Data-driven Efficient Solvers and Predictions of Conformational Transitions for Langevin Dynamics on Manifold in High Dimensions

We work on dynamic problems with collected data $\{\mathsf{x}_i\}$ that distributed on a manifold $\mathcal{M}\subset\mathbb{R}^p$. Through the diffusion map, we first learn the reaction coordinates $\{\mathsf{y}_i\}\subset \mathcal{N}$ where $\mathcal{N}$ is a manifold isometrically embedded into an Euclidean space $\mathbb{R}^\ell$ for $\ell \ll p$. The reaction coordinates enable us to obtain an efficient approximation for the dynamics described by a Fokker-Planck equation on the manifold $\mathcal{N}$. By using the reaction coordinates, we propose an implementable, unconditionally stable, data-driven upwind scheme which automatically incorporates the manifold structure of $\mathcal{N}$. Furthermore, we provide a weighted $L^2$ convergence analysis of the upwind scheme to the Fokker-Planck equation. The proposed upwind scheme leads to a Markov chain with transition probability between the nearest neighbor points. We can benefit from such property to directly conduct manifold-related computations such as finding the optimal coarse-grained network and the minimal energy path that represents chemical reactions or conformational changes. To establish the Fokker-Planck equation, we need to acquire information about the equilibrium potential of the physical system on $\mathcal{N}$. Hence, we apply a Gaussian Process regression algorithm to generate equilibrium potential for a new physical system with new parameters. Combining with the proposed upwind scheme, we can calculate the trajectory of the Fokker-Planck equation on $\mathcal{N}$ based on the generated equilibrium potential. Finally, we develop an algorithm to pullback the trajectory to the original high dimensional space as a generative data for the new physical system.Comment: 59 pages, 16 figure

Frequency Detection and Change Point Estimation for Time Series of Complex Oscillation

We consider detecting the evolutionary oscillatory pattern of a signal when it is contaminated by non-stationary noises with complexly time-varying data generating mechanism. A high-dimensional dense progressive periodogram test is proposed to accurately detect all oscillatory frequencies. A further phase-adjusted local change point detection algorithm is applied in the frequency domain to detect the locations at which the oscillatory pattern changes. Our method is shown to be able to detect all oscillatory frequencies and the corresponding change points within an accurate range with a prescribed probability asymptotically. This study is motivated by oscillatory frequency estimation and change point detection problems encountered in physiological time series analysis. An application to spindle detection and estimation in sleep EEG data is used to illustrate the usefulness of the proposed methodology. A Gaussian approximation scheme and an overlapping-block multiplier bootstrap methodology for sums of complex-valued high dimensional non-stationary time series without variance lower bounds are established, which could be of independent interest

[1,1′-Bis(dicyclo­hexyl­phosphino)cobalto­cenium-κ2 P,P′]chlorido(η5-cyclo­penta­dien­yl)ruthenium(II) hexa­fluorido­phosphate

In the title structure, [CoRu(C5H5)(C17H26P)2Cl]PF6, the RuII atom is bonded to a cyclo­penta­dienyl ring, a Cl atom and two P atoms of the chelating 1,1′-bis­(dicyclo­hexyl­phosphino)cobaltocenium (di-cypc) ligand, leading to a three-legged piano-stool coordination. Part of the PF6 − counter-anion is disordered over two positions, with a site-occupancy ratio of 0.898 (7):0.102 (7). The components are linked by C—H⋯F and C—H⋯Cl hydrogen bonds

Coalescence of Carbon Atoms on Cu (111) Surface: Emergence of a Stable Bridging-Metal Structure Motif

By combining first principles transition state location and molecular dynamics simulation, we unambiguously identify a carbon atom approaching induced bridging metal structure formation on Cu (111) surface, which strongly modify the carbon atom coalescence dynamics. The emergence of this new structural motif turns out to be a result of the subtle balance between Cu-C and Cu-Cu interactions. Based on this picture, a simple theoretical model is proposed, which describes a variety of surface chemistries very well

Building quantum neural networks based on swap test

Artificial neural network, consisting of many neurons in different layers, is an important method to simulate humain brain. Usually, one neuron has two operations: one is linear, the other is nonlinear. The linear operation is inner product and the nonlinear operation is represented by an activation function. In this work, we introduce a kind of quantum neuron whose inputs and outputs are quantum states. The inner product and activation operator of the quantum neurons can be realized by quantum circuits. Based on the quantum neuron, we propose a model of quantum neural network in which the weights between neurons are all quantum states. We also construct a quantum circuit to realize this quantum neural network model. A learning algorithm is proposed meanwhile. We show the validity of learning algorithm theoretically and demonstrate the potential of the quantum neural network numerically.Comment: 10 pages, 13 figure

Single-walled carbon nanotube bundle under hydrostatic pressure studied by the first-principles calculations

The structural, electronic, optical and vibrational properties of the collapsed (10,10) single-walled carbon nanotube bundle under hydrostatic pressure have been studied by the first-principles calculations. Some features are observed in the present study: First, a collapsed structure is found, which is distinct from both of the herringbone and parallel structures obtained previously. Secondly, a pseudo-gap induced by the collapse appears along the symmetry axis \textit{$\Gamma$X}. Thirdly, the relative orientation between the collapsed tubes has an important effect on their electronic, optical and vibrational properties, which provides an efficient experimental method to distinguish unambiguously three different collapsed structures.Comment: 14 pages, 6 figure