Aiming at the problem that existing matching localization algorithms rely on high-memory and dense point cloud map, a light detection and ranging (LiDAR)/inertial measurement unit (IMU) matching localization algorithm based on an implicit neural map is proposed. Firstly, a lightweight and high-resolution implicit neural map is constructed using a shallow perceptron to predict the signed distance field. Secondly, low-frequency state estimation based on the lightweight implicit neural map is realized through a point and implicit neural model registration method. Meanwhile, to address the challenge of handling aggressive motion during single LiDAR implicit registration, an IMU pre-integration method is introduced. This provides predictive state estimation for implicit registration, reducing the number of iterations required during the registration process. Finally, robust high-frequency state estimation is realized by fusing the LiDAR odometry factor and IMU pre-integration factor based on the factor graph. Experimental results in the KITTI dataset, as well as in real-world indoor corridor and outdoor campus environments, demonstrate the effectiveness of the proposed algorithm. The memory usage of the implicit neural map is reduced by 87% compared to traditional point cloud maps, enabling a more lightweight map representation. In the KITTI dataset, the proposed LiDAR implicit registration algorithm improves the positioning accuracy by 43.4% compared to the traditional normal distributions transform (NDT) algorithm. Furthermore, the fusion algorithm, which incorporates IMU data, achieves a 60% improvement in positioning accuracy compared to the NDT-IMU algorithm. The centimeter-level real-time positioning capability of the proposed algorithm in small scenes is also verified in both practical indoor and outdoor campus. Meanwhile, analysis reveals that the integration of IMU significantly reduces the computational time required for implicit neural map registration, effectively enhancing real-time localization performance.