Dam deformation prediction and interpretability analysis method based on mathematical-deep learning fusion

Abstract

The invention relates to a mathematical-deep learning fused dam deformation prediction and interpretability analysis method, and the method comprises the steps: firstly carrying out the preprocessing of a dam deformation monitoring time sequence, and reconstructing the time sequence into a trend term, a periodic term and a residual term through multi-scale feature decomposition; aiming at physical characteristics of different components, a trend prediction model based on a mathematical model, a period prediction model based on time sequence analysis and a residual prediction model based on deep learning are respectively constructed, and physical mechanism constraints are introduced in a deep learning model training process so as to improve engineering rationality and interpretability of a prediction result; furthermore, through a multi-objective optimization and dynamic weight updating mechanism, weighted integration is carried out on prediction results of all the components, a final prediction result of dam deformation is obtained, prediction precision, long-term stability and physical interpretability can be considered, meanwhile, the adaptive capacity of the model to different operation working conditions is improved, and the prediction accuracy of dam deformation is improved. The method is suitable for dam operation period safety monitoring and risk analysis.

Description

Technical Field

[0001] This application relates to the field of safety monitoring and intelligent analysis technology for water conservancy and hydropower projects, and in particular to a method for predicting and interpreting dam deformation by integrating mathematical and deep learning. Background Technology

[0002] As crucial water conservancy and hydropower infrastructure, the safe operation of dams directly impacts the safety of life and property downstream and the stability of the regional economy and society. With the increasing service life of dams and the growing complexity of their operating conditions, dam structures often exhibit complex deformation evolution characteristics under the coupled effects of long-term loads, water level fluctuations, temperature changes, and material aging. Accurately predicting key effects such as dam deformation is one of the core technical challenges in dam safety monitoring and risk control.

[0003] Existing methods for predicting dam deformation can be mainly divided into three categories: traditional models based on mathematical statistics, statistical learning models based on time series analysis, and data-driven models based on deep learning.

[0004] On the one hand, traditional mathematical models (such as regression models and grey models) have clear structures and parameters with certain physical meanings, making them suitable for describing the long-term evolution trend of dam deformation. However, such models usually rely on strong prior assumptions, making it difficult to simultaneously characterize the nonlinear characteristics and sudden disturbances in the deformation process, and limiting their adaptability to complex operating conditions.

[0005] On the other hand, time series models such as ARIMA and SARIMA have certain advantages in handling periodic changes and can effectively describe periodic responses caused by reservoir water level regulation or seasonal temperature changes. However, these models are usually based on the assumption of stationarity and are insufficient in characterizing non-stationary signals and sudden abnormal responses. Furthermore, the model order and parameters are highly sensitive, limiting their engineering generalization ability.

[0006] In recent years, with the development of artificial intelligence technology, deep learning models (such as recurrent neural networks and convolutional neural networks) have been gradually introduced into the field of dam deformation prediction. These methods have significant advantages in uncovering complex nonlinear relationships, but they still face the following prominent problems in practical engineering applications: (1) The model is highly dependent on the data, which makes it prone to physical distortion and the prediction results lack engineering interpretability; (2) It is sensitive to extreme working conditions or abnormal data, and its long-term extrapolation stability is insufficient; (3) Different models have significant performance differences under different working conditions, and a single model is difficult to maintain optimal performance across the entire working condition range.

[0007] To overcome these shortcomings, existing research has attempted to improve prediction accuracy using model fusion or signal decomposition methods, such as combining empirical mode decomposition, wavelet analysis, and machine learning models. However, current fusion methods mostly focus on empirical superposition, lacking a systematic multi-scale feature-model matching mechanism, and the weights are often fixed during model fusion, making it difficult to dynamically adjust them according to changes in operating conditions. Furthermore, most methods fail to effectively integrate engineering physics mechanisms into the model training and fusion process, still posing a risk of "black box" operation.

[0008] Therefore, there is an urgent need for an intelligent prediction method that can address the multi-scale characteristics of dam deformation monitoring data, integrate the advantages of mathematical models and deep learning models, and introduce physical constraints and multi-objective dynamic weight optimization mechanisms to achieve a balance between prediction accuracy, engineering applicability, and interpretability, thereby better supporting the long-term operational safety monitoring and risk early warning of dams. Summary of the Invention

[0009] This application provides a mathematical-deep learning-integrated method for dam deformation prediction and interpretability analysis to address the problem in related technologies that dam deformation prediction models struggle to simultaneously characterize multi-scale deformation features, achieve long-term prediction stability, and provide engineering-physical interpretability.

[0010] Firstly, a mathematical-deep learning integrated method for dam deformation prediction and interpretability analysis is provided, which includes the following steps: a. Obtain time series data of deformation monitoring during the dam's operation period, and perform anomaly handling, scale unification, and time alignment preprocessing on the monitoring data; b. Perform multi-scale feature decomposition on the preprocessed deformation monitoring time series, and decompose and reconstruct the time series into trend terms, periodic terms and residual terms with clear physical meaning; c. Construct a trend prediction model based on a mathematical model for the aforementioned trend term, which is used to describe the slow evolution and deformation process of the dam structure under long-term load, material aging, or foundation settlement. d. Construct a time series prediction model for the periodic term to describe the periodic deformation response caused by reservoir water level scheduling or changes in ambient temperature; e. Construct a deep learning prediction model for the residual term to capture high-frequency deformation features caused by sudden loads, nonlinear disturbances, or measurement noise; f. Introduce physical mechanism constraints during the training process of the deep learning prediction model to ensure that the prediction results meet the requirements of engineering physical rationality; g. Based on multi-objective optimization and dynamic weight update mechanism, the prediction results of trend term, periodic term and residual term are weighted and integrated to obtain the final prediction result of dam deformation; h. Based on the multi-scale component prediction results, model weight changes, and physical constraint information, perform interpretability analysis on the prediction results.

[0011] In some embodiments, the multi-scale feature decomposition step employs a time-frequency joint analysis method to decompose the preprocessed dam deformation monitoring time series. This time-frequency joint analysis method includes wavelet packet decomposition, empirical mode decomposition, or a combination thereof. Through recursive decomposition of the original time series at different frequency bands and time scales, multiple sub-component sequences with different frequency characteristics and time scale properties are obtained. Furthermore, the multi-scale feature decomposition process comprehensively considers the sampling frequency of the monitoring data, the deformation evolution cycle characteristics, and engineering interpretability requirements when selecting the number of decomposition layers. This avoids noise amplification or weakening of physical meaning caused by excessive decomposition, thereby providing a structured, multi-scale feature input foundation for subsequent prediction model matching based on physical characteristics.

[0012] In some embodiments, after completing the multi-scale feature decomposition, the sub-components are physically mapped and reconstructed based on the differences in energy proportion, dominant frequency range, and statistical characteristics of each sub-component sequence. Low-frequency sub-components with high energy proportion and gradual changes are reconstructed as trend terms to characterize the long-term evolution deformation characteristics caused by dam creep or foundation settlement. Mid-frequency sub-components with obvious periodic fluctuations are reconstructed as periodic terms to characterize the periodic deformation response caused by reservoir water level regulation or changes in ambient temperature. High-frequency sub-components with relatively small energy proportion but significant abrupt changes are reconstructed as residual terms to characterize the high-frequency deformation characteristics caused by sudden loads, local nonlinear disturbances, or measurement noise. This achieves the transformation from pure mathematical decomposition to components with clear engineering physical meaning.

[0013] In some embodiments, the trend prediction model constructed for the trend term includes a grey prediction model, a nonlinear function fitting model, or a combination of both. The nonlinear function fitting model selects at least one candidate model from logarithmic function models, exponential function models, power function models, and polynomial function models, and estimates the model parameters using weighted least squares. After multiple candidate trend prediction models are constructed, a comprehensive evaluation criterion is built based on fitting error, correlation indicators, and stability indicators. Each candidate model is compared and screened, and the trend prediction model that performs best in terms of long-term extrapolation stability and engineering rationality is selected for predicting the trend term.

[0014] In some embodiments, when the trend prediction model adopts a gray prediction model, the gray prediction model is a GM(1,1) model, and the traditional gray model is improved by introducing a background value adaptive optimization mechanism. In this model, the background value in the gray model is represented in the form of adjustable weight parameters, and the background value weight parameters are searched and updated using a particle swarm optimization algorithm with the goal of minimizing the prediction error. This overcomes the problem of insufficient fitting accuracy caused by the fixed value of the background value in the traditional gray model, thereby improving the fitting accuracy and extrapolation stability of the trend prediction results when describing the long-term slow deformation process of the dam body.

[0015] In some embodiments, the periodic prediction model constructed for the periodic term is an autoregressive integral moving average (ARIMA) model or a seasonal autoregressive integral moving average (SARIMA) model. Before model construction, the stationarity of the periodic term sequence is tested, and the difference order is determined based on the test results. After the difference processing is completed, the candidate range of the model order is determined by autocorrelation function and partial autocorrelation function analysis, and the model parameters are selected and optimized in combination with information criteria, so that the periodic prediction model can effectively characterize the periodic response features of dam deformation caused by reservoir water level scheduling or seasonal temperature changes.

[0016] In some embodiments, the deep learning prediction model constructed for the residual term is a temporal convolutional neural network (TCN). The TCN adopts a causal convolutional structure to ensure that the prediction process relies only on historical information, and expands the temporal receptive field without significantly increasing the network depth through an expanded convolutional mechanism. At the same time, a residual connection mechanism is introduced into the network structure to enhance the stability of model training, enabling the deep learning prediction model to effectively capture the high-frequency nonlinear perturbations and sudden deformation response features contained in the residual term.

[0017] In some embodiments, the physical mechanism constraints are implemented by constructing a joint loss function during the training phase of the deep learning prediction model. The joint loss function includes a fitting error term based on measured data and a physical constraint loss term constructed based on engineering mechanics laws. The physical constraint loss term is used to restrict the prediction results to satisfy the boundedness of deformation, the consistency of deformation response direction, or the stress-strain relationship, thereby suppressing the risk of the prediction results deviating from the basic engineering physics laws while maintaining the prediction accuracy, and improving the engineering interpretability and credibility of the prediction results.

[0018] In some embodiments, when integrating the prediction results of the trend term, the prediction results of the period term, and the prediction results of the residual term, the weight solving problem corresponding to each prediction model is constructed as a multi-objective optimization problem. The multi-objective optimization problem considers at least the prediction error index and the prediction stability index simultaneously. The multi-objective optimization problem is transformed into a single-objective optimization problem through the weighting method or equivalent transformation method. The optimal weight vector is solved under the condition that the weights satisfy the non-negativity and normalization constraints, so as to achieve reasonable fusion of the prediction results of different prediction models.

[0019] In some embodiments, the dynamic weight update mechanism continuously monitors changes in prediction error, operating conditions, and model credibility based on a rolling time window. When any prediction model shows a significant change in error level, stability index, or consistency of physical constraints within a preset time window, a weight reallocation process is triggered. During the weight update process, the weight configuration of the previous stage is used as the initial constraint, and the weight vector is resolved within a limited adjustment range. This enables the integrated prediction model to adaptively adjust the contribution of each prediction model as the dam's operating conditions change, thereby maintaining the stability of prediction performance and engineering applicability.

[0020] The beneficial effects of the technical solution provided in this application include: 1. This invention uses multi-scale feature decomposition to separate the dam deformation monitoring sequence into trend terms, periodic terms, and residual terms, and matches the optimal prediction model for each scale characteristic. This avoids the accuracy reduction problem caused by a single model fitting multiple time series features at the same time, and significantly improves the accuracy and stability of the prediction results as a whole.

[0021] 2. By introducing the particle swarm optimization algorithm into the gray model GM(1,1) to adaptively optimize the background value, the problem of the traditional gray model being highly dependent on the fixed value of the background value is effectively overcome, and the model's fitting ability and extrapolation stability for the slow evolution trend of the dam such as long-term creep and settlement are significantly improved.

[0022] 3. The use of ARIMA / SARIMA time series models to specifically model the periodic terms can effectively characterize the periodic deformation response caused by factors such as reservoir water level scheduling and seasonal temperature changes, improve the model's adaptability to periodic characteristic changes under different operating conditions, and reduce the accumulation of periodic errors.

[0023] 4. By introducing a temporal convolutional neural network (TCN) to model the residual term, and utilizing causal convolution and dilated convolution structures, the model can effectively capture high-frequency nonlinear disturbances and sudden abnormal responses while ensuring the causality of the prediction, thereby improving the robustness of the model under extreme working conditions and abnormal data conditions.

[0024] 5. This invention embeds physical mechanism constraints during the training process of deep learning models, integrating engineering mechanics laws into the model optimization process in the form of soft constraints. This effectively suppresses the risk of prediction results deviating from physical laws and significantly improves the rationality, interpretability, and credibility of model output in engineering applications.

[0025] 6. By constructing a multi-objective optimization model and introducing a dynamic weight integration mechanism, the contribution of different base predictors can be adaptively adjusted according to the operating conditions and prediction error feedback, avoiding the problem of fixed weight combinations failing when switching operating conditions. This ensures that the prediction performance remains superior in different operating stages, thereby improving the engineering applicability and practical value of the system. Attached Figure Description

[0026] To more clearly illustrate the technical solutions in the embodiments of this application, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0027] Figure 1 This is a diagram of the intelligent combined prediction model architecture in this invention; Figure 2 A schematic diagram of multi-scale decomposition and physical component reconstruction of dam deformation monitoring data; Figure 3 A schematic diagram of intelligent matching relationships for prediction models driven by multi-scale component characteristics; Figure 4 A schematic diagram illustrating the process of building and optimizing a trend prediction model; Figure 5 A schematic diagram of the modeling process for a periodic time series forecasting model; Figure 6 This is a schematic diagram of the prediction structure of a temporal convolutional neural network (TCN) for residual terms. Figure 7 A schematic diagram illustrating the training of a deep learning model with embedded physical constraints; Figure 8 A schematic diagram of a multi-objective optimization and dynamic weighted integrated prediction mechanism; Figure 9 This is a schematic diagram of the intelligent combination prediction and interpretability closed-loop process of physical mechanism-data driven dual-engine. Detailed Implementation

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

[0029] This application provides a mathematical-deep learning-integrated method for dam deformation prediction and interpretability analysis, which can solve the problem in related technologies that dam deformation prediction models are difficult to simultaneously take into account multi-scale deformation characteristics, long-term prediction stability and engineering physical interpretability.

[0030] Please see Figures 1-9 The principle of intelligent combination model construction: 1) Overall Model Architecture This research aims to develop and systematically validate an intelligent combined prediction framework that integrates classical mathematical models and advanced deep learning models. This framework performs multi-scale physical decomposition and characteristic identification on dam deformation sequences, intelligently matches and adaptively fuses the optimal prediction model for each component, and ultimately constructs an intelligent dam deformation prediction system that combines physical interpretability with data-driven high accuracy. Please refer to [link / reference]. Figure 1 The intelligent combined prediction model proposed in this study adopts a dual-engine fusion architecture of "physical mechanism-data driven", which aims to provide more reliable and sensitive technical support for engineering safety management and risk prevention and control.

[0031] Its core idea is as follows: First, the original monitoring sequence is decomposed into multiple scales to separate the trend term, periodic term and residual term with clear physical meaning; second, the optimal base predictor is intelligently matched and constructed according to the variation characteristics of different components; finally, the results of each base predictor are integrated through multi-objective optimization and dynamic weighting strategy to form the final prediction output.

[0032] The core mathematical expression of the model is shown in the following equation: In the formula: For the i-th mathematical model (such as gray GM (1,1), ARIMA, etc.), it is responsible for fitting the trend and periodic components; The dynamic weight coefficients are optimized in real time through the LSTM-Attention mechanism to enhance the contribution of key models. g(·) is the nonlinear compensation function of the temporal convolutional network (TCN), used to capture residual terms caused by sudden disturbances (such as extreme weather, material nonlinearity); It is a historical state vector (containing the environmental loads and effects at the previous time t-1). For TCN network parameters; The random error term (follows a normal distribution with a mean of 0).

[0033] 2) Multi-scale feature decomposition and model matching Given that dam deformation monitoring sequences exhibit multi-scale mixed characteristics, including long-term evolution trends, seasonal periodic fluctuations, and short-term random disturbances, single-time-scale modeling methods cannot simultaneously ensure both long-term stability and short-term response sensitivity.

[0034] Therefore, this study introduces the concept of multi-scale signal analysis and uses time-frequency joint analysis techniques such as wavelet packet decomposition (WPD) to perform multi-scale and multi-frequency adaptive decomposition of the original monitoring sequence, providing a structured feature basis for the targeted construction and matching of subsequent models.

[0035] (1) Wavelet packet decomposition principle and decomposition level setting Wavelet packet decomposition, by recursively decomposing both low-frequency and high-frequency components simultaneously, can achieve finer frequency band division across the entire frequency domain, making it suitable for processing non-stationary and nonlinear dam deformation monitoring signals. Let the original monitoring sequence be... ,through After layer wavelet packet decomposition, it can be represented as: In the formula, This represents the sub-signal component corresponding to the j-th layer and k-th frequency band node.

[0036] In engineering applications, a larger wavelet packet decomposition level J is not always better; a balance must be struck between frequency resolution, physical interpretability, and model complexity. Based on the sampling frequency of long-term dam operation monitoring data (typically diurnal or lower) and the typical deformation response periodic characteristics, this study adopts the following engineering-representative decomposition level setting strategy: J = 2 layers: The original sequence is divided into 4 frequency bands, which is suitable for monitoring sequences with limited data length and a focus on long-term trend analysis. J = 3 layers (typical recommended value): Dividing the signal into 8 frequency bands can better distinguish annual cycle changes, seasonal responses and short-term disturbances, which is the default setting for most monitoring points in this study; J = 4 layers (high-precision analysis scenario): The signal is subdivided into 16 frequency bands, which is suitable for use when the sampling rate is high or when it is necessary to focus on identifying responses to sudden emergencies (such as earthquakes or sudden rises in water levels).

[0037] In practical implementation, the results under different decomposition levels can be compared through the energy entropy criterion or frequency band energy ratio analysis. Finally, the optimal decomposition scheme is selected as the number of levels that can both maintain the clarity of the main physical components and avoid excessive decomposition leading to noise amplification.

[0038] (2) Multi-scale component reconstruction and physical meaning mapping After wavelet packet decomposition, based on the energy distribution characteristics and corresponding time scales of each frequency band component, they are physically reconstructed and merged to form three types of characteristic components with clear engineering interpretations: In the formula, , , These represent the trend term (low frequency), the periodic term (medium frequency), and the residual term (high frequency), respectively. This component reconstruction process realizes the transformation from pure mathematical decomposition to mapping of engineering physical meaning, providing a clear basis for subsequent model selection.

[0039] Based on the physical characteristics and change patterns of each component, a dedicated prediction model is intelligently matched to achieve accurate prediction through a "divide and conquer" approach.

[0040] (3) Component-driven model matching strategy In response to the significant differences in the variation mechanism, stationarity, and nonlinearity of components at different scales, this study adopts the intelligent matching concept of "component-model one-to-one correspondence" to construct a differentiated base predictor system, achieving a true "divide and conquer" approach.

[0041] The matching relationships are shown in Table 1.

[0042] Table 1. Overview of Component Characteristics and Optimal Model Matching For trend items Its changes are gradual and its monotonicity is relatively strong, making it suitable for modeling with a gray model GM(1,1) optimized by background values ​​or a low-order nonlinear regression model to enhance the ability to characterize long-term evolution patterns. For periodic terms It is significantly driven by external environmental factors and has strong periodicity and quasi-stationarity. The seasonal ARIMA (SARIMA) model is preferred, and the periodic influence can be effectively eliminated through seasonal difference. For the residual term Its nonlinearity is strong and its mutation is significant. By using a temporal convolutional neural network (P-TCN) with embedded physical constraints, it can capture local mutation features while avoiding violations of basic mechanical laws.

[0043] Through the aforementioned multi-scale feature decomposition and model matching mechanism, the original complex problem of dam deformation prediction is transformed into several sub-problems with clear physical meaning and relatively simple statistical characteristics, laying a solid foundation for subsequent dynamic weight integration and interpretability analysis.

[0044] (4) Intelligent matching decision-making mechanism for component characteristic determination and prediction model After completing the multi-scale component reconstruction, in order to avoid relying on human experience for model selection, this embodiment further constructs a component characteristic-driven intelligent matching decision mechanism for prediction models, which is used to automatically determine the prediction model type suitable for each component during the engineering process.

[0045] Specifically, for each reconstructed component sequence, its set of statistical characteristic indicators is first calculated. These indicators include at least the stationarity test result, coefficient of variation, autocorrelation length, and component energy proportion. The stationarity test characterizes whether the component sequence satisfies the basic assumptions of time series modeling; the coefficient of variation reflects the intensity and stability of component fluctuations; the autocorrelation length characterizes the time dependence range of the component sequence; and the component energy proportion measures the relative contribution of the component to the original deformed sequence.

[0046] Based on this, and according to the combined judgment results of statistical characteristic indicators, model matching decisions are performed on each component: when the component sequence satisfies the stationarity condition and exhibits significant periodic characteristics after differencing, the component is determined to be a periodic dominant component and matched with a seasonal time series prediction model; when the component sequence exhibits monotonically slow-changing characteristics, a significant trend after accumulation, and weak random disturbances, the component is determined to be a long-term evolution component and matched with a grey prediction model or a nonlinear function fitting model; when the component sequence has high-frequency fluctuation characteristics, obvious abrupt changes, or a high proportion of residual energy, the component is determined to be a nonlinear disturbance component and matched with a deep learning prediction model.

[0047] When a component simultaneously satisfies multiple judgment conditions, it is included in the candidate model parallel prediction set. The subsequent dynamic weight integration mechanism coordinates and adjusts the contribution of each prediction model based on prediction error feedback and operating condition information, thereby avoiding the failure of single rule decision-making under complex operating conditions.

[0048] Through the above-mentioned component characteristic determination and prediction model intelligent matching mechanism, the prediction model selection process is automated and engineered, ensuring that features at different physical scales can always be characterized by the most adaptive prediction model, providing a stable and reliable decision-making basis for subsequent multi-model fusion and dynamic weight optimization.

[0049] 3) Base predictor construction (1) Nonlinear fitting model Trend terms obtained from multi-scale decomposition This mainly reflects the slow changes in dam body creep, foundation settlement, and long-term material evolution, characterized by stable changes, low dominant frequency, and minimal impact from random disturbances. To fully characterize this type of long-term evolution, this study constructs a nonlinear fitting prediction model based on multiple candidate functions, and models the trend term using functional modeling.

[0050] In selecting the model form, considering both engineering experience and mathematical expression capabilities related to dam deformation evolution, a family of candidate models is constructed from the following typical function sets: Logarithmic function model: suitable for creep processes that change rapidly in the early stage and gradually slow down in the later stage; Exponential function model: Applicable to long-term evolutionary trends with obvious acceleration or decay characteristics; Power function model: Applicable to situations where the scaling effect is significant and the rate of change evolves nonlinearly over time; Polynomial function model: used to describe general smooth trends, balancing fitting flexibility and computational stability.

[0051] Let the general form of the candidate model be: In the formula, It is in the form of a candidate nonlinear function. Let be the vector of parameters to be estimated. This is the random error term.

[0052] To reduce the interference of outlier observations or local fluctuations on the trend modeling results, this study uses weighted least squares (WLS) to estimate the model parameters. The weighting coefficients are determined based on the following factors: Time location weighting of monitoring data (giving higher weight to recent data to enhance the real-world applicability of predictions); Statistical characteristics of observation residuals (appropriately reduce the weight of observation points with large fluctuations); Monitor data quality or credibility indicators (such as manual verification marks, anomaly identification results).

[0053] After estimating the parameters of each candidate model, a multi-index comprehensive evaluation strategy is used to select the best model. Specifically, each model is quantitatively evaluated using the following evaluation index system: Standard deviation (SD): measures the dispersion of the fit residuals and reflects the stability of the model; Correlation coefficient (R): Measures the consistency between the model output and the measured trend term; Coefficient of determination ( ): Measures the model’s ability to explain the variance of the trend term.

[0054] By constructing a multi-objective evaluation function or a weighted scoring mechanism, the above indicators are comprehensively ranked, and the nonlinear function model that performs best in terms of fitting accuracy, stability, and engineering rationality is selected as the trend term. () is the base predictor.

[0055] This nonlinear fitting model, as a long-term trend prediction module in the intelligent combined prediction framework, has the advantages of clear model structure, strong parameter interpretability, and good extrapolation stability, providing a reliable low-frequency prediction benchmark for subsequent multi-model fusion and dynamic weight integration.

[0056] (2) Grey model GM(1,1) For the trend term sequence obtained by multi-scale decomposition, after completing the necessary data preprocessing (such as outlier removal, smoothing and scale unification), a GM(1,1) grey prediction model is constructed to model and predict its long-term evolution law.

[0057] Let the original trend sequence be: After one cumulative generation (AGO), we get: Based on this, construct the GM(1,1) grey differential equation: In the formula, a is the development coefficient, reflecting the growth or decay rate of the trend term; b is the gray action amount. This is a background value used to approximate the average state of the system within the interval [k-1,k].

[0058] To overcome the problem that the traditional GM(1,1) model, with its fixed background value (usually 0.5), is not adaptable to non-ideal monotonic sequences in engineering applications, this study introduces the Particle Swarm Optimization (PSO) algorithm to adaptively optimize the construction method of the background value, thereby improving the model's fitting accuracy and stability for slowly changing trend terms.

[0059] Specifically, the background value is represented as: Where λ∈(0,1) is the background value weight parameter to be optimized. The PSO algorithm uses λ (or its vectorized extension) as the optimization variable and searches for the optimal background value parameter by minimizing the prediction error objective function (such as RMSE or MAE).

[0060] In typical engineering applications, the PSO algorithm can be configured with the following representative parameter settings: Particle swarm size: 20–40; Maximum number of iterations: 50-100; Inertia weight w: 0.4~0.9 (can decrease linearly with iteration); Individual learning factors : 1.5~2.0; Group learning factor : 1.5~2.0; Background value weight Search range: 0.1, 0.9.

[0061] The above parameter range is only a typical example in engineering implementation. The specific settings can be adjusted according to the length of the monitoring data, the complexity of the trend, and the requirements of calculation efficiency.

[0062] By adaptively optimizing the background values ​​using the PSO algorithm, the constructed GM(1,1) model significantly enhances its adaptability to non-strictly exponential, slowly evolving trend sequences while maintaining the simplicity of the gray model structure and the clarity of the physical meaning of the parameters. As a long-term trend prediction base model in the intelligent combined prediction framework, this model provides stable and interpretable low-frequency prediction results for subsequent dynamic weight integration.

[0063] (3) Time series model ARIMA / SARIMA For the periodic terms obtained from multi-scale decomposition Its changes are mainly driven by external periodic loads such as reservoir water level regulation and seasonal temperature changes, and usually exhibit periodic fluctuations in mean or variance over time, but can approximately satisfy the stationarity assumption after appropriate differencing. Therefore, this study uses the Autoregressive Integral Moving Average (ARIMA) model and its seasonally extended model (SARIMA) to model and predict the periodic term.

[0064] ① Model Form and Applicable Conditions For periodic term sequences with no obvious seasonal cycle or weak periodic characteristics, construct... The model, in its general form, can be represented as: In the formula, Let the order be the autoregressive order. For non-seasonal difference order, The moving average order is... For lag operators, This is the white noise term.

[0065] When the periodic term exhibits significant annual or monthly periodic characteristics (such as the annual period corresponding to daily-scale monitoring data) =365, monthly data corresponding to an annual cycle When =12), the seasonal ARIMA model is used. Its expression is: in, For seasonal autoregression, differencing, and moving average orders, This represents the length of the seasonal cycle.

[0066] ②Stationarity test and determination of difference order Before building the model, the stationarity of the periodic term series should be tested. Unit root tests (such as the ADF test) can be used to determine whether the series is stationary, and the difference order can be determined accordingly. Non-seasonal difference order : Usually takes the value 0 or 1; Seasonal difference order When there are significant seasonal fluctuations, the typical value is 0 or 1.

[0067] In engineering practice, to avoid introducing additional noise through excessive differentiation, the minimum difference principle is generally preferred, and the lowest possible difference order is selected while meeting the stationarity requirements.

[0068] ③ Model order selection and typical value examples After differencing, the range of model order is initially determined using autocorrelation function (ACF) and partial autocorrelation function (PACF) analysis, and then finalized using information criteria. Common principles for selecting model order include: Non-seasonal portion: Generally, a low-order integer between 0 and 3 is selected; Seasonal section: Generally, the value is between 0 and 2 to avoid making the model structure too complex.

[0069] In practical implementation, the optimal order combination can be automatically searched by minimizing the Akaike Information Criterion (AIC) or the Bayesian Information Criterion (BIC). Common effective model forms in typical engineering applications include, but are not limited to: ARIMA(1,1,1), ARIMA(2,1,0); ; .

[0070] The above model form is only a representative example, and the specific order can be adaptively determined according to the data characteristics of different monitoring points.

[0071] ④ Model estimation and prediction output After determining the model order, the model parameters are solved using maximum likelihood estimation or least squares method, and the residual sequence is subjected to white noise testing to verify the model's rationality. When the residuals approximately follow a zero-mean white noise distribution, the model is considered to adequately characterize the main time-series structure of the periodic terms.

[0072] The trained ARIMA / SARIA model is used to generate periodic terms. The prediction results, as the periodic prediction base model in the intelligent combined prediction framework, provide stable and interpretable mid-frequency prediction components for subsequent multi-model dynamic weight fusion.

[0073] (4) Temporal Convolutional Neural Network (TCN) For the residual terms obtained after multi-scale decomposition These complex temporal patterns typically contain high-frequency information such as sudden loads, local nonlinear responses, or measurement noise, exhibiting significant nonlinearity, nonstationarity, and short-time correlation characteristics. To effectively capture such complex temporal patterns, this study introduces a Temporal Convolutional Network (TCN) as a deep learning predictor for the residual terms.

[0074] TCN employs a causal convolutional structure to ensure that predictions at any given time point rely solely on historical information, structurally preventing the leakage of future information. Simultaneously, through the dilated convolution mechanism, it significantly expands the temporal receptive field while maintaining a relatively shallow network layer count, making it suitable for long-term sequence modeling tasks.

[0075] For a one-dimensional input sequence f{Z} and a convolution kernel f{g}, the dilation convolution operation at time point t is defined as: Where d is the dilation factor and k is the kernel size. The dilation factor increases exponentially with network depth, enabling the network to obtain a large temporal receptive field at a relatively shallow depth.

[0076] In network architecture design, the kernel size *k* directly determines the model's ability to perceive local temporal patterns. Considering the sampling frequency and residual term variation characteristics of dam deformation monitoring data, the kernel size is typically chosen as a small odd number to balance local feature extraction capability with computational stability. In typical engineering applications, the following representative values ​​can be used: k=2~3: Suitable for scenarios where the main focus is on capturing extremely short-term abrupt changes or high-frequency noise features; k=3~5 (commonly recommended range): When the residual term contains a mixture of short-term disturbances and local response characteristics, it can better balance sensitivity and robustness; k=5~7: Applicable to situations where the residual terms still contain certain short- to medium-term correlation structures and the single-layer perception range needs to be expanded.

[0077] The above convolution kernel size is only a typical example in engineering implementation. The specific value can be adjusted according to the sampling interval of the monitoring data, the statistical characteristics of the residual term, and the computing resource conditions.

[0078] Regarding the inflation factor setting, an exponentially increasing strategy (e.g., d=1,2,4,8,...) is typically adopted, enabling the network to obtain receptive fields covering multiple time scales with a relatively small number of layers, thereby simultaneously capturing local mutations and relatively slowly changing residual information. To enhance the stability of network training, a residual connection mechanism is introduced into the TCN structure to short-circuit the input and output, effectively alleviating the gradient vanishing problem in deep network training.

[0079] Through the above structural design, the TCN model can efficiently model the nonlinear perturbation mode in the residual term without significantly increasing computational complexity, providing a high-sensitivity and high-robust high-frequency prediction compensation component for the intelligent combinatorial prediction framework.

[0080] (5) Physical constraint embedding To address the issues of physical distortion, extrapolation instability, and unreasonable predictions under extreme conditions that may occur in pure data-driven deep learning models in engineering applications, this study introduces a physical mechanism constraint embedding mechanism into the residual predictor of a temporal convolutional neural network (TCN) to achieve synergistic constraints between data-driven modeling and engineering physical laws.

[0081] This physical constraint embedding does not replace the data-driven model with explicit analytical solutions. Instead, it introduces constraints that conform to the understanding of engineering mechanics during the model training phase, integrating physical laws into the network optimization process in the form of "soft constraints." This guides the model to output physically reasonable and engineering-reliable prediction results while maintaining prediction accuracy.

[0082] ①Physical constraint embedding location and method Physical constraints are primarily embedded in the loss function layer of the TCN network. By constructing a physical constraint loss term and combining it with the data fitting error term to form a joint loss function, the network parameter learning process is constrained and guided. This approach does not alter the network's forward propagation structure and is characterized by flexibility and adaptability, making it suitable for widespread application in complex engineering scenarios.

[0083] ② Typical physical constraint forms and their engineering implications Taking the stress-strain relationship of a dam structure as an example, in the process of predicting the residual term, a physical consistency constraint based on the assumption of linear elasticity is introduced, and its loss function can be expressed as: In the formula, To predict stress for the network, To observe strain, This is the constitutive matrix constructed based on the elastic modulus and Poisson's ratio.

[0084] This constraint is used to suppress predictions that deviate significantly from the fundamental mechanical relationships in the network output, thus avoiding physically unacceptable stress responses.

[0085] In addition to the stress-strain constraints mentioned above, other forms of physical constraints can be introduced in the modeling of different monitored effect quantities, depending on the actual engineering situation. For example: Deformation monotonicity or boundedness constraint (to prevent non-physical oscillations in long-term forecasts); Upper limit constraint on deformation amplitude (based on engineering experience or design allowable value); Load-response direction consistency constraint (ensuring that the direction of the influence of water level and temperature changes on deformation is consistent with engineering understanding).

[0086] The above-mentioned constraint forms can be used individually or in combination, and can be flexibly configured according to the type of monitoring object and engineering requirements.

[0087] ③ Construction of joint loss function and setting of weights Taking into account both prediction accuracy and physical plausibility, the following joint loss function is constructed: In the formula, This refers to the fitting loss (such as mean square error) based on measured data. For physical constraint loss terms, This is the physical constraint weighting coefficient, used to balance the influence between data-driven approaches and physical constraints.

[0088] In engineering applications, The weighting coefficient can be adjusted based on the quality of monitoring data, the reliability of the physical model, and the required predictive stability. Typical values ​​can be determined through cross-validation or empirical analysis. This weighting coefficient is not fixed, in order to enhance the method's adaptability to different engineering scenarios.

[0089] ④ Technical effects and engineering advantages By embedding physical constraints into the TCN residual predictor, this study achieves the following technical effects: Effectively suppresses non-physical results that occur in extrapolation predictions by deep learning models; Significantly improves the predictive stability and robustness of the model under extreme working conditions and anomalous data conditions; It retains the strong fitting ability of deep learning models to complex nonlinear patterns, while enhancing the engineering interpretability of prediction results.

[0090] This physical constraint embedding mechanism, as an important component of the intelligent combinatorial prediction framework, provides key technical support for the reliable application of deep learning models in the field of safety monitoring of major engineering projects.

[0091] (6) Multi-objective optimization and dynamic weight integration After modeling the trend, periodic, and residual components, this study integrates the outputs of each base predictor to obtain a final prediction result that combines high accuracy and high stability. Considering that the advantages of different base predictors are not constant under different operating conditions (e.g., the periodic component contributes more during high water levels, while the residual compensation model is more critical during periods of abnormal disturbances), this study constructs a fusion mechanism of "multi-objective optimization + dynamic weight integration" to achieve adaptive determination and online updating of the integrated weights.

[0092] ① The meaning of modeling and constraints in multi-objective optimization problems During the integration phase, the weight vector The solution is transformed into a multi-objective optimization problem, aiming to simultaneously minimize multiple statistics of the prediction error (such as RMSE, MAE). This can be further transformed into a single-objective problem using a weighted method: In the formula, For the first One evaluation indicator, The corresponding weights are assigned. An intelligent optimization algorithm can be used to solve for the optimal weight vector. .

[0093] ② Typical values ​​of the objective function index system To balance the accuracy, robustness, and long-term extrapolation stability of engineering predictions, It can consist of multiple evaluation indicators. Typical optional indicators include, but are not limited to: RMSE: Emphasizes the penalty for large errors and is suitable for safety-sensitive scenarios; MAE: Reflects the average deviation and has better robustness; MAPE: Reflects relative error, facilitating horizontal comparison between different units or different measuring points; Long-term stability metrics (such as the mean of the sliding correlation coefficient, the mean of the rolling window error, etc.): enhance the reliability of extrapolation over long prediction steps.

[0094] In engineering applications, The weighting can be set according to management objectives. For example, "prioritizing safety early warning" can increase the weighting of RMSE, while "daily operation management" can increase the weighting of MAE or long-term stability. The above are just representative examples to avoid rigidly limiting the specific components of the indicators, thereby enhancing the applicability of the method.

[0095] ③ Dynamic weight update triggering mechanism In actual engineering operations, the characteristics of dam operating conditions and monitoring data exhibit significant phased and uncertain features. The applicability and predictive advantages of different baseline prediction models are not constant across different time periods. To avoid the decline in predictive performance during operating condition transitions caused by fixed weights or periodic manual adjustments, this embodiment further constructs a dynamic weight update triggering mechanism to determine the timing and conditions for weight reallocation.

[0096] The dynamic weight update triggering mechanism continuously monitors prediction errors and operating conditions based on a rolling time window, and makes judgments based on preset threshold conditions. Specifically, it includes the following triggering scenarios: In practice, to avoid frequent adjustments caused by occasional fluctuations, the above triggering conditions can only take effect when the preset judgment criteria are met within multiple consecutive prediction periods. Firstly, the triggering condition is based on prediction error feedback. During operation, the prediction errors of each base prediction model within a preset rolling time window are statistically analyzed. When the error index of any base prediction model deviates significantly from its historical stable level, a weight update process is triggered to reduce the weight ratio of that model in the current prediction stage.

[0097] Secondly, the triggering conditions are based on changes in operating conditions. When a significant change is detected in external operating condition indicators closely related to dam deformation, the weight update process is triggered. External operating condition indicators may include, but are not limited to, rapid rises and falls in reservoir water level, sudden changes in ambient temperature, switching of operation and scheduling status, or other operating condition characteristics that may cause changes in the deformation response mechanism.

[0098] Third, based on the triggering conditions of changes in model credibility. When a base prediction model exhibits decreased prediction stability, abnormal residual distribution, or increased deviation from physical constraints over multiple consecutive prediction periods, it is determined that the credibility of the model under the current operating conditions has decreased, and the weight update process is triggered.

[0099] When any of the above triggering conditions are met, the system initiates a dynamic weight redistribution process to re-optimize the weights of each base prediction model. This re-optimization is carried out while maintaining the continuity of weights and limiting the adjustment range, so that the weight distribution can reflect the changes in the current operating conditions and the model's predictive capabilities in a timely manner. When no update conditions are triggered, the system maintains the weight configuration of the previous stage to avoid unnecessary fluctuations caused by frequent adjustments.

[0100] Through the aforementioned dynamic weight update triggering mechanism, the adaptive adjustment of the weights of the multi-model integration is realized, enabling the prediction model combination to maintain stable and reliable prediction performance as the engineering operation status changes, and providing continuous and effective technical support for the long-term safety monitoring of the dam.

[0101] ④ Weighting strategy and feasible process In practical solutions, two complementary strategies can be used to obtain the optimal weights: Offline global optimization: Based on historical training and validation sets, the optimal weight vector is solved using a weighted method or intelligent optimization algorithm. This is used to form the basic weights and initial fusion configuration; Online dynamic correction: When performing online dynamic correction, the weights are updated with the weight configuration of the previous stage as the initial state or constraint center, and constrained optimization is performed within the preset adjustment range to ensure the continuity of weight changes and the stability of prediction results. During the running period or rolling prediction process, the weights are adaptively updated according to the current working condition characteristics and model error feedback to avoid fixed weights becoming invalid when the working condition changes.

[0102] To ensure the feasibility of the project, the weight calculation process can typically be organized into the following steps: (I) Obtain the prediction sequences of each base predictor on the validation set; (II) Constructing a multi-objective error index ; (III) Solving under convex constraints The optimal solution; (IV) Output the fused prediction value and enter the rolling update stage.

[0103] ⑤ Dynamic weight integration mechanism and adaptive interpretation of operating conditions Considering the significant operational switching characteristics of dams (seasonal temperature variations, high water level scheduling, data anomalies, etc.), this study introduces the concept of dynamic weight updates during the integration phase, making the weights a function that changes over time. This achieves adaptive matching of "working condition-model contribution". Dynamic weights can be updated driven by the following information: Historical state vector: includes historical sequence characteristics of environmental loads and effects such as reservoir water level, temperature, and time duration; Component confidence / error feedback: for example, the error statistics of the base predictor within the recent rolling window; Operating condition identification results: such as labels or discrimination signals for periods of high water level, cold wave and temperature drop, and sensor abnormality.

[0104] In typical engineering scenarios, the dynamic weighting mechanism can exhibit interpretable adjustment patterns: During periods of rapid changes in high water levels: the weights of periodic term models (such as SARIMA) can be increased to enhance the ability to track scheduling cycles and water pressure responses; During periods of data anomaly interference: The weights of residual compensation models (such as TCN / P-TCN) can be increased to suppress the bias caused by spike noise and local abrupt changes; Long-term stable operation period: The weights of the trend term model (such as GM(1,1) or nonlinear fitting) are more stable to ensure the smoothness of extrapolation and long-term consistency.

[0105] By correlating the weight changes with the physical response mechanisms corresponding to deformation components at different scales, the dynamic weight integration process not only reflects changes in statistical error but also has a clear engineering physics interpretation basis.

[0106] ⑥ Integrated Output and Engineering Advantages Through "multi-objective optimization + dynamic weight integration", this study achieved the following engineering results: A controllable trade-off is achieved among different error indices to avoid insufficient engineering applicability caused by the optimization of a single index; To achieve adaptive weight adjustment during operating condition switching, thereby improving model robustness and long-term stability; The weights satisfy convex constraints, and the integration results are stable, interpretable, and easy for engineers to understand and review.

[0107] The above description is merely a specific embodiment of this application, enabling those skilled in the art to understand or implement this application. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of this application. Therefore, this application is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features claimed herein.

Claims

1. A mathematical-deep learning integrated method for dam deformation prediction and interpretability analysis, characterized in that... It includes the following steps: a. Obtain time series data of deformation monitoring during the dam's operation period, and perform anomaly handling, scale unification, and time alignment preprocessing on the monitoring data; b. Perform multi-scale feature decomposition on the preprocessed deformation monitoring time series, and decompose and reconstruct the time series into trend terms, periodic terms and residual terms with clear physical meaning; c. Construct a trend prediction model based on a mathematical model for the aforementioned trend term, which is used to describe the slow evolution and deformation process of the dam structure under long-term load, material aging, or foundation settlement. d. Construct a time series prediction model for the periodic term to describe the periodic deformation response caused by reservoir water level scheduling or changes in ambient temperature; e. Construct a deep learning prediction model for the residual term to capture high-frequency deformation features caused by sudden loads, nonlinear disturbances, or measurement noise; f. Introduce physical mechanism constraints during the training process of the deep learning prediction model to ensure that the prediction results meet the requirements of engineering physical rationality; g. Based on multi-objective optimization and dynamic weight update mechanism, the prediction results of trend term, periodic term and residual term are weighted and integrated to obtain the final prediction result of dam deformation; h. Based on the multi-scale component prediction results, model weight changes, and physical constraint information, perform interpretability analysis on the prediction results.

2. The dam deformation prediction and interpretability analysis method integrating mathematical physics and deep learning as described in claim 1, characterized in that: The multi-scale feature decomposition step employs a time-frequency joint analysis method to decompose the preprocessed dam deformation monitoring time series. This time-frequency joint analysis method includes wavelet packet decomposition, empirical mode decomposition, or a combination thereof. Through recursive decomposition of the original time series at different frequency bands and time scales, multiple sub-component sequences with different frequency characteristics and time scale properties are obtained. Furthermore, the multi-scale feature decomposition process comprehensively considers the sampling frequency of the monitoring data, the deformation evolution cycle characteristics, and engineering interpretability requirements when selecting the number of decomposition layers. This avoids noise amplification or weakening of physical meaning caused by excessive decomposition, thereby providing a structured, multi-scale feature input foundation for subsequent prediction model matching based on physical characteristics.

3. The dam deformation prediction and interpretability analysis method integrating mathematical physics and deep learning as described in claim 2, characterized in that: After completing the multi-scale feature decomposition, the sub-components are physically mapped and reconstructed based on the differences in energy proportion, dominant frequency range and statistical characteristics of each sub-component sequence. The low-frequency, high-energy-proportion and slowly changing sub-components are reconstructed into trend terms to characterize the long-term evolution deformation characteristics caused by dam creep or foundation settlement. The mid-frequency components with obvious periodic fluctuations are reconstructed into periodic terms to characterize the periodic deformation response caused by reservoir water level regulation or changes in ambient temperature. The high-frequency components with relatively small energy proportions but significant abrupt changes are reconstructed into residual terms to characterize the high-frequency deformation characteristics caused by sudden loads, local nonlinear disturbances, or measurement noise, thereby realizing the transformation from pure mathematical decomposition to components with clear engineering physical meaning.

4. The dam deformation prediction and interpretability analysis method integrating mathematical physics and deep learning as described in claim 1, characterized in that: The trend prediction model constructed for the trend term includes a grey prediction model, a nonlinear function fitting model, or a combination of both. The nonlinear function fitting model selects at least one candidate model from logarithmic function models, exponential function models, power function models, and polynomial function models, and estimates the model parameters using the weighted least squares method. After multiple candidate trend prediction models are constructed, a comprehensive evaluation criterion is constructed based on fitting error, correlation index, and stability index. The candidate models are compared and screened, and the trend prediction model that performs best in terms of long-term extrapolation stability and engineering rationality is selected for predicting the trend term.

5. The dam deformation prediction and interpretability analysis method integrating mathematical physics and deep learning as described in claim 4, characterized in that: When the trend prediction model adopts a gray prediction model, the gray prediction model is a GM(1,1) model, and the traditional gray model is improved by introducing an adaptive optimization mechanism for background values. The background values ​​in the gray model are represented in the form of adjustable weight parameters, and the optimization objective is to minimize the prediction error. The particle swarm optimization algorithm is used to search and update the background value weight parameters to overcome the problem of insufficient fitting accuracy caused by the fixed value of the background value in the traditional gray model, thereby improving the fitting accuracy and extrapolation stability of the trend prediction results when describing the long-term slow deformation process of the dam body.

6. The dam deformation prediction and interpretability analysis method integrating mathematical physics and deep learning as described in claim 1, characterized in that: The periodic prediction model constructed for the periodic term is either an autoregressive integral moving average model (ARIMA) or a seasonal autoregressive integral moving average model (SARIMA). Before constructing the model, the stationarity of the periodic term sequence is tested, and the difference order is determined based on the test results. After completing the difference processing, the candidate range of the model order is determined by autocorrelation function and partial autocorrelation function analysis, and the model parameters are selected and optimized in combination with information criteria, so that the periodic prediction model can effectively characterize the periodic response characteristics of dam deformation caused by reservoir water level scheduling or seasonal temperature changes.

7. The dam deformation prediction and interpretability analysis method integrating mathematical and deep learning as described in claim 1, characterized in that: The deep learning prediction model constructed for the residual term is a temporal convolutional neural network (TCN). The TCN adopts a causal convolutional structure to ensure that the prediction process relies only on historical information, and expands the temporal receptive field without significantly increasing the network depth through an expanded convolutional mechanism. At the same time, a residual connection mechanism is introduced into the network structure to enhance the stability of model training, enabling the deep learning prediction model to effectively capture the high-frequency nonlinear perturbations and sudden deformation response features contained in the residual term.

8. The dam deformation prediction and interpretability analysis method integrating mathematical physics and deep learning as described in claim 7, characterized in that: The physical mechanism constraints are implemented by constructing a joint loss function during the training phase of the deep learning prediction model. The joint loss function includes a fitting error term based on measured data and a physical constraint loss term constructed based on engineering mechanics laws. The physical constraint loss term is used to restrict the prediction results to satisfy the boundedness of deformation, the consistency of deformation response direction, or the stress-strain relationship, thereby suppressing the risk of the prediction results deviating from the basic engineering physics laws while maintaining the prediction accuracy, and improving the engineering interpretability and credibility of the prediction results.

9. The dam deformation prediction and interpretability analysis method integrating mathematical physics and deep learning as described in claim 1, characterized in that: When integrating the prediction results of the trend term, the period term, and the residual term, the weight solution problem corresponding to each prediction model is constructed as a multi-objective optimization problem. The multi-objective optimization problem considers at least the prediction error index and the prediction stability index simultaneously. The multi-objective optimization problem is transformed into a single-objective optimization problem through the weighting method or equivalent transformation. The optimal weight vector is solved under the condition that the weights satisfy the non-negativity and normalization constraints, so as to achieve reasonable integration of the prediction results of different prediction models.

10. The dam deformation prediction and interpretability analysis method integrating mathematical and deep learning as described in claim 1, characterized in that: The dynamic weight update mechanism continuously monitors changes in prediction error, operating conditions, and model credibility based on a rolling time window. When any prediction model shows a significant change in error level, stability index, or consistency of physical constraints within a preset time window, a weight reallocation process is triggered. During the weight update process, the weight configuration of the previous stage is used as the initial constraint, and the weight vector is resolved within a limited adjustment range. This enables the integrated prediction model to adaptively adjust the contribution of each prediction model as the dam's operating conditions change, thereby maintaining the stability of prediction performance and engineering applicability.

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