Water body algae three-dimensional spatio-temporal distribution prediction method based on physical information neural network

Abstract

This invention discloses a method for predicting the three-dimensional spatiotemporal distribution of algae in aquatic bodies based on a physical information neural network (PINN), belonging to the field of algal bloom prediction and control technology. Addressing the problems of poor generalization of purely data-driven models, difficulty in fitting purely mechanistic models, and insufficient vertical fitting ability in existing algal distribution prediction methods, this invention integrates the advantages of physical mechanisms and data-driven algorithms. It embeds the advection-diffusion-response (ADR) equation of algal growth into the loss function of a physical information neural network (PINN), constructing a hybrid prediction model that combines data fitting accuracy with physical constraints. This achieves accurate prediction of the three-dimensional spatiotemporal distribution of algae and inversely obtains physically meaningful ecological parameters. This invention improves the accuracy, generalization, and physical interpretability of algal distribution prediction and can be effectively applied to the monitoring, early warning, and aquatic ecological environment management of harmful algal blooms.

Description

Technical Field

[0001] This invention relates to the field of algal bloom control technology, specifically a method for predicting the three-dimensional spatiotemporal distribution of algae in water bodies based on physical information neural networks, applicable to algal bloom prediction, early warning, and risk control in rivers, lakes, reservoirs, and other water bodies. Background Technology

[0002] Rivers, lakes, and reservoirs provide water resources for humankind. Rapidly and accurately predicting the three-dimensional spatiotemporal distribution of algal biomass in aquatic environments has become a key technological requirement for improving the aquatic ecological environment, promoting ecological civilization, and achieving high-quality development of water conservancy.

[0003] Currently, methods for predicting algal biomass are mainly divided into two categories: one is numerical mechanism models based on hydrodynamic changes, such as EFDC and Delft3D. Although their physical meaning is clear, they have shortcomings such as large computational resource consumption and extremely difficult parameter calibration; the other is data-driven artificial intelligence models, such as LSTM and CNN. Although they are fast in prediction, they often produce results that violate the law of conservation of mass, physical laws, and biological mechanisms due to the lack of actual mechanism constraints. Moreover, since the prediction results come from a "black box," it is difficult to explain the mechanism, which is not conducive to exploring the root cause of algal blooms and making targeted prevention and control in advance.

[0004] In addition, the existing practical applications and patents have the following limitations: (1) Based on time series frameworks such as LSTM, they focus on the change of biomass over time, making it difficult to restore the continuous distribution profile of algae in the vertical (Z-axis) of the water body and to capture the aggregation phenomenon of algae at a specific depth; (2) They only consider the biological proliferation process and lack partial differential equations (PDEs) to describe the physical-biological coupling processes such as active vertical migration of algae (air sac regulation, sedimentation) and spatial turbulent diffusion; (3) The prediction results are mostly single concentration values, lacking dynamic diagnosis of the degree of restriction of different environmental factors (light, temperature, nutrients, etc.) and unable to provide specific ecological control thresholds for water management. Summary of the Invention

[0005] This invention provides a method for predicting the three-dimensional spatiotemporal distribution of algae in water bodies based on a physical information neural network (PINN). It solves the problems of existing models ignoring differences in vertical depth, low prediction accuracy, poor physical consistency, and lack of causal explanation. It has application value for realizing rapid prediction of algal bloom processes in aquatic environments and for intervening in the mechanism of algal bloom outbreaks.

[0006] A method for predicting the three-dimensional spatiotemporal distribution of algae in aquatic bodies based on physical information neural networks includes the following steps:

[0007] S1. Construct a four-dimensional spatiotemporal feature tensor: acquire multi-source monitoring data of the target water area, perform data preprocessing on the multi-source monitoring data, and construct a four-dimensional spatiotemporal feature tensor with the three-dimensional spatial coordinates (x, y, z) of the water body and the time dimension t as indexes. The tensor includes the four-dimensional spatiotemporal coordinates (x, y, z, t), the values ​​of key environmental factors affecting algal growth, and the logarithmically transformed algal biomass index C.

[0008] S2. Establish a partial differential equation mechanism model for the spatiotemporal changes of algae: Based on the hydrodynamic characteristics of water bodies and the ecological laws of algal growth and migration, construct a partial differential equation mechanism model (PDE) to describe the advection transport, spatial diffusion, active vertical migration and coupled growth of algae in the four-dimensional spatiotemporal domain.

[0009] S3. Construct and train the Physical Information Neural Network (PINN) model with fused mechanistic constraints: Using the four-dimensional spatiotemporal coordinates (x,y,z,t) obtained in step S1 as input and the logarithmically transformed algal biomass index C as output, construct a physical information neural network. Calculate the partial derivative of the output with respect to the input using automatic differentiation technology. Combine the mechanistic residual of the partial differential equation mechanism model PDE obtained in step S2 with the measured data error to construct a composite loss function containing data fidelity terms and mechanistic constraint terms. Use an adaptive gradient balancing strategy to train the model, obtaining the trained PINN model M. Simultaneously, inversely obtain a series of algal ecological parameters that conform to reality and common sense.

[0010] S4. Diagnosis of growth limiting factors and analysis of ecological control threshold intervals: Using the PINN model M trained in step S3, the partial derivatives of algal biomass C with respect to the characteristics of each environmental factor Xi are calculated to generate the Jacobian matrix J. By analyzing the distribution characteristics of J at vertical depth z, the dominant growth limiting factor F at different depths is identified. A gradient perturbation sequence is set for the target environmental factors, and the first and second partial derivatives of algal biomass C with respect to the target environmental factors are calculated. By identifying the extreme points of the second partial derivative and the saturation points where the first partial derivative approaches zero, the ecological control threshold interval TS for preventing algal blooms is determined.

[0011] S5. Prediction and Model Evaluation of Three-Dimensional Spatiotemporal Distribution of Algae in Aquatic Bodies: The standardized aquatic environmental data of the area to be predicted are input into the PINN model M trained in step S3, and the three-dimensional spatiotemporal distribution prediction result R of algal biomass is output. At the same time, the algal ecological parameters retrieved during the model training process are extracted. The prediction results are ecologically interpreted and verified by combining the dominant growth limiting factor F and the ecological control threshold interval TS obtained in step S4, and the physical consistency of the model is evaluated and verified.

[0012] Furthermore, in step S1, the multi-source monitoring data includes in-situ observation data, satellite remote sensing data, and water environment reanalysis data; the in-situ observation data includes algae concentration, water temperature, nutrient concentration, and flow velocity data at different depths obtained by a profiler; the satellite remote sensing data includes sea surface temperature, satellite-retrieved algae concentration, and photosynthetically active radiation data; the water environment reanalysis data includes water mixing layer depth, three-dimensional flow field, three-dimensional nutrient concentration field, and vertical distribution data of water irradiance.

[0013] Furthermore, in step S1, the data preprocessing includes: cleaning records containing null values, outliers exceeding 3σ, duplicates, or inconsistencies; filling missing values ​​using the optimal interpolation method and the empirical orthogonal function method for data interpolation; interpolating satellite remote sensing data and water environment reanalysis data to the spatiotemporal nodes of in-situ observations to achieve spatiotemporal registration; performing standard scaling and standardization on all data; and performing a logarithmic transformation on the algae concentration C=ln(1+Cg), where Cg is the measured algae concentration.

[0014] Furthermore, in step S1, the method for constructing the four-dimensional spatiotemporal feature tensor is to expand the two-dimensional surface data into three-dimensional layered data based on the vertical stratification characteristics of algae in the water body and the vertical monitoring data of some representative points, divide the data into several depth layers, and obtain a dataset containing four-dimensional information of longitude, latitude, depth, and time.

[0015] Furthermore, the partial differential equation mentioned in step S2 is:

[0016] ;

[0017] Where C represents algal biomass, u, v, and w are the longitudinal, lateral, and vertical velocities of water flow, respectively, and w s denoted as net vertical migration velocity of algae, D represents diffusion coefficient, µ represents physiological growth rate, I represents light intensity, N and P represent the concentrations of representative nutrients such as nitrogen and phosphorus, and T represents water temperature.

[0018] Furthermore, in step S3, the expression for the composite loss function is:

[0019] ;

[0020] in This is the data fidelity term, which is the mean square error between the model's predicted algal biomass and the measured value. The mechanism constraint term is the mean square error of the difference between the left and right sides of the partial differential equation. These are weighting coefficients used to balance the contributions of data fitting and physical mechanism constraints.

[0021] Furthermore, in step S3, the adaptive gradient balancing strategy is as follows: the weight coefficient λ is designed as the ratio of the exponential moving average of the L2 norm of the gradient of the data fidelity term and the mechanistic constraint term with respect to the model parameters, expressed as:

[0022] ;

[0023] in L is the gradient of the loss with respect to the model parameters θ. It uses an exponential moving average, and this strategy ensures that the gradient magnitude of the data loss and the mechanistic loss during training are comparable.

[0024] The network weight parameters are initialized using Xavier uniform initialization with a bias of 0. The Optuna automatic hyperparameter optimization framework is used to search for the optimal number of hidden layers, number of neurons per layer, activation function, learning rate, and dropout rate. The dataset is divided into training and validation sets in an 8:2 ratio. The root mean square error of the validation set is used as the optimization objective, and MedianPruner is used to prune poorly performing experiments in advance.

[0025] Furthermore, in step S4, the identification logic of the dominant limiting factor is as follows: at a specific depth z, compare the partial derivatives of the characteristics of each environmental factor. The absolute value of, where As environmental factors affecting algal growth, the largest term is the dominant limiting factor at depth z.

[0026] Furthermore, the analysis of the ecological control threshold interval TS includes: ① second-order partial derivatives The local maxima are the ecological warning thresholds, representing the areas near which algae are most sensitive to factor Xi; ② First-order partial derivatives The saturation zone is defined as the region where algal biomass C approaches zero and is above 75% of its predicted full range.

[0027] Furthermore, the algal ecological parameters include the maximum algal growth rate µ. max Vertical net migration velocity w s Light half-saturation constant K I The nitrogen half-saturation constant KN and the phosphorus half-saturation constant KP, all algal ecological parameters converge to the physically reasonable range defined by the partial differential equation mechanism model PDE in steps S2 and S3, and have practical ecological interpretation significance.

[0028] Furthermore, in step S5, the evaluation and verification of the physical consistency of the model specifically includes: ① Physical consistency verification: analyzing the training evolution of the weight coefficient λ, verifying whether the output results conform to the mass conservation law of the advection-diffusion-reaction equation, and checking the analytical continuity of the vertical distribution profile in areas without measured data; ② Quantitative accuracy evaluation: calculating the RMSE and R between the predicted and measured values. 2 Indicators to ensure accuracy meets early warning requirements; ③ Spatiotemporal generalization verification: evaluate the model's predictive robustness in different time periods or adjacent unsampled water areas.

[0029] Compared with the prior art, the present invention has the following beneficial effects:

[0030] 1. Integrating physical mechanisms with data-driven approaches enhances model generalization and interpretability. This invention embeds the advection-diffusion-reaction partial differential equation of algal growth as a mechanistic constraint term into the loss function, overcoming the shortcomings of pure data-driven models, such as overfitting and poor generalization. It also solves the problems of difficult parameter calibration and high computational resource consumption in pure mechanistic models. Through physical constraints, the model output naturally satisfies mass conservation and biological mechanisms, and the prediction results have clear physical interpretability.

[0031] 2. Achieving accurate reconstruction of the vertical distribution of algae, breaking through the dimensional limitations of traditional models. This invention calculates and outputs the continuous distribution of algal biomass C at different depths (z-axis direction) in water by performing a logarithmic transformation on the algal biomass index and combining it with automatic differentiation technology. This allows the model to generate a realistic vertical distribution profile even in deep waters lacking the constraints of measured samples, through the constraints of surface boundary conditions and derivatives of physical equations. This solves the technical problem that existing models struggle to capture the phenomenon of algal aggregation at specific depths.

[0032] 3. Dynamic diagnosis of dominant limiting factors provides a basis for precise prevention and control decisions. This invention uses a trained PINN model to calculate the partial derivatives of algal biomass with respect to various environmental factors, generates a Jacobian matrix, and analyzes its distribution characteristics at vertical depth. It can accurately locate the dominant growth limiting factors (light limitation, nitrogen limitation, phosphorus limitation, or temperature limitation, etc.) under different water depth conditions, breaking through the limitations of traditional methods that can only perform global or surface analysis.

[0033] 4. Analyzing ecological control threshold ranges to achieve early warning of algal blooms. This invention calculates the first and second partial derivatives of algal biomass with respect to target factors by setting an environmental factor gradient perturbation sequence, and identifies sensitive inflection points of the response curve: the local maximum point of the second partial derivative is defined as the "ecological warning threshold", and the region where the first partial derivative approaches zero and biomass remains high is defined as the "saturation plateau point". This threshold analysis method based on "virtual experiments" avoids the drawback of traditional models requiring large-scale recalculation of the flow field, and provides quantifiable control targets for water management.

[0034] 5. Constructing an end-to-end prediction device to achieve rapid prediction of three-dimensional spatiotemporal distribution. This invention inputs standardized data of the area to be predicted into a trained PINN model, which is then converted to algal biomass physical concentration via inverse exponential transformation. This data is then integrated to obtain the three-dimensional spatiotemporal distribution characteristics of the study area in both horizontal space and vertical depth over time. Simultaneously, ecological parameters such as the maximum growth rate, net vertical migration velocity, and half-saturation constant retrieved during model training converge to physically reasonable ranges, possessing practical ecological interpretive significance and providing a complete technical solution for the management of aquatic ecological environments. Attached Figure Description

[0035] Figure 1 This is a flowchart of the method for predicting the three-dimensional spatiotemporal distribution of algae in water bodies based on physical information neural networks, as described in this invention. Detailed Implementation

[0036] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, 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, 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.

[0037] like Figure 1 As shown, this invention proposes a method for predicting the three-dimensional spatiotemporal distribution of algae in water bodies based on a physical information neural network, comprising:

[0038] S1. Construct a four-dimensional spatiotemporal feature tensor: Obtain multi-source monitoring data of the target water area, perform data preprocessing on the multi-source monitoring data, and construct a four-dimensional spatiotemporal feature tensor with the three-dimensional spatial coordinates (x, y, z) of the water body and the time dimension t as indexes. The tensor includes the four-dimensional spatiotemporal coordinates (x, y, z, t), the values ​​of key environmental factors affecting algal growth, and the logarithmically transformed algal biomass index C.

[0039] The multi-source monitoring data includes in-situ observation data, satellite remote sensing data, and water environment reanalysis data. The in-situ observation data includes algae concentration, water temperature, nutrient concentration, and flow velocity data at different depths obtained by a profiler. The satellite remote sensing data includes sea surface temperature, satellite-retrieved algae concentration, and photosynthetically active radiation data. The water environment reanalysis data includes water mixing layer depth, three-dimensional flow field, three-dimensional nutrient concentration field, and vertical distribution data of water irradiance.

[0040] The data preprocessing includes: cleaning records containing null values, outliers exceeding 3σ, duplicates, or inconsistencies; filling missing values ​​using optimal interpolation and empirical orthogonal function interpolation; interpolating satellite remote sensing data and water environment reanalysis data to in-situ observation spatiotemporal nodes to achieve spatiotemporal registration; standardizing all data by standard scaling; and performing a logarithmic transformation on algae concentration C=ln(1+Cg), where Cg is the measured algae concentration.

[0041] The method for constructing the four-dimensional spatiotemporal feature tensor is as follows: based on the vertical stratification characteristics of algae in the water body and the vertical monitoring data of some representative points, the two-dimensional surface data is expanded into three-dimensional stratified data, and several depth layers are divided to obtain a dataset containing four-dimensional information of longitude, latitude, depth and time.

[0042] S2. Establish a partial differential equation mechanism model for the spatiotemporal changes of algae: Based on the hydrodynamic characteristics of water bodies and the ecological laws of algal growth and migration, construct a partial differential equation mechanism model (PDE) to describe the advection transport, spatial diffusion, active vertical migration and coupled growth of algae with multiple environmental factors in the four-dimensional spatiotemporal domain.

[0043] The partial differential equation is:

[0044] ;

[0045] The left side of the equation represents the change in algal biomass plus spatial migration, while the right side represents the diffusion term plus the growth term. The main emphasis is on considering the periodic diurnal vertical migration of algae themselves in the vertical direction, and the inclusion of multiple measured environmental factors in the algal growth term.

[0046] Where C represents algal biomass, u, v, and w are the longitudinal, lateral, and vertical velocities of water flow, respectively, and w s denoted as net vertical migration velocity of algae, D represents diffusion coefficient, µ represents physiological growth rate, I represents light intensity, N and P represent the concentrations of representative nutrients such as nitrogen and phosphorus, and T represents water temperature.

[0047] Where w sLet be the net vertical migration velocity of algae, defined as a periodic adaptive function of time t, representing the diurnal vertical migration characteristics of algal cells due to pseudo-empty cell regulation:

[0048]

[0049] Where V amp This represents the amplitude of the algal migration rate, where t represents time in hours. This is a phase factor, related to sunrise / noon time; V offset This represents the basic sedimentation velocity of algal cells.

[0050] µ is controlled by multiple factors:

[0051] ;

[0052] Where µ max Indicates the maximum growth rate. The effect of temperature on microbial enzyme activity is represented by θ(L,z), which represents vertical light limitation, and η(N,P) which represents nutrient limitation.

[0053] The effect of water temperature T on microbial enzyme activity and reaction rate can be expressed by a modified Arrhenius formula:

[0054]

[0055] in Reference temperature T ref The reaction rate at time θ T (T-Tref) This is the temperature coefficient.

[0056] Vertical light limiting factor The value ranges between 0 and 1, where I(z) represents the light intensity at depth z, and K... I This represents the light half-saturation constant.

[0057] Illumination intensity follows the Beer-Lambert law, which decreases with depth z:

[0058]

[0059] Where I0 represents the incident light intensity, i.e., the illumination intensity at depth 0, and K e The overall extinction coefficient is:

[0060]

[0061] k w Extinction of pure water; k c For the self-shading of algae; k dSS represents the extinction of suspended particles, and SS represents the concentration of suspended solids (such as silt).

[0062] S3. Construct and train a Physical Information Neural Network (PINN) model incorporating mechanistic constraints: Using the four-dimensional spatiotemporal coordinates (x, y, z, t) obtained in step S1 as input and the logarithmically transformed algal biomass index C as output, construct a physical information neural network. Utilize automatic differentiation techniques to calculate the partial derivative of the output with respect to the input. Combine the mechanistic residual of the partial differential equation (PDE) mechanistic model obtained in step S2 with the measured data error to construct a composite loss function containing data fidelity terms and mechanistic constraint terms. Employ an adaptive gradient balancing strategy for model training to obtain the trained PINN model M. Simultaneously, inversely obtain a series of algal ecological parameters that conform to reality and common sense. These algal ecological parameters include the maximum algal growth rate µ. max Vertical net migration velocity w s Light half-saturation constant K I The nitrogen half-saturation constant KN and the phosphorus half-saturation constant KP, all algal ecological parameters converge to the physically reasonable range defined by the partial differential equation mechanism model PDE in steps S2 and S3, and have practical ecological interpretation significance.

[0063] In the process of constructing the loss function, a composite loss function is designed as follows: L d The data fidelity term represents the mean square error between the model's predicted algal biomass and the measured value, while L... m The mechanistic constraint term is the mean square error of the difference between the left and right sides of the partial differential equation. These are weighting coefficients used to balance the contributions of data fitting and physical mechanism constraints.

[0064] Among them, the mechanism constraint term L m The calculation formula is:

[0065]

[0066] In the loss function L m If no measured values ​​are available during the calculation, the mechanism parameter values ​​can be set with reference to Table 1.

[0067] Table 1

[0068]

[0069] S4. Diagnosis of growth limiting factors and analysis of ecological control threshold intervals: Using the PINN model M trained in step S3, the partial derivatives of algal biomass C with respect to the characteristics of each environmental factor Xi are calculated to generate the Jacobian matrix J. By analyzing the distribution characteristics of J at vertical depth z, the dominant growth limiting factor F at different depths is identified. A gradient perturbation sequence is set for the target environmental factors, and the first and second partial derivatives of algal biomass C with respect to the target environmental factors are calculated. By identifying the extreme points of the second partial derivative and the saturation points where the first partial derivative approaches zero, the ecological control threshold interval TS for preventing algal blooms is determined.

[0070] By fully utilizing the differentiability of the prediction chain in PINN, the Jacobian matrix J is obtained with extremely low computational cost. Different water depths z are then substituted to analyze the variation of J at different depths. The absolute values ​​of the partial derivatives of each factor are identified and compared to determine the dominant limiting factor for algal biomass growth at the current depth. For example, at a depth z = 5 (m), the following calculations are performed: This means that the region is under phosphorus limitation rather than nitrogen limitation.

[0071] Perturbation tests were conducted on phosphorus concentration P, and response curves were plotted. Analysis revealed that when P was 0.075 mg / L, its second derivative... Since a local peak value was obtained, 0.075 mg / L was set as the warning threshold for the reservoir in the current season, and the ecological control threshold range for phosphorus concentration P is 0~0.075 mg / L.

[0072] S5. Three-dimensional spatiotemporal distribution prediction and model evaluation of algae in aquatic bodies: Standardized aquatic environmental data of the area to be predicted are input into the PINN model M trained in step S3, outputting the three-dimensional spatiotemporal distribution prediction result R of algal biomass. Simultaneously, algal ecological parameters retrieved during model training are extracted. The prediction results are ecologically interpreted and validated using the dominant growth limiting factor F and ecological control threshold interval TS obtained in step S4. The physical consistency of the model is also evaluated and validated. Specifically, standardized data of the area to be predicted are input into the trained PINN model M to obtain the prediction index C, which is then converted to algal biomass physical concentration through inverse exponential transformation. This is integrated to obtain the three-dimensional spatiotemporal distribution results of the study area in horizontal space and vertical depth. Simultaneously, physically meaningful algal ecological parameters converged during model training are extracted, including the maximum algal growth rate µ. max Vertical net migration velocity w s Light half-saturation constant K I Nitrogen half-saturation constant KN, phosphorus half-saturation constant KP, etc.

[0073] The model was evaluated and validated from the following dimensions: ① Physical consistency validation: Analyze the training evolution of the weight coefficient λ, verify whether the output results conform to the mass conservation law of the advection-diffusion-reaction equation, and check the analytical continuity of the vertical distribution profile in areas without measured data; ② Quantitative accuracy evaluation: Calculate the RMSE and R-value between the predicted and measured values. 2 Indicators such as [list of indicators] are used to ensure that the accuracy meets the early warning requirements; ③ Spatiotemporal generalization verification: The predictive robustness of the model in different time periods or adjacent unsampled water areas is evaluated. For problems found during verification, feedback is used to optimize the model structure or parameters to ensure that the output results not only conform to the algal growth and migration mechanism but also possess practical application accuracy.

[0074] This invention integrates the growth and vertical migration mechanisms of algae into a physical information neural network, which not only enhances the realism and reliability of the data model, but also constructs a loss function that considers the mechanism. By solving the first and second-order partial derivatives, it analyzes the dominant limiting factors of the reservoir in the current season and their corresponding early warning thresholds, which helps to assist in the prevention and control of lake eutrophication through the application of artificial intelligence models.

[0075] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A method for predicting the three-dimensional spatiotemporal distribution of algae in aquatic bodies based on a physical information neural network, characterized in that, Includes the following steps: S1. Construct a four-dimensional spatiotemporal feature tensor: acquire multi-source monitoring data of the target water area, perform data preprocessing on the multi-source monitoring data, and construct a four-dimensional spatiotemporal feature tensor with the three-dimensional spatial coordinates (x, y, z) of the water body and the time dimension t as indexes. The tensor includes the four-dimensional spatiotemporal coordinates (x, y, z, t), the values ​​of key environmental factors affecting algal growth, and the logarithmically transformed algal biomass index C. S2. Establish a partial differential equation mechanism model for the spatiotemporal changes of algae: Based on the hydrodynamic characteristics of water bodies and the ecological laws of algal growth and migration, construct a partial differential equation mechanism model (PDE) to describe the advection transport, spatial diffusion, active vertical migration and coupled growth of algae in the four-dimensional spatiotemporal domain. S3. Construct and train the Physical Information Neural Network (PINN) model with fused mechanistic constraints: Using the four-dimensional spatiotemporal coordinates (x,y,z,t) obtained in step S1 as input and the logarithmically transformed algal biomass index C as output, construct a physical information neural network. Calculate the partial derivative of the output with respect to the input using automatic differentiation technology. Combine the mechanistic residual of the partial differential equation mechanism model PDE obtained in step S2 with the measured data error to construct a composite loss function containing data fidelity terms and mechanistic constraint terms. Use an adaptive gradient balancing strategy to train the model, obtaining the trained PINN model M. Simultaneously, inversely obtain a series of algal ecological parameters that conform to reality and common sense. S4. Diagnosis of growth limiting factors and analysis of ecological control threshold intervals: Using the PINN model M trained in step S3, the partial derivatives of algal biomass C with respect to the characteristics of each environmental factor Xi are calculated to generate the Jacobian matrix J. By analyzing the distribution characteristics of J at vertical depth z, the dominant growth limiting factor F at different depths is identified. A gradient perturbation sequence is set for the target environmental factors, and the first and second partial derivatives of algal biomass C with respect to the target environmental factors are calculated. By identifying the extreme points of the second partial derivative and the saturation points where the first partial derivative approaches zero, the ecological control threshold interval TS for preventing algal blooms is determined. S5. Prediction and Model Evaluation of Three-Dimensional Spatiotemporal Distribution of Algae in Aquatic Bodies: The standardized aquatic environmental data of the area to be predicted are input into the PINN model M trained in step S3, and the three-dimensional spatiotemporal distribution prediction result R of algal biomass is output. At the same time, the algal ecological parameters retrieved during the model training process are extracted. The prediction results are ecologically interpreted and verified by combining the dominant growth limiting factor F and the ecological control threshold interval TS obtained in step S4, and the physical consistency of the model is evaluated and verified.

2. The method according to claim 1, characterized in that, The partial differential equation mentioned in step S2 is: ； Where C represents algal biomass, u, v, and w are the longitudinal, lateral, and vertical velocities of water flow, respectively, and w s denoted as net vertical migration velocity of algae, D represents the diffusion coefficient, µ represents the physiological growth rate, I represents the light intensity, N and P represent the concentrations of representative nutrients such as nitrogen and phosphorus, respectively, and T represents the water temperature.

3. The method according to claim 1, characterized in that, In step S3, the expression for the composite loss function is: ； in This is the data fidelity term, which is the mean square error between the model's predicted algal biomass and the measured value. The mechanism constraint term is the mean square error of the difference between the left and right sides of the partial differential equation. These are weighting coefficients used to balance the contributions of data fitting and physical mechanism constraints.

4. The method according to claim 2, characterized in that, In step S3, the adaptive gradient balancing strategy is as follows: the weight coefficient λ is designed as the ratio of the exponential moving average of the L2 norm of the gradient of the data fidelity term and the mechanistic constraint term with respect to the model parameters, expressed as: ； in L is the gradient of the loss with respect to the model parameters θ. It uses an exponential moving average and an adaptive gradient balancing strategy to ensure that the gradient magnitudes of data loss and mechanistic loss are comparable during training. The network weight parameters were initialized using Xavier uniform initialization with a bias of 0. The Optuna automatic hyperparameter optimization framework was used to search for the optimal number of hidden layers, neurons per layer, activation function, learning rate, and dropout rate. The dataset was divided into training and validation sets in an 8:2 ratio. The root mean square error of the validation set was used as the optimization objective, and MedianPruner was used to prune poorly performing experiments in advance.

5. The method according to claim 1, characterized in that, In step S4, the identification logic of the dominant growth limiting factor is as follows: at a specific depth z, compare the partial derivatives of the characteristics of each environmental factor. The absolute value of, where The largest term represents the dominant growth limiting factor at depth z, representing the environmental factors that influence algal growth.

6. The method according to claim 5, characterized in that, The analysis of the ecological control threshold interval TS includes: ① second-order partial derivatives The local maxima are the ecological warning thresholds, representing the areas near which algae are most sensitive to factor Xi; ② First-order partial derivatives The saturation zone is defined as the region where algal biomass C approaches zero and is above 75% of its predicted full range.

7. The method according to claim 1, characterized in that, The algal ecological parameters include the maximum algal growth rate µ. max Vertical net migration velocity w s Light half-saturation constant K I The nitrogen half-saturation constant KN and the phosphorus half-saturation constant KP, all algal ecological parameters converge to the physically reasonable range defined by the partial differential equation mechanism model PDE in steps S2 and S3, and have practical ecological interpretation significance.

8. The method according to claim 1, characterized in that, Step S5, specifically the evaluation and verification of the physical consistency of the model, includes: ① Physical consistency verification: analyzing the training evolution of the weight coefficient λ, verifying whether the output results conform to the mass conservation law of the advection-diffusion-reaction equation, and checking the analytical continuity of the vertical distribution profile in areas without measured data; ② Quantitative accuracy evaluation: calculating the RMSE and R-value between the predicted and measured values. 2 Indicators to ensure accuracy meets early warning requirements; ③ Spatiotemporal generalization verification: evaluate the model's predictive robustness in different time periods or adjacent unsampled water areas.

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