用神经算子理论统一建模动态系统平滑与预测，为数据驱动方法提供数学基础。
Operator Learning for Smoothing and Forecasting
- 基于神经算子构建连续时间映射，学习数据同化与预测的隐含规律
- 证明了纯数据驱动方法在洛伦兹63、96及库拉莫托-西瓦辛斯基系统中的通用逼近能力
- 适用于缺乏解析模型的复杂动力系统，适合研究气象、气候等领域的学者
机器学习为数据同化和动态系统预测开辟了全新的纯数据驱动算法路径，展现出一定前景。然而，与基于模型的方法相比，这些数据驱动方法的理论分析仍不充分。本文针对这一问题，构建了一个理论框架，支撑纯数据驱动方法解决数据同化中的平滑问题与预测问题。该框架依赖两个关键要素：(i) 待学习映射的存在性；(ii) 用于近似该映射的算子学习架构的性质。通过联合分析这两者，我们建立了针对动态系统平滑与预测的全新通用逼近定理。研究在连续时间设定下进行，采用神经算子架构。理论结果通过在洛伦兹'63、洛伦兹'96和库拉莫托-西瓦辛斯基系统上的实验得到验证。
原文摘要 · Abstract (English)
Machine learning has opened new frontiers in purely data-driven algorithms for data assimilation in, and for forecasting of, dynamical systems; the resulting methods are showing some promise. However, in contrast to model-driven algorithms, analysis of these data-driven methods is poorly developed. In this paper we address this issue, developing a theory to underpin data-driven methods to solve smoothing problems arising in data assimilation and forecasting problems. The theoretical framework relies on two key components: (i) establishing the existence of the mapping to be learned; (ii) the properties of the operator learning architecture used to approximate this mapping. By studying these two components in conjunction, we establish novel universal approximation theorems for purely data driven algorithms for both smoothing and forecasting of dynamical systems. We work in the continuous time setting, hence deploying neural operator architectures. The theoretical results are illustrated with experiments studying the Lorenz `63, Lorenz `96 and Kuramoto-Sivashinsky dynamical systems.
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