An algal growth simulation method based on causal time lag optimization and PINN

Abstract

The application discloses an algal growth simulation method based on causal time lag optimization and PINN, belongs to the technical field of water eutrophication research, and comprises the steps of data acquisition and preprocessing, time lag optimization based on causal discovery, feature construction and data set division, construction of an algal growth mechanism module, construction of a parameter generation network, construction of a physical information neural network, stage-by-stage training, and model evaluation. The method significantly improves the model accuracy, and the net specific growth rate presents stable Monod saturation characteristics with the nutrient salt concentration, and the parameter change conforms to the ecology priori; sensitivity analysis can clearly indicate the dominant role of water temperature and wind speed on the maximum specific growth rate, thereby providing quantitative basis for water bloom early warning and nutrient salt reduction decision.

Description

Technical Field

[0001] This invention belongs to the field of water eutrophication research technology, specifically involving an algal growth simulation method based on causal time delay optimization and PINN. Background Technology

[0002] Eutrophication leading to abnormal algal proliferation is a prominent problem in current water environment management. Accurate simulation of chlorophyll concentration is of great value for algal bloom early warning and aquatic ecosystem regulation. Existing algal simulation technologies are mainly divided into two categories: one is purely data-driven statistical models or neural network models, such as multiple linear regression and long short-term memory networks. These methods rely entirely on historical data to fit mapping relationships, resulting in poor interpretability and a tendency to produce predictions that violate physical laws under conditions of sparse data or distributional deviations. The other is mechanism-based kinetic models, which use Monod equations or Droop equations to describe the limiting effect of nutrients on algal growth. Their parameters are usually set as fixed constants, which cannot reflect the dynamic response of algal physiological parameters (growth rate, respiration rate, etc.) when environmental factors such as wind speed, water temperature, and pH change continuously, leading to a significant decrease in simulation accuracy when environmental fluctuations are large.

[0003] Furthermore, existing methods often rely on empirical settings or trial-and-error calculations to determine the time delays of driving factors, lacking causal discrimination criteria. This can easily lead to the omission of key time delay information or the introduction of irrelevant lag terms, affecting the rationality of feature construction. While existing physical information neural networks can incorporate differential equation constraints into the loss function, in algal growth simulations, they often directly use growth equations with fixed parameters as constraints, failing to adaptively correlate mechanistic parameters with environmental conditions, thus limiting the expressive power of physical constraints.

[0004] Therefore, there is an urgent need for an algal growth simulation technology that can integrate causal inference, adaptive mechanism parameters, and physical constraints, while taking into account both prediction accuracy and physical interpretability. Summary of the Invention

[0005] To address the technical problems in existing algal growth simulation methods, such as the neglect of causal time-delay effects of environmental factors, the inability of physiological parameters to dynamically respond to continuous environmental changes, and the lack of quantitative analysis of physiological parameter sensitivity under multi-factor coupling effects, the present invention aims to provide an algal growth simulation method based on causal time-delay optimization and PINN (Physical Information Neural Network).

[0006] The objective of this invention is achieved by including the following steps:

[0007] Step 1: Data Acquisition and Preprocessing: Acquire daily monitoring data of nutrient and organic matter indicators, environmental factor indicators, and chlorophyll concentration. Nutrient and organic matter indicators include total nitrogen (TN), total phosphorus (TP), and permanganate index (COD). MnEnvironmental factors include water temperature, pH, dissolved oxygen, transparency, solar shortwave radiation, wind speed, and rainfall; missing values ​​were filled forward and backward, and all variables were standardized using Z-score.

[0008] Step 2: Time-Lap Optimization Based on Causal Discovery: Daily monitoring data of standardized nutrient and organic matter indicators, environmental factor indicators, and chlorophyll concentration are constructed into a time series data frame. The PCMCI (Partial Correlation Matrix Inference) causal discovery method is used, with chlorophyll concentration as the causal factor. () is the target variable, and the maximum time delay is set. (e.g., 60 days) and the significance level of the conditional independence test. (e.g., 0.05), run PCMCI to obtain the cause-effect graph matrix; for driving variables If there is a time lag Existing For direct connections, the minimum time delay among significant connections is selected as the optimal time delay for that variable. If no direct connection is detected, the following formula is used to calculate. Different time delays Mutual information below:

[0009] ;

[0010] In the formula, Represents mutual information value; This represents the mutual information calculation function, specifically the `mutual_info_regression` function in the Python library. For driving variables (Nutrient and organic matter indicators, environmental factor indicators) in Time series data at a given moment; For chlorophyll in Time series data at a given moment;

[0011] Select the time delay that maximizes mutual information as This yields the optimal time-delay set for each nutrient and organic matter variable, as well as the environmental variable. ;

[0012] Step 3: Feature Construction and Dataset Partitioning: Based on the optimal time lag, each variable is shifted to its corresponding time lag and aligned with the original time to construct the design matrix. ; The first column is the number of days from the start date. The subsequent columns are the shifted TN, TP, and COD, respectively. Mn And 7 environmental factors; target variable The unshifted chlorophyll concentration; after removing samples with missing values, the previous values ​​are arranged in chronological order. % of the samples were divided into the training set, and then % represents the test set; multiply the chlorophyll target value by the scaling factor using the following formula. (For example, S=100) Perform scaling transformation:

[0013] ;

[0014] In the formula, This is the scaled-down chlorophyll concentration, in mg / L; The chlorophyll concentration before scaling is in mg / L; This is the scaling factor;

[0015] The environmental factors are standardized to ensure that the input features have similar scales.

[0016] Step 4: Constructing the algal growth mechanism module: Define the vectors of nutrients and organic matter. The corresponding half-saturation constants are respectively ,Right now The maximum specific growth rate is respiratory rate is The mortality rate is First, calculate the restriction functions for each nutrient and organic matter. :

[0017] ;

[0018] In the formula, Use small positive numbers (such as 10^{-8}) to prevent division by zero; take the minimum of the three as the combined nutrient limiting factor:

[0019] ;

[0020] In the formula, This is a function of the combined nutrient limiting factors;

[0021] The growth term is The loss term is Net growth rate is expressed as:

[0022] ;

[0023] In the formula, Net specific growth rate, d -1 ;

[0024] Step 5: Construct the parameter generation network: Design a fully connected neural network The normalized seven-dimensional environmental factor vector As input, output the original values ​​of the six-dimensional parameters:

[0025] ;

[0026] In the formula, A vector composed of seven environmental factor indicators; Six-dimensional parameters Corresponding Output value;

[0027] The network structure consists of three fully connected layers (e.g., dimensions 128, 128, 64 respectively), each followed by batch normalization and ReLU activation, with the final layer outputting 6 dimensions; a boundary mapping is applied to the original output. Ensure it is greater than For bounded parameters , , , The Sigmoid mapping is applied to a preset range; simultaneously, a biological rationality constraint is imposed: if Then let Finally, a reasonable dynamic parameter vector is obtained. ;

[0028] Step Six: Construct a Physical Information Neural Network: The Physical Information Neural Network uses input vectors... For input, where, For the time series, day, Seven environmental factors are first identified through a multi-layer fully connected network. Simulated scaled chlorophyll concentration:

[0029] ;

[0030] In the formula, To simulate scaled chlorophyll concentration, mg / L;

[0031] At the same time, the nutrient salt part and environmental aspects The input parameter generation network obtains the dynamic parameters. Then, the data is passed to the algae growth mechanism module to calculate the net specific growth rate under the current conditions. ;

[0032] Introducing physical residual loss and using automatic differentiation to obtain Output time partial derivative:

[0033] ;

[0034] In the formula, For sample size;

[0035] This loss forces the temporal variation of chlorophyll to satisfy... The exponential growth / decline pattern;

[0036] Simultaneously set constraint losses:

[0037] ;

[0038] In the formula, For sample size;

[0039] punish A combination of parameters that violates the laws of physics, such as those less than the sum of respiration and death, and those with excessively small half-saturation constants.

[0040] The data fitting loss is the mean squared error:

[0041] ;

[0042] In the formula, For sample size; The measured chlorophyll concentration is in mg / L.

[0043] The total loss function is:

[0044] ;

[0045] In the formula, The weights for the physical residual loss; The weights are used to constrain the loss;

[0046] Step 7: Phased Training: The training is divided into a pre-training phase and a combined training phase.

[0047] Specifically, the pre-training phase involves freezing the chlorophyll simulation branch. The parameters are optimized only in the parameter generation network. Loss of use This stage enables the parameter network to learn to output dynamic parameters that satisfy the growth kinetic equation and parameter rationality constraints without the guidance of observation data; it terminates early when the total loss does not decrease for several consecutive rounds on the validation set, and saves the optimal parameter network weights.

[0048] The joint training phase specifically involves: unfreezing all parameters to obtain the complete loss. Simultaneously optimize and The optimizer is Adam, with learning rate decay (ReduceLROnPlateau); training stops when the validation loss (MSE) does not improve for several consecutive rounds, and the model with the optimal validation loss is loaded as the final model.

[0049] Step 8: Model Evaluation: Simulate the final model on the test set and inversely scale it to obtain the original scale chlorophyll values.

[0050] ;

[0051] In the formula, The simulated chlorophyll concentration before scaling is in mg / L;

[0052] Calculate the coefficient of determination R 2 Model evaluation metrics such as mean absolute percentage error (MAPE) are used; output nutrient growth curves, parameter sensitivity maps, model prediction performance maps, and model iteration loss curves are plotted.

[0053] Preferably, step eight further includes a physical interpretability analysis, specifically:

[0054] Nutrient growth curve: fixed environmental factors To obtain the median of each dimension of the training set, each nutrient was scanned one by one within its actual measurement range to calculate the net specific growth rate and the changes in each kinetic parameter, and to plot the growth rate-nutrient concentration curve.

[0055] Parameter sensitivity map: For each environmental factor, samples are taken at equal intervals within its measured range. The median of the remaining environmental variables is fixed. The change range of each dynamic parameter (maximum value minus minimum value) is calculated. After normalization by range, a grouped bar chart is plotted to reflect the regulatory intensity of environmental factors on each parameter.

[0056] The beneficial effects of this invention: The improved multiple linear regression on the test set has R0. 2 The value was -0.121, and the MAPE was 46.37%. While pure black-box neural networks can improve the fit, they cannot guarantee the physical consistency of the growth rate (e.g., the appearance of a negative half-saturation constant). After using the method of this invention, the test set R... 2 The net specific growth rate (NPG) significantly increased to 0.438, with a MAPE of 48.53%. Furthermore, the NPG exhibited a stable Monod saturation characteristic with increasing nutrient concentration, and the parameter changes were consistent with ecological priors. Traditional deep learning models often only predict chlorophyll concentration, failing to explain "why it increases" and "how it is regulated." This invention's sensitivity analysis clearly indicates the dominant role of water temperature and wind speed in influencing the maximum specific growth rate. When simulations show that the NPG is consistently greater than 0, an early warning of algal bloom risk can be issued within the lag period, providing quantitative evidence for algal bloom early warning and nutrient reduction decisions. Attached Figure Description

[0057] Figure 1 This is a schematic flowchart of the method of the present invention;

[0058] Figure 2 The graph shows the performance of the multiple linear regression simulation in Example 1.

[0059] Figure 3 The performance graph of the model based on causal time delay optimization and PINN in Example 1 is shown.

[0060] Figure 4 This is a schematic diagram illustrating the sensitivity of each environmental factor to the parameters of Model 6 in Example 1;

[0061] Figure 5 This is a graph showing the effect of TP on net growth rate in Example 1;

[0062] Figure 6 The training and validation loss curves are shown in Example 1. Detailed Implementation

[0063] The present invention will be further described below with reference to the embodiments and accompanying drawings, but this does not limit the present invention in any way. Any changes or substitutions made based on the teachings of the present invention shall fall within the protection scope of the present invention.

[0064] Example 1

[0065] This example uses daily monitoring data from a eutrophic lake / reservoir from 2021 to 2023; see attached. Figure 1 The algae growth simulation method shown in this embodiment, based on causal time-delay optimization and PINN, includes the following steps:

[0066] Step 1: Data Acquisition and Preprocessing: Acquire daily monitoring data of nutrient and organic matter indicators, environmental factor indicators, and chlorophyll concentration. Nutrient and organic matter indicators include total nitrogen (TN), total phosphorus (TP), and permanganate index (COD). Mn Environmental factors include water temperature, pH, dissolved oxygen, transparency, solar shortwave radiation, wind speed, and rainfall; missing values ​​were filled forward and backward, and all variables were standardized using Z-score.

[0067] Step 2: Time-Lap Optimization Based on Causal Discovery: Daily monitoring data of standardized nutrient and organic matter indicators, environmental factor indicators, and chlorophyll concentration are constructed into a time series data frame. The PCMCI causal discovery method is used, with chlorophyll as the target variable, and a maximum time lag is set. (This embodiment) The significance level of the conditional independence test and the conditional independence test. (In this example, the value is 0.05), running PCMCI yields the cause-effect graph matrix; for the driving variables... If there is a time lag Existing For direct connections, the minimum time delay among significant connections is selected as the optimal time delay for that variable. If no direct connection is detected, the following formula is used to calculate. Different time delays Mutual information below:

[0068]

[0069] In the formula, Represents mutual information value; This represents the mutual information calculation function, specifically the `mutual_info_regression` function in the Python library. For driving variables (Nutrient and organic matter indicators, environmental factor indicators) in Time series data at a given moment; For chlorophyll in Time series data at a given moment;

[0070] Select the time delay that maximizes mutual information as This yields the optimal time-delay set for each nutrient and organic matter variable, as well as the environmental variable. ;

[0071] In actual calculations, the optimal time lag for nutrients TN is 2 days, and for TP it is 2 days; the time lag for environmental factors such as water temperature is 2 days; for wind speed and rainfall, which did not show significant connections, the time lags were determined to be 6 days and 19 days, respectively, by maximizing mutual information.

[0072] Step 3: Feature Construction and Dataset Partitioning: Based on the optimal time lag, each variable is shifted to its corresponding time lag and aligned with the original time to construct the design matrix. ; The first column is the number of days from the start date. The subsequent columns are the shifted TN, TP, and COD, respectively. Mn And 7 environmental factors; target variable The unshifted chlorophyll concentration; after removing samples with missing values, the previous values ​​are arranged in chronological order. % of the samples were divided into the training set, and then % represents the test set; multiply the chlorophyll target value by the scaling factor using the following formula. (In this embodiment, S=100) Scale transformation is performed:

[0073] ;

[0074] In the formula, This is the scaled-down chlorophyll concentration, in mg / L; The chlorophyll concentration before scaling is in mg / L; This is the scaling factor;

[0075] The environmental factors are standardized to ensure that the input features have similar scales.

[0076] Step 4: Constructing the algal growth mechanism module: Define the vectors of nutrients and organic matter. The corresponding half-saturation constants are respectively ,Right now The maximum specific growth rate is respiratory rate is The mortality rate is First, calculate the restriction functions for each nutrient and organic matter. :

[0077] ;

[0078] In the formula, The factor is a small positive number (10^{-8} in this example) to prevent division by zero; the minimum value of the three factors is taken as the combined nutrient limiting factor.

[0079] ;

[0080] In the formula, This is a function of the combined nutrient limiting factors;

[0081] The growth term is The loss term is Net growth rate is expressed as:

[0082] ;

[0083] In the formula, Net specific growth rate, d -1 ;

[0084] Step 5: Construct the parameter generation network: Design a fully connected neural network The normalized seven-dimensional environmental factor vector As input, output the original values ​​of the six-dimensional parameters:

[0085] ;

[0086] In the formula, A vector composed of seven environmental factor indicators; Six-dimensional parameters Corresponding Output value;

[0087] The network structure consists of three fully connected layers (in this embodiment, the dimensions are 128, 128, and 64 respectively), each followed by batch normalization and ReLU activation, with the last layer outputting 6 dimensions; a boundary mapping is applied to the original output: Ensure it is greater than For bounded parameters , , , The Sigmoid mapping is applied to a preset range; simultaneously, a biological rationality constraint is imposed: if Then let Finally, a reasonable dynamic parameter vector is obtained. ;

[0088] Step Six: Construct a Physical Information Neural Network: The Physical Information Neural Network uses input vectors... For input, where, For the time series, day, Seven environmental factors are first identified through a multi-layer fully connected network. Simulated scaled chlorophyll concentration:

[0089] ;

[0090] In the formula, To simulate scaled chlorophyll concentration, mg / L;

[0091] At the same time, the nutrient salt part and environmental aspects The input parameter generation network obtains the dynamic parameters. Then, the data is passed to the algae growth mechanism module to calculate the net specific growth rate under the current conditions. ;

[0092] Introducing physical residual loss and using automatic differentiation to obtain Output time partial derivative:

[0093] ;

[0094] In the formula, For sample size;

[0095] This loss forces the temporal variation of chlorophyll to satisfy... The exponential growth / decline pattern;

[0096] Simultaneously set constraint losses:

[0097] ;

[0098] In the formula, For sample size;

[0099] punish A combination of parameters that violates the laws of physics, such as those less than the sum of respiration and death, and those with excessively small half-saturation constants.

[0100] The data fitting loss is the mean squared error:

[0101] ;

[0102] In the formula, For sample size; The measured chlorophyll concentration is in mg / L.

[0103] The total loss function is:

[0104] ;

[0105] In the formula, The weights for the physical residual loss; The weights are used to constrain the loss;

[0106] Step Seven: Phased Training: The training is divided into two phases: Phase One (pre-training) and Phase Two (joint training). During training... , ;

[0107] Specifically, the pre-training phase involves freezing the chlorophyll simulation branch. The parameters are optimized only in the parameter generation network. Loss of use This stage enables the parameter network to learn to output dynamic parameters that satisfy the growth kinetic equation and parameter rationality constraints without the guidance of observation data; it terminates early when the total loss does not decrease for several consecutive rounds on the validation set, and saves the optimal parameter network weights; the first stage pre-training lasts for 200 rounds.

[0108] The joint training phase specifically involves: unfreezing all parameters to obtain the complete loss. Simultaneously optimize and The optimizer is Adam, and the learning rates are respectively... (Adjustable) and (Adjustable), and combined with learning rate decay (ReduceLROnPlateau); training stops when the validation loss (MSE) does not improve for a maximum of 800 consecutive rounds. In fact, it was terminated early at the 168th round because the validation loss no longer decreased, and the model with the optimal validation loss was loaded as the final model.

[0109] Step 8: Model Evaluation: Simulate the final model on the test set and inversely scale it to obtain the original scale chlorophyll values.

[0110] ;

[0111] In the formula, The simulated chlorophyll concentration before scaling is in mg / L;

[0112] Calculate the coefficient of determination R 2And model evaluation metrics such as Mean Absolute Percentage Error (MAPE). The original multiple linear regression showed R1 on the test set. 2 The value was -0.121, and the MAPE was 46.37%. Figure 2 While pure black-box neural networks can improve the fit, they cannot guarantee the physical consistency of the growth rate (e.g., the appearance of a negative half-saturation constant). After adopting the method of this invention, the test set R... 2 Significantly improved to 0.438, MAPE was 48.53% (see...) Figure 3 The model method of this invention exhibits excellent performance in both the training and validation phases. Both the training and validation loss curves decrease, and the final loss value reaches a low point and remains stable, indicating that the model has converged well. (See...) Figure 6 .

[0113] Physical interpretability analysis, specifically:

[0114] Nutrient growth curve: fixed environmental factors To obtain the median of each dimension of the training set, each nutrient was scanned within its measured range, and the net specific growth rate and changes in each kinetic parameter were calculated. A growth rate-nutrient concentration curve was then plotted. The nutrient growth curve showed that when TP increased from 0.016 mg / L to 0.032 mg / L, From -0.0298 d -1 Rising to 0.0876 d -1 See details Figure 5 .

[0115] Parameter sensitivity plot: For each environmental factor, samples are taken at equal intervals within its measured range. With the median of other environmental variables fixed, the variation amplitude of each kinetic parameter (maximum minus minimum) is calculated. After amplitude normalization, a grouped bar chart is plotted to reflect the regulatory intensity of environmental factors on each parameter. The parameter sensitivity plot reveals the effect of water temperature on... The regulation amplitude was the largest (normalized sensitivity 0.99), followed by wind speed (0.80), which is highly consistent with the understanding of algal physiology and ecology; see details. Figure 4 .

[0116] In summary, the method of this invention successfully integrates causal inference, adaptive mechanism analysis, and physical constraints, maintaining high prediction accuracy while outputting dynamic parameter maps with clear ecological significance. It is suitable for algal bloom prediction and nutrient management decision support in lakes and reservoirs.

Claims

1. A method for simulating algal growth based on causal time-delay optimization and PINN, characterized in that... Includes the following steps: Step 1: Data Acquisition and Preprocessing: Acquire daily monitoring data of nutrient and organic matter indicators, environmental factor indicators, and chlorophyll concentration. Nutrient and organic matter indicators include total nitrogen (TN), total phosphorus (TP), and permanganate index (COD). Mn Environmental factors include water temperature, pH, dissolved oxygen, transparency, solar shortwave radiation, wind speed, and rainfall; missing values ​​were filled forward and backward, and all variables were standardized using Z-score. Step 2: Time-Lap Optimization Based on Causal Discovery: Daily monitoring data of standardized nutrient and organic matter indicators, environmental factor indicators, and chlorophyll concentration are constructed into a time series data frame. The PCMCI causal discovery method is used, with chlorophyll as the target variable, and a maximum time lag is set. significance level of conditional independence test Running PCMCI yields the cause-effect graph matrix; for driving variables... If there is a time lag Existing For direct connections, the minimum time delay among significant connections is selected as the optimal time delay for that variable. If no direct connection is detected, the following formula is used to calculate. Different time delays Mutual information below: ； In the formula, Represents mutual information value; This represents the mutual information calculation function, specifically the `mutual_info_regression` function in the Python library. For driving variables exist Time series data at a given moment; For chlorophyll in Time series data at a given moment; Select the time delay that maximizes mutual information as This yields the optimal time-delay set for each nutrient and organic matter variable, as well as the environmental variable. ; Step 3: Feature Construction and Dataset Partitioning: Based on the optimal time lag, each variable is shifted to its corresponding time lag and aligned with the original time to construct the design matrix. ; The first column is the number of days from the start date. The subsequent columns are the shifted TN, TP, and COD, respectively. Mn And 7 environmental factors; target variable The unshifted chlorophyll concentration; after removing samples with missing values, the previous values ​​are arranged in chronological order. % of the samples were divided into the training set, and then % represents the test set; Multiply the chlorophyll target value by a scaling factor using the following formula. Perform scaling transformation: ； In the formula, This is the scaled-down chlorophyll concentration, in mg / L; The chlorophyll concentration before scaling is in mg / L; This is the scaling factor; The environmental factors are standardized to ensure that the input features have similar scales. Step 4: Constructing the algal growth mechanism module: Define the vectors of nutrients and organic matter. The corresponding half-saturation constants are respectively ,Right now The maximum specific growth rate is respiratory rate is The mortality rate is First, calculate the restriction functions for each nutrient and organic matter. : ； In the formula, It is a small positive number; the minimum value of the three is taken as the combined nutrient limiting factor: ； In the formula, This is a function of the combined nutrient limiting factors; The growth term is The loss term is Net growth rate is expressed as: ； In the formula, Net specific growth rate, d -1 ; Step 5: Construct the parameter generation network: Design a fully connected neural network The normalized seven-dimensional environmental factor vector As input, output the original values ​​of the six-dimensional parameters: ； In the formula, A vector composed of seven environmental factor indicators; Six-dimensional parameters Corresponding Output value; The network structure consists of three fully connected layers, each followed by batch normalization and ReLU activation, with the last layer outputting a 6-dimensional value; a boundary mapping is applied to the original output. Ensure it is greater than For bounded parameters , , , The Sigmoid mapping is applied to a preset range; simultaneously, a biological rationality constraint is imposed: if Then let Finally, a reasonable dynamic parameter vector is obtained. ; Step Six: Construct a Physical Information Neural Network: The Physical Information Neural Network uses input vectors... For input, where, For the time series, day, Seven environmental factors are first identified through a multi-layer fully connected network. Simulated scaled chlorophyll concentration: ； In the formula, To simulate scaled chlorophyll concentration, mg / L; At the same time, the nutrient salt part and environmental aspects The input parameter generation network obtains the dynamic parameters. Then, the data is passed to the algae growth mechanism module to calculate the net specific growth rate under the current conditions. ; Introducing physical residual loss and using automatic differentiation to obtain Output time partial derivative: ； In the formula, For sample size; This loss forces the temporal variation of chlorophyll to satisfy... The exponential growth / decline pattern; Simultaneously set constraint losses: ； In the formula, For sample size; punish A combination of parameters that violates the laws of physics, such as those less than the sum of respiration and death, and those with excessively small half-saturation constants. The data fitting loss is the mean squared error: ； In the formula, For sample size; The measured chlorophyll concentration is in mg / L. The total loss function is: ； In the formula, The weights for the physical residual loss; The weights are used to constrain the loss; Step 7: Phased Training: The training is divided into a pre-training phase and a combined training phase. Specifically, the pre-training phase involves freezing the chlorophyll simulation branch. The parameters are optimized only in the parameter generation network. Loss of use This stage enables the parameter network to learn to output dynamic parameters that satisfy the growth kinetic equation and parameter rationality constraints without the guidance of observation data; it terminates early when the total loss does not decrease for several consecutive rounds on the validation set, and saves the optimal parameter network weights. The joint training phase specifically involves: unfreezing all parameters to obtain the complete loss. Simultaneously optimize and The optimizer is Adam, with learning rate decay; training stops when the validation loss fails to improve for several consecutive rounds, and the model with the optimal validation loss is loaded as the final model. Step 8: Model Evaluation: Simulate the final model on the test set and inversely scale it to obtain the original scale chlorophyll values. ； In the formula, The simulated chlorophyll concentration before scaling is in mg / L; Calculate the coefficient of determination R 2 The mean absolute percentage error (MAPE) model evaluation index was used; output nutrient growth curves, parameter sensitivity plots, model prediction performance plots, and model iteration loss curves were plotted.

2. The algal growth simulation method based on causal time-delay optimization and PINN according to claim 1, characterized in that... Step eight also includes a physical interpretability analysis, specifically: Nutrient growth curve: fixed environmental factors To obtain the median of each dimension of the training set, each nutrient was scanned one by one within its actual measurement range, and the changes in net specific growth rate and each kinetic parameter were calculated. The growth rate-nutrient concentration curve was then plotted. Parameter sensitivity map: For each environmental factor, samples are taken at equal intervals within its measured range. The median of the remaining environmental variables is fixed. The variation range of each dynamic parameter is calculated. After normalization by range, a grouped bar chart is plotted to reflect the regulatory intensity of environmental factors on each parameter.

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