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Marco Romito

Publications and source records attributed to Marco Romito.

At least 19 recordsLinked to original sources
arXiv2026

Delayed Blow-up in 3D Fluids via Pseudo-transport Noise

We establish scaling limit results for fluid dynamics equations driven by pseudo-transport noise. The behaviour of noise at small scales is governed by a parameter a. This extends previous results by Flandoli and Luo (2020) and Galeati (2020), which correspond to a=0 in our setting. Depending on the value of a, we prove that the noise delays the potential blow-up of both the 3D Euler and Navier-Stokes (NS) equations with high probability.

math.PR↗
arXiv2026

Universality in Deep Neural Networks: An approach via the Lindeberg exchange principle

We consider the infinite-width limit of a fully connected deep neural network with general weights, and we prove quantitative general bounds on the $2$-Wasserstein distance between the network and its infinite-width Gaussian limit, under appropriate regularity assumptions on the activation function. Our main tool is a Lindeberg principle for Deep Neural Networks, which we use to successively replace the weights on each layer by Gaussian random variables.

math.PR↗
arXiv2026

Stochastic Scaling Limits and Synchronization by Noise in Deep Transformer Models

We prove pathwise convergence of the layerwise evolution of tokens in a finite-depth, finite-width transformer model with MultiLayer Perceptron (MLP) blocks to a continuous-time stochastic interacting particle system. We also identify the stochastic partial differential equation describing the evolution of the tokens' distribution in this limit and prove propagation of chaos when the number of such tokens is large. The bounds we establish are quantitative and the limits we consider commute. We further prove that the limiting stochastic model displays synchronization by noise and establish exponential dissipation of the interaction energy on average, provided that the common noise is sufficiently coercive relative to the deterministic self-attention drift. We finally characterize the activation functions satisfying the former condition.

math.PR↗
arXiv2025

Gaussian Fluctuations for the Stochastic Landau-Lifshitz Navier-Stokes Equation in Dimension $D\geq2$

We revisit the large-scale Gaussian fluctuations for the stochastic Landau-Lifshitz Navier-Stokes equation (LLNS) at and above criticality, using the method in \cite{CGT24}. With the classical diffusive scaling in $d\geq 3$ and weak coupling scaling in $d=2$, we obtain the convergence of the regularised LLNS to a stochastic heat equation with a non-trivially renormalized coefficient. Moreover, we obtain an asymptotic expansion of the effective coefficient when $d\geq3$, and show that the one in \cite[Conjecture 6.5]{JP24} is incorrect. The new ingredient in our proof is a case-by-case analysis to track the evolution of the vector under the action of the Leray projection, combined with the use of the anti-symmetric part of the generator and a rotational change of coordinates to derive the desired decoupled stochastic heat equation from the original coupled system.

math.PR↗
arXiv2025

Instantaneous Total Enhanced Dissipation For Very Rough Shear Flows

This paper investigates enhanced dissipation for a passive scalar advected by "very rough" horizontal shear flows, described by an advection-diffusion equation on the 2D torus. The authors extend results of Galeati and Gubinelli (2023) to generic flows in negative Besov spaces, proving that the dissipation rate increases to infinity as viscosity vanishes. This is obtained by deriving (non-sharp) upper and lower bounds on the dissipation rate. The upper bound holds for truly irregular velocities, namely those verifying a suitable version of the Wei irregularity index (Wei (2021)). As a by-product, it follows that for truly rough shear flows the vanishing viscosity solution to the corresponding inviscid equation is trivial.

math.AP↗
arXiv2025

Effective continuous equations for adaptive SGD: a stochastic analysis view

We present a theoretical analysis of some popular adaptive Stochastic Gradient Descent (SGD) methods in the small learning rate regime. Using the stochastic modified equations framework introduced by Li et al., we derive effective continuous stochastic dynamics for these methods. Our key contribution is that sampling-induced noise in SGD manifests in the limit as independent Brownian motions driving the parameter and gradient second momentum evolutions. Furthermore, extending the approach of Malladi et al., we investigate scaling rules between the learning rate and key hyperparameters in adaptive methods, characterising all non-trivial limiting dynamics.

stat.ML↗
arXiv2025

Global Optimization via Softmin Energy Minimization

Global optimization, particularly for non-convex functions with multiple local minima, poses significant challenges for traditional gradient-based methods. While metaheuristic approaches offer empirical effectiveness, they often lack theoretical convergence guarantees and may disregard available gradient information. This paper introduces a novel gradient-based swarm particle optimization method designed to efficiently escape local minima and locate global optima. Our approach leverages a "Soft-min Energy" interacting function, $J_\beta(\mathbf{x})$, which provides a smooth, differentiable approximation of the minimum function value within a particle swarm. We define a stochastic gradient flow in the particle space, incorporating a Brownian motion term for exploration and a time-dependent parameter $\beta$ to control smoothness, similar to temperature annealing. We theoretically demonstrate that for strongly convex functions, our dynamics converges to a stationary point where at least one particle reaches the global minimum, with other particles exhibiting exploratory behavior. Furthermore, we show that our method facilitates faster transitions between local minima by reducing effective potential barriers with respect to Simulated Annealing. More specifically, we estimate the hitting times of unexplored potential wells for our model in the small noise regime and show that they compare favorably with the ones of overdamped Langevin. Numerical experiments on benchmark functions, including double wells and the Ackley function, validate our theoretical findings and demonstrate better performance over the well-known Simulated Annealing method in terms of escaping local minima and achieving faster convergence.

cs.LG↗
arXiv2024

$2D$ vorticity Euler equations: Superposition solutions and nonlinear Markov processes

In this note we contribute two results to the theory of the $2D$ Euler equations in vorticity form on the full plane. First, we establish a generalized Lagrangian representation of weak (in general measure-valued) solutions, which includes and extends classical results on the Lagrangianity of weak solutions. Second, we construct nonlinear Markov processes which are uniquely determined by a selection of weak solutions from initial data in $L^1\cap L^p$, $p \geq 2$, and related spaces such as the classical and uniformly localized Yudovich space. It is well-known that for $p <\infty$ weak solutions are in general not unique, which renders a suitable selection nontrivial.

math.AP↗
arXiv2024

Non-uniqueness of weak solutions for a logarithmically supercritical hyperdissipative Navier-Stokes system

The existence of non-unique solutions of finite kinetic energy for the three dimensional Navier-Stokes equations is proved in the slightly supercritical hyper-dissipative setting introduced by Tao. The result is based on the convex integration techniques of Buckmaster and Vicol and extends Luo and Titi result in the slightly supercritical setting. To be able to be closer to the threshold identified by Tao, we introduce the impulsed Beltrami flows, a variant of the intermittent Beltrami flows of Buckmaster and Vicol.

math.AP↗
arXiv2023

Gaussian processes based data augmentation and expected signature for time series classification

The signature is a fundamental object that describes paths (that is, continuous functions from an interval to a Euclidean space). Likewise, the expected signature provides a statistical description of the law of stochastic processes. We propose a feature extraction model for time series built upon the expected signature. This is computed through a Gaussian processes based data augmentation. One of the main features is that an optimal feature extraction is learnt through the supervised task that uses the model.

cs.LG↗
arXiv2023

Gibbs Equilibrium Fluctuations of Point Vortex Dynamics

We consider a system of N point vortices in a bounded domain with null total circulation, whose statistics are given by the Canonical Gibbs Ensemble at inverse temperature $\beta\geq 0$. We prove that the space-time fluctuation field around the (constant) Mean Field limit satisfies when $N\to\infty$ a generalized version of 2-dimensional Euler dynamics preserving the Gaussian Energy-Enstrophy ensemble.

math.PR↗
arXiv2023

Zero-Noise Selection for Point Vortex Dynamics after Collapse

The continuation of point vortex dynamics after a vortex collapse is investigated by means of a regularization procedure consisting in introducing a small stochastic diffusive term, that corresponds to a vanishing viscosity. In contrast with deterministic regularization, in which a cutoff interaction selects in the limit a single trajectory of the system after collapse, the zero-noise method produces a probability distribution supported by trajectories satisfying relevant conservation laws of the point vortex system.

physics.flu-dyn↗
arXiv2022

Yet another notion of irregularity through small ball estimates

We introduce a new notion of irregularity of paths, in terms of control of growth of the size of small balls by means of the occupation measure of the path. This notion ensures Besov regularity of the occupation measure and thus extends the analysis of Catellier and Gubinelli (2016) to general Besov spaces. On stochastic processes this notion is granted by suitable properties of local non-determinism.

math.PR↗
arXiv2021

Curious Explorer: a provable exploration strategy in Policy Learning

Having access to an exploring restart distribution (the so-called wide coverage assumption) is critical with policy gradient methods. This is due to the fact that, while the objective function is insensitive to updates in unlikely states, the agent may still need improvements in those states in order to reach a nearly optimal payoff. For this reason, wide coverage is used in some form when analyzing theoretical properties of practical policy gradient methods. However, this assumption can be unfeasible in certain environments, for instance when learning is online, or when restarts are possible only from a fixed initial state. In these cases, classical policy gradient algorithms can have very poor convergence properties and sample efficiency. In this paper, we develop Curious Explorer, a novel and simple iterative state space exploration strategy that can be used with any starting distribution $ρ$. Curious Explorer starts from $ρ$, then using intrinsic rewards assigned to the set of poorly visited states produces a sequence of policies, each one more exploratory than the previous one in an informed way, and finally outputs a restart model $μ$ based on the state visitation distribution of the exploratory policies. Curious Explorer is provable, in the sense that we provide theoretical upper bounds on how often an optimal policy visits poorly visited states. These bounds can be used to prove PAC convergence and sample efficiency results when a PAC optimizer is plugged in Curious Explorer. This allows to achieve global convergence and sample efficiency results without any coverage assumption for REINFORCE, and potentially for any other policy gradient method ensuring PAC convergence with wide coverage. Finally, we plug (the output of) Curious Explorer into REINFORCE and TRPO, and show empirically that it can improve performance in MDPs with challenging exploration.

cs.LG↗
arXiv2021

The Gaussian Structure of the Singular Stochastic Burgers Equation

We consider the stochastically forced Burgers equation with an emphasis on spatially rough driving noise. We show that the law of the process at a fixed time $t$, conditioned on no explosions, is absolutely continuous with respect to the stochastic heat equation obtained by removing the nonlinearity from the equation. This establishes a form of ellipticity in this infinite dimensional setting. The results follow from a recasting of the Girsanov Theorem to handle less spatially regular solutions while only proving absolute continuity at a fixed time and not on path-space. The results are proven by decomposing the solution into the sum of auxiliary processes which are then shown to be absolutely continuous in law to a stochastic heat equation. The number of levels in this decomposition diverges to infinite as we move to the stochastically forced Burgers equation associated to the KPZ equation, which we conjecture is just beyond the validity of our results (and certainly the current proof). The analysis provides insights into the structure of the solution as we approach the regularity of KPZ. A number of techniques from singular SPDEs are employed as we are beyond the regime of classical solutions for much of the paper.

math.PR↗
arXiv2020

A Central Limit Theorem for Gibbsian Invariant Measures of 2D Euler Equations

We consider Canonical Gibbsian ensembles of Euler point vortices on the 2-dimensional torus or in a bounded domain of R 2 . We prove that under the Central Limit scaling of vortices intensities, and provided that the system has zero global space average in the bounded domain case (neutrality condition), the ensemble converges to the so-called Energy-Enstrophy Gaussian random distributions. This can be interpreted as describing Gaussian fluctuations around the mean field limit of vortices ensembles. The main argument consists in proving convergence of partition functions of vortices and Gaussian distributions.

math.PR↗
arXiv2020

Decay of Correlation Rate in the Mean Field Limit of Point Vortices Ensembles

We consider the Mean Field limit of Gibbsian ensembles of 2-dimensional point vortices on the torus. It is a classical result that in such limit correlations functions converge to 1, that is, point vortices decorrelate: we compute the rate at which this convergence takes place by means of Gaussian integration techniques, inspired by the correspondence between the 2-dimensional Coulomb gas and the Sine-Gordon Euclidean field theory.

math-ph↗