Qinghai Zhang

Yan Tan, Yixiao Qian, Zhiqi Li et al.

For interface tracking of an arbitrary number of materials in two dimensions, we propose a multiphase cubic MARS method that (a) represents the topology and geometry of the interface via graphs, cycles, and cubic splines, (b) applies to any number of materials with arbitrarily complex topology and geometry, (c) maintains an $(r,h)$-regularity of the interface so that the distance between any pair of adjacent markers is within a user-specified range, (d) distributes the markers adaptively along the interface so that arcs with high curvature are resolved by densely populated markers, and (e) achieves fourth-, sixth-, and eighth-order accuracy both in time and in space.} In particular, all possible types of junctions, which pose challenges to VOF methods and level-set methods, are handled with ease. Results of a variety of benchmark tests confirm the analysis and demonstrate the superior accuracy, efficiency, and versatility of the proposed method.

Kejie Chen, Lin Wang, Qinghai Zhang et al.

Recent studies have highlighted the limitations of large language models in mathematical reasoning, particularly their inability to capture the underlying logic. Inspired by meta-learning, we propose that models should acquire not only task-specific knowledge but also transferable problem-solving skills. We introduce MetaRuleGPT, a novel Transformer-based architecture that performs precise numerical calculations and complex logical operations by learning and combining different rules. In contrast with traditional training sets, which are heavily composed of massive raw instance data, MetaRuleGPT is pre-trained on much less abstract datasets containing basic, compound, and iterative rules for mathematical reasoning. Extensive experimental results demonstrate MetaRuleGPT can mimic human's rule-following capabilities, break down complexity, and iteratively derive accurate results for complex mathematical problems. These findings prove the potential of rule learning to enhance the numerical reasoning abilities of language models.