss, mainly used for robust training under noisy and missing data conditions and for consistency distillation constraints between the prediction results and the pure input branch. When multi-branch consistency distillation robust training is not enabled, the final total training loss of the model degenerates into the pure input branch loss, i.e.: In one implementation, the Safe-PINN physical loss primarily affects the prediction results of the pure input branch; the perturbation input branch and the missing input branch are mainly used for robust training and consistency distillation constraints. This invention does not limit the specific combination of the physical loss with each training branch.

[0109] Through the aforementioned multi-branch consistency distillation robust training module, the model can simultaneously learn prediction patterns under normal input, noisy input, and missing input conditions during training. Furthermore, by suppressing prediction drift caused by abnormal inputs through teacher prediction consistency constraints, the model enhances its prediction stability under conditions of sensor noise, random packet loss, and local network outages in engineering settings.

[0110] Example 2 like Figure 6 As shown, this invention proposes a method for constructing an oxygen concentration field in liquid lead-bismuth, comprising: The oxygen electromotive force and temperature data in liquid lead-bismuth under the test condition are obtained. The oxygen concentration prediction method for liquid lead-bismuth described above is then used to predict the future oxygen concentration. After obtaining the oxygen concentration prediction results at each oxygen electromotive force measurement point under each prediction time step, the following operations are performed: Step S8: Construct a continuous oxygen concentration field reconstruction module In step S7, the future is obtained. After predicting the oxygen concentration at each oxygen electromotive force measurement point under each prediction time step, a continuous oxygen concentration field reconstruction module is constructed to obtain the continuous oxygen concentration field in the liquid lead-bismuth experimental device or liquid metal circuit based on the spatial location of the measurement point, the graph structure propagation relationship, and physical feasibility constraints.

[0111] Let the first Spatial position vector of each oxygen electromotive force measuring point for: ; in, , , Indicates the first The position coordinates of each measuring point in the loop space coordinate system. (The remaining text appears to be incomplete and requires further context.) The discrete oxygen concentration prediction vector at all measuring points under each prediction time step is represented as follows: .

[0112] (1) Graph-based discrete oxygen concentration enhancement To utilize the spatial topological relationships and higher-order indirect associations between measurement points, a three-source fusion adjacency matrix can be used. Graph propagation enhancement is applied to the discrete oxygen concentration prediction vector. In one implementation, a multi-order graph propagation method is used: ; in, Indicates the future number The discrete oxygen concentration vector after graph propagation enhancement at each prediction time step; This represents the upper limit of the graph propagation order in the oxygen concentration field reconstruction; Indicates the first Graph propagation weights; This represents the three-source fusion adjacency matrix corresponding to the current historical input window; This indicates that the predicted oxygen concentration value of the measuring point itself is retained.

[0113] Through the graph propagation enhancement described above, neighborhood information related to the spatial topology, flow relationships, dynamic states, and implicit coupling relationships of the measurement points can be introduced into the discrete oxygen concentration prediction results, thereby improving the spatial discontinuity problem caused by noise or missing values ​​in the local measurement point predictions. Alternatively, without graph propagation enhancement, one could directly... .

[0114] (2) Construction of continuous fields based on spatial interpolation In obtaining the future After predicting the discrete oxygen concentration or the enhanced graph propagation value at each prediction time step, and combining this with the spatial location of each oxygen electromotive force measurement point, the result is used to determine the position of any location in the loop space region. Spatial interpolation is performed to obtain a continuous oxygen concentration field. : ; in, Indicates the future number Spatial location at each predicted time step Predicted oxygen concentration at the location; Represents a spatial interpolation function; express Middle The graph propagation-enhanced oxygen concentration value corresponding to each measuring point.

[0115] In one implementation, the spatial interpolation function may employ inverse distance weighted interpolation, which has the following form: ; The weighting function is: ; To prevent extremely small positive numbers with a denominator of zero, , where is the distance attenuation coefficient. Besides inverse distance-weighted interpolation, the spatial interpolation function can also employ radial basis function interpolation, kriging interpolation, spline interpolation, finite element interpolation, path distance interpolation based on loop geometry topology, or other methods suitable for reconstructing the spatial field of liquid metal loops.

[0116] (3) Physical feasibility projection and field constraints To avoid significant non-physical values ​​in the continuous oxygen concentration field, a physically feasible projection can be performed on the interpolated oxygen concentration field. Let the future... Spatial location at each predicted time step The physical lower boundary of oxygen concentration at that location is The upper boundary of saturation is The projected oxygen concentration field can then be expressed as: ; in, This represents a projection operator that constrains the oxygen concentration field to a physically feasible interval. This projection can be achieved through truncation, soft constraint optimization, smoothing regularization, or other physically consistent reconstruction methods. Furthermore, by combining factors such as loop flow direction, pipe segment connectivity, oxygen concentration saturation boundary, mass transfer smoothness, or local conservation approximation, the continuous oxygen concentration field can be spatially smoothed and physically reconstructed, making the oxygen concentration distribution in adjacent pipe segments or flow-related regions more continuous, stable, and in accordance with physical laws.

[0117] Step S9: Continuous oxygen concentration field output For the future Perform the discrete oxygen concentration enhancement, spatial interpolation, and physical feasibility projection at each prediction time step to obtain the future multi-step continuous oxygen concentration field. : ; in, Indicates the future number A continuous oxygen concentration field satisfying physical feasibility constraints at each prediction time step. Through the aforementioned continuous oxygen concentration field reconstruction module, this invention can extend the discrete oxygen concentration prediction results at multiple oxygen electromotive force measurement points into a continuous spatial distribution field in a liquid lead-bismuth experimental device or liquid metal circuit. This continuous oxygen concentration field can be used for oxygen control regulation, abnormal trend early warning, structural material corrosion risk assessment, sensor status assessment, and intelligent operation and maintenance of liquid metal circuits.

[0118] A readable storage medium having a computer program stored thereon, which, when executed, implements the method for predicting oxygen concentration in liquid lead-bismuth.

[0119] In summary, this invention constructs a method for predicting oxygen concentration and an oxygen concentration field in liquid lead-bismuth. Based on historical oxygen electromotive force (EMF) data, temperature or other external state characteristics, and spatial correlations between measuring points, it predicts the oxygen EMF for multiple future time steps. When sensor monitoring data experiences noise interference, abnormal fluctuations, local missing data, or short-term failures, the prediction results can serve as auxiliary monitoring information for the oxygen control system, providing a reference for judging oxygen concentration change trends, issuing anomaly warnings, assessing sensor status, and adjusting oxygen control. Furthermore, based on the predicted multi-measuring-point oxygen EMF, corresponding temperature information, and the physical conversion relationship between oxygen EMF and oxygen concentration, discrete oxygen concentration prediction results at each measuring point in multiple future time steps can be obtained. These results are then combined with the spatial location of the measuring points, loop topology, and spatial interpolation methods to construct a continuous oxygen concentration field in the liquid lead-bismuth loop. Thus, the multi-measuring-point oxygen EMF prediction results can be further converted into oxygen concentration field information with spatial distribution characteristics, improving the continuous monitoring capability, spatial sensing capability, and operational reliability of the oxygen control system under complex operating conditions.

[0120] Of course, those skilled in the art will recognize that the present invention is not limited to the details of the exemplary embodiments described above, but also includes the same or similar structures that can be implemented in other specific forms without departing from the spirit or essential characteristics of the invention. Therefore, the embodiments should be considered illustrative and non-limiting in all respects, and the scope of the invention is defined by the appended claims rather than the foregoing description. Thus, all variations falling within the meaning and scope of equivalents of the claims are intended to be included within the present invention. No reference numerals in the claims should be construed as limiting the scope of the claims.

[0121] Furthermore, it should be understood that although this specification describes embodiments, not every embodiment contains only one independent technical solution. This narrative style is merely for clarity. Those skilled in the art should consider the specification as a whole, and the technical solutions in each embodiment can also be appropriately combined to form other embodiments that can be understood by those skilled in the art.

[0122] The technologies, shapes, and structures not described in detail in this invention are all known technologies.

Claims

1. A method for predicting oxygen concentration in liquid lead-bismuth, characterized in that, include: Step S1: Using several oxygen electromotive force sensor measurement points set in the lead-bismuth experimental device as graph structure nodes, collect the oxygen electromotive force time series data and temperature time series data of each measurement point, and fuse the static physical prior graph, dynamic state graph and adaptive learning graph to construct a three-source fusion adjacency matrix. Step S2: Based on the three-source fusion adjacency matrix, perform K-order diffusion graph convolution on the node input feature matrix composed of the node input features of all nodes at each time step within the historical input window, extract the spatial coupling features between each measurement point, and obtain the spatially enhanced node feature sequence composed of the spatially enhanced node features of all historical time steps under this window; the node input features include one or more of the following: oxygen electromotive force data, valid data labeling, measurement point location embedding, node identity embedding, temperature time series data mapping results, and node features after local time series enhancement; Step S3: Perform bidirectional temporal modeling and temporal attention weighted aggregation on the spatially enhanced node feature sequence to extract the temporal dependencies and key time step information within the historical input window, and obtain comprehensive temporal features; Step S4: Based on the comprehensive time series features, perform spatiotemporal fusion and multi-step prediction to obtain the oxygen electromotive force prediction results of each measuring point under the future P prediction time steps; Step S5: Based on the physical conversion relationship between oxygen electromotive force, temperature and oxygen concentration, convert the oxygen electromotive force prediction result into the oxygen concentration prediction result at each measuring point. The oxygen concentration prediction model is constructed through the above steps S1-S5. The training process includes: Step S6: Construct a prediction time step adaptive physical constraint module, dynamically adjust the physical loss weights according to the prediction time step P, apply physical consistency constraints to the oxygen electromotive force prediction results, and use the total training loss. Train the model: The adaptive physical loss weights are used to predict the time step P and the number of training rounds e. Data-driven loss prediction; Total physical loss ; Step S7: Construct a multi-branch consistent distillation robust training module, using the prediction results of the pure input branch as the teacher signal, apply consistent distillation constraints to the perturbed input branch and the missing input branch, and perform robust training of the model; In step S6, the total physical loss of the prediction time step adaptive physical constraint module The calculation method is as follows: in, Indicates the loss of the saturation boundary barrier. Indicates loss of safety anchor points. Indicates a loss due to a safety trend. Indicates smoothing loss; , , , These represent the weighting coefficients corresponding to boundary fence loss, safety anchor point loss, safety trend loss, and smoothing loss, respectively. The adaptive physical loss weight The predicted step size P is dynamically determined based on the current training epoch e: when the predicted step size P is less than or equal to the first preset threshold... =0; when the prediction step size P is greater than the first preset threshold. After the preheating rounds, the weight is gradually increased to the preset upper limit of the corresponding prediction step size interval.

2. The method for predicting oxygen concentration in liquid lead-bismuth as described in claim 1, characterized in that, In step S1, the three-source fusion adjacency matrix A is represented as follows: in, This is a static physical prior diagram; This is a dynamic state diagram; For adaptive learning graphs; , , These are the learnable fusion weights for the three types of adjacency matrices.

3. The method for predicting oxygen concentration in liquid lead-bismuth as described in claim 2, characterized in that, The static physical prior map The construction method is as follows: construct the flow direction topology matrix based on the spatial arrangement of the oxygen electromotive force sensor in the lead-bismuth experimental device loop, the pipe segment it is located in, the upstream and downstream relationship and the flow direction; calculate the historical correlation matrix between the measuring points based on the oxygen electromotive force time series data and temperature time series data of each measuring point in the training set. The flow topology matrix and the historical correlation matrix are fused element-wise and normalized to obtain the static physical prior graph. ; The dynamic state diagram The construction method is as follows: the temperature time series data in the current historical input window is mapped to hidden space state features, their statistical state variables are calculated and concatenated, and a dynamic adjacency matrix corresponding to the current sample is generated through a nonlinear mapping network, thereby obtaining the dynamic state diagram. ; The adaptive learning graph The construction method is as follows: assign a first learnable node embedding matrix and a second learnable node embedding matrix to each oxygen electromotive force measurement point, which are used to characterize the hidden features of the receiving end and the hidden features of the transmitting end of the node, respectively. The inner product of the two sets of learnable node embedding matrices is calculated to obtain the correlation score between the measurement points, and the non-negative correlation strength matrix is ​​obtained by ReLU activation. Using the mean of each row in the non-negative correlation strength matrix as a threshold for mask sparsity filtering, and then performing row-by-row normalization, we obtain the adaptive learning graph formed by the adjacency matrix of the adaptive learning graph. .

4. The method for predicting oxygen concentration in liquid lead-bismuth as described in claim 1, characterized in that, In step S2, the spatial augmentation node features at the t-th historical time step in the current historical input window. Represented as: Where σ(·) represents the nonlinear activation function; K represents the upper bound of the diffusion convolution order; and k represents the diffusion order number. A is the three-source fusion adjacency matrix; , where I is the identity matrix, representing the information of the node itself preserved in the 0th-order diffusion; This represents the node input feature matrix of all oxygen electromotive force measurement points at the t-th historical time step in the current historical input window; The learnable weight parameters represent the k-th order diffusing convolution branch; This represents the bias term corresponding to the k-th order diffusion characteristic; Step S3 includes: The spatial augmentation node features of each historical time step are aggregated by node dimension to obtain the temporal input feature sequence F; The temporal input feature sequence F is input into a bidirectional long short-term memory network, and temporal dependencies are extracted from the forward temporal order and the reverse temporal order, respectively, to obtain the temporal output feature sequence L; Attention weights for each historical time step are calculated based on the temporal output feature sequence L. The temporal output features of all historical time steps are then weighted and aggregated to obtain the global temporal features. ; The temporal output features of the last historical time step of the bidirectional long short-term memory network. With the aforementioned time-series global features By splicing the data, we obtain the comprehensive temporal features. .

5. The method for predicting oxygen concentration in liquid lead-bismuth as described in claim 1, characterized in that, Step S4 includes: The comprehensive time series features The spatiotemporal fusion prediction features, serving as the current historical input window, are input to the multi-step prediction output network. , to obtain the depth prediction term ; Simultaneously, the change ΔX of the oxygen electromotive force observation value at each measuring point within the current historical input window relative to the historical anchor point B is extracted and processed through an autoregressive residual mapping network. Obtain the residual prediction term ; The depth prediction term D, the residual prediction term R, and the historical anchor point B are added together, and then reconstructed to obtain the oxygen electromotive force prediction results for all N measurement points under the future P prediction steps: in, ; p represents the future prediction time step number; i represents the oxygen electromotive force measurement point number. This represents the predicted oxygen electromotive force at the i-th measurement point in the p-th prediction step. This indicates that the output result will be reconstructed into a prediction matrix form; the historical anchor point B is calculated based on the effective historical observation values ​​of the oxygen electromotive force of each measuring point within the current historical input window.

6. The method for predicting oxygen concentration in liquid lead-bismuth as described in claim 1, characterized in that, Step S5 includes: Obtain the predicted oxygen electromotive force value at the i-th measurement point in the p-th prediction step. and their corresponding temperatures Using physical conversion functions Convert the predicted oxygen electromotive force and corresponding temperature into predicted oxygen concentration characterization values: in, The oxygen electromotive force, temperature, and oxygen concentration are determined based on the Nernst equation or oxygen sensor calibration relationship; the predicted oxygen concentration is characterized by the oxygen concentration itself or the logarithm of the oxygen concentration; the corresponding temperature... The temperature sequence obtained is extrapolated from the historical input window, or provided by an external temperature prediction model, or obtained from a set operating curve or measured temperature data. Using the predicted oxygen concentration characterization value as the predicted oxygen concentration, the discrete oxygen concentration prediction results for all N measurement points under the next P prediction steps are obtained. : in, , This represents the predicted oxygen concentration at the i-th measurement point in the p-th prediction step.

7. The method for predicting oxygen concentration in liquid lead-bismuth as described in claim 1, characterized in that, Step S7 includes: using the undisturbed historical input window as the pure input branch to obtain the pure prediction result. Random noise perturbation is applied to the oxygen electromotive force time series data in the historical input window to obtain the perturbation input branch and perturbation prediction results. A missing data mask matrix is ​​constructed to mask some measurement point data in the historical input window, thus obtaining the missing input branch and the missing prediction result. Based on the pure prediction results As a teacher signal, the disturbance prediction result and the missing prediction results Apply a uniform distillation constraint; the uniform distillation loss is expressed as: in, The distillation consistency metric function can be represented by mean square error or other distance metrics. This indicates that the gradient operation is stopped.

8. The method for predicting oxygen concentration in liquid lead-bismuth as described in claim 1, characterized in that, In step S6, the saturation boundary barrier loss for: Where N represents the number of oxygen electromotive force measurement points; P represents the prediction time step; i represents the oxygen electromotive force measurement point number; and p represents the future prediction time step number. This represents the predicted oxygen concentration at the i-th oxygen electromotive force measurement point in the p-th prediction time step. This represents the physical lower boundary of oxygen concentration at the i-th oxygen electromotive force measurement point in the future p-th predicted time step; This represents the upper boundary of oxygen concentration saturation at the i-th oxygen electromotive force measurement point in the p-th predicted time step in the future; Represents a linear activation function; Safety anchor point loss for: in, This represents the predicted oxygen concentration characterization value at the i-th oxygen electromotive force measurement point at the p-th prediction time step; The oxygen concentration characterization value at the historical physical anchor point of the i-th oxygen electromotive force measurement point at the p-th prediction time step; This represents the safety aiming point gating variable for the i-th oxygen electromotive force measurement point in the p-th predicted time step. Safety Trend Loss Defined as: in, The predicted oxygen concentration trend at the i-th oxygen electromotive force measurement point in the p-th prediction time step; This represents the safety trend gating variable for the i-th oxygen electromotive force measurement point at the p-th prediction time step in the future; This indicates the oxygen concentration trend at the corresponding historical anchor point.

9. A method for constructing an oxygen concentration field in liquid lead-bismuth, characterized in that, Obtain the oxygen electromotive force and temperature data in liquid lead-bismuth under the test conditions, and use the oxygen concentration prediction method in liquid lead-bismuth as described in any one of claims 1-8 to obtain the oxygen concentration prediction result. Based on the spatial location of each measuring point and the oxygen concentration prediction result, perform spatial interpolation reconstruction to obtain the continuous oxygen concentration field in the liquid lead-bismuth experimental device. The spatial interpolation employs any one of the following: inverse distance weighted interpolation, radial basis function interpolation, Kriging interpolation, spline interpolation, and path distance interpolation based on loop geometry topology; and the oxygen concentration field obtained by interpolation is subjected to physical feasibility projection, constraining it to a preset physical feasibility interval.

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