  We present the performances of our mathematical deep learning (MathDL) models
for D3R Grand Challenge 4 (GC4). This challenge involves pose prediction,
affinity ranking, and free energy estimation for beta secretase 1 (BACE) as
well as affinity ranking and free energy estimation for Cathepsin S (CatS). We
have developed advanced mathematics, namely differential geometry, algebraic
graph, and/or algebraic topology, to accurately and efficiently encode high
dimensional physical/chemical interactions into scalable low-dimensional
rotational and translational invariant representations. These representations
are integrated with deep learning models, such as generative adversarial
networks (GAN) and convolutional neural networks (CNN) for pose prediction and
energy evaluation, respectively. Overall, our MathDL models achieved the top
place in pose prediction for BACE ligands in Stage 1a. Moreover, our
submissions obtained the highest Spearman correlation coefficient on the
affinity ranking of 460 CatS compounds, and the smallest centered root mean
square error on the free energy set of 39 CatS molecules. It is worthy to
mention that our method for docking pose predictions has significantly improved
from our previous ones.
