## 4 Discussion  
### Model Prediction and Output  
To avoid an overlap between the training and prediction sets, the algorithm employed a ten-fold cross-validation, which partitioned the data into ten subsets, each in turn serving as a validation set while the remainder formed the training set. This approach reduced the bias and prevented the training set from coincidentally resembling the data condition, enhancing the reliability of the learning outcomes.  
The BP neural network optimization model used the number of commercial POI facilities in each grid as the dependent variable and other functional POI facilities as independent variables.  
The research adopted a grid-based approach, selecting square grids as the focal point of study due to the convenience they offerred in conducting spatial analysis and the reliability they provided in terms of data support. The study area was methodically subdivided into  
Figure 5: R\({}^{2}\) score and error line in momentum-learning rate.  
a grid of 1 km \(\times\) 1 km squares using the fishing net tool, resulting in a comprehensive total of 1097 individual units for a detailed analysis. This resulted in an optimized commercial facility with an R\({}^{2}\) = 0.7046, and an increase in the total number of commercial facilities from 44,034 to 49,871.  
Figure 6 displays the static density map of both the existing and optimized commercial spaces, with the density of commercial facilities in each cell indicated. The optimization results, which were categorized and visualized by the same criteria, were exported to the field properties in the ArcGIS software. This revealed a prospective development of commercial space in Weinan City's center, which was focused around two agglomeration points in a multi-core pattern, with commercial complexes propelling the growth. Such a spatial representation not only encapsulated the current state but also forecasted the evolution of commercial zones, signifying where growth is likely to intensify.  
The envisioned commercial expansion, as it unfurls along the main transportation routes, commands a wider influence than the currently observed commercial footprint. Significant developments are poised to stretch westward into the dynamic high-tech industrial development zone, as well as northward, bridging the river into the economic and technological development zone, reflecting strategic foresight in urban commercial growth.  
Figure 7 deftly superimposes the projected commercial facilities upon the existing commercial landscape, spotlighting the strategic areas of Chaoyang Street and Lotian Street at the heart of Weinan City's center. Chaoyang Street is already home to sizable commercial complexes, whereas Lotian Street, despite its paucity of large-scale commercial entities, demonstrates a dense congregation of commercial activity.  
These observations are astutely aligned with the principles of traditional agglomeration theory and central place theory. Chaoyang Street and Lotian Street are identified as nascent hubs of urban commerce, with Chaoyang Street already exhibiting a pronounced degree of commercial clustering. The forecasted growth in commercial space dovetails with the optimized projections, thereby affirming the validity of agglomeration tendencies and central place theory within the urban planning context.  
Figure 6: Static density variation diagram.