Structure-aware learning of Poincaré return dynamics via symbolic decomposition

We develop a data-driven framework for approximating the first same-direction return time and the associated Poincaré map of chaotic flows. Learning these quantities is challenging because the return-time field may exhibit sharp localized variations, and nearby initial conditions may undergo qualitatively different pre-return excursions. We introduce a symbolic residual-learning strategy based on finite itineraries of transverse section crossings. A global neural network captures the dominant trend, and a classifier routes class-dependent residual corrections through soft or hard rules. Reflection symmetry is incorporated using a symmetry-aware input representation and symmetry-reduced symbolic classes, while the predicted return time is supplied as an auxiliary feature to the Poincaré-map model. Experiments on the Lorenz and Shimizu-Morioka systems show that the global configuration using a logarithmic return-time target and a symmetry-aware input representation improves accuracy, while symbolic residual correction further reduces errors, especially in high-gradient regions. The return-time input also provides useful information for learning the Poincaré map, although the resulting gains are system dependent. The results reveal a qualitative correspondence between prominent return-time gradients and finite symbolic-class boundaries, supporting symbolic decomposition as a structured approach to learning chaotic return dynamics.