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Motivation: Despite its great success in various physical modeling,differential geometry (DG) has rarely been devised as a versatile tool foranalyzing large, diverse and complex molecular and biomolecular datasets due tothe limited understanding of its potential power in dimensionality reductionand its ability to encode essential chemical and biological information indifferentiable manifolds. Results: We put forward a differential geometry basedgeometric learning (DG-GL) hypothesis that the intrinsic physics ofthree-dimensional (3D) molecular structures lies on a family of low-dimensionalmanifolds embedded in a high-dimensional data space. We encode crucialchemical, physical and biological information into 2D element interactivemanifolds, extracted from a high-dimensional structural data space via amultiscale discrete-to-continuum mapping using differentiable densityestimators. Differential geometry apparatuses are utilized to construct elementinteractive curvatures %Gaussian curvature, mean curvature, minimum curvatureand maximum curvature in analytical forms for certain analyticallydifferentiable density estimators. These low-dimensional differential geometryrepresentations are paired with a robust machine learning algorithm to showcasetheir descriptive and predictive powers for large, diverse and complexmolecular and biomolecular datasets. Extensive numerical experiments arecarried out to demonstrated that the proposed DG-GL strategy outperforms otheradvanced methods in the predictions of drug discovery related protein-ligandbinding affinity, drug toxicity, and molecular solvation free energy.

DG-GL: Differential geometry based geometric learning of molecular datasets