In this paper we investigate the differential topological properties of a subclass of singular space--subcarteisan spaces. First, we get further result on partition of unity for differential spaces. Second, we establish the tubular neighborhood theorem for subcartesian spaces with constant structural dimension. Third, we generalize the concept of Morse functions on smooth manifolds to differential spaces. For subcartesian space with constant structural dimension, we provide examples of Morse functions; With the assumption that the subcartesian space can be embedded as a bounded subset of an Euclidean space, we prove that smooth bounded functions on this space can be approximated by Morse functions; We study the infinitesimal stability of Morse functions on subcartesian spaces. Classical results on Morse functions on smooth manifolds can be treated directly as corollaries of our results here.

With the rapid development of artificial intelligence, machine learning is gradually becoming popular in predictions in all walks of life. In meteorology, It is gradually competing with traditional climate predictions dominated by physical models. This survey aims to consolidate the current understanding of Machine Learning (ML) applications in weather and climate prediction—a field of growing importance across multiple sectors including agriculture and disaster management. Building upon an exhaustive review of more than 20 methods highlighted in existing literature, this survey pinpointed eight techniques that show particular promise for improving the accuracy of both short-term weather and medium-to-long-term climate forecasts. According to the survey, while ML demonstrates significant capabilities in short-term weather prediction, its application in medium-to-long-term climate forecasting remains limited, constrained by factors such as intricate climate variables and data limitations. Current literature tends to focus narrowly on either short-term weather or medium-to-long-term climate forecasting, often neglecting the relationship between the two, as well as general neglect of modelling structure and recent advances. By providing an integrated analysis of models spanning different time scales, this survey aims to bridge these gaps, thereby serving as a meaningful guide for future interdisciplinary research in this rapidly evolving field.