Detecting unstable periodic orbits (UPOs) based solely on time series is an essential data-driven problem, attracting a great deal of attention and arousing numerous efforts, in nonlinear sciences. Previous efforts and their developed algorithms, though falling into a category of model-free methodology, dealt with the time series mostly with a regular sampling rate. Here, we develop a data-driven and model-free framework for detecting UPOs in chaotic systems using the irregularly sampled time series. This framework articulates the neural differential equations (NDEs), a recently developed and powerful machine learning technique, with the adaptive delayed feedback (ADF) technique. Since the NDEs own the exceptional capability of accurate reconstruction of chaotic systems based on the observational time series with irregular sampling rates, UPOs detection in this scenario could be enhanced by an integration of the NDEs and the ADF technique. We demonstrate the effectiveness of the articulated framework on representative examples.