arXiv:2606. 29679v1 Announce Type: new Abstract: Observable Matrix Dynamics (OMD) is a diagnostic framework that probes the dynamics of high-dimensional internal representations of inputs by a neural network via a fixed-size $N \times N$ distance matrix $M(t)$ on a held set of $N$ inputs.
arXiv:2606. 29675v1 Announce Type: new Abstract: Bogomolny, Bohigas and Schmit (BBS) found that the spectrum of the pairwise distance matrix on N points sampled from a smooth d-dimensional manifold encodes a signature of the underlying geometry.
arXiv:2606. 28486v1 Announce Type: cross Abstract: The emergence of low-dimensional structures in the spectra of neural network weight matrices is a common empirical feature of trained models, but the dynamical origin of this phenomenon during learning remains an open problem.
arXiv:2602. 02908v2 Announce Type: replace-cross Abstract: Diffusion models trained on different, non-overlapping subsets of a dataset often produce strikingly similar outputs when given the same noise seed.
arXiv:2610.01522v1 Announce Type: cross Abstract: Many scientific and machine learning systems, from molecular dynamics to diffusion models and beyond, are governed by stochastic dynamics with low-di...
arXiv:2607. 02937v1 Announce Type: new Abstract: Reduced-order models compress high-dimensional dynamics into low-dimensional representations that can be evaluated rapidly, but they lose accuracy when online dynamics drift beyond the training data.
The paper introduces DOODL, a framework that learns a dictionary of spectral dynamics to represent related dynamical systems as points on a low‑dimensional manifold in operator space. By constraining operator estimation to this learned manifold, DOODL provides compact, interpretable embeddings and enables fast, accurate operator estimation from short, partially observed trajectories. Experiments on metastable Langevin dynamics and turbulent plasma simulations show that DOODL achieves one to two orders of magnitude lower errors than independent estimation methods, especially in low‑data regimes.
The paper introduces an in‑span adaptation technique for reduced‑order models, where the reduced subspace is continually updated using the model’s own predictions via an incremental singular‑value decomposition with a forgetting factor. This creates a trajectory‑informed spectral preconditioner that reweights and realigns the basis without changing the subspace, enabling the model to better absorb future out‑of‑span corrections. The authors demonstrate the method on a 3‑D spiral example and nonlinear PDEs such as viscous Burgers and Fisher–KPP, and relate the approach to in‑context learning in dynamical systems.
arXiv:2609.14712v1 Announce Type: cross Abstract: Metastability---a phenomenon where systems remain trapped in quasi-stable states before abruptly transitioning under rare perturbations---is ubiquito...
Training in artificial neural networks can be viewed as a trajectory evolving through a high-dimensional loss landscape. However, the large number of trainable parameters makes the direct analysis of these dynamics challenging.
arXiv:2609.37392v1 Announce Type: new Abstract: Modeling the temporal evolution of macroscopic properties of complex systems is an important scientific task. To predict this evolution without full mi...
arXiv:2607. 00063v1 Announce Type: cross Abstract: This paper studies how spectral geometry emerges in quantum learning models and how it can be diagnosed with physically grounded probes.
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.