发表机构
University of Washington; AI Institute in Dynamic Systems（华盛顿大学; 动态系统人工智能研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究在数据-噪声平均理论框架下，揭示降阶建模中双重下降现象的起源，提出缓解该不稳定性的正则化机制，并通过海面温度重构与PDE降阶模型时间积分验证方法有效性。
AI 中文摘要
自然与工程系统数据集的潜在低维结构可实现稀疏感知，即从历史数据及少量精心选择的局部测量值中重构全状态。根据重构算法、传感器位置及测量噪声的不同，重构风险曲线呈现出多种模式，包括机器学习文献中被称为双重下降的误差骤增峰。本文在统一的“数据-噪声平均”理论框架下探究此类场景：定性层面，通过重构中病理信号的灾难性放大，明确双重下降出现的充分条件；定量层面，以极低计算成本预测详细风险曲线，将重构不稳定性追溯至单个传感器及其组合，并提供正则化机制以缓解该不稳定性。本文针对海面温度模式的静态重构及某偏微分方程（PDE）降阶模型的时间积分，均验证了上述结果。
英文摘要
Latent low-dimensional structure in datasets of natural and engineered systems enables their sparse sensing, or full-state reconstruction from historical data and very few carefully chosen localized measurements. Depending on the reconstruction algorithm, sensor locations, and measurement noise, the reconstruction risk curves demonstrate a diversity of patterns including a dramatic peak in error known as double descent in Machine Learning literature. Here we explore those scenarios under a unified Data-Noise Averaging theory. Qualitatively, we formulate sufficient criteria for double descent to emerge through a catastrophic amplification of a pathological signal in reconstruction. Quantitatively, we predict the detailed risk curves at a fraction of computational cost, trace reconstruction instability to individual sensors and their combinations, and provide regularization mechanisms to mitigate the instability. We demonstrate results for both static reconstruction of Sea Surface Temperature patterns and time integration of a reduced order model of a PDE.
Comments20 RevTeX pages, 11 figures
