半直线上波动方程的Carleman估计：AI辅助权重与应用
Carleman Estimates for Wave Equations on the Half-Line: AI-Assisted Weights and Applications
- University of California, San Diego（加州大学圣地亚哥分校）
- Southwest Jiaotong University（西南交通大学）
- Worcester Polytechnic Institute（伍斯特理工学院）
- University of Missouri-Kansas City（密苏里大学堪萨斯城分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出AI辅助搜索与人工验证相结合的流程，构造半直线上波动方程的Carleman权重，建立全局估计并应用于半线性波动方程的重建问题，数值实验验证了鲁棒性。
AI中文摘要：
我们开发了一种AI辅助的搜索与验证工作流程，用于在半无限域上构造波动方程的Carleman权重。从用于推导Carleman估计的加权交叉项恒等式出发，我们推导出显式的解析筛选条件，并使用AI系统提出满足这些条件的符号候选权重。搜索识别出一种具有具体参数值的对数权重结构；随后作者进行人工验证，保留该结构，推导出允许的参数范围，并通过验证所需的定量条件严格认证所选权重。基于已识别并认证的Carleman权重，后续的证明、分析和应用完全由作者完成：我们建立了半直线上波动算子的全局Carleman估计，并推导出加权条件侧向Cauchy稳定性结果。所提出的权重进一步用于半直线上半线性波动方程的有限深度单侧重建问题，其中初始时刻通量诱导端点图稳定子和一个收缩的冻结非线性重建映射。数值实验支持预测的收缩行为，并显示Carleman加权对噪声数据具有改进的鲁棒性，且在高噪声水平下诱导的端点图项提供进一步的稳定化。
英文摘要：
We develop an AI-assisted search-and-certification workflow for constructing Carleman weights for wave equations on semi-infinite domains. Starting from a weighted cross-term identity used to derive the Carleman estimate, we derive explicit analytical screening conditions and use an AI system to propose symbolic candidate weights subject to those conditions. The search identifies a logarithmic weight structure with concrete parameter values; the subsequent human verification by the authors then retains this structure, derives the admissible parameter range, and rigorously certifies the selected weight by verifying the required quantitative conditions. Based on the identified and certified Carleman weight, the subsequent proofs, analysis, and applications are carried out entirely by the authors: we establish a global Carleman estimate for the wave operator on the half-line and derive a weighted conditional lateral Cauchy stability result. The proposed weight is further used in a finite-depth one-sided reconstruction problem for semilinear wave equations on the half-line, where the initial-time flux induces an endpoint graph stabilizer and a contractive frozen nonlinear reconstruction map. Numerical experiments support the predicted contraction behavior and show improved robustness to noisy data from the Carleman weighting, with the induced endpoint graph term providing further stabilization at higher noise levels.
