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We introduce a novel approach called Locally-Subdivided Neural Intersection Function (LSNIF) that replaces bottom-level BVHs used as traditional geometric representations with a neural network. Our method introduces a sparse hash grid encoding scheme incorporating geometry voxelization, a scene-agnostic training data collection, and a tailored loss function. It enables the network to output not only visibility but also hit-point information and material indices. LSNIF can be trained offline for a single object, allowing us to use LSNIF as a replacement for its corresponding BVH. With these designs, the network can handle hit-point queries from any arbitrary viewpoint, supporting all types of rays in the rendering pipeline. 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