# Error-Bounded Learned Scientific Data Compression with Preservation of Derived Quantities
## 1 Introduction  
As high-performance computing (HPC) enters the exascale era, scientific simulations are becoming capable of predicting physics phenomena with high accuracy, while generating extremely large volumes of data at high velocity. The increase in data volumes is rapidly outpacing the growth of storage and I/O capacities. For example, climate simulations [1] can generate 260 TB every 16 s when running large ensembles of high-fidelity 1 km \(\times\) 1 km simulations of 15 years of climate. This necessitates the development of data compression algorithms for scientific applications that are reliable and can achieve a large reduction ratio. There have been successful approaches for traditional image and video compression [2], but they are not directly applicable to scientific data compression because of scientific post-processing requirements, which markedly differ from visual perception. Unlike image and video processing, where the dimensionality of data is two or three, scientific datasets are typically high-dimensional tensors and have underlying spatial and temporal structure. Compressing scientific data is challenging because we need a significant reduction of the Primary Data (PD) with a guaranteed error bound, and we need to preserve the downstream Quantities Of Interest (QoI) as accurately as possible. The necessity of guaranteed error bounds of scientific data is due to the need for high accuracy after compression. For example, in turbulence simulations, the paths of turbulence can be estimated by properties such as latitude, longitude, pressure, and so on, which should be preserved to high fidelityduring data reduction. This has to be achieved while ensuring that compression algorithms continue to cater to high-performance computing standards to reduce the time spent storing and retrieving the data. The mathematical relationships between the point-wise errors of PD and the errors of QoI are central to scientific data compression and create unique challenges for the reduction community. For this reason, we focus on simultaneously achieving lossy data compression while preserving the QoI to within specific error tolerances. Data reduction for scientific applications has been extensively studied. Error-bounded lossy compression techniques are considered to be one of the most successful for these applications. Compared to lossless compressors [3; 4] that achieve limited compression [5], error-bounded lossy compressors can significantly reduce the data and, moreover, achieve high fidelity of the reconstructed data according to the user's required data distortion bounds. The most well-known and state-of-the-art error-bounded lossy compressors in the literature are based on prediction (SZ [6; 7; 8]), block transform (ZFP [9]), and multilevel decomposition (MGARD [10; 11; 12]). SZ is designed based on data prediction models. It first performs a preprocessing step such as linearization of multi-dimensional data and predicts each data point based on its neighboring values using a Lorenzo predictor [13]. ZFP is based on orthogonal block transforms and embedded coding. It breaks the data into a set of blocks (4 \(\times\) 4 \(\times\) 4 for 3D data) and then applies an orthogonal block transformation to decorrelate the data. The generated coefficients are encoded to produce compressed data. Most error-bounded lossy compression algorithms such as SZ [6; 7; 8] and ZFP [9] just focus on preserving point-wise values and provide guaranteed error control on the primary data. However, this does not ensure QoI preservation. Derived quantities are critical to accurate post-analysis. Inconsistent moments and energy in XGC applications that simulate gyrokinetic particle motion, for example, cause totally different fusion simulation analysis. MGARD [10; 11; 12] is currently the leading error-controlled lossy compression technique with QoI preservation. MGARD can bound the errors of linear QoI by using a non-uniform multilevel subgrid algorithm and providing a mathematical error analysis.