The flexible four-finger gripper, as a specialized robotic end-effector, is highly valued for its ability to passively adapt to the shape of objects and perform non-destructive grasping. However, the development of grasping detection algorithms for flexible four-finger grippers remains relatively unexplored. This paper addresses the unique characteristics of the flexible four-finger gripper by proposing a grasping detection method based on deep learning. Firstly, the Acute Angle Representation model (AAR-model), which is based on the structure of the flexible four-finger gripper and consists of grasp points and angles, is designed as the grasping representation model that reduces unnecessary rotations of the gripper and improves its versatility in grasping objects. Then, the Flexible Gripper Adaptive Attribute model (FGAA-model) is proposed to represent the grasping attributes of objects, calculate the grasp angles that meet the criteria of the AAR-model, and aggregate the AAR-models on the image data into a unified set, thereby circumventing the time-consuming process of pixel-level annotation. Finally, the Adaptive Grasping Neural Net (AGNN), which is based on Adaptive Feature Fusion and the Grasp Aware Network (AFFGA), is introduced by eliminating redundant network detection headers, fusing color and depth images as inputs, and incorporating a Series Atrous Spatial Pyramid (SASP) structure to produce more accurate grasp poses. Our method not only attains a remarkable accuracy of 97.62% on the Cornell dataset but also swiftly completes grasping detection within 25 ms. In practical robotic arm grasping tests, where a robot is outfitted with a flexible four-finger gripper, it successfully grasps unknown objects with a 96% success rate. These results underscore the reliability and real-time performance of our method, significantly enhancing the gripper's adaptability and precision when handling objects of varying sizes and shapes. This advancement provides a powerful technical solution for robots utilizing flexible four-finger grippers, enabling autonomous, real-time, and highly accurate grasping maneuvers. Moreover, it addresses the persistent challenge of the scarcity of efficient grasping detection techniques tailored for flexible four-finger grippers.