揭示了降维建模中双下降现象的成因并提出抑制方法
Origins and mitigation of double descent in reduced order modeling
- 基于数据-噪声平均理论统一分析双下降成因
- 可精准预测风险曲线，计算成本仅为传统方法的几分之一
- 定位问题传感器组合，提供有效正则化缓解策略
自然与工程系统数据中的潜在低维结构使得稀疏传感成为可能，即仅通过历史数据和少量精心选择的局部测量即可重建全状态。根据重建算法、传感器位置和测量噪声的不同，重建风险曲线呈现出多样化模式，包括机器学习文献中著名的双下降现象——误差出现显著峰值。本文在统一的数据-噪声平均理论框架下研究这些情形。定性上，我们提出了双下降出现的充分条件：重建中病态信号被灾难性放大。定量上，我们以极低计算成本预测详细的风险曲线，追溯重建不稳定的根源至单个传感器及其组合，并提供正则化机制以缓解不稳定性。实验验证涵盖海表温度静态重构以及偏微分方程降阶模型的时间积分。
原文摘要 · Abstract (English)
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.
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