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We introduce NeuralVDB, which improves on an existing industry standard for efficient storage of sparse volumetric data, denoted VDB [Museth 2013], by leveraging recent advancements in machine learning. Our novel hybrid data structure can reduce the memory footprints of VDB volumes by orders of magnitude, while maintaining its flexibility and only incurring small (user-controlled) compression errors. Specifically, NeuralVDB replaces the lower nodes of a shallow and wide VDB tree structure with multiple hierarchical neural networks that separately encode topology and value information by means of neural classifiers and regressors respectively. This approach is proven to maximize the compression ratio while maintaining the spatial adaptivity offered by the higher-level VDB data structure. For sparse signed distance fields and density volumes, we have observed compression ratios on the order of 10x to more than 100x from already compressed VDB inputs, with little to no visual artifacts. Furthermore, NeuralVDB is shown to offer more effective compression performance compared to other neural representations such as Neural Geometric Level of Detail [Takikawa et al. 2021], Variable Bitrate Neural Fields [Takikawa et al. 2022a], and Instant Neural Graphics Primitives [Müller et al. 2022]. Finally, we demonstrate how warm-starting from previous frames can accelerate training, i.e., compression, of animated volumes as well as improve temporal coherency of model inference, i.e., decompression.