Long Chen | Search arXiv

arXiv2026

Tangential-Normal Decompositions of the Second Family of Finite Element Differential Forms

This paper introduces a novel tangential-normal ($t$-$n$) decomposition for the second family of finite element differential forms, presenting a new framework for constructing bases in finite element exterior calculus. The main contribution is the development of a $t$-$n$ basis in which degrees of freedom and shape functions are explicitly dual, a property that streamlines stiffness matrix assembly and enhances the efficiency of interpolation and numerical integration. Additionally, the integration of the well-documented Lagrange element basis supports practical implementation of finite element differential forms in applications.

math.NA↗

arXiv2026

From Sampled Outcomes to Capability Distributions: Rethinking Supervision for LLM Routing

Existing LLM routing methods often construct supervision from a single sampled response for each query--model pair. Because LLM generation is stochastic, however, such an observation can be an unstable estimate of model capability: semantically equivalent query formulations and repeated decoding may yield different scores and even different model preferences. We show that this instability can further propagate from routing labels to learned routing policies. To address this issue, we propose DARS (Distribution-Aware Routing Supervision), which estimates query-level model capability from repeated observations spanning semantics-preserving query rewrites and stochastic decoding. DARS summarizes expected quality, expected cost, and performance variability to construct risk-aware supervision without changing the downstream router architecture. Experiments across diverse tasks and routing methods show that DARS generally improves routing utility and cost--quality trade-offs over single-shot supervision. Further analyses show that its benefits persist under moderate sampling budgets and different decoding temperatures. These results suggest that reliable LLM routing should move beyond individual sampled outcomes and instead model query-level capability distributions.

cs.LG↗