The paper investigates how machine learning models can regress scale‑free processes, such as earthquakes or avalanches, focusing on predicting rare, large events that require extrapolation. It studies two self‑similar systems: a 2‑dimensional fractional Gaussian field and the Abelian sandpile model. Experiments compare existing architectures (U‑net, Riesz network) with new proposals (wavelet‑based Graph Neural Network, Fourier embedding, Fourier‑Mellin Neural Operator) to identify spectral bias and coarse‑graining challenges and suggest inductive biases to address them.
arXiv:2607. 03347v1 Announce Type: new Abstract: We consider the Multiscale Single-Index Model (MSIM), first introduced in \cite{oymak2021learning}, as a stylized model for hierarchical learning with \emph{scale separation}.
arXiv:2510. 24616v4 Announce Type: replace-cross Abstract: For four decades statistical physics has been providing a framework to analyse neural networks.
arXiv:2608. 13335v1 Announce Type: new Abstract: Neural networks trained by gradient descent on a smooth cost function can nevertheless learn in steps: the cost holds on long plateaus and then drops abruptly.
arXiv:2608. 08003v1 Announce Type: cross Abstract: As machine learned models increase in complexity and expressive power, features of simpler models, such as interpretability and control over the shape of the modeled function are lost.
arXiv:2603.02843v2 Announce Type: replace Abstract: Generalisation across image scales remains a fundamental challenge for deep networks, which often fail to handle images at scales not seen during t...
arXiv:2510.05606v2 Announce Type: replace Abstract: Fundamental limits to predictability are central to our understanding of many physical and computational systems. In deep learning, training outcom...
arXiv:2507. 05164v2 Announce Type: replace-cross Abstract: In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms.
arXiv:2606. 16575v2 Announce Type: replace Abstract: Deep neural networks (DNNs) have achieved remarkable success in scientific computing, yet they often suffer from spectral bias in capturing oscillatory and multiscale behaviors.
arXiv:2609.36890v1 Announce Type: new Abstract: In scientific machine learning, physical fields governed by partial differential equations exhibit low-rank structure and scale invariance. When solvin...
arXiv:2505. 24849v2 Announce Type: replace-cross Abstract: For three decades statistical mechanics has been providing a framework to analyse neural networks.
arXiv:2607. 07127v1 Announce Type: cross Abstract: Lattice field theory is the workhorse of non-perturbative physics, used to simulate phenomena from the strong nuclear force to critical phenomena in materials.
arXiv:2206. 04359v3 Announce Type: replace Abstract: One of the fundamental challenges in the deep learning community is to theoretically understand how well a deep neural network generalizes to unseen data.