Traditional approaches in hydrological modeling typically rely on parameter calibration within prior empirical functional forms and an integrated subsurface bucket. While effective, these fixed mathematical representations can sometimes limit flexibility in capturing complex and localized behaviors. To complement these approaches, we proposed a temporal difference loss regularization (TDLR) framework. Using meteorological and streamflow observations, this framework enables a Long Short-Term Memory (LSTM) network to extract interpretable functional representations of distinct hydrological processes in parallel within a mass-conserving, conceptual bucket architecture. Applications of TDLR across three hillslope catchments (Shihmen, Deji, and Jiji) yielded stable re-extracted function representations under the given structural constraints, offering strong internal consistency between extracted representations of processes. For the integrated subsurface bucket, the LSTM extracted patterns consistent with the transition from saturation-excess to infiltration-excess runoff. Furthermore, results from the Shihmen catchment captured localized variations that plausibly suggest preferential flow dynamics during heavy rainfall – patterns that are often challenging to represent using fixed functional forms. The framework also presents daily variations consistent with transitions from water- to energy-limited regimes in evapotranspiration. Canopy interception exhibited the expected asymptotic plateau, showing high sensitivity to precipitation and its covariance with wind speed. Additionally, the approach captured dynamic routing relationships that align with local morphological and meteorological characteristics. In our study area, the reconstruction accuracy of the extracted functional representations compares favorably with or exceeds both baseline hybrid models and pure black-box LSTMs. By demonstrating that catchment-scale behaviors can be effectively represented as emergent combinations of stable, data-driven relationships, this study provides a complementary perspective for exploring functional representations to advance hydrological modeling.