Determining the similarity between dynamical systems remains a long-standing challenge in both machine learning and neuroscience. Recent works based on Koopman operator theory have proven effective in analyzing dynamical similarity by examining discrepancies in the Koopman spectrum. Nevertheless, existing similarity metrics can be severely constrained when systems exhibit complex nonlinear behaviors across multiple temporal scales. In this work, we propose KoopSTD, a dynamical similarity measurement framework that precisely characterizes the underlying dynamics by approximating the Koopman spectrum with explicit timescale decoupling and spectral residual control. We show that KoopSTD maintains invariance under several common representation-space transformations, which ensures robust measurements across different coordinate systems. Our extensive experiments on physical and neural systems validate the effectiveness, scalability, and robustness of KoopSTD compared to existing similarity metrics. We also apply KoopSTD to explore two open-ended research questions in neuroscience and large language models, highlighting its potential to facilitate future scientific and engineering discoveries. Code is available at link.

Pansharpening is a special image fusion task of reconstructing a high-resolution multispectral (HRMS) image by integrating a panchromatic (PAN) image of high spatial resolution and a low-resolution multispectral (LRMS) image. To handle such an ill-posed multi-modal fusion task, in this paper, we propose a novel pansharpening method, referred to as model-driven and data-driven network (MD3Net), which combines model-driven and data-driven approaches. The architecture design of MD3Net is inspired from the traditional model constructed based on domain knowledge and thus making its network topology explainable and its input/output predictable. In order to further explore the powerful learning ability of deep learning based approaches, we introduce the deep prior into the MD3Net as its implicit regularization, thus improving its data adaptability and representation capability. Comprehensive experiments conducted on both reduced and full resolution of several acknowledged datasets have qualitatively and quantitatively verified the superiority of our network compared to a benchmark consisting of several state-of-the-art approaches. The code can be downloaded from https://github.com/YinsongYan/M3DNet..