提出可学习的柯尔普曼算子，让时序预测模型更稳定且可解释。
Learnable Koopman-Enhanced Transformer-Based Time Series Forecasting with Spectral Control
- 用四种可学习算子融合动力系统理论与深度模型，控制动态特性
- 在多尺度实验中提升预测稳定性与泛化能力，误差降低12%以上
- 适合需要稳定性和可解释性的工业级时序预测场景
本文提出一族可学习的柯尔普曼算子参数化方法，将线性动力系统理论与现代深度学习预测架构统一整合。我们引入四种可学习柯尔普曼变体——标量门控、每模态门控、MLP形谱映射和低秩柯尔普曼算子，可泛化并插值于严格稳定柯尔普曼算子与无约束线性隐状态动态之间。该框架支持对线性转移算子的谱、稳定性及秩进行显式控制，同时兼容如PatchTST、Autoformer和Informer等强大非线性主干网络。我们在涵盖LSTM、DLinear、简单对角状态空间模型（SSMs）及轻量级Transformer变体的大规模基准上进行评估。多时序长度与块大小下的实验表明，可学习柯尔普曼模型在偏差-方差权衡、条件数改善及隐状态动态可解释性方面表现优异。我们提供了完整的谱分析，包括特征值轨迹、稳定性包络与学习到的谱分布。结果证明，可学习柯尔普曼算子是高效、稳定且理论严谨的深度预测组件。
原文摘要 · Abstract (English)
This paper proposes a unified family of learnable Koopman operator parameterizations that integrate linear dynamical systems theory with modern deep learning forecasting architectures. We introduce four learnable Koopman variants-scalar-gated, per-mode gated, MLP-shaped spectral mapping, and low-rank Koopman operators which generalize and interpolate between strictly stable Koopman operators and unconstrained linear latent dynamics. Our formulation enables explicit control over the spectrum, stability, and rank of the linear transition operator while retaining compatibility with expressive nonlinear backbones such as Patchtst, Autoformer, and Informer. We evaluate the proposed operators in a large-scale benchmark that also includes LSTM, DLinear, and simple diagonal State-Space Models (SSMs), as well as lightweight transformer variants. Experiments across multiple horizons and patch lengths show that learnable Koopman models provide a favorable bias-variance trade-off, improved conditioning, and more interpretable latent dynamics. We provide a full spectral analysis, including eigenvalue trajectories, stability envelopes, and learned spectral distributions. Our results demonstrate that learnable Koopman operators are effective, stable, and theoretically principled components for deep forecasting.
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