用少量数据结合方程先验，实现系统预测与参数反演
Integrated utilization of equations and small dataset in the Koopman operator: applications to forward and inverse problems
- 将未知参数的微分方程作为模糊先验融入EDMD算法
- 仅用少量数据即可完成杜芬与范德波尔系统的准确预测
- 适用于数据稀缺但有物理方程线索的建模场景
近年来，数据驱动方法在物理学中日益受到关注，如扩展动态模态分解（EDMD）。EDMD算法聚焦非线性时序系统，通过构建柯尔莫哥洛夫矩阵，仅用线性矩阵乘法即可实现下一时刻的预测。然而，数据驱动方法通常需要大量数据。若已有部分先验知识，即使不明确，也可借助其从少量数据中实现充分学习。本文提出将模糊先验知识融入EDMD的方法，其中先验对应未知参数的时序演化方程。首先，将该方法应用于前向问题（预测任务）；其次，提出一种用于反向问题（参数估计任务）的方案。通过杜芬系统和范德波尔系统等引导示例，验证了仅用少量数据即可实现有效学习。
原文摘要 · Abstract (English)
In recent years, there has been a growing interest in data-driven approaches in physics, such as extended dynamic mode decomposition (EDMD). The EDMD algorithm focuses on nonlinear time-evolution systems, and the constructed Koopman matrix yields the next-time prediction with only linear matrix-product operations. Note that data-driven approaches generally require a large dataset. However, assume that one has some prior knowledge, even if it may be ambiguous. Then, one could achieve sufficient learning from only a small dataset by taking advantage of the prior knowledge. This paper yields methods for incorporating ambiguous prior knowledge into the EDMD algorithm. The ambiguous prior knowledge in this paper corresponds to the underlying time-evolution equations with unknown parameters. First, we apply the proposed method to forward problems, i.e., prediction tasks. Second, we propose a scheme to apply the proposed method to inverse problems, i.e., parameter estimation tasks. We demonstrate the learning with only a small dataset using guiding examples, i.e., the Duffing and the van der Pol systems.
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