用柯尔莫哥洛夫表示法提升混沌系统预测与控制的通用性
On the Generalisation of Koopman Representations for Chaotic System Control
- 通过自编码器学习柯尔莫哥洛夫嵌入，再预训练变换器进行状态预测
- 嵌入方法在数据效率和精度上优于传统与物理信息主成分分析
- 固定预训练参数微调仍保持性能，说明表示具有可迁移的动力学结构
本文研究基于柯尔莫哥洛夫表示法在混沌动力系统中的泛化能力，重点考察其在预测与控制任务间的迁移性。以洛伦兹系统为测试平台，提出三阶段方法：通过自编码器学习柯尔莫哥洛夫嵌入，利用变换器在下一状态预测任务上预训练，再针对安全关键控制任务进行微调。结果表明，柯尔莫哥洛夫嵌入在准确性和数据效率上均优于标准及物理信息主导的主成分分析基线。值得注意的是，在微调过程中固定预训练变换器权重未导致性能下降，说明所学表示捕捉的是可复用的动力学结构而非特定任务模式。这些发现支持将柯尔莫哥洛夫嵌入作为物理信息机器学习中多任务学习的基础。项目页面见 https://kikisprdx.github.io/。
原文摘要 · Abstract (English)
This paper investigates the generalisability of Koopman-based representations for chaotic dynamical systems, focusing on their transferability across prediction and control tasks. Using the Lorenz system as a testbed, we propose a three-stage methodology: learning Koopman embeddings through autoencoding, pre-training a transformer on next-state prediction, and fine-tuning for safety-critical control. Our results show that Koopman embeddings outperform both standard and physics-informed PCA baselines, achieving accurate and data-efficient performance. Notably, fixing the pre-trained transformer weights during fine-tuning leads to no performance degradation, indicating that the learned representations capture reusable dynamical structure rather than task-specific patterns. These findings support the use of Koopman embeddings as a foundation for multi-task learning in physics-informed machine learning. A project page is available at https://kikisprdx.github.io/.
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