The paper introduces K$^2$SVD, a method that learns the leading singular functions of the Koopman operator by optimizing a Hilbert-Schmidt objective, producing a low‑rank, interpretable Koopman representation with a compact latent space. In this space, temporal evolution is modeled with a linear Gaussian state‑space model and inference is performed via Kalman filtering to reduce noise accumulation in multi‑step predictions. Experiments demonstrate that K$^2$SVD outperforms state‑of‑the‑art methods on multiple datasets, achieving faster prediction speeds and lower computational cost.
By Ruiquan Li, Yuheng Bu
FreKoo++ is a continuous spectral-dynamical framework designed for Temporal Domain Generalization (TDG). It unifies continuous Koopman modal dynamics with adaptive spectral disentanglement, mapping source-domain parameters into a latent space and modeling their evolution as a superposition of learnable continuous modes. The method handles irregular timestamps, supports arbitrary horizon extrapolation, and introduces an adaptive soft spectral weighting mechanism that isolates persistent dynamics from transient noise, achieving state‑of‑the‑art performance on discrete and continuous TDG benchmarks.
By En Yu, Xiaoyu Yang, Wei Duan, Guangquan Zhang, Jie Lu
arXiv:2608. 04471v1 Announce Type: cross Abstract: Time series in real-world applications are often generated by nonlinear dynamical systems, making accurate forecasting challenging.
By Mengzhou Gao, Huangqian Yu, Pengfei Jiao
The paper presents an ensemble Kalman–Bucy smoother (EnKBS) for continuous‑time data assimilation of nonlinear dynamical systems, reconstructing conditional distributions from ensemble moments without needing tangent‑linear or adjoint models. It demonstrates that EnKBS achieves exact smoothing mean and covariance in the infinite‑ensemble limit for linear‑Gaussian systems and incorporates regularization techniques like covariance localization and inflation for high‑dimensional problems. The method is applied to Bayesian inference of causal relationships in a dyadic trigger‑feedback model and to an iterative learning algorithm that uncovers the structure and hidden parameters of a reduced‑order model of midlatitude atmospheric circulation, all with small ensembles under partial observations.
By Zhang Jiang (University of Wisconsin-Madison), Marios Andreou (University of Wisconsin-Madison), Sebastian Reich (University of Potsdam), Nan Chen (University of Wisconsin-Madison)
arXiv:2609.15488v1 Announce Type: new
Abstract: Machine learning models process large amounts of data, and Principal Component Analysis (PCA) is a widely used technique to reduce the dimensionality o...
By Kris Lokere, John Fossaceca
arXiv:2608. 08788v1 Announce Type: cross Abstract: Koopman theory offers a linear-operator view of nonlinear sequence dynamics by lifting observations into a space where evolution is governed by a linear time-invariant Koopman operator.
By De-Yan Lu, Xugang Lu, Yu Tsao, Jian-Jiun Ding
arXiv:2606. 17070v1 Announce Type: cross Abstract: Accurate oceanic forecasting is critical for climate monitoring and disaster early warning.
By Qinghui Chen, Zekai Zhang, Hailong Liu, Jinglin Zhang, Cong Bai
arXiv:2607. 29036v1 Announce Type: new Abstract: Sparse identification of nonlinear dynamics (SINDy) and PDE functional identification (PDE-FIND) recover parsimonious ordinary and partial differential equations (ODEs and PDEs) from data.
By Pongpisit Thanasutives, Yoshinobu Kawahara
arXiv:2602.02832v4 Announce Type: replace
Abstract: Forward forecasting and data assimilation are the two important aspects in physical simulation: one propagates the state forward, the other recover...
By Rares Grozavescu, Etienne Meunier, Pengyu Zhang, Mark Girolami
arXiv:2607. 28035v1 Announce Type: new Abstract: Irregular multivariate time series are widely encountered in applications such as healthcare monitoring, human activity recognition, and environmental sensing.
By Tianen Shen, Zhengyu Li, Yutong Li, Xiangfei Qiu, Xingjian Wu, Bin Yang, Jilin Hu
arXiv:2407. 06312v2 Announce Type: replace-cross Abstract: Many systems resist analytical modeling, making data-driven inference of dynamics important.
By Matthew J. Colbrook, Igor Mezi\'c, Alexei Stepanenko
arXiv:2605. 05540v2 Announce Type: replace Abstract: Fast surrogate modeling for high-dimensional physical dynamics requires more than low short-term error: useful models must roll out efficiently while preserving the statistical structure of long trajectories.
By Tianyue Yang, Xiao Xue