arXiv subjects

Victor Armegioiu

Publications and source records attributed to Victor Armegioiu.

10 recordsLinked to original sources

arXiv2026

Exact Asymptotics for the 2D Euclidean Random Matching Problem

We determine the exact first-order asymptotics of the expected optimal cost in two-dimensional random bipartite matching, for every finite power cost $q \ge 1$, on the flat torus. In the endpoint case $q=1$, this answers a question by Talagrand, in the periodic case. The argument involves the closely related asymptotics of the energy of the solution of the $p$-Poisson equation with a regularized white-noise source. In the limit of vanishing regularization parameter, we identify this energy as the solution of a Variational Martingale Problem on a limiting Gaussian filtration, whose value is characterized by a parabolic Monge-Amp\`ere flow.

math.PR↗

arXiv2026

Delayed Dissipation for Two-Dimensional Vortex Sheets

We quantify viscous energy loss for two-dimensional Delort vortex sheets. Let $u^\nu$ be Leray-Hopf solutions on $\mathbb{T}^2$ with uniformly bounded kinetic energy and total vorticity variation, and write $\omega_0^\nu=\mu_0^\nu+f_0^\nu$, where $\mu_0^\nu\geq0$ and $f_0^\nu$ is bounded in $L^p$, $p>1$. For every fixed $0<\delta 0$. Previous estimates covered only $T_\nu=o(\exp(|\log\nu|^\kappa))$, $\kappa<1/2$, so this gives a polynomial lower bound on the energetic lifetime of the inviscid vortex-sheet model. If instead $f_0^\nu$ is bounded in $L(\log L)^\alpha$, the rate is $O(|\log\nu|^{-q_\alpha})$, $q_\alpha=\min\{2\alpha,1\}$, and the loss vanishes when $\log T_\nu=o(|\log\nu|^{q_\alpha})$. On $\mathbb{R}^2$, exact radial solutions attain these exponents for $0<\alpha\leq1/2$. At the endpoint, a bounded-energy $L^p$ family attains the rate $1/|\log\nu|$, while every fixed radial datum dissipates $o(1/|\log\nu|)$ and can lose a fixed amount of energy only on the diffusive scale $1/\nu$.

math.AP↗

arXiv2026

Projected Inverse Iteration: An Eigenvalue Approach to Ground-State Computation with Neural Quantum States

Deep learning offers a powerful approach to quantum many-body problems via neural network wavefunctions, but their optimization remains a severe bottleneck. Existing optimization methods, including natural gradient descent and stochastic reconfiguration, suffer from spectral gap-dependent convergence that limits their effectiveness on systems fraught with competing orders and nearly degenerate ground states, such as frustrated magnets and strongly correlated electron materials. Here, we introduce Projected Inverse Iteration (PII) by re-framing the ground-state search as an eigenvalue problem. PII achieves rapid, gap-insensitive convergence while preserving the favorable polynomial computational scaling of stochastic reconfiguration. Demonstrated on challenging two-dimensional spin systems, including the highly frustrated $J_1$-$J_2$ model, PII outperforms standard optimization techniques and presents a promising algorithmic strategy for discovering complex quantum states in the presence of small spectral gaps. More broadly, PII can be interpreted as a novel natural gradient method tailored for eigenvalue problems, opening up its application to related challenges within deep learning.

quant-ph↗

arXiv2026

Statistical Error Bounds for Generative Solvers of Chaotic PDEs: Wasserstein Stability, Generalization, and Turbulence

Statistical solutions of incompressible Euler describe turbulent dynamics as time-parameterized laws on $L^2$ whose multi-point correlations satisfy an infinite hierarchy of weak identities. Modern generative samplers for PDE forecasting (flow matching, rectified flows, diffusion via probability-flow ODEs) are measure-transport mechanisms and therefore induce Markov operators on laws. We develop a law-level analysis compatible with the correlation-measure framework of Lanthaler--Mishra--Parés-Pulido (LM): convergence in $d_T(μ,ν)=\int_{0}^{T} W_{1}\!\bigl(μ_t,ν_t\bigr)\,\mathrm{d}t$, compactness controlled by structure functions, and identification of limits through hierarchy identities. Quantitatively, we prove a $W_2$ stability estimate whose growth rate is a distance-weighted average strain under optimal couplings, and a one-step error decomposition into a resolved mismatch term and an unavoidable high-frequency coverage tail controlled by structure-function (spectral) bounds. These inputs propagate through multi-step rollouts via a discrete Grönwall recursion with amplification governed by the average-strain exponent rather than a worst-case Lipschitz constant. On the qualitative side, sampler-native path controls yield LM time regularity; together with uniform energy and structure-function bounds this gives precompactness in $d_T$ and strong convergence of LM-admissible observables. If hierarchy residuals vanish along a sequence, every limit is an LM statistical solution, with residuals bounded by training-native drift/score regression errors. Finally, we show how common finite-grid diagnostics--proper distributional scores and likelihood-style certificates--admit principled interpretations as resolved observables within the same statistical-solution framework.

math.AP↗

arXiv2026

Rectified Flows for Fast Multiscale Fluid Flow Modeling

Statistical surrogate modeling of fluid flows is hard because dynamics are multiscale and highly sensitive to initial conditions. Conditional diffusion surrogates can be accurate, but usually need hundreds of stochastic sampling steps. We propose a rectified-flow surrogate that learns a time-dependent conditional velocity field transporting input-to-output laws along near-straight trajectories. Inference is then a deterministic ODE solve, making each function evaluation more informative: on multiscale 2D benchmarks, we match diffusion-class posterior statistics with only (8) ODE steps versus (\ge 128) for score-based diffusion. Theoretically, we give a law-level analysis for conditional PDE forecasting. We (i) connect one-point Wasserstein field metrics to the (k=1) correlation-marginal perspective in statistical solutions, (ii) derive a one-step error split into a **coverage** term (high-frequency tail, controlled by structure functions/spectral decay) and a **fit** term (controlled by the training objective), and (iii) show that rectification-time **straightness** controls ODE local truncation error, yielding practical step-size/step-count guidance. Motivated by this, we introduce a curvature-aware sampler that uses an EMA straightness proxy to adapt blending and step sizes at inference. Across incompressible and compressible multiscale 2D flows, it matches diffusion baselines in Wasserstein statistics and spectra, preserves fine-scale structure beyond MSE surrogates, and significantly reduces inference cost.

cs.LG↗

arXiv2026

Memory-Conditioned Flow-Matching for Stable Autoregressive PDE Rollouts

Autoregressive generative PDE solvers can be accurate one step ahead yet drift over long rollouts, especially in coarse-to-fine regimes where each step must regenerate unresolved fine scales. This is the regime of diffusion and flow-matching generators: although their internal dynamics are Markovian, rollout stability is governed by per-step \emph{conditional law} errors. Using the Mori--Zwanzig projection formalism, we show that eliminating unresolved variables yields an exact resolved evolution with a Markov term, a memory term, and an orthogonal forcing, exposing a structural limitation of memoryless closures. Motivated by this, we introduce memory-conditioned diffusion/flow-matching with a compact online state injected into denoising via latent features. Via disintegration, memory induces a structured conditional tail prior for unresolved scales and reduces the transport needed to populate missing frequencies. We prove Wasserstein stability of the resulting conditional kernel. We then derive discrete Grönwall rollout bounds that separate memory approximation from conditional generation error. Experiments on compressible flows with shocks and multiscale mixing show improved accuracy and markedly more stable long-horizon rollouts, with better fine-scale spectral and statistical fidelity.

cs.LG↗

arXiv2026

The Semigeostrophic--Euler Limit via Perturbative Monge--Amp\`ere Estimates

We study the two-dimensional semigeostrophic system on the flat torus in the small-amplitude regime. We formulate the rescaled dynamics as the Lie--Poisson flow of a renormalized optimal-transport energy and expand this Hamiltonian in $C^1$. The leading term is the Euler Hamiltonian, while the first correction is an explicit cubic Monge--Amp\`ere functional. We then derive quantitative consequences for the semigeostrophic--Euler limit: a perturbative scale-uniform endpoint Monge--Amp\`ere estimate under Hessian pinching, an explicit logarithmic perturbative lifespan for the strong branch, fixed-slow-time $O(\eps)$ velocity convergence for canonically prepared strong branches, a conditional weak--strong rate-transfer corollary, and an $O(\eps^2)$ Wasserstein comparison for the physical densities.

math.AP↗

arXiv2025

Out-of-Distribution Detection in Molecular Complexes via Diffusion Models for Irregular Graphs

Predictive machine learning models generally excel on in-distribution data, but their performance degrades on out-of-distribution (OOD) inputs. Reliable deployment therefore requires robust OOD detection, yet this is particularly challenging for irregular 3D graphs that combine continuous geometry with categorical identities and are unordered by construction. Here, we present a probabilistic OOD detection framework for complex 3D graph data built on a diffusion model that learns a density of the training distribution in a fully unsupervised manner. A key ingredient we introduce is a unified continuous diffusion over both 3D coordinates and discrete features: categorical identities are embedded in a continuous space and trained with cross-entropy, while the corresponding diffusion score is obtained analytically via posterior-mean interpolation from predicted class probabilities. This yields a single self-consistent probability-flow ODE (PF-ODE) that produces per-sample log-likelihoods, providing a principled typicality score for distribution shift. We validate the approach on protein-ligand complexes and construct strict OOD datasets by withholding entire protein families from training. PF-ODE likelihoods identify held-out families as OOD and correlate strongly with prediction errors of an independent binding-affinity model (GEMS), enabling a priori reliability estimates on new complexes. Beyond scalar likelihoods, we show that multi-scale PF-ODE trajectory statistics - including path tortuosity, flow stiffness, and vector-field instability - provide complementary OOD information. Modeling the joint distribution of these trajectory features yields a practical, high-sensitivity detector that improves separation over likelihood-only baselines, offering a label-free OOD quantification workflow for geometric deep learning.

cs.LG↗

arXiv2025

Functional Neural Wavefunction Optimization

We propose a framework for the design and analysis of optimization algorithms in variational quantum Monte Carlo, drawing on geometric insights into the corresponding function space. The framework translates infinite-dimensional optimization dynamics into tractable parameter-space algorithms through a Galerkin projection onto the tangent space of the variational ansatz. This perspective unifies existing methods such as stochastic reconfiguration and Rayleigh-Gauss-Newton, provides connections to classic function-space algorithms, and motivates the derivation of novel algorithms with geometrically principled hyperparameter choices. We validate our framework with numerical experiments demonstrating its practical relevance through the accurate estimation of ground-state energies for several prototypical models in condensed matter physics modeled with neural network wavefunctions.

cond-mat.str-el↗

arXiv2025

Generative AI for fast and accurate statistical computation of fluids

We present a generative AI algorithm for addressing the pressing task of fast, accurate, and robust statistical computation of three-dimensional turbulent fluid flows. Our algorithm, termed as GenCFD, is based on an end-to-end conditional score-based diffusion model. Through extensive numerical experimentation with a set of challenging fluid flows, we demonstrate that GenCFD provides an accurate approximation of relevant statistical quantities of interest while also efficiently generating high-quality realistic samples of turbulent fluid flows and ensuring excellent spectral resolution. In contrast, ensembles of deterministic ML algorithms, trained to minimize mean square errors, regress to the mean flow. We present rigorous theoretical results uncovering the surprising mechanisms through which diffusion models accurately generate fluid flows. These mechanisms are illustrated with solvable toy models that exhibit the mathematically relevant features of turbulent fluid flows while being amenable to explicit analytical formulae. Our codes are publicly available at https://github.com/camlab-ethz/GenCFD.

cs.LG↗