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版本更新 · v2arXiv:2608.17032

验证牛顿引力N体问题的直接求解器，并系统比较IEEE浮点与Posits

Validating direct solvers for Newton's gravitational N-body problem, and the systematic comparison between IEEE floating point and Posits

Simon Portegies Zwart

AI 速读 基于摘要

系统比较任意精度、IEEE-754浮点（fp16/bfp16/fp32/fp64/fp128）与两种Posits实现在求解牛顿混沌N体问题中的精度与速度表现。

physics.comp-ph · astro-ph.IM · cs.PL · nlin.CD

arXiv ↗PDF ↗
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We present a systematic comparison between arbitrary precise arithmetic and integration, IEEE-754 compliant floating point arithmetic (fp16, bfp16, fp32, double precision fp64, and quadruple precision fp128), and two implementations of Posits (type III unum) for solving Newton's chaotic N-body problem. Each implementation is benchmarked with arbitrary precise calculations to objectively evaluate their performance in precision as well as speed. We rely on hardware and compiler implementations for fp64, and software implementations for arbitrary-precision arithmetic and Posits. Half precision arithmetic (fp16, bfp16, and Posits$<16,1>$) are insufficiently precise for solving Newton's equations of motion. Single precision (fp32, and Posits$<32,2>$) could be used for statistical ensemble calculations, but lead to relatively large errors in any individual strong encounter. All 64-bit implementations fp64 as well as Posits (Posits$<64,3>$) experience difficulty in our tests. One of the implementations of Posits (Universal) gives precision comparable to fp64 but is slow (by at least an orders of magnitude compared to fp64 after correcting for the more efficient hardware support for the latter). The other (CPPPosits) has a speed comparable to fp64 but has systematically larger errors (by about an order of magnitude compared to fp64 with excesses exceeding two orders of magnitude). As a consequence, this implementation leads to a systematic drift in the result space and has difficulty resolving close encounters. Posits and fp64 have difficulty when integrating a dynamical system in a moving reference frame; testing Galileo invariancy. In their current implementation, Posits do not seem to be the ideal alternative for fp64 when integrating chaotic or stiff ordinary differential equations, such as Newton's equations of motion.

版本更新 · v2arXiv:2607.10430

Emergent Generalization by Representation Learning in Artificial Neural Networks

Hardik Rajpal, Dan Goodman

原文摘要

Dimensionality reduction has proven powerful for identifying neural manifolds, which are low-dimensional structures underlying high-dimensional neural activity. These low-dimensional representations have improved the interpretability of population-level coding. Yet whether such low-dimensional representations are biologically relevant and confer functional advantages in learning systems, or merely reflect neuron-level activity, remains contested in neuroscience. We show that an explicit information bottleneck forcing a recurrent neural network to learn a low-dimensional representation is necessary for rotational and out-of-distribution generalisation in a time-series prediction task. Using information-theoretic measures of causal emergence, we characterise the dynamics of this representation across the memorisation-to-generalisation transition, finding a non-monotonic trajectory which shows an initial decrease, a minimum, and a subsequent rise to a maximum, even as prediction loss falls monotonically. This trajectory scales with task complexity, and the magnitude of emergent structure reliably predicts generalisation performance. Analysis of hippocampal CA1 and medial prefrontal activity in rats learning an alternating maze task reveals analogous non-monotonic emergence dynamics that track behavioural performance. Together, these findings indicate that the ability of neural networks to learn compact, distributed and emergent representations confers a functional advantage for generalisation, supporting a causal role for learned representations in cognition.

q-bio.NC · cs.IT · cs.LG · cs.NE · nlin.CD

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arXiv ↗PDF ↗
版本更新 · v3arXiv:2604.23952

Score-Based Modeling of Effective Langevin Dynamics with State-Dependent Mobility

Ludovico T. Giorgini

原文摘要

In a previous work [L. T. Giorgini, Phys. Rev. E 114, L012102 (2026)], we introduced stochastic reduced-order models that reproduce by construction the stationary density and short-time correlations of the observed variables, using a constant mobility matrix and the score, the gradient of the logarithm of this density. Here we generalize the framework to state-dependent mobilities, calibrated so that the model also reproduces the finite-lag correlation functions of chosen target observables. The calibration uses exact finite-lag forms of the generalized fluctuation-dissipation theorem, which express time derivatives of lagged correlations as averages over observed pairs of states and are linear in the mobility. When the mobility is expanded on a basis of known fields, its coefficients follow from a least-squares problem involving only the stationary score, and the observables and basis are selected by data-driven diagnostics. Otherwise, a neural-network mobility is calibrated through the conditional score of the finite-time transition density. We validate the approach analytically on a Cox-Ingersoll-Ross process and numerically on a two-dimensional nonequilibrium diffusion with multiplicative noise and a 36-dimensional soft-spin Landau-Lifshitz chain. Both methods reproduce the stationary statistics and the target correlations accurately. Because the identities can be integrated between consecutive lags, so that no time derivatives need to be estimated, the recovered drift and diffusion are much less sensitive to coarse sampling than Kramers-Moyal estimates, which require short sampling intervals. Calibration requires no simulation of the model and applies to high-dimensional systems.

stat.ML · cs.LG · nlin.CD

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2610.27321

Physics-Inspired Optimizers beyond the Edge of Stability: Period-Two Orbits and Delayed Instability

Alexander Itin

原文摘要

Gradient descent with momentum can be viewed as a particle moving with friction in the loss landscape. In deep learning the learning rate is usually large, so training is a discrete dynamical system: in many full-batch experiments the sharpness grows to the optimizer's stability limit, the edge of stability (EoS), and training continues there. Physics-inspired optimizers change this motion; Velocity-Regularized Adam (VRAdam), for example, shrinks the step at high speed. We study kinetic maps, in which the step is the gradient of a kinetic energy of the momentum, and ask what they do beyond the EoS. (i) In a reduced model, a kinetic map creates a symmetric period-two orbit beyond the edge, strictly linearly stable if and only if the map's elasticity (the log-log slope of step length versus momentum) lies between 0 and 1; on it, training sits at an effective edge of stability. VRAdam's damper fails above twice the critical learning rate, where its kinetic energy is not convex; the Legendre dual of VRAdam's own Lagrangian is convex and passes at every learning rate. (ii) Because the sharpness rises gradually, the loss of stability is delayed, as in Neishtadt's theory of slowly swept bifurcations. In a network with a controlled sweep, the dynamics match this theory without fitted parameters, including the memory of where the sweep started and its erasure by noise. (iii) With momentum, reaching the oscillation also requires elasticity at large momentum: maps that bound the step have competing period-four and chaotic attractors already at beta1 = 0.9, while the cubic-Legendre map reached the oscillation from every sampled start up to beta1 = 0.99. Experiments on an exact quadratic problem, a full-batch MLP and ResNet-32 support the theory; on ResNet-18, much of the apparent benefit of the maps is reproduced by matching the effective learning rate.

nlin.CD · cond-mat.dis-nn · cs.LG

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.35123

Generative AI in Turbulence: Foundations, Applications, Validations and Benchmarks

Luca Biferale, Michele Buzzicotti

原文摘要

Generative AI offers new approaches to turbulence problems in which incomplete observations or unresolved d.o.f. admit multiple physically plausible solutions. Potential applications include stochastic closures, super-resolution, data assimilation, uncertainty quantification, and the generation of Eulerian and Lagrangian flow ensembles. However, turbulence is not a conventional image-generation problem. Its multiscale dynamics, intermittency, extreme events, rough velocity fields and intrinsically stochastic transport make visual realism, pointwise error, and low-order spectral agreement insufficient measures of physical fidelity. We examine where generative AI provide novel capabilities with emphasis on diffusion models, score-based methods, and related probability-transport formulations. We discuss their use in unconditional generation, probabilistic reconstruction, reduced-order modeling, temporal evolution, and mesh-free representations. We argue that validation must address turbulence properties: energy transfer, structure functions, anomalous scaling, pdf tails, coherent structures, conservation laws, temporal consistency, ensemble diversity and generalization. Finally, we discuss potential limitations in validation, baseline, data availability and reproducibility, and the need for a benchmark ecosystem combining open data, standardized challenges, hidden tests, physics-aware metrics, and documented failures. This paper serves as a bridge between two communities. For AI practitioners, we provide a fair and accessible introduction to turbulence, essential for appreciating the challenges posed by the field and for assessing the physical relevance of generative models. For turbulent practitioners, we provide an introduction to the key technological aspects of Generative AI, clarifying what these tools actually do, and where their potential and current limitations lie.

cs.AI · cs.LG · nlin.CD

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.35121

Revisiting entropy and entropy-related magnitudes of a symbolic sequence and their estimation

Efren Aragon-Perez, Ania Mesa-Rodriguez, Karel Garcia-Medina, Daniel Estevez-Moya, Amanda Mollineda-Rodriguez, Ernesto Estevez-Rams

原文摘要

This article revisits entropy and related magnitudes (entropy density, effective measure complexity, and informational distance) for the analysis of sequential data. We present their mathematical foundations within information theory and ergodic theory, establishing connections between probabilistic Shannon entropy and algorithmic Kolmogorov complexity. A central result, Zivs theorem, demonstrates the asymptotic equivalence between the normalised number of distinct words and entropy density, providing the theoretical basis for estimation via Lempel-Ziv factorisation. We examine the convergence properties of these magnitudes and introduce estimation procedures for finite sequences, including a method for the effective measure complexity and the transfer entropy rate. An open-source software library implementing parallel algorithms for efficient Lempel-Ziv factorisation is presented. The practical utility of the framework is illustrated through three case studies: functional brain connectivity analysis from fMRI data, characterisation of spatiotemporal dynamics in coupled nonlinear oscillators, and electrocardiogram-based classification of atrial fibrillation, demonstrating how entropic measures capture meaningful structural and dynamical properties across diverse domains.

nlin.CD · cs.IT

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.34471

Extrapolating learned dynamics across bifurcations based on generalized synchronization

Lingfeng Chen, Masanobu Inubushi

原文摘要

Predicting dynamics across bifurcations by extrapolating trained models is crucial in diverse fields of physics. Generalized synchronization provides a framework for understanding dynamical reconstruction by neural networks, especially in the context of reservoir computing. Building on this framework, we derive a formula combining pretrained readouts to reconstruct unseen chaotic dynamics. Using classical Lagrange interpolation theory, we derive rigorous bounds on readout errors and conjugacy residuals. Numerical examples demonstrate accurate prediction of chaotic dynamics across bifurcations.

nlin.CD · cs.LG · math.DS

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arXiv ↗PDF ↗