深度Koopman感知
Deep Koopman Sensing
- Rensselaer Polytechnic Institute（伦斯勒理工学院）
- University of Central Florida（中佛罗里达大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
提出深度Koopman感知框架，结合自编码器与Koopman线性潜动力学，在四个流体基准上实现最低同化误差，证明潜动力学应针对估计任务设计而非仅预测精度。
AI中文摘要：
从稀疏传感器测量中实时重建流体流动对于物理理解和流动控制都至关重要。当基于第一性原理的模型因计算成本过高而无法用于在线数据同化（DA）时，学习型降阶模型提供了一种高效的替代方案，但这些模型通常针对前向预测而非状态估计进行优化。我们提出了深度Koopman感知（Deep Koopman Sensing），一种数据驱动的降阶数据同化框架，它将非线性自编码器与近似Koopman算子的参数条件线性潜变量动力学相结合。我们将所提出的模型与参数化动态模式分解（pDMD）、多层感知机（MLP）和xLSTM在四个基准上进行了比较：一维粘性Burgers方程、二维圆柱绕流、二维溃坝和三维球体绕流。我们的结果揭示了预测与感知之间的显著区别：开环精度不能可靠地预测同化性能，而深度Koopman感知在所有四个基准上均实现了最低的同化误差。更重要的是，使用扩展卡尔曼滤波器时，融合传感器测量改进了所有四个基准上两个线性潜变量模型的估计，而尽管非线性模型具有强大的开环性能，融合却使其性能退化。使用集合滤波时，非线性模型不再因同化而退化，而Koopman模型仍然达到最低的同化误差。这些结果表明，潜变量动力学应针对下游估计任务进行设计，而非仅根据预测精度进行选择，并展示了基于Koopman的降阶建模作为从稀疏、流式测量中实时重建流动的有效方法。
英文摘要：
Real-time reconstruction of fluid flows from sparse sensor measurements is important for both physical understanding and flow control. When first-principles models are too expensive for online data assimilation (DA), learned reduced-order models provide an efficient alternative, but are commonly optimized for forward prediction rather than state estimation. We propose Deep Koopman Sensing, a data-driven reduced-order DA framework that combines a nonlinear autoencoder with parameter-conditioned linear latent dynamics approximating the Koopman operator. We compare the proposed model with parametric dynamic mode decomposition (pDMD), a multilayer perceptron (MLP), and xLSTM across four benchmarks: 1D viscous Burgers, 2D flow past a cylinder, 2D dambreak, and 3D flow past a sphere. Our results reveal a marked distinction between forecasting and sensing: open-loop accuracy does not reliably predict assimilation performance, while Deep Koopman Sensing achieves the lowest assimilation error across all four benchmarks. More importantly, with an extended Kalman filter, incorporating sensor measurements improves the estimates of both linear latent models across all four benchmarks, whereas it degrades the nonlinear models, despite their strong open-loop performance. With ensemble filtering, the nonlinear models are no longer degraded by assimilation, while the Koopman model still attains the lowest assimilation error. These results show that latent dynamics should be designed for the downstream estimation task rather than selected solely for forecast accuracy, and demonstrate Koopman-based reduced-order modeling as an effective approach for real-time flow reconstruction from sparse, streaming measurements.
