To study complex real-world phenomena using computer simulations, scientists often rely on ensemble datasets generated from multiple simulation runs with varying parameter configurations. This process can produce ensemble datasets with many members, making traditional data analysis pipelines impractical due to limited I/O bandwidth and disk capacity. Distribution-based data representations have been proposed as a promising solution. Processing data in situ to generate compact distribution-based representations not only alleviates the challenges of limited I/O bandwidth and disk capacity but also enables uncertainty quantification, thus mitigating the risk of misinterpretation. Nevertheless, distribution-based methods inherently sacrifice spatial information of data samples within the distribution, potentially reducing precision in the data analysis pipeline. To address this issue, we introduce a deep learning model to reconstruct data volume from the distribution representation. Instead of using a model that predicts a data block directly from its distribution representation, we propose a deep learning model based on the Sinkhorn operator and Gumbel trick that learns to map samples drawn from a distribution to spatial locations within the block. The deep learning model can support high-quality downstream data analysis and visualization, provide point-wise uncertainty quantification, and guarantee the distribution of the reconstructed data block follows the block's distribution representation.