We propose spectral methods for long-term forecasting of temporal signals stemming from linear and nonlinear quasi-periodic dynamical systems. For linear signals, we introduce an algorithm with similarities to the Fourier transform but which does not rely on periodicity assumptions, allowing for forecasting given potentially arbitrary sampling intervals. We then extend this algorithm to handle nonlinearities by leveraging Koopman theory. The resulting algorithm performs a spectral decomposition in a nonlinear, data-dependent basis. The optimization objective for both algorithms is highly non-convex. However, expressing the objective in the frequency domain allows us to compute global optima of the error surface in a scalable and efficient manner, partially by exploiting the computational properties of the Fast Fourier Transform. Because of their close relation to Bayesian Spectral Analysis, uncertainty quantification metrics are a natural byproduct of the spectral forecasting methods. We extensively benchmark these algorithms against other leading forecasting methods on a range of synthetic experiments as well as in the context of real-world power systems and fluid flows.
Meta-Learning for Koopman Spectral Analysis with Short Time-series
NTT Communication Science Laboratories, NTT Corporation（NTT公司NTT通信科学实验室）；Institute of Mathematics for Industry, Kyushu University（九州大学产业数学研究所）；Center for Advanced Intelligence Project, RIKEN（理化学研究所高级智能项目中心）
SKOLR: Structured Koopman Operator Linear RNN for Time-Series Forecasting
Department of Electrical and Computer Engineering, McGill University, Montreal, Canada（麦吉尔大学电气与计算机工程系）；Amazon Science. This work does not relate to the author’s position at Amazon（亚马逊科学）；Mila - Quebec Artificial Intelligence Institute, Montreal, Canada（魁北克人工智能研究所）；ILLS - International Laboratory on Learning Systems, Montreal, Canada（学习系统国际实验室）
End-to-End Neural Decomposition with Koopman Operators for Time-Series Forecasting
National Taiwan University（台湾大学）；National Institute of Information and Communications Technology（信息通信研究机构）；Research Center for Information Technology Innovation（资讯科技创新研究中心）
Learnable Koopman-Enhanced Transformer-Based Time Series Forecasting with Spectral Control
Max Planck Institute of Geoanthropology（马克斯·普朗克地理人类学研究所）；Department of Electrical Engineering and Information Technology, University of Naples（那不勒斯大学电气工程与信息学院）
Koopman-theoretic Approach for Identification of Exogenous Anomalies in Nonstationary Time-series Data
University of Washington（华盛顿大学）；NASA Global Modeling and Assimilation Office, Goddard Space Flight Center（美国国家航空航天局戈达德太空飞行中心全球建模与同化办公室）；Morgan State University（摩根州立大学）
在许多情况下，有必要通过时间序列观测来监测复杂系统，并确定异常外源事件何时发生，以便采取相关行动。确定当前观察结果是否异常具有挑战性。它需要从历史数据中学习动态的外推概率模型，并使用有限数量的当前观测进行分类。我们利用长期概率预测的最新进展，即{ \em Deep ProbabilityKoopman}，建立一个通用的方法来分类多维时间序列数据中的异常。我们还展示了如何利用模型与域知识的动态，以减少I型和II型错误。我们证明了我们提出的方法对全球大气污染监测的重要现实世界的任务，将其与美国宇航局的全球地球系统模型。该系统成功检测到因COVID-19封锁和野火等事件而导致的局部空气质量异常。
DeepKoopFormer: A Koopman Enhanced Transformer Based Architecture for Time Series Forecasting
Helmholtz Center for Environmental Research-UFZ（海德堡环境研究中心-UFZ）；Delft Center for Systems and Control（代尔夫特系统与控制中心）；School of Engineering, University of Pittsburgh（匹兹堡大学工程学院）
Munich Data Science Institute（慕尼黑数据科学研究所）；School of Computation, Information and Technology, Technical University of Munich（技术大学慕尼黑计算、信息与技术学院）；Interdisciplinary Center for Scientific Computing (IWR)（跨学科科学计算中心）；Department of Mathematics and Computer Science, Heidelberg University（海德堡大学数学与计算机科学系）
School of Electrical and Mechanical Engineering, The University of Adelaide（电气与机械工程学院，阿德莱德大学）；Information Systems Technology and Design Pillar, Singapore University of Technology and Design（信息系统技术与设计支柱，新加坡科技设计大学）
A Time-Series Foundation Model by Universal Delay Embedding
Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences（中国科学院大学杭州高等研究院）；Shanghai Jiao Tong University（上海交通大学）；Fudan University（复旦大学）；Ocean University of China（中国海洋大学）；South China University of Technology（华南理工大学）
