## 4 Discussion
BP neural network algorithms use single hidden layer feedforward neural networks (SLFNs) as universal approximators, but their parameter optimization is complicated [92]. ELM just solves this problem, and its hidden layer parameters do not need to be optimized. At the same time, the approximation capability of SLFNs can be maintained. Therefore, the correlation between SLN and NMP created an overlap of known information in CNN and ELM. Thus, the accuracy of CNN and ELM decreased, so it was better to use only one kind of physiological information. In the verification using the RO1 site, the addition of SLN and NMP enabled different ML methods to obtain different results using the unbiased estimator (URMSE) for evaluation. All ML methods had a high degree of fit of scatter points after considering physiological information, which proved its effectiveness. In the process of evaluating the accuracy of GPP prediction results, data uncertainty has impacts on the verification of results. First, inaccurate GPP observations in flux towers will produce errors [75], and there are uncertainties in NDVI remote sensing observation data sources. The GPP at three Nebraska sites ranges from 0 and 32 gC\(\cdot\)m\({}^{-2}\).day\({}^{-1}\), which leads to the saturation of NDVI [93]. The reconstruction of NDVI using a 8-day MOD09Q1 product and a daily MOD09Q1 product can generally obtain higher accuracy. However, in cloudy conditions, the 8-day composite product still contains continuous noise [53]. Secondly, the maximum allowable SLN we considered was obtained at leaf scale, which has the problem of scale mismatch with meteorological data and remote sensing data. Moreover, the SLN of the RO1 site was derived from the polynomial fitting relationship between NMP and SLN at three sites in Nebraska, and the points in Figure 3 are still discrete to some extent, which also brings uncertainty. Thirdly, when RO1 was used for verification, all models produced a high RMSE, which was probably due to the differences in farmland management between the ROI site and the three sites in Nebraska. Differences in moisture and soil brought about spatial consistency. Finally, ML method sensitivity is highly dependent on the amount of data and its accuracy [94]. While the data set used in this study has high accuracy, errors in the training data set and correlations between input variables (such as SLN and NMP) can affect GPP estimation. Through the results, the advantages of SLN and NMP in improving maize daily GPP estimation via four ML methods have been demonstrated. In the future, it would be interesting to consider other vegetation indices in the input to correct the saturation phenomenon of NDVI in high-value GPP. Moreover, certain important factors associated with site meteorological data, such as vapor pressure deficit (VPD) and soil moisture, are anticipated to be incorporated into the model. In addition, utilizing longer time spans of data to increase data volume to enhance data alignment is another method to enhance accuracy. Note that, owing to the fixed polynomial function between SLN and NMP, data redundancy seemed to occur. On the one hand, in this case, the introduction of SLN or NMP alone, rather than both, may guarantee the robustness of ML methods, such as ELM and CNN (Table 3). On the other hand, direct measurement or high-frequency remote sensing inversion of SLN is needed in the future to further study the value of SLN. However, the big gap between satellite pixel and leaf blade areas will make it a big challenge for SLN inversion via satellite platforms. Fortunately, low-altitude unmanned aerial vehicles (UAVs) provide an available way at the regional scale.