In real-world application scenarios, training an effective prediction model relies on a large number of measured samples, whereas accurately measuring large-scale data is often time-consuming, labor-intensive, and costly. Active learning is one of the main ways to reduce the cost of sample collection. Therefore, we applied an active learning method, namely the sampling strategy of NRBO-AL. We iteratively selected the most valuable simulation data and enhanced the generalization ability of the training model based on the simulation dataset to actual scenarios to improve the performance of the prediction model with minimum cost.

The specific implementation steps are as follows.

The shortest distance from X t to X m is represented as follows:

3. We established a multiple linear regression model f(x) based on representative samples and selected simulated data samples with a smaller distance from the simulated biomass y t as the updated training dataset. We also used the updated training dataset to train regression models for estimating biomass in the measured dataset.

Among them, the distance from f ( x t ) to y t is represented as follows:

We selected a new sample from the measurement dataset for each iteration to update the training dataset until the maximum number of samples (K) was reached. To balance sampling costs and model performance, we usually set the maximum number of new samples to 10% of the measured datasets (Wan et al., 2020, 2022). However, for small datasets, K will be set higher (Wan et al., 2023), and if too few samples are selected, effective features cannot be learned.

The traditional random selection method is based on the maximum number of new samples, and it directly selects samples from the measurement dataset, which cannot guarantee the quality of new samples. Contrary to this random method, the greedy algorithm based on NRBO can impose constraints on the quality of new samples, thereby enhancing the potential to improve model performance beyond random selection.

To achieve high-precision prediction mechanisms for maize SPU, we comprehensively explored and analyzed the use of small-scale remote sensing data ( Figure 2 ). We established two empirical models based solely on actual ground measurement data based on two-dimensional correlation spectra and a hybrid model based on Newton–Raphson-based optimizer’s active learning (NRBO-AL) optimization and a combined prediction model combining mixed models and partial empirical models.

We used 2T2DCOS (see Appendix S2.3.1 ) to couple the spectra of the filling and milk stages to establish an empirical model. We selected new spectral features based on peak position screening for accurate AGB prediction. We also used first-order differentially enhanced two-dimensional correlation spectroscopy (1Der-2DCOS) (see Appendix S2.3.2 ) to analyze the spectral changes under P concentration disturbance, thereby predicting the P concentration in maize canopy leaves. We multiplied the two as SPU, except for the grains. For predicting P uptake by grains, we utilized 1Der-2DCOS to analyze the spectral changes under yield disturbances, thus establishing a prediction model for maize kernel biomass. We multiplied this by the P concentration in the seeds to determine the amount of P absorbed by the seeds.

To establish a hybrid model for predicting leaf biomass, we conducted a sensitivity analysis of the parameters in the PROSAIL-5B model (Jacquemoud et al., 2009) (see Appendix S2.3.3 ). We determined the parameter’s sensitivity through EFAST sensitivity analysis (Qiao et al., 2020) (see Appendix S2.3.4 ) and using the mixing sine and cosine algorithm with the Lévy flying chaotic sparrow algorithm (Qinghua et al., 2021) (see Appendix S2.3.5 ) to optimize the model parameters and determine the range of high-sensitivity parameters and the values of other low-sensitivity parameters. Subsequently, we screened the simulation dataset generated by the PROSAIL-5B model using the NRBO-AL method, which comprehensively improved the PROSAIL-5B model’s effectiveness, achieving localization of model parameters and high-precision prediction of leaf biomass. We multiplied this by the P concentration in the canopy leaves to determine the amount of P absorbed by corn leaves. Subsequently, we established a prediction model for P uptake by stems and grains based on P uptake by corn leaves.

Using actual observed data, we evaluated the accuracy of the four SPU combination prediction models by determining the coefficient (R ²), root mean square error (RMSE), and relative percentage difference (RPD) (see Appendix S2.3.6 ) and selecting the best prediction model from them.

Taking the prediction of whole reproductive period LPC as an example, we selected different sample sizes to investigate the possibility of stable prediction on small datasets. We selected samples consistent with the entire dataset’s distribution, namely 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, and 100% samples (n=252), and compared and analyzed the stability of different dimensionality reduction methods and the combination effect of prediction methods.

We applied derivative transformation to the spectrum to eliminate background noise, and compared the stability of different dimensionality reduction methods such as Successive Projections Algorithm (SPA), Least Absolute Shrinkage and Selection Operator (LASSO), (Elastic Net) EN, and 2DCOS. As the number of repetitions decreased from 10 to 5, Table 1 recorded the changes in the number of bands under different methods. The SPA method has almost no repeated band selection, while both the EN and 2DCOS methods have repeated band selection. However, at a repetition rate of 10, the 2DCOS method shows a higher number of bands. When the number of repetitions decreases, the number of bands in the EN method slightly increases, while the 2DCOS method shows a stable trend. This set of data reveals the dynamic variation characteristics of the number of bands under different methods as the number of repetitions changes. The selected bands for 2DCOS dimensionality reduction are independent of the amount of data, exhibiting strong stability and unique advantages (Yue et al., 2021).

In this study, we selected bands with more than 8 datasets that were repeatedly selected under 6 different preprocessing methods to test the LPC prediction performance. Among them, we compared the prediction performance of selecting different bands and the same band under different data partitioning (as shown in Figure 3 ).

From Figure 3 , it can be observed that for the raw data, the prediction performance when selecting the same sensitive bands is inferior to when selecting different sensitive bands (-0.1275 [-0.1022, -0.1529]). This suggests that the sensitive bands in the raw data are unstable and susceptible to interference from external factors such as background noise, leading to shifts in the sensitive bands. Therefore, after removing the impact of background noise using derivative transformation, the prediction performance significantly improves. Under the first-order derivative preprocessing, for datasets with non-uniform bands, EN’s dimensionality reduction performs better than 2DCOS. This is because the sensitive bands after EN dimensionality reduction have lower redundancy, indicating that EN dimensionality reduction has poor universality and is more influenced by the heterogeneity of the dataset. In contrast, on a uniform dataset, 2DCOS shows better prediction performance because it has higher universality in dimensionality reduction, performing well across different datasets with minimal model tuning required. Additionally, under 2DCOS dimensionality reduction, regardless of whether fixed bands are selected, the prediction performance difference is very small (-0.0033 [-0.0119, 0.0052]). Therefore, combining derivative transformation with 2DCOS dimensionality reduction provides the most stable regression model.

We compared three different types of regression methods—Support Vector Regression (SVR), Back-Propagation Neural Network (BPNN), and Gradient Boosting Regression Tree (GBRT)—under varying data sizes. While the SVR and BPNN models perform well on small datasets, their predictive accuracy significantly decreases with larger datasets (as shown in Figure 4 ). When the data volume is less than 100, the results of 1Der-2DCOS-SVR are higher and the model is more stable. However, when the amount of data is large, the prediction effect is average. The 1Der-2DCOS-GBRT model is the most stable among these models. No matter how the size of the dataset changes, its prediction accuracy remains consistent, which can achieve more robust and accurate predictions. This result also solves the problem of using small data to predict.

We established a combined model to predict corn SPU based on the above research results. Combining the advantages of empirical and hybrid models in SPU prediction, the combined model consists of a leaf P uptake prediction model, a stem P uptake prediction model in the hybrid model, and a grain P uptake model in the empirical model. Figure 12 shows that the combination model achieved the best performance. Although the introduction of the NRBO-AL sampling strategy greatly increases the computational complexity and time cost of the model, the accuracy of the prediction model has also been considerably improved (R ² = 0.8707, RPD = 2.7140), a 24.63% increase.

Based on the combined prediction model, it is possible to achieve nondestructive prediction of SPU in maize plants based on hyperspectral data, which will provide important guidance for timely P nutrition management decisions.

Compared with visible and near-infrared spectra, 2DCOS and 2T2DCOS are advanced spectroscopic analysis techniques with wider applications in infrared, fluorescence, and Raman spectra (Noda, 2017; Li et al., 2021a). Nevertheless, researchers such as Zhang et al. (2021a) have proposed using shape- and amplitude-enhanced 2DCOS combined with transfer learning to estimate chlorophyll content in winter wheat. Zhang et al. (2024) used a 2DCOS analysis method based on first-order differential enhancement to highlight chlorophyll information in the spectrum. These results indicate that 2DCOS has enormous potential in visible/near-infrared remote sensing.

Our study found that compared to the Raw-2DCOS, baseline-corrected 2DCOS can effectively improve the accuracy of the prediction model (15–17%) ( Supplementary Figure S8 ). This improvement is due to the narrowing of the absorption peak in the first-order derivative spectrum and the sharpening of the peak shape after derivative conversion, this method extends traditional spectra in two dimensions and improves the screening ability of sensitive bands based on first-order differential spectra, thereby obtaining more effective sample information (Dong et al., 2021). Although the peak of Raw-2DCOS is clear and the feature information is relatively intuitive, based on the full band spectrum, 1Der-2DCOS displays subtler feature information, including the number of self-peaks and cross-peaks, peak intensity, and their correlation ( Figure 13 ). These are helpful for the attribution analysis and sample identification analysis of different peaks.

Although biomass prediction models vary under the influence of different external factors ( Figures 6 , 7 ), small sample datasets can still provide predictions with sufficient accuracy in this variability. To better characterize the relationship between spectra at different growth stages and AGB, we proposed a 2DCOS analysis method that couples filling and milk stages. Supplementary Figure S9 shows the autocorrelation peak extracted from the filling-milk-2T2DCOS synchronous spectrum, where the peak position is relatively stable. Therefore, this method performs well on small datasets, with a high repetition rate of the selected sensitive bands and good universality.

Our experiment is based only on ground-scale data from the fields used to construct the models. In the next step, we will use the small dataset model of this study to accurately monitor and predict satellite remote sensing data, reduce the cost of on-site sampling, provide a scientific basis for management decision-making in agriculture, the environment, and other fields, and promote the application and development of remote sensing technology in P uptake in aboveground parts of farmland.

Research has shown a synergistic effect between leaves and stems in absorbing and using P elements (Wu et al., 2015). For example, increasing the P uptake capacity of leaves may promote the transportation and use of P in stems, thereby improving the overall P utilization efficiency of plants. In contrast, the storage and transportation of P in stems may also affect the uptake and utilization of P by leaves, and there is an interactive relationship between the two. Therefore, we conducted a correlation analysis between the two to predict the P uptake of corn stalks without any damage.

However, the relationship between P uptake by maize leaves and stems is not constant but varies with growth stage and environmental conditions (Zhang et al., 2022).

First, there are substantial differences in the function and tissue structure of leaves and stems during plants’ growth and development stages, which leads to different ways in which they absorb and utilize P (Adam et al., 2018). In general, leaves are the main organs for photosynthesis (Wu et al., 2019); therefore, a high demand for P is required for photosynthesis. Relatively, the stem mainly plays a role in supporting and transporting nutrients (Wang and Ning, 2019), and its demand for P is relatively low. Therefore, during the growth process of plants, leaves usually absorb and utilize P more actively than stems. Second, plant growth stage and external environmental factors influence the relationship between leaf P uptake and stem P uptake (Marschner et al., 1996) and external environmental factors (Bi et al., 2020; Mao et al., 2021). In the early plant growth stages, the growth rate of leaves is relatively fast; therefore, their demand for P is also relatively high (Kerkhoff et al., 2006). As the plant grows, the growth rate of the stem gradually accelerates, and its demand for P will also gradually increase. Therefore, to understand the relationship between P uptake by corn leaves and straw, we must consider other factors, such as plant growth stage and environmental factors.

We only investigated the P uptake by leaves and stems during one stage of growth and development without considering the influence of external factors; thus, we could not fully interpret the complex and diverse relationship between leaf P uptake and stem P uptake. Although the stem P uptake prediction results in this study are generally low, the impact on the overall SPU is relatively small. If this relationship can be further studied in the future, it will help to better understand the demand and utilization patterns of P in plants, provide a scientific basis and technical support for agricultural production, and have important significance in plant growth and development and P nutrition management.

P plays a crucial role in various physiological processes in plants (Lambers, 2022; Khan et al., 2023), including energy transfer and nucleic acid synthesis. However, its availability in soil is often low (Gao and DeLuca, 2018; Cao et al., 2021), which limits the efficiency of plant uptake. As a result, environmental factors (Song et al., 2020) and cultivation practices (Zhang et al., 2016) significantly influence P uptake. Additionally, the impact of interannual planting, as a relatively long-term cultivation factor, can also affect P uptake and utilization.

With interannual planting, the availability of P in the soil may be affected by processes such as annual fertilization (He et al., 2018), adaptability of plant roots (Liu, 2021) and soil P conversion (Hussain et al., 2023; He et al., 2018). Long-term application of chemical phosphorus fertilizers can lead to the accumulation of phosphorus in the soil, particularly when soil phosphorus levels reach saturation. This may reduce the plant’s ability to absorb phosphorus. Additionally, the continuous cultivation of certain crops, especially those with high phosphorus requirements, can result in the gradual depletion of soil phosphorus over the years, potentially impacting phosphorus uptake in subsequent seasons.

Therefore, proper cultivation practices such as planting density (Luo et al., 2023; Yan et al., 2024), irrigation methods (Li et al., 2021b; Wang et al., 2021), and fertilization management can improve P availability in the soil and enhance plant P uptake. When planting density is high, root space is limited, which can increase competition among roots and reduce their ability to absorb P. Proper density management can mitigate this issue and improve P use efficiency in the soil. Irrigation has a significant impact on P uptake (Wang et al., 2021); excessive irrigation can lead to P leaching or loss, particularly when soil P levels are high. On the other hand, insufficient irrigation can cause soil water stress, restricting the plant’s ability to absorb P. Thus, effective irrigation management not only improves water use efficiency but also optimizes P uptake. The application method and timing of P fertilizer directly affect plant P uptake. Therefore, it is necessary to forecast P uptake in the above-ground plant parts before the season and implement integrated measures based on local soil conditions and crop types (Lyu et al., 2016) to ensure the efficient use of P fertilizers.

In this study, we used a combined prediction model to improve the accuracy of inverting maize SPU from field spectral data and compared its performance with that of empirical methods and a hybrid model. Overall, the combination model has the highest accuracy and is superior to the empirical models ( Figure 7 ) and hybrid models ( Figure 11 ). This may be because, in the empirical model, we established a relationship between the sum of the biomass of organs (such as leaves, stems, and axes) and the spectra. Because of crop canopy obstruction, the response of spectral reflectance to AGB may only be concentrated on the biomass of canopy leaves and cannot fully extract effective information from other organs, resulting in slightly poor prediction performance.

The hybrid model based on NRBO-AL constraints also performs better than the general hybrid models ( Supplementary Figures S7 , S8 ). When there are many unknown variables in the search space of the PROSAIL model, RTM inversion makes it challenging to accurately infer the biophysical variables of corn (Liang et al., 2015; Danner et al., 2021). Although prior knowledge under the constraint of actual measurement range increases the feasibility of hybrid methods for simulating plant growth states, it cannot replace real datasets. This may also be why despite limiting the search space for highly sensitive parameters in the PROSAIL model, the results are still not as accurate as those of the hybrid model based on active learning correction ( Supplementary Figure S7 ).

To address this issue, we combined active learning with hybrid methods to improve the estimation of leaf biomass through canopy reflectance. Our results indicate that compared with random selection, the NRBO-AL sampling strategy can select more representative samples in each iteration and exhibit better distribution in the sample space, with higher diversity and representativeness ( Supplementary Figure S10 ). As the number of simulated data increases, this difference may become more pronounced, and the larger the randomly selected sample size, the more errors and uncertainties it may bring (Verrelst et al., 2016).

Féret et al. (2021) reported that the scale, reflectance anisotropy, and canopy structure from leaves to canopy can affect inversion accuracy. However, we improved the prediction accuracy of SPU and reduced the uncertainty level of the simulated dataset using a hybrid model based on active learning correction. This indicates that the model’s predictive performance can achieve consistent results under different environmental conditions.

Therefore, using high-quality and informative samples is crucial instead of numerous training datasets (Verrelst et al., 2016), as this will reduce costs and computational requirements and improve model performance. This requires selecting sufficiently representative data (Hauser et al., 2021) to fully cover the possible range of these variables, i.e., an active learning method based on the NRBO-AL sampling strategy and selecting high-quality training sets from a large number of simulated datasets. Our work will drive further research on biomass prediction models using the PROSAIL model.

Although in situ samples are still required for predicting P uptake in combined models, our results show that active learning and two-dimensional spectral analysis require fewer samples to achieve excellent model performance compared with traditional empirical prediction models. Therefore, this study can reduce sampling costs and establish an estimation model by reducing the number of in situ samples. In practical situations, it is sufficient to select representative samples at critical reproductive stages; therefore, there is no need to accumulate many on-site samples for model training.

In addition, our model performs very well in monitoring P uptake in maize using remote sensing data from scientific experimental area-scale experiments. If new samples from different experiments can be combined, active learning can continuously improve the model. This continuous updating mechanism is highly robust under different plant species and environmental conditions. Therefore, in the future, we will collect ground measurement samples and remote sensing satellite images from different regions, experimental treatments, and future crops, calibrate the model to local crops, and provide spatial distribution predictions of P uptake over a large area. However, an important source of uncertainty in applying this method to large-scale farmland is the relatively large spatial variability of P observed in small-scale crops on the planting site.

To train and validate models using satellite remote sensing data in this situation, in our future work, we will (i) determine pixel sizes smaller than the spatial variability, (ii) collect accurate samples and image localization, and (iii) select sampling areas with minimal spatial variability. We also suggest that further research should include the evaluation of P uptake prediction models for large-scale farmland to evaluate the accuracy of these models in practical applications.