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At least 163 records · Page 9Linked to original sources
arXiv2026

The electrostatic-hydrodynamic interaction of two spheres in a linear compressional flow

We investigate the collision dynamics of two like-charged dielectric spheres in a uniaxial compressional flow, accounting for non-continuum lubrication interactions, van der Waals attraction, and the complete finite-dielectric electrostatic interaction. Mutual polarization causes the electrostatic force to behave substantially differently from the point-charge approximation and can produce strong near-field attraction despite far-field Coulomb repulsion, depending on the particle size and charge ratios. We identify a critical electrostatic strength above which particles approaching from the far field cannot reach contact. This threshold corresponds to a saddle-node bifurcation of the relative-trajectory field on the compressional axis. Below this threshold, near-field polarization-induced attraction can increase the upstream interception area and substantially enhance the collision efficiency. We report that electrostatic interactions lead to non-monotonic variations of collision efficiency with electrostatic strength, particle size ratio, and charge ratio. When electrostatic interactions remain repulsive at all separations, increasing their strength suppresses collisions monotonically. Non-continuum lubrication and van der Waals interactions further modify the collision efficiency, although sufficiently strong polarization-induced attraction can dominate these near-field effects. For weak electrostatic interactions, a perturbation analysis expresses the leading correction to the collision efficiency as a weighted average of the electrostatic force along the limiting colliding trajectory, with the weight concentrated in the lubrication region; this prediction agrees closely with the numerical results. These results demonstrate that finite-size polarization and near-field hydrodynamic interactions jointly govern the collision behaviour of like-charged particles in compressional flows.

physics.flu-dyn↗
arXiv2026

Point-Group Equivariant Neural Networks for Local Dynamics in Lattice Systems

We develop point-group equivariant neural networks for learning local mappings used in dynamical simulations of lattice systems. The same network acts on a finite neighborhood around each site, respecting lattice translations through shared parameters. The local configuration is decomposed into features that transform according to irreducible representations of the point group, and hidden features retain this organization throughout the network. Equivariant linear maps and symmetry-allowed tensor products then learn nonlinear spatial correlations while preserving the required transformation laws. This approach avoids constructing a fixed set of hand-crafted invariant descriptors, whose selection requires choices about which local correlations to retain. It also allows the complexity of the learned mapping to grow without enlarging its spatial cutoff. We demonstrate the construction by learning the forces governing adiabatic dynamics in the square-lattice Holstein model. Comparisons with exact diagonalization test the accuracy of the predicted forces and the evolution of spatial correlations. Force evaluation scales linearly with system size, allowing the trained model to simulate charge-density-wave ordering on large lattices. These simulations reproduce anomalously slow coarsening of the charge-density-wave domains, illustrating the use of the learned local mapping for collective dynamics.

cond-mat.str-el↗
arXiv2026

Arboreal Networks and Ultrametrics

Ultrametrics are an important class of distances used in applications such as phylogenetics, clustering and classification theory. Ultrametrics are essentially distances that can be represented by an edge-weighted rooted tree so that all of the distances in the tree from the root to any leaf of the tree are equal. In this paper, we introduce a generalization of ultrametrics called arboreal ultrametrics which have applications in phylogenetics and also arise in the theory of distance-hereditary graphs. These are partial distances, that is distances that are not necessarily defined for every pair of elements in the groundset, that can be represented by an ultrametric arboreal network, that is, an edge-weighted rooted network whose underlying graph is a tree. As with ultrametrics all of the distances in the ultrametric arboreal network from any root to any leaf below it are are equal but, in contrast, the network may have more than one root. In our two main results we characterize when a partial distance is an arboreal ultrametric as well as proving that, somewhat surprisingly, given any unrooted edge-weighted phylogenetic tree there is a necessarily unique way to insert roots into this tree so as to obtain an arboreal ultrametric.

math.CO↗
arXiv2026

Parallelizable memory recurrent units

With the emergence of massively parallel processing units, parallelization has become a desirable property for new sequence models. The ability to parallelize the processing of sequences with respect to the sequence length during training is one of the main factors behind the uprising of the Transformer architecture. However, Transformers lack efficiency at sequence generation, as they need to reprocess all past timesteps at every generation step. Recently, state-space models (SSMs) emerged as a more efficient alternative. These new kinds of recurrent neural networks (RNNs) keep the efficient update of the RNNs while gaining parallelization by getting rid of nonlinear dynamics (or recurrence). SSMs can reach state-of-the-art performance through the efficient training of potentially very large networks, but still suffer from limited representation capabilities. In particular, SSMs cannot exhibit persistent memory, or the capacity of retaining information for an infinite duration, because of their monostability. In this paper, we introduce a new family of RNNs, the memory recurrent units (MRUs), that combine the persistent memory capabilities of nonlinear RNNs with the parallelizable computations of SSMs. These units leverage multistability as a source of persistent memory, while getting rid of transient dynamics for efficient computations. We then derive a specific implementation as proof-of-concept: the bistable memory recurrent unit (BMRU). This new RNN is compatible with the parallel scan algorithm. We show that BMRU achieves good results in tasks with long-term dependencies, and can be combined with SSMs to create parallelizable hybrid networks that have both transient dynamics and persistent memory.

cs.LG↗
arXiv2026

Penalty-Based Smoothing and Accelerated Inertial Dynamics for Convex Supremum Optimization

We study dual-penalty smoothing for convex functions with a finite-dimensional supremum representation. The regularization yields a differentiable surrogate with controlled approximation error, a single-valued dual maximizer, and explicit gradient formulas. We then study an accelerated inertial dynamic with vanishing damping driven by a time-dependent regularized function whose parameter decreases to zero. The global well-posedness of this system is not an immediate consequence of standard ODE theory, since the gradient of the regularized objective is in general only locally Lipschitz and its local Lipschitz modulus may deteriorate as the smoothing parameter vanishes. We therefore establish existence and uniqueness of global trajectories through a local well-posedness analysis and a continuation argument. Then, assuming that the original problem has a minimizer and under mild integrability conditions on the regularization schedule, we establish an accelerated $\mathcal{O}(t^{-2})$ decay estimate for the regularized and original objective values and, in the regime $α>3$, a sharper $o(t^{-2})$ decay together with weak convergence of trajectories to a minimizer of the original nonsmooth problem via an Opial-type argument. Applications to multiobjective optimization (through Chebyshev/max scalarization) and to distributionally robust optimization (via entropic regularization over ambiguity sets) illustrate the scope of the framework.

math.OC↗
arXiv2026

Multigrade Neural Network Approximation

We study multigrade deep learning (MGDL) as a principled framework for structured error refinement in deep neural networks. While the approximation power of neural networks is now relatively well understood, training very deep architectures remains challenging due to highly nonconvex and often ill-conditioned optimization landscapes. In contrast, for relatively shallow networks, most notably certain one-hidden-layer ReLU models, training admits convex reformulations with global guarantees under appropriate settings, motivating learning paradigms that improve stability while scaling to depth. MGDL builds on this insight by training deep networks grade by grade: previously learned grades are frozen, and each newly added grade-wise subnetwork is composed on top of the previously learned grades and trained to fit the residual left by the current approximation, yielding a structured and interpretable hierarchical refinement process. We develop an operator-theoretic foundation for MGDL and prove that, for any continuous target function defined on a hypercube, there exists a fixed-width multigrade ReLU scheme whose residuals are pointwise nonincreasing in magnitude and converge uniformly to zero, with strict $L^p$-norm decay at every nontrivial grade for $p\in [1,\infty)$. To the best of our knowledge, this work provides the first rigorous constructive approximation guarantee showing that a grade-wise residual refinement scheme can achieve vanishing error in a fixed-width multigrade ReLU architecture.

cs.LG↗
arXiv2026

Stochstic Sampling for Generative Diffusion Models: From Euler-Maruyama to Higher-Order Schemes

We develop a convergence analysis for generative diffusion models that simultaneously accounts for the three principal error sources in stochastic sampling: initialization, score matching, and discretization of the reverse-time stochastic differential equation (SDE). Our central tool is the notion of a iterative-wise general strong scheme, a broad class of discretization methods for the reverse dynamics defined via explicit, index-wise tolerances on their Itô--Taylor coefficients. This notion refines Kloeden and Platen's general strong schemes into an iterate-wise formulation whose per-step conditions imply the classical remainder bound. We prove a 2-Wasserstein convergence theorem that applies to this entire class, reducing the analysis of any concrete sampler to a finite checklist, and covers general forward processes with time-dependent, spatially linear drift and spatially independent diffusion coefficient. We focus on dissipative reverse-time SDEs, which allows us to relax the $T$-dependence of the discretization and score-matching errors from exponential to polynomial. We instantiate our theorem for Euler--Maruyama, the exponential integrator, and, as our main application, derivative-free stochastic Runge--Kutta schemes of strong order $1.5$. To our knowledge, these are the first full convergence guarantees for solvers whose strong order strictly exceeds that of Euler--Maruyama. We further derive the resulting iteration complexity and a parameter-selection rule for the terminal time, score accuracy, and step size. Numerical experiments on Gaussian toy models and CIFAR-10 are consistent with the predicted convergence orders, and on ImageNet, out-of-the-box higher-order SDE solvers can yield substantial speed-ups, requiring at least four times fewer score evaluations than Euler--Maruyama. Code available at: https://github.com/emanuelpfarr/SSGDM.

math.NA↗
arXiv2026

On decay of solutions to the anisotropic Boussinesq equations near the hydrostatic balance in half space $\mathbb{R}_+^3$

The system of the Boussinesq equations is one of the most important models for geophysical fluids. This paper focuses on the initial-boundary problem of the 3D incompressible anisotropic Boussinesq system with horizontal dissipation. The goal here is to assess the stability property and large-time behavior of perturbations near the hydrostatic balance. By utilizing the structure of the system, the energy methods and the means of bootstrapping argument, we prove the global stability property in the Sobolev space $H^3(\mathbb{R}^3_+)$. After taking a Fourier transform in $x_h = (x_1, x_2)$ and Fourier cosine and sine transforms in $x_3$ for the system, we obtain the decay rates for the global solution itself as well as its derivatives.

math.AP↗
arXiv2026

Sampling Sphere Packings with Continuum Glauber Dynamics

Continuum Glauber dynamics is a spatial birth-death process whose stationary distribution is a Gibbs distribution. We establish a spectral gap for Continuum Glauber dynamics applied to Gibbs point processes with repulsive pair potentials, a well-known special case of which is the hard sphere model. For arbitrary-range repulsive pair potentials, we show that a continuous version of Spectral Independence suffices to establish a spectral gap. This extends the regime of activity for which Continuum Glauber dynamics is known to mix, yielding a simple efficient sampling algorithm for arbitrary-range pair potentials that matches the known efficient sampling regime for finite-range pair potentials currently based on specialized algorithms. As a consequence, we also improve the threshold up to which packings of fixed size/density can be efficiently sampled from a bounded domain, the first improvement since Kannan, Mahoney and Montenegro (2003). To prove these results, we develop continuous analogs of Spectral Independence and negative fields localization. We show that a stronger variant of zero-freeness implies Spectral Independence, which in turn allows us to run the localization scheme to boost the spectral gap of Continuum Glauber dynamics from smaller activity to larger activity. While this follows the high-level blueprint of Chen and Eldan (2022) for the discrete setting, we have to address several novel difficulties due to the continuous setting. Notably, we avoid discretization in the algorithm and the analysis and work directly in the continuous setting.

cs.DS↗
arXiv2026

Efficient Evaluation of LLM Performance with Statistical Guarantees

Exhaustively evaluating many large language models (LLMs) on a large suite of benchmarks is expensive. We cast benchmarking as finite-population inference and, under a fixed query budget, seek tight confidence intervals (CIs) for model accuracy with asymptotic frequentist coverage. We propose Factorized Active Querying (FAQ), which (a) leverages historical information through a Bayesian factor model; (b) adaptively selects questions using a hybrid variance-reduction/active-learning sampling policy; and (c) maintains validity through Pro-Active Inference---a finite-population extension of active inference (Zrnic & Candès, 2024) that enables direct question selection while preserving asymptotic frequentist coverage. In our experiments, FAQ delivers up to $5\times$ effective sample size gains over strong baselines on two benchmark suites, across varying historical-data missingness levels, with negligible computational overhead: this means that it matches the CI width of uniform sampling while using up to $5\times$ fewer queries. We release our source code and our curated datasets to support reproducibility and future research.

stat.ML↗
arXiv2026

Gravitational wave detection via photon-graviton scattering and quantum interference

We present a fully quantum field-theoretic framework for gravitational wave (GW) detection, describing the interaction as inelastic photon-graviton scattering. We demonstrate that for \textit{a coherent graviton state}, this microscopic model explicitly reproduces the standard classical geometric-optics time delay, providing a rigorous consistency test of the correspondence limit. We propose a detection scheme based on Hong-Ou-Mandel interference, where the GW-induced spectral shifts render entangled photon pairs distinguishable. This encodes the gravitational signal into the modulation of photon coincidence rates --- a quantum observable robust against laser intensity noise. Furthermore, we show that non-classical graviton states, such as squeezed states, induce unique decoherence signatures, establishing theoretical bounds on gravitational decoherence and defining the operational limits for probing the quantum nature of gravity.

gr-qc↗
arXiv2026

TabClustPFN: A Prior-data Fitted Network for Tabular Data Clustering

Prior-data Fitted Networks (PFNs) have reframed supervised tabular learning as single-pass in-context inference without per-dataset optimization. Extending this paradigm to clustering is appealing yet fundamentally more challenging, due to absent supervision, unknown cluster cardinality, and label switching inherent to partition outputs. Existing PFN-based clustering methods address these challenges only partially, either requiring known cardinality as input or relying on unstable label-ordering conventions and overly restrictive synthetic priors. We introduce TabClustPFN, a clustering PFN that resolves all challenges jointly through co-designed prior, objective, and architecture. Our hybrid pretraining prior captures heterogeneous real-tabular geometry; our decoupled partition inference network--cardinality inference network architecture jointly infers cluster assignments and cardinality in a single pass; and our SoftARI training objective is permutation-invariant by construction, eliminating the need for any label-ordering convention. On a 44 curated real-world tabular benchmark, TabClustPFN achieves state-of-the-art clustering performance against classical, deep, and amortized baselines, with runtime comparable to efficient baselines. Code is available at https://github.com/Tianqi-Zhao/TabClustPFN.

cs.LG↗
arXiv2026

LoRIF: Pushing the Scalability Frontier of Training Data Attribution

Training data attribution (TDA) identifies which training examples most influenced a model's prediction. Influence function methods are a theoretically grounded family of TDA methods and exploit gradients. To overcome the scalability challenge arising from gradient computation, the most popular strategy is random projection (e.g., TRAK, LoGRA). However, this still faces two bottlenecks when scaling to large training sets and high-quality attribution: \emph{(i)} storing and loading projected per-example gradients for all $N$ training examples, where query latency is dominated by I/O; and \emph{(ii)} forming the $D \times D$ inverse Hessian approximation, which costs $O(D^2)$ memory. Both bottlenecks scale with the projection dimension $D$, yet increasing $D$ is necessary for attribution quality -- creating a quality--scalability tradeoff. We introduce \textbf{LoRIF} (\textbf{Lo}w-\textbf{R}ank \textbf{I}nfluence \textbf{F}unctions), which exploits low-rank structures of gradient to address both bottlenecks. First, we store rank-$c$ factors of projected per-example gradients rather than full matrices, reducing storage and query-time I/O from $O(D)$ to $O(c\sqrt{D})$ per layer per sample. Second, we use truncated SVD with the Woodbury identity to approximate the inverse Hessian term in an $r$-dimensional subspace, reducing memory from $O(D^2)$ to $O(Dr)$. On models from 0.1B to 70B parameters trained on datasets with millions of examples, LoRIF achieves up to 20$\times$ storage reduction and query-time speedup compared to LoGRA, while matching or exceeding its attribution quality. LoRIF makes gradient-based TDA practical at frontier scale. Code is available at https://github.com/doub7e/LoRIF.

cs.LG↗
arXiv2026

Riemannian Lyapunov Framework: Optimization as Closed Loop Control on Riemannian Manifolds

We introduce the Riemannian Lyapunov Framework (RLF), a unified framework that represents first order optimization as a closed loop controlled dynamical system on a Riemannian parameter manifold. Within this framework, optimizers are specified by a quadruple $(M, g, Φ, η)$ consisting of a Riemannian metric, an internal state transition map, a direction field generator, and a lifting parameter. Classical and modern methods including SGD, Adam, AdamW, Lion and Shampoo are recovered as specific instantiations of this quadruple. The central structural object of the framework is a time indexed family of target submanifolds in the extended state space of parameters, velocities, and internal memory, which we call the Normally Attracting Invariant Manifold (NAIM) family. These submanifolds organize training dynamics into two timescales: a fast contraction of the velocity state onto the current target submanifold, followed by a slow descent along it. Using Riemannian backstepping, we construct a strict Lyapunov function on the extended state space and prove Uniform Ultimate Boundedness (UUB) of the optimization trajectory under a descent alignment assumption and bounded stochastic disturbances. Under the additional Polyak-Lojasiewicz (PL) condition, the Lyapunov function converges linearly to a bounded neighborhood of the optimum, defined as the drift of the target direction field between consecutive iterates after parallel transport. To validate the framework's generativity, we derive three concrete instantiations of the quadruple, each exercising a distinct design axis, and show on geometric diagnostics, image classification benchmarks, and language modeling that their empirical behavior matches the theoretical predictions. The framework casts optimizer design as controller synthesis, complementing heuristic perspectives by providing a constructive template with provable guarantees.

cs.LG↗
arXiv2026

zkCraft: Prompt-Guided LLM as a Zero-Shot Mutation Pattern Oracle for TCCT-Powered ZK Fuzzing

Zero-knowledge circuits enable privacy-preserving and scalable systems but are difficult to implement correctly due to the tight coupling between witness computation and circuit constraints. We present zkCraft, a practical framework that combines deterministic, R1CS-aware localization with proof-bearing search to detect semantic inconsistencies. zkCraft encodes candidate constraint edits into a single Row-Vortex polynomial and replaces repeated solver queries with a Violation IOP that certifies the existence of edits together with a succinct proof. Deterministic LLM-driven mutation templates bias exploration toward edge cases while preserving auditable algebraic verification. Evaluation on real Circom code shows that proof-bearing localization detects diverse under- and over-constrained faults with low false positives and reduces costly solver interaction. Our approach bridges formal verification and automated debugging, offering a scalable path for robust ZK circuit development.

cs.CR↗
arXiv2026

Leave-One-Out Neighborhood Smoothing for Graphons: Berry--Esseen Bounds and Edge-Wise Confidence Intervals

We develop a framework for edge-wise inference in graphon-based random networks. The proposed leave-one-out neighborhood procedure excludes the target vertex when selecting neighbors, then averages the selected vertices' edges to that target. Conditional on the latent probabilities and selected neighborhood, these edges are independent. This structure yields finite-sample concentration inequalities and a Berry--Esseen bound for fluctuations about the selected-neighborhood mean. Under blockwise graphon regularity, quantitative two-hop identifiability, and across-block separation, we control the smoothing bias uniformly over latent positions, including positions near block boundaries. Together with an observable variance estimator, this gives Gaussian confidence intervals for individual edge probabilities with asymptotically correct pointwise coverage. We also establish finite-sample empirical Bernstein coverage for the selected-neighborhood mean, Gaussian inference under weaker bandwidth conditions on identical-profile blocks, and inference for target pairs chosen measurably from the latent positions. Finally, we derive an expected-risk identity for leave-one-out cross-validation at each fixed neighborhood size. Simulations assess interval coverage, variance estimation, and estimation error.

stat.ME↗
arXiv2026

Saturn's Evolutionary History and Seismology: Survival of Deep Stably Stratified Regions in Evolutionary Models of Saturn Consistent with Ring Seismology

With recent advances in the modeling of the solar system giant planets, rapid progress has been made in understanding the remaining questions pertaining to their formation and evolution. However, this progress has largely neglected the significant constraints on the interior of Saturn's structure imposed by the observed oscillation frequencies in its rings. Here, we study initial conditions for Saturn's evolution that, after $4.56\;\mathrm{Gyr}$ of evolution, give rise to planetary structures admitting oscillation frequencies consistent with those observed via Saturn's ring seismology. Restricting our attention to models without compact rocky cores, we achieve simultaneous good agreement with most observed properties of Saturn at the level of current evolutionary models and with key frequencies in the observed oscillation spectrum. Our preliminary work suggests that Saturn's interior stably stratified region may be moderately less extended ($\sim 0.4$--$0.5R_{\rm Sat}$) than previously thought, which is important for reconciling the seismic constraints with evolutionary models. We also tentatively find that the deep helium gradients inferred by previous, static structural modelling of Saturn's ring seismology may not be required to reproduce the observed seismology data.

astro-ph.EP↗
arXiv2026

Origin Lens: A Privacy-First Mobile Framework for Cryptographic Image Provenance and AI Detection

The proliferation of generative AI poses challenges for information integrity assurance, requiring systems that connect model governance with end-user verification. We present Origin Lens, a privacy-first mobile framework that targets visual disinformation through a layered verification architecture. Unlike server-side detection systems, Origin Lens performs cryptographic image provenance verification and metadata-based AI detection locally on the device via a Rust/Flutter hybrid architecture. Our system integrates multiple signals - including cryptographic provenance, generative-model metadata and watermarks, and optional retrieval-augmented verification - to provide users with graded confidence indicators at the point of consumption. We discuss the framework's alignment with regulatory requirements (EU AI Act, DSA) and its role in verification infrastructure that complements platform-level mechanisms.

cs.CR↗