We introduce a simple yet powerful framework for training large language models. In contrast to the standard autoregressive next-token prediction based on an exact prefix, we propose a perturbation-based procedure that first transforms the prefix into a semantic neighbor and then conditions on this perturbed variant for next-token prediction. This yields a hierarchical model with a pre-post-additive noise structure. Within this framework, we develop a rigorous theory of extrapolability, namely, the capacity of a model class to make reliable predictions for token sequences that lie outside the empirical support of the training corpus. We evaluate the finite-sample performance of the proposed procedure using both synthetic and real-world language data. Results show that the proposed method consistently improves out-of-support prediction while maintaining competitive in-support performance, demonstrating that perturbation offers a practical route to language modeling.

The application of process-based and data-driven hydrological models is crucial in modern hydrological research, especially for predicting key water cycle variables such as runoff, evapotranspiration (ET), and soil moisture. These models provide a scientific basis for water resource management, flood forecasting, and ecological protection. Process-based models simulate the physical mechanisms of watershed hydrological processes, while data-driven models leverage large datasets and advanced machine learning algorithms. This paper reviewed and compared methods for assessing and enhancing the extrapolability of both model types, discussing their prospects and limitations. Key strategies include the use of leave-one-out cross-validation and similarity-based methods to evaluate model performance in ungauged regions. Deep learning, transfer learning, and domain adaptation techniques are also promising in their potential to improve model predictions in data-sparse and extreme conditions. Interdisciplinary collaboration and continuous algorithmic advancements are also important to strengthen the global applicability and reliability of hydrological models.