用拓扑分析改进数据同化，无需噪声信息也能精准预测动态系统。
Topological Approach for Data Assimilation
- 基于拓扑数据分析，通过梯度下降优化模型参数。
- 在洛伦兹63和96系统中均实现高精度状态估计。
- 适合缺乏测量噪声统计的复杂系统预测任务。
许多动力系统难以用高保真物理模型描述，研究者越来越依赖数据驱动模型进行预测。但受限于训练数据，机器学习模型随时间会偏离真实状态，需借助数据同化不断更新。传统方法通常需要测量噪声统计信息，而这些信息可能未知。本文提出一种基于拓扑数据分析的新同化算法，利用持久性函数的可微性，通过梯度下降最小化观测与预报之间的拓扑差异，从而调整数据驱动模型系数，无需测量噪声信息。我们详细描述该方法，并以混沌的Lorenz 63系统为例展示其性能；同时验证其在更高维的Lorenz 96系统上的有效性。
原文摘要 · Abstract (English)
Many dynamical systems are difficult or impossible to model using high fidelity physics based models. Consequently, researchers are relying more on data driven models to make predictions and forecasts. Based on limited training data, machine learning models often deviate from the true system states over time and need to be continually updated as new measurements are taken using data assimilation. Classical data assimilation algorithms typically require knowledge of the measurement noise statistics which may be unknown. In this paper, we introduce a new data assimilation algorithm with a foundation in topological data analysis. By leveraging the differentiability of functions of persistence, gradient descent optimization is used to minimize topological differences between measurements and forecast predictions by tuning data driven model coefficients without using noise information from the measurements. We describe the method and focus on its capabilities performance using the chaotic Lorenz 63 system as an example and we also show that the method works on a higher dimensional example with the Lorenz 96 system.
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