## 3 Results
### Calibrate and Evaluate the 3 PG-MT-LSTM Model  
For the present section, we initially employed the residual index to assess the efficacy of 3 PG-LSTM, 3 PG-MT-LSTM without compatibility rules, and 3 PG-MT-LSTM with compatibility rules. Next, we analyzed whether there was a significant difference between the model residuals (Figure 4). Our findings indicate that incorporating biomass compatibility rule into the model can substantially enhance its precision. Simultaneously, the difference between the different models obtained by the Solling site with a large dataset and the joint training of multi-site data is more significant. When the datasets are small, like at the Hyytiala or Bily Kriz sites, the difference in performance between the single-task LSTM models and the multi-task LSTM models is not statistically significant. Following the incorporation of compatibility regulations, in contrast to the solitary-task LSTM model, the performance of the four models was substantially enhanced to differing extents.  
Our assessment of the model was conducted based on the metrics outlined in Section 2.3 (Figure 5). The compatibility rule makes the model more capable of explaining the growth process and variation trend of distinct Norway spruce components (R\({}^{2}\) > 0.9). We observed that there is essentially no change in the prediction ability of the model for samples of varying forest ages since MAPE is comparable to the product of MAE and the average value of the data. This demonstrates that the coupling model described in this study performs better and is less susceptible to influence from outside sources. From the ratio of RMSE and MAE (RMSE/MAE\({}_{\text{solling\_stem}}\) = 6.96, RMSE/MAE\({}_{\text{hybrid\_stem}}\) = 6.45, RMSE/MAE\({}_{\text{pilly\_stem}}\) = 4.95, RMSE/MAE\({}_{\text{multi-model\_stem}}\) = 7.97), it is clear that adding the biomass compatibility rule makes it harder for the model's forecast process to avoid residual extremes. The model prioritizes minimizing the penalty values of the biomass compatibility loss function and the mean squared error (MSE) loss function. This results in models that sometimes sacrifice the accuracy of individual variables to improve overall model accuracy. Because of the volume and range of the data, the Solling site and multi-site estimation models have larger RMSE and MAE values. The inaccuracy is fairly substantial as a result of the wide variance in the stem biomass.  
Figure 3: Taking the Solling site as an example, (**a**) comparison of the prediction results of the traditional 3-PG model and the staged 3-PG model. Among them, the 3-PG simulation and 3-PG staged simulation are estimated values after optimizing the physiological and ecological parameters of the model using all observation data and stages, respectively. Observation is the value of the observation data. The black dashed line indicates that the stand was thinned in its corresponding year; (**b**) the changes in the optimal value of the main sensitive parameters—PF20, PRn, and alphaCx—of the 3-PG model at different stages (stand density).  
Figure 4: Residual distribution and significance analysis of three models in different sites. (a–c) represent the results obtained from training at the Solling site, Hyytiala site, and Bily Kriz site, while (d) represents the results obtained from training on data from all three sites combined. Asterisks refer to the significant differences in the models’ residuals (NS, no significant difference; *, \(p<0.05\); **, \(p<0.01\); ***, \(p<0.001\)).  
Figure 5: 3 PG-MT-LSTM model evaluation results. (**a–d**) represent the results obtained from training at the Solling site, Hyytiala site, Bily Kriz site, and all three sites combined. Different colors represent different evaluation indicators. It should be noted that the unit of RMSE is 10 t.