  Motivation: Despite its great success in various physical modeling,
differential geometry (DG) has rarely been devised as a versatile tool for
analyzing large, diverse and complex molecular and biomolecular datasets due to
the limited understanding of its potential power in dimensionality reduction
and its ability to encode essential chemical and biological information in
differentiable manifolds. Results: We put forward a differential geometry based
geometric learning (DG-GL) hypothesis that the intrinsic physics of
three-dimensional (3D) molecular structures lies on a family of low-dimensional
manifolds embedded in a high-dimensional data space. We encode crucial
chemical, physical and biological information into 2D element interactive
manifolds, extracted from a high-dimensional structural data space via a
multiscale discrete-to-continuum mapping using differentiable density
estimators. Differential geometry apparatuses are utilized to construct element
interactive curvatures %Gaussian curvature, mean curvature, minimum curvature
and maximum curvature in analytical forms for certain analytically
differentiable density estimators. These low-dimensional differential geometry
representations are paired with a robust machine learning algorithm to showcase
their descriptive and predictive powers for large, diverse and complex
molecular and biomolecular datasets. Extensive numerical experiments are
carried out to demonstrated that the proposed DG-GL strategy outperforms other
advanced methods in the predictions of drug discovery related protein-ligand
binding affinity, drug toxicity, and molecular solvation free energy.
