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Aqueous solubility and partition coefficient are important physicalproperties of small molecules. Accurate theoretical prediction of aqueoussolubility and partition coefficient plays an important role in drug design anddiscovery. The prediction accuracy depends crucially on molecular descriptorswhich are typically derived from theoretical understanding of the chemistry andphysics of small molecules. The present work introduces an algebraic topologybased method, called element specific persistent homology (ESPH), as a newrepresentation of small molecules that is entirely different from conventionalchemical and/or physical representations. ESPH describes molecular propertiesin terms of multiscale and multicomponent topological invariants. Suchtopological representation is systematical, comprehensive, and scalable withrespect to molecular size and composition variations. However, it cannot beliterally translated into a physical interpretation. Fortunately, it is readilysuitable for machine learning methods, rendering topological learningalgorithms. Due to the inherent correlation between solubility and partitioncoefficient, a uniform ESPH representation is developed for both properties,which facilitates multi-task deep neural networks for their simultaneouspredictions. This strategy leads to more accurate prediction of relativelysmall data sets. A total of six data sets is considered in the present work tovalidate the proposed topological and multi-task deep learning approaches. Itis demonstrate that the proposed approaches achieve some of the most accuratepredictions of aqueous solubility and partition coefficient. Our software isavailable online at {\url{http://weilab.math.msu.edu/TopP-S/}}

TopP-S: Persistent homology based multi-task deep neural networks for simultaneous predictions of partition coefficient and aqueous solubility