Crop single-leaf photosynthetic rate estimation method considering small sample learning and uncertainty prediction

Abstract

The invention discloses a crop single-leaf photosynthetic rate estimation method considering small sample learning and uncertainty prediction, and the method comprises the steps: obtaining photosynthetic rate data of different crops under different environment conditions to construct a data set, and dividing the data set of each crop into subsets; constructing a basic prediction model based on a multi-layer perceptron, taking the divided subsets as a test domain and a training domain, and updating model parameters by utilizing meta-learning training; aiming at an unknown task, constructing a loss function in combination with a quantile regression method to finely adjust the basic prediction model so as to realize interval prediction of the photosynthetic rate; and finally, introducing Gaussian kernel density to estimate probability distribution of the photosynthetic rate under different environmental conditions, and synchronously realizing point prediction and interval prediction visualization of the photosynthetic rate by taking a median as a point prediction value. Compared with the prior art, the method has the advantages that the photosynthetic rate in an unknown environment can be accurately predicted only by using a small amount of sample data for training, and the interval estimation method is adopted, so that the method can adapt to various application scenes.

Description

Technical Field

[0001] The present invention belongs to the technical field of planting, relates to the application of artificial intelligence in planting, and particularly relates to a method for estimating the photosynthetic rate of a single crop leaf taking into account small sample learning and uncertainty prediction. Background Art

[0002] Multiple environmental factors, including light intensity (LI), air temperature (Ta), and carbon dioxide concentration (CO2 concentration, [CO2]), jointly affect the single-leaf photosynthetic rate (A L ). Accurately and efficiently evaluate A L It is crucial for applications in crop yield assessment, stress perception and biological breeding. L Prediction methods typically require large amounts of training data and often rely on point estimates, which poses challenges for small-sample learning and high-reliability predictions. Summary of the Invention

[0003] In order to overcome the shortcomings of the above-mentioned prior art, the purpose of the present invention is to provide a method for estimating the photosynthetic rate of a single crop leaf that takes into account small sample learning and uncertainty prediction, so as to use a small amount of data for training to predict and evaluate the photosynthetic rate in an unknown environment, and expand the traditional photosynthetic rate point estimation strategy to interval estimation to adapt to a variety of application scenarios.

[0004] In order to achieve the above object, the technical solution adopted by the present invention is:

[0005] A method for estimating the photosynthetic rate of a single crop leaf considering small sample learning and uncertainty prediction includes the following steps:

[0006] Step 1: Obtain photosynthetic rate data of different crops under different environmental conditions to construct a data set, and divide the data set of each crop into subsets. The subsets are photosynthetic rate data obtained by fixing some environmental factors and changing other environmental factors;

[0007] Step 2: Build a basic prediction model based on a multi-layer perceptron, use the divided subsets as test domains and training domains, and use meta-learning training to update the model parameters;

[0008] Step 3: For unknown tasks, a loss function is constructed by combining the quantile regression method to fine-tune the basic prediction model to achieve interval prediction of photosynthetic rate;

[0009] Step 4: Based on the interval prediction, Gaussian kernel density is introduced to estimate the probability distribution of photosynthetic rate under different environmental conditions, and the median is used as the point prediction value to simultaneously realize the visualization of photosynthetic rate point prediction and interval prediction.

[0010] In one embodiment, the different environmental conditions refer to a combination of multiple environmental factors that affect the photosynthetic rate of the crop. The size of the dataset subset depends on the situation, but generally does not need to exceed 5000. The dataset size used in the present invention is 104 to 270, which is a very small number of samples and can be expanded to 50 to 5000. The present invention fixes one or several identical environmental factors for the same crop and classifies the resulting subsets by varying other environmental factors into the same subset category. When performing prediction task training, the subsets are divided into a test domain and a training domain.

[0011] In one embodiment, step 2 divides the dataset of the same crop into N learning problems, further dividing the dataset into a training domain and a test domain, and dividing each training domain and test domain into a support set and a query set, respectively. The meta-learning method for updating model parameters includes an inner loop and an outer loop. The inner loop promotes the update of a specific task by calculating losses, adjusting gradients, and determining optimal task parameters. The outer loop coordinates cross-task backpropagation, uses the aggregated gradients of the inner loop training parameters to optimize global parameters, and then uses the global parameters to fine-tune the test domain task parameters.

[0012] In one embodiment, the method of dividing a dataset of the same crop into N learning problems specifically involves dividing a dataset of photosynthetic rates of the same crop under different combinations of environmental factors into a number of subsets (learning problems) of the same type with specific environmental characteristics, based on one or more environmental factors. These subsets are then divided into training domains and test domains to simulate the generalization ability of the model under cross-environmental conditions. Among these subsets, subsets with environmental factors in non-boundary intervals are preferentially selected as test domains to improve the model's adaptability in representative areas, thereby enhancing the stability and transferability of the overall learning problem.

[0013] In one embodiment, the inner loop loss function is defined as:

[0014]

[0015] Among them, x i and y i Respectively represent Input and output, is the training domain support set, T i is the i-th learning problem, the superscript r indicates the training domain, and f θ (x i ) represents the calculated value of xi;

[0016] The process of adjusting the gradient in the inner loop is expressed as:

[0017]

[0018] in, represents the training domain task parameters updated by the inner loop training, β represents the learning rate of the inner loop training, θ represents the MLP initialization trainable parameters, Indicates the gradient descent of θ, typically using the Adam strategy; represents the inner loop loss function;

[0019] The outer loop is expressed as:

[0020]

[0021] Among them, θ * is the global parameter of meta-learning, I is the number of training domain data, is the training domain query set, Denotes the outer loop loss function, which has the same structure as the inner loop loss function. The test domain support set is used to fine-tune the meta-learning global parameters to obtain the test domain task parameters, which are expressed as:

[0022]

[0023] Where α represents the learning rate of the fine-tuning process, is the training domain support set, Represents the loss function during fine-tuning.

[0024] In one embodiment, in step 3, a quantile regression method is introduced into the basic prediction model, and the loss function of the fine-tuning model under different quantiles is expressed as:

[0025]

[0026] where x j and y j represent Input and output, represents the x calculated by the model j Value, e j represents the difference between the predicted value and the observed value, τ is the preset quantile.

[0027] In one embodiment, the preset quantile τ ranges from 0.05 to 0.95, representing a 90% confidence interval. This threshold reflects a balance between reliability and sensitivity—enough to adequately cover actual results while avoiding an excessively wide interval that could reduce the accuracy of photosynthetic rate predictions. This choice is consistent with mainstream approaches to uncertainty-aware modeling in agricultural and environmental systems.

[0028] In one embodiment, in step 4, the probability density of the predicted value under the confidence interval is:

[0029]

[0030] Where x represents the target position of the estimated probability density; x k It represents the kth photosynthetic rate sample value falling within the selected confidence interval under a certain environmental condition, K is the total number of photosynthetic rate samples in the confidence interval, h is the bandwidth, which is used to control the smoothness of the estimate, and the Gaussian kernel function is used to perform weighted summation on the samples to obtain the probability density estimate of all photosynthetic rate values ​​within the confidence interval under this environmental condition.

[0031] In one embodiment, k=0 represents the photosynthetic rate value calculated when τ is 0.05, k=1 represents τ is 0.06, and so on, k=90 represents the photosynthetic rate value calculated when τ is 0.95.

[0032] Compared with the existing technology, the present invention only needs to use a small amount of sample data for training to accurately predict the photosynthetic rate in an unknown environment, and adopts an interval estimation method to adapt to various application scenarios. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] Figure 1 It is a structural schematic diagram of the present invention.

[0034] Figure 2 A is a subset of three datasets L (a) is the Dataset-I subset, (b) is the Dataset-II subset, and (c) is the Dataset-III subset.

[0035] Figure 3 The different environmental factors in the three data sets and A L correlation.

[0036] Figure 4These are the prediction results of ML-MLP. (a) is Dataset-I-II-I, (b) is Dataset-I-II-II, (c) is Dataset-I-II-III, (d) is Dataset-II-V, (e) is Dataset-II-VI, (f) is Dataset-III-V, and (g) is Dataset-III-VI.

[0037] Figure 5 These are the RMSE and MAE of the prediction models using different methods for Dataset-I. (a) is the RMSE of Dataset-I-II-I, (b) is the RMSE of Dataset-I-II-II, (c) is the RMSE of Dataset-I-II-III, (d) is the MAE of Dataset-I-II-I, (e) is the MAE of Dataset-I-II-II, and (f) is the MAE of Dataset-I-II-III.

[0038] Figure 6 The RMSE and MAE of the prediction models using different methods for Dataset-II are shown in Figure 2. (a) is the RMSE of Dataset-II-V, (b) is the RMSE of Dataset-II-VI, (c) is the MAE of Dataset-II-V, and (d) is the MAE of Dataset-II-VI.

[0039] Figure 7 The RMSE and MAE of the prediction models using different methods for Dataset-III are shown in Figure 2. (a) is the RMSE of Dataset-III-V, (b) is the RMSE of Dataset-III-VI, (c) is the MAE of Dataset-III-V, and (d) is the MAE of Dataset-III-VI.

[0040] Figure 8 Here are the RMSE and MAE of the prediction models using ML-MLP, ANIL, and TF for the three datasets.

[0041] Figure 9 is the A of the seven test domains from 0.05 to 0.95 quantiles LInterval prediction. (a) is Dataset-I-II-I, (b) is Dataset-I-II-II, (c) is Dataset-I-II-III, (d) is Dataset-II-V, (e) is Dataset-II-VI, (f) is Dataset-III-V, (g) is Dataset-III-VI, and (h) is the R2 of fitting curves for the 0.05quantile and 0.95quantile.

[0042] Figure 10 Comparison of prediction performance of different internal prediction methods, where (a) is PICP and (b) is PINAW.

[0043] Figure 11 It's a certain A L Probability density curves in seven test domains. (a) is Dataset-I-II-I, (b) is Dataset-I-II-II, (c) is Dataset-I-II-III, (d) is Dataset-II-V, (e) is Dataset-II-VI, (f) is Dataset-III-V, (g) is Dataset-III-VI, and (h) is the impact of bandwidth changes on the probability density curves. DETAILED DESCRIPTION

[0044] The embodiments of the present invention are described in detail below with reference to the accompanying drawings and examples.

[0045] To alleviate the current difficulty in achieving small-sample learning and high-reliability predictions for photosynthetic rate, the present invention combines meta-learning (ML) and a multilayer perceptron (MLP) to first construct a basic prediction model and then fine-tune it for unknown tasks. Compared with currently popular methods, the fine-tuned model achieves higher prediction accuracy with fewer training samples. Satisfactory prediction performance can be achieved by introducing only approximately 5% of new samples from the dataset. In addition, the present invention introduces the quantile regression (QR) method to obtain the 95% confidence interval of the photosynthetic rate, thereby alleviating the limitations of high-reliability predictions. Finally, the present invention uses Gaussian kernel density estimation (GKDE) to derive the probability density under various environmental conditions, constructing a crop photosynthetic rate prediction model with reliable uncertainty estimation capabilities. Detailed validation and interpretability analysis using multiple datasets from different species demonstrate the universal applicability and rationality of the present invention. This paper demonstrates the high generalization capability of ML-MLP for unknown datasets and extends photosynthetic rate prediction from point estimation to interval prediction through QR-GKDE, thereby promoting crop cultivation research. This paper provides a robust method for crop monitoring and stress perception, significantly advancing the state of precision agriculture.

[0046] like Figure 1 As shown in the figure, the present invention integrates the ML-MLP-QR-GKDE method to construct a crop photosynthetic rate (A L ) interval prediction model, which is used to estimate the photosynthetic process of crops under different environmental and physiological conditions. The main steps include: 1) Selecting a variety of crops cultivated under different environmental conditions, using their photosynthetic rate data to construct a data set, and dividing the data set according to their characteristics; 2) Building a basic prediction model based on ML-MLP and verifying its superiority by comparing with other algorithms; 3) For unknown tasks, combining the quantile regression method to construct a loss function to fine-tune the basic prediction model and achieve A L Interval prediction; 4) Gaussian kernel density estimation was used to estimate the probability distribution of photosynthetic rate under different environmental conditions, using the median as the point prediction value. This allowed for simultaneous visualization of both point and interval predictions of photosynthetic rate. 5) The interpretability of the ML-MLP method was verified by combining sample physiological characteristics and model structure. This is explained in detail below.

[0047] Step 1: Data collection and preprocessing.

[0048] To address the challenges of few-sample learning and uncertainty estimation in photosynthetic rate prediction, photosynthetic rate data for a range of different crop varieties under different environmental conditions were collected. Data cleaning, normalization, feature analysis, and dataset partitioning were then performed to prepare the dataset for modeling. The present invention partitions the dataset for each crop into subsets, where different environmental conditions refer to combinations of multiple environmental factors that affect the crop's photosynthetic rate. A subset consists of photosynthetic rate data obtained by fixing some environmental factors and varying others.

[0049] In this embodiment, three datasets were constructed: a lettuce dataset under different environmental factors, a cucumber dataset under different environmental factors, and a cucumber dataset under different environmental factors, Dataset-III.

[0050] For Dataset-I, experimental data were collected on lettuce samples (Lactuca sativa L.cv Italian lettuce) in three controlled climate chambers (RGL-P500D-CO2, Hefei Darth Carter, China). Twelve experimental conditions were designed using three light intensities (LI) and four nutrient solution nitrogen levels (NCNS) (Table 1). The experiment was conducted over a period of 20 days, with sample A being tested every four days. L Following the procedure of Gao et al. (2021), A was obtained under controlled conditions on the leaf surface using a portable photosynthetic rate device (LI-6800, LI-COR, USA). L Measurement. Multiple submodules of the LI-6800 were selected to control the blade chamber environmental parameters as needed. Four experimental temperature gradients (18°C, 23°C, 28°C, and 33°C) were set in the temperature control module, and the [CO2] temperature was set in five modules (300 μmol·mol -1 , 700 μmol·mol -1 , 1100 μmol·mol -1 , 1400 μmol·mol -1 , and 1800 μmol·mol -1 ) was set up in the CO2 gradient. During the measurement, the photosynthetic photon flux density (PPFD) was set to 500 μmol·m -2 ·s -1 , set the flow rate to 500 μmol·s -1 The relative humidity is set to 50% in the moisture control module. In addition, other lettuce samples are used to verify the method proposed in the present invention.

[0051] Table 1. Sample growth environment of Dataset-I

[0052] NCNS (mmol / L) <![CDATA[LI(μmol·m –2 ·s –1 )]]> serial number Dataset size 1.75 100 / 200 / 300 II / II / III 104 / 104 / 104 7 100 / 200 / 300 II-I / II / III 104 / 104 / 104 12.25 100 / 200 / 300 III-I / II / III 104 / 104 / 104 17.5 100 / 200 / 300 IV-I / II / III 104 / 104 / 104

[0053] Dataset-II studies the combined effects of Ta, [CO2], PPFD, and light quality (LQ) on cucumber samples. This dataset is derived from Gao et al. (2021) and includes seven subsets (Dataset-II-I / II / II / IV / V / VI / VII), each corresponding to a different LQ value (0.1, 0.3, 0.4, 0.5, 0.6, 0.7, and 0.9). Multiple submodules of the LI-6800 were selected to control the leaf chamber environmental parameters. Six Ta gradients (18°C, 21°C, 24°C, 27°C, 30°C, and 33°C) were set in the temperature control module, and 15 PPFD gradients (0, 30, 75, 150, 300, 500, 600, 700, 800, 1000, 1200, 1600, and 1800 μmol·m -2 ·s -1 Three [CO2] gradients (400 μmol·mol -1 , 700 μmol·mol -1 and 1000 μmol·mol -1 The environment settings of other variables are consistent with those in Dataset-I. Each subset of Dataset-II has a total of 270 data.

[0054] Dataset-III studies the combined effects of Ta, [CO2], PPFD, and soil moisture (SM) on cucumber samples. This dataset is derived from Hou et al. (2023) and contains eight subsets (Dataset-III-I / II / III / IV / V / VI / VII / VIII), each corresponding to a different SM (5%, 10%, 15%, 20%, 25%, 30%, 35%, and 40%). Multiple submodules of the LI-6800 were selected to control the environmental parameters of the leaf chamber. Five experimental temperature gradients (18°C, 22°C, 26°C, 30°C, and 34°C) were set in the temperature control module, and nine PPFD gradients (0, 100, 200, 300, 500, 700, 800, 1000, and 1200 μmol·m -2 ·s -1 ), the CO2 control module set 4 [CO2] gradients (400 μmol·mol -1 , 600 μmol·mol -1 , 800 μmol·mol -1 and 1000 μmol·mol-1 ), and the environment settings of other variables are consistent with Dataset-I. Each subset of Dataset-III contains a total of 160 data.

[0055] According to the above examples, the subset size of the dataset of the present invention only ranges from 104 to 270, which greatly reduces the number of samples compared with the existing model. In practical applications, this can be expanded on this basis.

[0056] The present invention fixes one or several identical environmental factors for the same crop and classifies the subsets obtained by changing other environmental factors into the same type of subsets. When performing prediction task training, the same type of subsets are preferably used as the test domain and training domain.

[0057] The detailed information of the three data sets obtained in the experiment is shown in Table 2. After unified preprocessing, the proposed ML-MLP-QR-GKDE algorithm was used to perform A L Prediction. Accordingly, the following four prediction tasks can be constructed:

[0058] 1) Task I: Dataset-I-II is used as the test domain, and Dataset-II / III / IV is used as the training domain, with a total of 9 training domains and 3 test domains.

[0059] 2) Task II: Dataset-II-V / VI is used as the test domain, and Dataset-II-I / II / III / IV / VII is used as the training domain, with a total of 5 training domains and 2 test domains.

[0060] 3) Task III: Dataset-III-V / VI is used as the test domain, and Dataset-III-I / II / III / IV / VII / VIII is used as the training domain, with a total of 6 training domains and 2 test domains.

[0061] Table 2 Detailed information of the three datasets

[0062] Dataset / No. Preset variables enter Output Dataset-I / 12 LI and NCNS (I / II / III / VI and I / II / III) <![CDATA[ITAT, [CO2] and T a > <![CDATA[A L ]]> Dataset-II / 7 LQ(I~VII) <![CDATA[T a , [CO2] and PPFD]]> <![CDATA[A L ]]> Dataset-III / 8 SM (I~VIII) <![CDATA[T a , [CO2] and PPFD]]> <![CDATA[A L ]]>

[0063] Since the dimensions and units of the three data sets are different, their differences bring huge challenges to data analysis. To alleviate this problem, the present invention uses maximum and minimum normalization to eliminate the dimensional interactions between variables. Afterwards, the present invention conducts Pearson correlation analysis to test the relationship between environmental variables and A. L For each task's training domain, the normalized dataset is split into a support set and a query set in a ratio of 4:1. For each task's test domain, the query set ratios are 0.80, 0.85, 0.90, and 0.95, respectively.

[0064] Step 2: Build a basic model, use the divided subsets as test domains and training domains, and use meta-learning training to update the model parameters.

[0065] This step divides the photosynthetic rate dataset for the same crop under different environmental factor combinations into N subsets of the same type (learning problem) with specific environmental characteristics, based on one or more environmental factors. These subsets are then divided into training and test domains to simulate the model's generalization ability across environmental conditions. Within these subsets, we prioritize those with environmental factors in the non-boundary range as test domains to improve the model's adaptability to representative regions, thereby enhancing the stability and transferability of the overall learning problem. Each training and test domain is then divided into a support set and a query set, respectively.

[0066] In the basic prediction model of the present invention, meta-learning aims to discover generalizable knowledge across tasks and quickly adapt to new tasks using limited data. L Prediction problem, and use the dataset Dataset-I to demonstrate the algorithm workflow. First, divide Dataset-I into N learning problems, denoted as Training domain Further divided into support set and queryset Test domain The support set is further divided into and queryset It can be expressed as The basic prediction model used here is the MLP model. The MLP architecture consists of a single hidden layer with a 3-10-1 structure. ML methods are used to update the parameters in the MLP to achieve accurate predictions using a small number of samples. Meta-learning to update model parameters specifically includes an inner loop and an outer loop. The inner loop promotes updates for specific tasks by calculating losses, adjusting gradients, and determining optimal task parameters. This example uses Adam optimization and performs 100 training iterations. The loss function is defined as:

[0067]

[0068] Among them, x i and y i Respectively represent Input and output, T i is the i-th learning problem, the superscript r indicates the training domain, and f θ (x i ) represents x i The calculated value obtained.

[0069] Then, the process of adjusting the gradient in the inner loop can be expressed as:

[0070]

[0071] in, represents the training domain task parameters updated through the inner loop training, and β represents the learning rate of the inner loop training (taken as 0.001 in this embodiment).

[0072] After the inner loop training is completed, the outer loop training coordinates cross-task backpropagation and uses the inner loop training parameters Aggregate gradient to optimize the global parameters (θ * ), θ represents the MLP initialization trainable parameters, represents the gradient descent on θ, represents the inner loop loss function.

[0073] The outer loop is represented as:

[0074]

[0075] Among them, θ * is the global parameter of meta-learning, I is the number of training domain data, is the training domain query set, The outer loop loss function has the same structure as the inner loop loss function. For the outer loop training, Adam optimization is also used, and 100 training iterations are performed with a learning rate of 0.01.

[0076] Finally, the test domain support set is used to fine-tune the meta-learning global parameters to obtain the test domain task parameters to achieve good prediction performance on the unknown dataset, which is expressed as:

[0077]

[0078] Where α represents the learning rate of the fine-tuning process (0.001 in this embodiment). This embodiment uses the Adam optimization algorithm for 200 training iterations. Represents the loss function during fine-tuning.

[0079] In order to verify the effectiveness of the proposed framework, this paper LSeveral state-of-the-art methods commonly used in prediction are compared and analyzed. In a traditional data-intensive setting, this paper selects four representative algorithms: BPNN, support vector regression (SVR), random forest (RF), and non-linear regression (NLR). These algorithms use the same input features as the ML-MLP model. To ensure a fair comparison, these models use the default hyperparameters of scikit-learn under the standard training process. In the setting of few-shot learning, this paper employs almost no inner loop (ANIL) and transfer learning (TL) methods. To ensure architectural consistency, the basic network structure of ANIL and TL remains the same as that of the ML-MLP model. Notably, their training processes exhibit fundamental differences: for ANIL, only the outer loop parameters are updated during training, while the inner loop updates are largely omitted, thereby reducing model complexity. In contrast, TL methods adopt a parameter freezing strategy, in which the weights from the input layer to the hidden layer are fixed, and only the parameters from the hidden layer to the output layer are trainable during fine-tuning.

[0080] Step 3, uncertainty assessment

[0081] For unknown tasks, in order to obtain A of different quantiles L For prediction, the QR technique can be applied to the construction of ML-MLP. That is, the loss function is constructed by combining the quantile regression method to fine-tune the basic prediction model to achieve interval prediction of photosynthetic rate.

[0082] This step introduces the quantile regression method into the basic prediction model. The loss function of the fine-tuning model under different quantiles is expressed as:

[0083]

[0084] where x j and y j represent Input and output, represents the x calculated by the model j Value, e j represents the difference between the predicted value and the observed value, and τ is the preset quantile. In this invention, the τ value ranges from 0.05 to 0.95, representing a 90% confidence interval. This invention uses a 90% confidence interval to characterize prediction uncertainty. To verify the performance of QR, this invention uses GPR and BNN for comparative analysis.

[0085] Step 4: Calculate the probability density.

[0086] Based on the interval prediction, this step introduces Gaussian kernel density to estimate the probability distribution of photosynthetic rate under different environmental conditions, and finally uses the median as the point prediction value to simultaneously realize the visualization of photosynthetic rate point prediction and interval prediction.

[0087] Specifically, based on GKDE, the 90% confidence interval of A L The probability density of the predicted value is:

[0088]

[0089] Where x represents the target position of the estimated probability density; x k represents the kth photosynthetic rate sample value falling within the selected confidence interval under a given environmental condition. K is the total number of photosynthetic rate samples within the confidence interval, and h is the bandwidth, which controls the smoothness of the estimate. A Gaussian kernel function is used to weight the sum of the samples to obtain the probability density estimate of all photosynthetic rate values ​​within the confidence interval under the given environmental condition. k = 0 represents the photosynthetic rate value calculated when τ is 0.05, k = 1 represents τ is 0.06, and so on. k = 90 represents the photosynthetic rate value calculated when τ is 0.95.

[0090] The present invention adopts the determination coefficient (R 2 ), root mean square error (RMSE), mean absolute error (MAE), prediction interval coverage probability (PICP), prediction interval normalized average width (PINAW) and other indicators are used to evaluate the model accuracy and determine the best A L Prediction method. The calculation method is as follows:

[0091]

[0092] Among them, Y m,p , Y m,o , and M represent the model's predicted value, measured value, measured mean, and dataset size, respectively. When the condition is met, Φ m (·)=1, otherwise Φ m (·)=0. and Represents the lower and upper bounds of the prediction interval for the mth sample. max and Y min Represents the maximum and minimum values ​​of the output variable.

[0093] Figure 2 A of three datasets (Dataset-I, Dataset-II and Dataset-III) subsets is given. L Under the complex interaction of various environmental factors, A Lshowed greater variability, but A L The change with the predetermined variable is not significant. Specifically, in Dataset-I, the present invention observed that A L At a constant NCNS level, when the LI was set to 100 μmol·m -2 ·s -1 When A L On the contrary, at a fixed LI level, with the increase of NCNS, A L The value first increases and then decreases. In Dataset-II, when LQ is set to 0.1, A L The value is significantly lower than that of A under other conditions. L values, but the differences were not significant under other LQ conditions, indicating that the specific LQ had a significant effect on A L In Dataset-III, with the increase of SM, A L Under the same environmental conditions, both too low and too high SM levels have a negative impact on photosynthetic rate, which highlights the importance of maintaining optimal SM levels to achieve efficient photosynthetic rate. L Models are typically constructed using all environmental conditions as input. Methods such as BPNN and SVR perform well in such data-rich conditions, but their generalization capabilities often suffer when applied beyond their training range. In contrast, ML techniques can exploit inherent regularities across different samples and environmental conditions to quickly generate accurate models from limited data, making them particularly advantageous in this context.

[0094] Analyze the correlation between input and output variables to construct and evaluate A L Prediction model. Due to the different sets of environmental variables used in different datasets, L There are also differences in the correlation, such as Figure 3 As shown. In Dataset-I, L The correlation from high to low is [CO2], LI, Ta, ITAT and NCNS. In Dataset-II, A L The correlation with PPFD is the highest, followed by Ta, [CO2] and LQ. Finally, in Dataset-III, A L The correlation with PPFD is the strongest, followed by [CO2], SM and Ta. The present invention uses long-term adaptive environmental parameters as predefined variables, specifically, ITAT and NCNS for Dataset-I, LQ for Dataset-II, and SM for Dataset-III. These parameters are related to A LThe correlation between the two subsets is relatively low, which helps to limit the differences between the subsets and enhance the L Applicability of the prediction model.

[0095] The present invention investigates the performance of ML-MLP on seven test domains and compares its prediction capabilities under different query set ratios (0.80, 0.85, 0.90 and 0.95). Figure 4 It shows that despite the large difference in the ratio of support set to query set, no underfitting phenomenon is observed. As the number of samples in the support set decreases, the prediction error gradually increases. Among the three datasets, Dataset-III performs best on unseen datasets, which shows that the quality of the original dataset also affects the prediction A L For Dataset-I, under four query sets in three test domains, R 2 The ranges of RMSE and MAE were 0.844-0.927 and 1.272-2.492 μmol·m -2 ·s -1 and 0.986~2.095μmol·m -2 ·s -1 . Accordingly, for the support set, R 2 The ranges of RMSE and MAE were 0.835-0.936 and 1.091-2.424 μmol·m -2 ·s -1 and 0.953~2.045μmol·m -2 ·s -1 Compared with Dataset-I-II-II and Dataset-I-II-III, Dataset-I-II-I has lower prediction error, which may be due to A L For Dataset-II, under the four query sets of two test domains, R 2 The ranges of RMSE and MAE were 0.924-0.938 and 1.791-2.016 μmol·m -2 ·s -1 and 1.432~1.600μmol·m -2 ·s -1 . Accordingly, for the support set, R 2 The ranges of RMSE and MAE were 0.921-0.979 and 1.048-2.313 μmol·m -2 ·s -1 and 0.799~1.846μmol·m -2 ·s -1 ; For Dataset-III, under four query sets across two test domains, R2 The ranges of RMSE and MAE were 0.959-0.966 and 1.377-1.557 μmol·m -2 ·s -1 and 1.112~1.237μmol·m -2 ·s -1 . Accordingly, for the support set, R 2 The ranges of RMSE and MAE were 0.971-0.989 and 0.913-1.437 μmol·m -2 ·s -1 and 0.7773~1.189μmol·m -2 ·s -1 The results show that there is no significant difference between the support set and the query set, indicating that the ML-MLP model can effectively identify patterns to match unknown datasets.

[0096] In addition, the present invention also makes a detailed comparison of the prediction results of different query set ratios. For Dataset-I-II-II, when the query set ratio is 0.8, R 2 , RMSE, and MAE were 0.928 and 1.671 μmol·m -2 ·s -1 and 1.366 μmol·m -2 ·s -1 When the query set ratio is 0.95, these values ​​become 0.844 and 2.492 μmol·m -2 ·s -1 and 1.760 μmol·m -2 ·s -1 For Dataset-II and Dataset-III, R 2 , RMSE, and MAE show the smallest changes in the four query set ratios. For Dataset-II-V, when the query set ratio is 0.80, R 2 , RMSE, and MAE were 0.933 and 1.824 μmol·m -2 ·s -1 and 1.448 μmol·m -2 ·s -1 , and when the query set ratio is 0.95, these indicators become 0.924 and 2.002 μmol·m -2 ·s -1 and 1.576 μmol·m -2 ·s -1 Similarly, for Dataset-III-VI, when the query set ratio is 0.8, R 2, RMSE, and MAE were 0.965 and 2.113 μmol·m -2 ·s -1 and 1.181 μmol·m -2 ·s -1 ; When the query set ratio is 0.95, R 2 , RMSE, and MAE were 0.960 and 2.424 μmol·m -2 ·s -1 and 1.228 μmol·m -2 ·s -1 When the update exceeds 5% of the new samples, A L The prediction accuracy has reached the requirements of practical application. These results show that the introduction of machine learning methods enhances the adaptability of the model to new experimental conditions and can achieve good results even with a small amount of updated data (5%).

[0097] In order to illustrate the superiority of the present invention, four common modeling methods (NLR, BPNN, RF and SVR) were compared and analyzed on Dataset-I-II. The results are as follows: Figure 5 Although these four methods perform well in data-rich situations, their performance is poor on the limited dataset used in this paper. When the query set ratio is 0.80, the RMSE of the four methods ranges from 1.94 μmol·m -2 ·s -1 to 3.84 μmol·m -2 ·s -1 , and the MAE range is 1.53 μmol·m -2 ·s -1 to 3.20 μmol·m -2 ·s -1 When the query set ratio is 0.95, the RMSE value is expanded to 3.89 μmol·m -2 ·s -1 and 7.49 μmol·m -2 ·s -1 The MAE value expanded to 2.80 μmol·m -2 ·s -1 to 5.49 μmol·m -2 ·s -1 These findings suggest that it is possible to establish accurate A LPredictive models require extensive data collection because models trained on limited samples have reduced generalization capabilities to unseen datasets. This paper evaluated the accuracy of models at four different query set ratios, finding that ML-MLP consistently outperformed other methods. The advantages of the ML-MLP model are clear. This consistent accuracy across different query set ratios highlights the robustness of ML-MLP in generalizing to unseen samples.

[0098] Figure 6 and Figure 7 The prediction results of the four methods on Dataset-II and Dataset-III are shown respectively. It can be seen that, like Dataset-I, the prediction effects of the four methods are poor. For Dataset-II, when the query set ratio is 0.80, the RMSE ranges from 1.86 to 2.84 μmol·m -2 ·s -1 The MAE ranged from 1.34 to 2.36 μmol·m -2 ·s -1 When the query set ratio is 0.95, the RMSE is between 3.53 and 5.59 μmol·m -2 ·s -1 The MAE ranged from 2.59 to 3.96 μmol·m -2 ·s -1 For dataset III, when the query set ratio is 0.80, the RMSE range is 1.74 μmol·m -2 ·s -1 to 3.26 μmol·m -2 ·s -1 , MAE range is 1.33 μmol·m -2 ·s -1 to 2.68 μmol·m -2 ·s -1 When the query set ratio is 0.95, the RMSE range is 2.93 μmol·m -2 ·s -1 to 4.72 μmol·m -2 ·s -1 , MAE range is 2.42 μmol·m -2 ·s -1 to 3.82 μmol·m -2 ·s -1. The prediction errors of the four methods all increase significantly with the decrease in the number of training samples. Although NLR provides an interpretable formula for physiological analysis, it lacks the flexibility to handle complex environmental changes. RF overfits the training data under conditions of few samples, has poor generalization ability, and performs worse than SVR and BPNN. Although SVR and BPNN perform well on larger data sets, they have poor generalization ability under conditions of limited data. In contrast, the ML-MLP model proposed in the present invention maintains a high prediction accuracy on unseen data sets. This excellent performance is attributed to its ability to extract a basic prediction model with a faster response speed, thereby achieving better adaptability and generalization ability in scenarios where data is scarce.

[0099] In addition, when the test domain query set ratio is 0.8, the prediction results of ML-MLP, ANIL and TF are as follows: Figure 8 As shown. Among the three methods, TL performs the worst because it is difficult for TL to capture features and patterns when there is a significant distribution shift between the source and target domains, which may lead to negative transfer. The present invention recommends using the TL method when the source and target domains have similar basic feature distributions. ANIL performs worse than ML-MLP because its framework only updates the parameters of the last layer during the inner loop, which reduces the computational complexity but limits feature reconstruction. This makes it more suitable for simpler tasks. In contrast, the ML-MLP proposed in the present invention performs well in handling complex target domain distributions and high-dimensional feature spaces, verifying its strong adaptability and robustness.

[0100] For uncertainty assessment, interval estimation can reflect the uncertainty of the estimate and improve the prediction accuracy by providing a prediction range. L In predictive model applications, interval estimates are often preferred over point estimates. After fine-tuning the ML-MLP-QR model on the support set of seven test domains, the present invention obtains quantile distribution information for all test domains. Using the estimated values ​​of the 0.05 quantile and the 0.95 quantile, a prediction interval with a 90% confidence level can be constructed. The interval prediction results correspond to the point estimate results, as shown in Figure 9 As shown in (a) to (g). Figure 9 (h) shows the fitting curves of the 0.05 quantile and 0.95 quantile prediction values. The width of the prediction interval is related to the prediction A L The uncertainty is proportional to the prediction interval. Specifically, Dataset-II has the widest prediction interval, followed by Dataset-I, and Dataset-III has the narrowest interval. Therefore, interval predictions estimate the range within which the predicted value may fall, and the width of the interval is related to the accuracy of the point estimate model.

[0101] In order to illustrate the superiority of the present invention, GPR and BNN are used for comparative analysis. Figure 10Comparative results of three prediction methods are presented. A higher PICP value indicates that the interval contains more true values, while a lower PINAW value indicates a narrower interval. ML-MLP-QR outperforms QPR and BNN on both metrics. For Dataset-I, the PICP values ​​of the three methods are far from satisfactory. For Dataset-II and Dataset-III, ML-MLP-QR's PICP values ​​exceed 0.80, exceeding those of GPR and BNN. For all three datasets, ML-MLP-QR's PINAW values ​​are significantly lower than those of GPR and BNN, indicating that the resulting model has a narrower interval range. The ML-MLP-QR method achieves reasonable interval optimization based on quantiles. Although the GPR method is effective in terms of PICP, it fundamentally assumes that the data follows a Gaussian distribution. Unbalanced distribution of model parameters or insufficient sample size may cause the samples to not follow a Gaussian distribution, thus affecting the GPR model fit. The BNN method performs poorly on both PICP and PINAW metrics. Although theoretically, uncertainty can be flexibly captured through parameter distributions, this approach may have limitations in high-dimensional problems. Therefore, the present invention implements QR-based uncertainty estimation, giving priority to the balance between interval quality and computational efficiency.

[0102] To further quantify the effect of environmental conditions on A L The present invention uses GKDE to derive and predict A L The probability density of the value. Figure 11 (h) The effect of different bandwidths on the smoothness and continuity of the probability density curve is studied. The larger the bandwidth, the smoother the curve and the wider the range of data points; the smaller the bandwidth, the steeper the curve and the closer it may be to the original data distribution. The present invention uses grid search to select the optimal bandwidth and obtains the optimal bandwidth for each A in the seven test domains. L The median of the internal forecast is used as a point estimate. Figure 11 (a)-(g) are marked with blue lines. This approach is proven to be effective because using the quantile (0.5) as a point estimate can alleviate the problem of A L While point estimates may not fully capture uncertainty, interval estimates provide richer predictions, which are crucial for environmental regulation, stress responses, and breeding strategies. A 90% confidence interval (10% to 20% above or below the observed value) demonstrates that reliable probabilistic forecasts reduce the random variability inherent in single-point forecasts, providing a more robust alternative.

[0103] In summary, the present invention built a basic prediction model based on ML-MLP and fine-tuned it using the support set of the test domain. Even if only 5% of the test domain data was used for adaptive adjustment, ML-MLP always maintained a high accuracy in all domains. Compared with several commonly used models, it showed better generalization ability and prediction performance. In addition, its performance on unknown data sets also confirmed its robustness and generalization ability. Based on the ML-MLP model, the present invention modified the loss function in combination with the QR method and used GKDE to calculate the predicted probability density. A values ​​of different quantiles were obtained. L The estimated value can be used as a point estimate to characterize and quantify uncertainty. Verification on seven test domains shows that the proposed method outperforms other methods such as BNN and GPR.

Claims

1. A method for estimating the photosynthetic rate of a single crop leaf considering small sample learning and uncertainty prediction, characterized by: The steps include: Step 1: Obtain photosynthetic rate data of different crops under different environmental conditions to construct a data set, and divide the data set of each crop into subsets. The subsets are photosynthetic rate data obtained by fixing some environmental factors and changing other environmental factors; Step 2: Build a basic prediction model based on a multi-layer perceptron, use the divided subsets as test domains and training domains, and use meta-learning training to update the model parameters; Step 3: For unknown tasks, a loss function is constructed by combining the quantile regression method to fine-tune the basic prediction model to achieve interval prediction of photosynthetic rate; Step 4: Based on the interval prediction, Gaussian kernel density is introduced to estimate the probability distribution of photosynthetic rate under different environmental conditions, and the median is used as the point prediction value to simultaneously realize the visualization of photosynthetic rate point prediction and interval prediction.

2. The method for estimating the photosynthetic rate of a single crop leaf considering small sample learning and uncertainty prediction according to claim 1, characterized in that: The different environmental conditions refer to a combination of multiple different environmental factors that affect the photosynthetic rate of crops. The subset size of the data set does not exceed 5000. The subsets obtained by fixing one or several identical environmental factors for the same crop and changing other environmental factors are classified as the same type of subsets. When performing prediction task training, the subsets are divided into test domains and training domains.

3. The method for estimating photosynthetic rate of a single crop leaf considering small sample learning and uncertainty prediction according to claim 1, characterized in that: In step 2, a dataset of the same crop is divided into N learning problems, and then divided into a training domain and a test domain, and each training domain and test domain is divided into a support set and a query set, respectively. The meta-learning model parameter update includes an inner loop and an outer loop. The inner loop promotes the update of a specific task by calculating losses, adjusting gradients, and determining optimal task parameters. The outer loop coordinates cross-task backpropagation, uses the aggregated gradients of the inner loop training parameters to optimize global parameters, and then uses the global parameters to fine-tune the test domain task parameters.

4. The method for estimating photosynthetic rate of a single crop leaf considering small sample learning and uncertainty prediction according to claim 3, characterized in that: The method of dividing the dataset of the same crop into N learning problems specifically refers to dividing the photosynthetic rate dataset of the same crop under different combinations of environmental factors into several subsets of the same type with specific environmental characteristics based on one or several environmental factors, and then dividing these subsets into training domains and test domains to simulate the generalization ability of the model under cross-environmental conditions.

5. The method for estimating photosynthetic rate of a single crop leaf considering small sample learning and uncertainty prediction according to claim 4, characterized in that: In the subsets, a subset whose environmental factors are in a non-boundary interval is selected as a test domain.

6. The method for estimating photosynthetic rate of a single crop leaf considering small sample learning and uncertainty prediction according to claim 3, characterized in that: In the inner loop, the loss function is defined as: Among them, x i and y i Respectively represent Input and output, is the training domain support set, T i is the i-th learning problem, the superscript r indicates the training domain, and f θ (x i ) represents x i The calculated value obtained; The process of adjusting the gradient in the inner loop is expressed as: in, represents the training domain task parameters updated by the inner loop training, β represents the learning rate of the inner loop training, θ represents the MLP initialization trainable parameters, represents the gradient descent on θ, represents the inner loop loss function; The outer loop is expressed as: Among them, θ * is the global parameter of meta-learning, I is the number of training domain data, is the training domain query set, Denotes the outer loop loss function, which has the same structure as the inner loop loss function. The test domain support set is used to fine-tune the meta-learning global parameters to obtain the test domain task parameters, which are expressed as: Where α represents the learning rate of the fine-tuning process, is the training domain support set, Represents the loss function during fine-tuning.

7. The method for estimating photosynthetic rate of a single crop leaf considering small sample learning and uncertainty prediction according to claim 6, characterized in that: In step 3, the quantile regression method is introduced into the basic prediction model, and the loss function of the fine-tuning model under different quantiles is expressed as: where x j and y j represent Input and output, represents the x calculated by the model j Value, e j represents the difference between the predicted value and the observed value, τ is the preset quantile.

8. The method for estimating photosynthetic rate of a single crop leaf considering small sample learning and uncertainty prediction according to claim 7, characterized in that: The value range of the preset quantile τ is 0.05 to 0.95, representing a 90% confidence interval, and the 90% confidence interval is used to characterize the prediction uncertainty.

9. The method for estimating photosynthetic rate of a single crop leaf considering small sample learning and uncertainty prediction according to claim 7 or 8, characterized in that: In step 4, the probability density of the predicted value under the confidence interval is: Where x represents the target position of the estimated probability density; x k It represents the kth photosynthetic rate sample value falling within the selected confidence interval under a certain environmental condition, K is the total number of photosynthetic rate samples in the confidence interval, h is the bandwidth, which is used to control the smoothness of the estimate, and the Gaussian kernel function is used to perform weighted summation on the samples to obtain the probability density estimate of all photosynthetic rate values ​​within the confidence interval under this environmental condition.

10. The method for estimating photosynthetic rate of a single crop leaf considering small sample learning and uncertainty prediction according to claim 9, characterized in that: k=0 represents the photosynthetic rate value calculated when τ is 0.05, k=1 represents τ is 0.06, and so on, k=90 represents the photosynthetic rate value calculated when τ is 0.95.

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