The exponential growth of geographic data reveals limitations in traditional spatial indices. Spatial learned indices that incorporate machine learning models have been proposed to enhance index performance. However, due to the considerable overhead of fine-grained data partitioning and the complexity of hierarchical model structures, existing spatial learned indices still exhibit bottlenecks in index construction and query processing. To address the aforementioned issues, we propose TBSI, an in-memory Transformer-based spatial learned index with an end-to-end structure. TBSI employs an enhanced quadtree to optimize data partitioning and utilizes a Transformer-based position prediction model to manage each data partition, preserving a simple yet effective index structure. TBSI exhibits superior performance in both index construction and query processing. We also design spatial query algorithms based on a filtering-refinement mechanism and data update algorithms based on buffers and flag arrays to support efficient query processing and index maintenance. Extensive experiments on real-world and synthetic datasets demonstrated that, compared to baselines, TBSI achieved up to 23.4 times speedup in build time, up to 24.3 times reduction in index size, up to 5.9 times improvement in range queries, and up to 4.5 times improvement in kNN queries. Also, TBSI exhibited robust adaptability to dynamic data updates.

Spatial databases are the main means to manage geo-big data, and learned spatial indices are a novel approach to improve the spatial retrieval performance of spatial databases by modeling the data distribution. However, the complex hierarchical structures in current learning models pose significant limitations, including prolonged construction times, slow data updates, and suboptimal dynamic query performance. Consequently, improving the efficiency of both index construction and updates is essential. We addressed these challenges by introducing a new method, the Spatial Uniform Partition Learned Index (SUPLI). SUPLI utilizes an iterative uniform partitioning algorithm that simplifies data distribution by uniformly segmenting space and applies a linear regression function—instead of a neural network model—to enable efficient index construction. Additionally, SUPLI incorporates query load optimization and historical query learning strategies, which dynamically adjust the spatial query algorithm to enhance query efficiency. Furthermore, a buffer structure is employed to store change information, facilitating efficient updates. Comparative evaluations conducted on three synthetic datasets and two real-world datasets show that SUPLI outperforms the classic R-tree by an order of magnitude in construction, query, and update performance, and demonstrates additional advantages over similar spatial learned indices, such as SPRIG and LISA.

The cross-fertilization of the fast-developing AI technology and spatial indexing has given rise to spatial learned indexes. However, these indexes rely on historical data distributions to build models, which limits their ability to anticipate data that has not yet arrived. To address this, we propose a novel Spatio-Temporal Update Method (STUM) that enhances conventional spatial learned indexes by introducing a Spatial Delta Area (SDA) for updates without altering their hierarchical structure. STUM learns spatio-temporal auto-correlation from historical data and integrates predicted future distributions. We apply STUM to the Spatial Learned Block Range INdex (SLBRIN), resulting in the development of the Spatio-Temporal Updatable learned Block Range INdex (STUBRIN), which adopts Revmap to integrate spatio-temporal sequence predictions with the spatial block range. STUBRIN optimizes the retraining process by learning the temporal continuity from spatial distribution and fusing it into the error threshold control mechanism and historical delta learning mechanism. Our results show that STUBRIN achieves 1.9-2.4×, 1.8-13.3×, 3.4-6.7× better build, query and update performance compared to state-ofthe-art methods. Additionally, STUBRIN offers superior query and update stability. For concurrent learned indexes, we have also designed parallel scheduling for STUBRIN, which improves build, query and update performance by 6.2-6.8×, 0.3-4.2×, 2.6-5.5×, without increasing the index size.

This article investigates Nash equilibrium seeking for nonzero-sum games of switched nonlinear systems. A novel cost function is presented that measures the system state cost and control cost while considering the dynamics under different switching modes. Then, a new coupled switching Hamilton-Jacobi (HJ) equation is derived. To address the challenge of directly solving the HJ equation, an event-triggered two-stage reinforcement learning strategy is proposed. Upon event triggering, each player’s switching law determines the optimal subsystem to switch to by minimizing the HJ equation. Subsequently, the corresponding learning law for each player updates its respective input via the determined optimal subsystem. The proposed algorithm achieves Nash equilibrium while ensuring system stability. Furthermore, Zeno behavior is avoided, and the computational and communication loads are reduced. Finally, the proposed algorithm’s efficacy is substantiated through two simulation examples.