Although deep learning approaches have had tremendous success in image, video and audio processing, computer vision, and speech recognition, their applications to three-dimensional (3D) biomolecular structural data sets have been hindered by the geometric and biological complexity. To address this problem we introduce the element-specific persistent homology (ESPH) method. ESPH represents 3D complex geometry by one-dimensional (1D) topological invariants and retains important biological information via a multichannel image-like representation. This representation reveals hidden structure-function relationships in biomolecules. We further integrate ESPH and deep convolutional neural networks to construct a multichannel topological neural network (TopologyNet) for the predictions of protein-ligand binding affinities and protein stability changes upon mutation. To overcome the deep learning limitations from small and noisy training sets, we propose a multi-task multichannel topological convolutional neural network (MM-TCNN). We demonstrate that TopologyNet outperforms the latest methods in the prediction of protein-ligand binding affinities, mutation induced globular protein folding free energy changes, and mutation induced membrane protein folding free energy changes. Availability: weilab.math.msu.edu/TDL/

Necessary conditions of optimality are derived for multiobjective optimal control problems with free end-time, in which the dynamics constraint is modeled as a nonconvex differential inclusion. The obtained results cover some previous results on necessary conditions for multiojective and single objective optimal control problems.

In this paper, we generalize the Gauduchon metrics on a compact complex manifold and define the _k functions on the space of its hermitian metrics.

Solitons of the power-energy saturation (PES) equation are studied using adiabatic perturbation theory. In the anomalous regime individual soliton pulses are found to be well approximated by solutions of the classical nonlinear Schrödinger (NLS) equation with the key parameters of the soliton changing slowly as they evolve. Evolution equations are found for the pulse amplitude(s), velocity(ies), position(s), and phase(s) using integral relations derived from the PES equation. The results from the integral relations are shown to agree with multi-scale perturbation theory. It is shown that the single soliton case exhibits mode-locking behavior for a wide range of parameters, while the higher states form effective bound states. Using the fact that there is weak overlap between tails of interacting solitons, evolution equations are derived for the relative amplitudes, velocities, positions, and phase differences. Comparisons of interacting soliton behavior between the PES equation and the classical NLS equation are also exhibited.

No abstract uploaded!