Weizhi Shao | Searcharxiv

arXiv2025

A Stochastic Gradient Descent Method for Globally Minimizing Nearly Convex Functions

This paper proposes a stochastic gradient descent method with an adaptive Gaussian noise term for the global minimization of nearly convex functions, which are nonconvex and possess multiple strict local minimizers. The noise term, independent of the gradient, is determined by the difference between the current function value and a lower bound estimate of the optimal value. In both probability space and state space, we show that the proposed algorithm converges linearly to a neighborhood of the global optimal solution. The size of this neighborhood depends on the variance of the gradient and the deviation between the estimated lower bound and the optimal value. In particular, when full gradient information is available and a sharp lower bound of the objective function is provided, the algorithm achieves linear convergence to the global optimum. Furthermore, we introduce a double-loop scheme that alternately updates the lower bound estimate and the optimization sequence, enabling convergence to a neighborhood of the global optimum that depends solely on the gradient variance. Numerical experiments on several benchmark problems demonstrate the effectiveness of the proposed algorithm.

math.OC↗

arXiv2024

Revealing the molecular structures of a-Al2O3(0001)-water interface by machine learning based computational vibrational spectroscopy

Solid-water interfaces are crucial to many physical and chemical processes and are extensively studied using surface-specific sum-frequency generation (SFG) spectroscopy. To establish clear correlations between specific spectral signatures and distinct interfacial water structures, theoretical calculations using molecular dynamics (MD) simulations are required. These MD simulations typically need relatively long trajectories (a few nanoseconds) to achieve reliable SFG response function calculations via the dipole-polarizability time correlation function. However, the requirement for long trajectories limits the use of computationally expensive techniques such as ab initio MD (AIMD) simulations, particularly for complex solid-water interfaces. In this work, we present a pathway for calculating vibrational spectra (IR, Raman, SFG) of solid-water interfaces using machine learning (ML)-accelerated methods. We employ both the dipole moment-polarizability correlation function and the surface-specific velocity-velocity correlation function approaches to calculate SFG spectra. Our results demonstrate the successful acceleration of AIMD simulations and the calculation of SFG spectra using ML methods. This advancement provides an opportunity to calculate SFG spectra for the complicated solid-water systems more rapidly and at a lower computational cost with the aid of ML.

cond-mat.mtrl-sci↗