The paper introduces kernel-based methods for learning Hamiltonian systems directly from trajectory data, offering both a 2‑step approach (reconstruct trajectories first, then learn the Hamiltonian) and a 1‑step approach (joint inference). Experiments on mass‑spring dynamics, a nonlinear pendulum, and the Henon‑Heiles system show that the methods achieve accurate, data‑efficient predictions, outperforming 2‑step baselines especially when data are scarce, while preserving the Hamiltonian structure. The authors also provide a priori error estimates and a general numerical framework applicable to arbitrary dynamical systems.
By Yasamin Jalalian, Mostafa Samir, Boumediene Hamzi, Peyman Tavallali, Houman Owhadi
arXiv:2606. 00988v1 Announce Type: new Abstract: Symbolic regression (SR) offers a route to scientific discovery by converting observations into interpretable governing equations.
By Simon De Reuver, Tamas Kristof Toth, Teddy Lazebnik
arXiv:2607. 06287v1 Announce Type: cross Abstract: We study kernel-based operator learning in a two-stage sampling framework, where an offline kernel regression operator learns a discretized representation of the target operator from input-output pairs and an online kernel reconstruction operator recovers the output function from predicted observations.
By R\"udiger Kempf
We study kernel-based operator learning in a two-stage sampling framework, where an offline kernel regression operator learns a discretized representation of the target operator from input-output pairs and an online kernel reconstruction operator recovers the output function from predicted observations. Our main theoretical contribution is an explicit budget allocation condition relating the number $N$ of training pairs, the number $n$ of input observations, and the output resolution $m$.
arXiv:2607. 09801v1 Announce Type: new Abstract: Governing equations provide compact descriptions of physical systems, yet the variables in which they are simple are often hidden in high-dimensional measurements.
By Yi Zhu, Su Chen, Xiaojun Li, Xiuli Du
arXiv:2607. 16251v1 Announce Type: new Abstract: Spatio-Temporal Foundation Models (STFMs) aim to learn generalizable representations of complex dynamical systems across space and time.
By Yutong Feng, Shiyuan Piao, Yutong Xia, Xu Liu, Wenqi Fan, Fugee Tsung, See-Kiong Ng, Yuxuan Liang
arXiv:2604. 24662v2 Announce Type: replace-cross Abstract: Identifying the dynamical state variables of a system from high-dimensional observations is a central problem across physical sciences.
By K. Michael Martini, Eslam Abdelaleem, Paarth Gulati, Ilya Nemenman
arXiv:2606. 01894v1 Announce Type: new Abstract: Accurate Remaining Useful Life prediction is critical for industrial predictive maintenance.
By Deyu Zhuang, Peiliang Gong, Yang Shao, Liyuan Shu, Qi Zhu, Xiaoli Li, Daoqiang Zhang
The paper introduces a learning method for identifying interaction kernels in particle systems using only single-snapshot observations of collective steady states, rather than trajectory data. By regularizing with empirical distributions from varied, unseen initial conditions, the authors address the ill‑posed inverse problem and demonstrate stable, accurate recovery of interaction laws across several models. The recovered laws enable faithful reproduction of both steady‑state patterns and, in many cases, the preceding dynamics.
By Baoli Hao, Mauro Maggioni, Ming Zhong
arXiv:2605. 19805v2 Announce Type: replace-cross Abstract: Irregular multivariate time series impose a trade-off for long-horizon forecasting: discrete methods can distort temporal structure via re-gridding, while continuous-time models often require sequential solvers prone to drift.
By Zinuo You, Jin Zheng, John Cartlidge
arXiv:2607. 23337v1 Announce Type: new Abstract: Neural operators provide data-driven mappings for modeling dynamical systems.
By Zituo Chen, Qiaofeng Li, Jiaxin Hu, Sili Deng
arXiv:2505. 07068v2 Announce Type: replace-cross Abstract: In this paper, we investigate the data-driven identification of asymmetric interaction kernels in the Motsch-Tadmor model based on observed trajectory data.
By Jinchao Feng, Sui Tang