arXiv subjects

Ziyao Sun

Publications and source records attributed to Ziyao Sun.

4 recordsLinked to original sources

arXiv2026

On mesh patterns of short length: Equidistribution and enumeration

The classification and enumeration of short mesh patterns have emerged as two central directions in the area. We make substantial progress on both fronts. We construct an involution and a bijection that establish distributional equivalences for two classes of length-$2$ mesh patterns, thereby resolving a conjecture from 2019 and a recent conjecture. As a consequence, the best known upper bounds for the numbers of distribution-equivalence and Wilf-equivalence classes drop to $106$ and $47$, respectively. Combined with the known lower bounds of 105 and 46, conjectured to be exact, these results leave both classifications hinging on a single distribution-equivalence question conjectured in 2019, whose resolution would at once settle the remaining Wilf-equivalence case. We further conjecture that this unresolved equidistribution also holds for involutions, a subclass of all permutations. We also determine the distributions of three additional classes of length-$2$ mesh patterns through a detailed structural analysis. Our work combines bijective techniques with generating-function methods, yielding new insights into the structure and enumeration of short mesh patterns.

math.CO↗

arXiv2025

FuXi-TC: A generative framework integrating deep learning and physics-based models for improved tropical cyclone forecasts

Tropical cyclones (TCs) are among the most devastating natural hazards, yet their intensity remains notoriously difficult to predict. NWP models are constrained by both computational demands and intrinsic predictability, while state-of-the-art deep learning-based weather forecasting models tend to underestimate TC intensity due to biases in reanalysis-based training data. Here, we present FuXi-TC, a diffusion-based generative forecasting framework that combines the track prediction strength of the FuXi model with the intensity representation of NWP simulations. By conditioning a diffusion model on the large-scale forecasts of the global FuXi model, FuXi-TC effectively downscales and delivers higher-accuracy forecasts of fine-grained variable fields such as wind speed and precipitation. In evaluations across the 2024 Western North Pacific, our approach matches the TC intensity forecast skill of the operational ECMWF deterministic model while delivering superior precipitation forecasts. Meanwhile this is achieved with significantly higher inference speeds and lower computational costs. Moreover, FuXi-TC demonstrates robust zero-shot generalization directly when applied to North Atlantic hurricanes without any fine-tuning. When applied to the FuXi ensemble model, this framework effectively yields well-dispersed probabilistic forecasts and refines the ensemble intensity predictions.

physics.ao-ph↗

arXiv2016

A note on implementation of boundary variation diminishing algorithm to high-order local polynomial-based schemes

A novel approach for selecting appropriate reconstructions is implemented to the hyperbolic conservation laws in the high-order local polynomial-based framework, e.g., the discontinuous Galerkin (DG) and flux reconstruction (FR) schemes. The high-order polynomial approximation generally fails to correctly capture a strong discontinuity inside a cell due to the Runge phenomenon, which is replaced by more stable approximation on the basis of a troubled-cell indicator such as that used in the total variation bounded (TVB) limiter. This paper examines the applicability of a new algorithm, so-called boundary variation diminishing (BVD) reconstruction, to the weighted essentially non-oscillatory (WENO) methodology in the FR framework including the nodal type DG method. The BVD reconstruction adaptively chooses a proper approximation for the solution function so as to minimize the jump between values at the left- and right-side of cell boundaries without any ad hoc constant such as the TVB parameter. Several numerical tests are conducted for a linear advection equation as well as the selection of an appropriate reconstruction, where the results of the BVD algorithm are comparable to those using the conventional TVB limiter. Note that the present work is limited to third or lower order polynomials, leaving the implementations for higher-order schemes a future work.

math.NA↗

arXiv2016

Boundary Variation Diminishing (BVD) reconstruction: a new approach to improve Godunov scheme

This paper presents a new approach, so-called boundary variation diminishing (BVD), for reconstructions that minimize the discontinuities (jumps) at cell interfaces in Godunov type schemes. It is motivated by the observation that diminishing the jump at the cell boundary might effectively reduce the dissipation in numerical flux. Different from the existing practices which seek high-order polynomials within mesh cells while assuming discontinuities being always at the cell interfaces, we proposed a new strategy that combines a high-order polynomial-based interpolation and a jump-like reconstruction that allows a discontinuity being partly represented within the mesh cell rather than at the interface. It is shown that new schemes of high fidelity for both continuous and discontinuous solutions can be devised by the BVD guideline with properly-chosen candidate reconstruction schemes. Excellent numerical results have been obtained for both scalar and Euler conservation laws with substantially improved solution quality in comparison with the existing methods. This work provides a simple and accurate alternative of great practical significance to the current Godunov paradigm which overly pursues the smoothness within mesh cell under the questionable premiss that discontinuities only appear at cell interfaces.

physics.comp-ph↗