Yan Wang

Yan Wang, Xiaofei Zhao

A group of high order Gautschi-type exponential wave integrators (EWIs) Fourier pseudospectral method are proposed and analyzed for solving the nonlinear Klein-Gordon equation (KGE) in the nonrelativistic limit regime, where a parameter $0<\varepsilon\ll1$ which is inversely proportional to the speed of light, makes the solution propagate waves with wavelength $O(\varepsilon^2)$ in time and $O(1)$ in space. With the Fourier pseudospectral method to discretize the KGE in space, we propose a group of EWIs with designed Gautschi's type quadratures for the temporal integrations, which can offer any intended even order of accuracy provided that the solution is smooth enough, while all the current existing EWIs offer at most second order accuracy. The scheme is explicit, time symmetric and rigorous error estimates show the meshing strategy of the proposed method is time step $τ=O(\varepsilon^2)$ and mesh size $h=O(1)$ as $0<\varepsilon\ll1$, which is `optimal' among all classical numerical methods towards solving the KGE directly in the limit regime, and which also distinguish our methods from other high order approaches such as Runge-Kutta methods which require $τ=O(\varepsilon^3)$. Numerical experiments with comparisons are done to confirm the error bound and show the superiority of the proposed methods over existing classical numerical methods.

Yachao Zhao, Bo Wang, Yan Wang et al.

Large Language Models (LLMs) have been shown to exhibit various biases and stereotypes in their generated content. While extensive research has investigated biases in LLMs, prior work has predominantly focused on explicit bias, with minimal attention to implicit bias and the relation between these two forms of bias. This paper presents a systematic framework grounded in social psychology theories to investigate and compare explicit and implicit biases in LLMs. We propose a novel self-reflection-based evaluation framework that operates in two phases: first measuring implicit bias through simulated psychological assessment methods, then evaluating explicit bias by prompting LLMs to analyze their own generated content. Through extensive experiments on advanced LLMs across multiple social dimensions, we demonstrate that LLMs exhibit a substantial inconsistency between explicit and implicit biases: while explicit bias manifests as mild stereotypes, implicit bias exhibits strong stereotypes. We further investigate the underlying factors contributing to this explicit-implicit bias inconsistency, examining the effects of training data scale, model size, and alignment techniques. Experimental results indicate that while explicit bias declines with increased training data and model size, implicit bias exhibits a contrasting upward trend. Moreover, contemporary alignment methods effectively suppress explicit bias but show limited efficacy in mitigating implicit bias.