arXiv subjects

Eric Simonnet

Publications and source records attributed to Eric Simonnet.

12 recordsLinked to original sources

arXiv2026

A Measure-Theoretic Approach to Spontaneous Stochasticity

Spontaneous stochasticity (SpSt), originating in Richardson's picture of turbulent dispersion and Lorenz's Eulerian view of finite-time loss of predictability, was later formulated under this name by Gaw\k{e}dzki and collaborators and developed in shell models by Mailybaev and collaborators. Whether it occurs in fully developed turbulence remains a major open question. Beyond a few specific classes of systems, however, SpSt has lacked a general mathematical definition. We introduce a measure-theoretic formalism in which it is understood as a measure-selection principle. Given an inviscid problem, a well-posed regularization, and an ambient measure, we study the pushforward of that measure by the regularized flow. Strong SpSt occurs when these pushforward measures converge to a non-Dirac probability law, replacing classical deterministic selection by statistical selection. For finite-dimensional systems, we establish several structural results. Our central attainability theorem shows that, whenever the inviscid problem is nonunique, any probability measure supported on the set of inviscid states can be selected as the limiting law of a suitable regularization. We also identify singular sets in the inviscid dynamics, detected through Dini-type directional growth, as necessary obstructions underlying nonuniqueness. We analyze the relation between SpSt and sensitivity to initial data, clarifying the scope and limitations of turbulence-inspired finite-time separation criteria. Finally, we develop a renormalization-(semi)group viewpoint in which limiting statistics arise as statistical attractors. Explicit examples illustrate how ambient measures, inviscid singularities, regularization scales, and initial-data sensitivity interact in the emergence of SpSt.

nlin.CD↗

arXiv2025

Spontaneous stochasticity in the Armstrong-Vicol passive scalar

Spontaneous stochasticity refers to the emergence of intrinsic randomness in deterministic systems under singular limits, a phenomenon conjectured to be fundamental in turbulence. Armstrong and Vicol recently constructed a deterministic, divergence-free multiscale vector field arbitrarily close to a weak Euler solution, proving that a passive scalar transported by this field exhibits anomalous dissipation and lacks a selection principle in the vanishing diffusivity limit. We show that this advection-diffusion PDE also selects a non-Dirac measure in the space of weak solutions in the inviscid limit, thereby exhibiting Eulerian spontaneous stochasticity. We further provide numerical evidence of Lagrangian spontaneous stochasticity, together with a numerical illustration of the Obukhov-Corrsin conjecture for this system. We formulate a general framework for spontaneous stochasticity in arbitrary finite dimensional systems under arbitrary regularizations, distinguishing two regimes: weak, where different probability measures may arise along subsequences of inviscid limits, and strong, where the limit measure is unique. The advection diffusion system of Armstrong and Vicol lies in the strong regime. We prove that the set of selected measures is compact and equals the closed convex hull of Dirac measures. Moreover, for any non-Dirac measure supported on the set of nonunique solutions of the inviscid system, there exists a regularization that produces strong spontaneous stochasticity. Finally, we relate this framework to renormalization-group methods \`a la Feigenbaum and examine how the underlying dynamical system influences the inviscid limit. The discussion is complemented by elementary finite-dimensional examples illustrating a variety of cases.

math-ph↗

arXiv2025

Non-unique self-similar blowups in shell models: insights from dynamical systems and machine-learning

Strong numerical hints exist in favor of a universal blowup scenario in the Sabra shell model, a popular cascade model of 3D turbulence, which features complex velocity variables on a geometric progression of scales $\ell_n \propto \lambda ^{-n}$. The blowup is thought to be of self-similar type and characterized by the finite-time convergence towards a universal profile with non-Kolmogorov (anomalous) small-scale scaling $\propto \ell_n^{x}$. Solving the underlying nonlinear eigenvalue problem has however proven challenging, and prior insights mainly used the Dombre-Gilson renormalization scheme, transforming self-similar solutions into solitons propagating over infinite rescaled time horizon. Here, we further characterize Sabra blowups by implementing two strategies targeting the eigenvalue problem. The first involves formal expansion in terms of the bookkeeping parameter $\delta = (1-x)\log \lambda$, and interpretes the self-similar solution as a (degenerate) homoclinic bifurcation. Using standard bifurcation toolkits, we show that the homoclinic bifurcations identified under finite-truncation of the series converge to the observed Sabra solution. The second strategy uses machine-learning optimization to solve directly for the Sabra eigenvalue. It reveals an intricate phase space, with the presence of a continuous family of non-universal blowup profiles, characterized by various number $N$ of pulses and exponents $x_N\ge x$.

physics.flu-dyn↗

arXiv2023

Unveiling the Phase Diagram and Reaction Paths of the Active Model B with the Deep Minimum Action Method

Nonequilibrium phase transitions are notably difficult to analyze because their mechanisms depend on the system's dynamics in a complex way due to the lack of time-reversal symmetry. To complicate matters, the system's steady-state distribution is unknown in general. Here, the phase diagram of the active Model B is computed with a deep neural network implementation of the geometric minimum action method (gMAM). This approach unveils the unconventional reaction paths and nucleation mechanism in dimensions 1, 2 and 3, by which the system switches between the homogeneous and inhomogeneous phases in the binodal region. Our main findings are: (i) the mean time to escape the phase-separated state is (exponentially) extensive in the system size $L$, but it increases non-monotonically with $L$ in dimension 1; (ii) the mean time to escape the homogeneous state is always finite, in line with the recent work of Cates and Nardini~[Phys. Rev. Lett. 130, 098203]; (iii) at fixed $L$, the active term increases the stability of the homogeneous phase, eventually destroying the phase separation in the binodal for large but finite systems. Our results are particularly relevant for active matter systems in which the number of constituents hardly goes beyond $10^7$ and where finite-size effects matter.

cond-mat.stat-mech↗

arXiv2022

Computing non-equilibrium trajectories by a deep learning approach

Predicting the occurence of rare and extreme events in complex systems is a well-known problem in non-equilibrium physics. These events can have huge impacts on human societies. New approaches have emerged in the last ten years, which better estimate tail distributions. They often use large deviation concepts without the need to perform heavy direct ensemble simulations. In particular, a well-known approach is to derive a minimum action principle and to find its minimizers. The analysis of rare reactive events in non-equilibrium systems without detailed balance is notoriously difficult either theoretically and computationally. They are described in the limit of small noise by the Freidlin-Wentzell action. We propose here a new method which minimizes the geometrical action instead using neural networks: it is called deep gMAM. It relies on a natural and simple machine-learning formulation of the classical gMAM approach. We give a detailed description of the method as well as many examples. These include bimodal switches in complex stochastic (partial) differential equations, quasi-potential estimates, and extreme events in Burgers turbulence.

physics.comp-ph↗

arXiv2022

Deep spectral computations in linear and nonlinear diffusion problems

We propose a flexible machine-learning framework for solving eigenvalue problems of diffusion operators in moderately large dimension. We improve on existing Neural Networks (NNs) eigensolvers by demonstrating our approach ability to compute (i) eigensolutions for non-self adjoint operators with small diffusion (ii) eigenpairs located deep within the spectrum (iii) computing several eigenmodes at once (iv) handling nonlinear eigenvalue problems. To do so, we adopt a variational approach consisting of minimizing a natural cost functional involving Rayleigh quotients, by means of simple adiabatic technics and multivalued feedforward neural parametrisation of the solutions. Compelling successes are reported for a 10-dimensional eigenvalue problem corresponding to a Kolmogorov operator associated with a mixing Stepanov flow. We moreover show that the approach allows for providing accurate eigensolutions for a 5-D Schr\"odinger operator having $32$ metastable states. In addition, we address the so-called Gelfand superlinear problem having exponential nonlinearities, in dimension $4$, and for nontrivial domains exhibiting cavities. In particular, we obtain NN-approximations of high-energy solutions approaching singular ones. We stress that each of these results are obtained using small-size neural networks in situations where classical methods are hopeless due to the curse of dimensionality. This work brings new perspectives for the study of Ruelle-Pollicot resonances, dimension reduction, nonlinear eigenvalue problems, and the study of metastability when the dynamics has no potential.

math.NA↗

arXiv2020

Multistability and rare spontaneous transitions in barotropic $\beta$-plane turbulence

We demonstrate that turbulent zonal jets, analogous to Jovian ones, which are quasi-stationary, are actually metastable. After extremely long times, they randomly switch to new configurations with a different number of jets. The genericity of this phenomenon suggests that most quasi-stationary turbulent planetary atmospheres might have many climates and attractors for fixed values of the external forcing parameters. A key message is that this situation will usually not be detected by simply running the numerical models, because of the extremely long mean transition time to change from one climate to another. In order to study such phenomena, we need to use specific tools: rare event algorithms and large deviation theory. With these tools, we make a full statistical mechanics study of a classical barotropic beta-plane quasigeostrophic model. It exhibits robust bimodality with abrupt transitions. We show that new jets spontaneously nucleate from westward jets. The numerically computed mean transition time is consistent with an Arrhenius law showing an exponential decrease of the probability as the Ekman dissipation decreases. This phenomenology is controlled by rare noise-driven paths called {\it instantons}. Moreover, we compute the saddles of the corresponding effective dynamics. For the dynamics of states with three alternating jets, we uncover an unexpectedly rich dynamics governed by the symmetric group ${\cal S}_3$ of permutations, with two distinct families of instantons, which is a surprise for a system where everything seemed stationary in the hundreds of previous simulations of this model. We discuss the future generalization of our approach to more realistic models.

physics.ao-ph↗

arXiv2018

A rare event algorithm links transitions in turbulent flows with activated nucleations

Many turbulent flows undergo drastic and abrupt configuration changes with huge impacts. As a paradigmatic example we study the multistability of jet dynamics in a barotropic beta plane model of atmosphere dynamics. It is considered as the Ising model for Jupiter troposphere dynamics. Using the adaptive multilevel splitting, a rare event algorithm, we are able to get a very large statistics of transition paths, the extremely rare transitions from one state of the system to another. This new approach opens the way for addressing a set of questions that are out of reach through direct numerical simulations. We demonstrate for the first time the concentration of transition paths close to instantons, in a numerical simulation of genuine turbulent flows. We show that the transition is a noise-activated nucleation of vorticity bands. We address for the first time the existence of Arrhenius laws in turbulent flows. The methodology we developed shall prove useful to study many other transitions related to drastic changes for the turbulent dynamics of climate, geophysical, astrophysical and engineering applications. This opens a new range of studies impossible so far, and bring turbulent phenomena in the realm of non-equilibrium statistical mechanics.

cond-mat.stat-mech↗

arXiv2015

Computing transition rates for the 1-D stochastic Ginzburg--Landau--Allen--Cahn equation for finite-amplitude noise with a rare event algorithm

In this paper we compute and analyse the transition rates and duration of reactive trajectories of the stochastic 1-D Allen-Cahn equations for both the Freidlin-Wentzell regime (weak noise or temperature limit) and finite-amplitude white noise, as well as for small and large domain. We demonstrate that extremely rare reactive trajectories corresponding to direct transitions between two metastable states are efficiently computed using an algorithm called adaptive multilevel splitting. This algorithm is dedicated to the computation of rare events and is able to provide ensembles of reactive trajectories in a very efficient way. In the small noise limit, our numerical results are in agreement with large-deviation predictions such as instanton-like solutions, mean first passages and escape probabilities. We show that the duration of reactive trajectories follows a Gumbel distribution like for one degree of freedom systems. Moreover, the mean duration growths logarithmically with the inverse temperature. The prefactor given by the potential curvature grows exponentially with size. The main novelty of our work is that we also perform an analysis of reactive trajectories for large noises and large domains. In this case, we show that the position of the reactive front is essentially a random walk. This time, the mean duration grows linearly with the inverse temperature and quadratically with the size. Using a phenomenological description of the system, we are able to calculate the transition rate, although the dynamics is described by neither Freidlin--Wentzell or Eyring--Kramers type of results. Numerical results confirm our analysis.

physics.flu-dyn↗

arXiv2014

Statistical behavior of adaptive multilevel splitting algorithms in simple models

Adaptive multilevel splitting algorithms have been introduced rather recently for estimating tail distributions in a fast and efficient way. In particular, they can be used for computing the so-called reactive trajectories corresponding to direct transitions from one metastable state to another. The algorithm is based on successive selection-mutation steps performed on the system in a controlled way. It has two intrinsic parameters, the number of particles/trajectories and the reaction coordinate used for discriminating good or bad trajectories. We investigate first the convergence in law of the algorithm as a function of the timestep for several simple stochastic models. Second, we consider the average duration of reactive trajectories for which no theoretical predictions exist. The most important aspect of this work concerns some systems with two degrees of freedom. They are studied in details as a function of the reaction coordinate in the asymptotic regime where the number of trajectories goes to infinity. We show that during phase transitions, the statistics of the algorithm deviate significatively from known theoretical results when using non-optimal reaction coordinates. In this case, the variance of the algorithm is peaking at the transition and the convergence of the algorithm can be much slower than the usual expected central limit behavior. The duration of trajectories is affected as well. Moreover, reactive trajectories do not correspond to the most probable ones. Such behavior disappears when using the optimal reaction coordinate called committor as predicted by the theory. We finally investigate a three-state Markov chain which reproduces this phenomenon and show logarithmic convergence of the trajectory durations.

math.NA↗

arXiv2010

Climate dynamics and fluid mechanics: Natural variability and related uncertainties

The purpose of this review-and-research paper is twofold: (i) to review the role played in climate dynamics by fluid-dynamical models; and (ii) to contribute to the understanding and reduction of the uncertainties in future climate-change projections. To illustrate the first point, we focus on the large-scale, wind-driven flow of the mid-latitude oceans which contribute in a crucial way to Earth's climate, and to changes therein. We study the low-frequency variability (LFV) of the wind-driven, double-gyre circulation in mid-latitude ocean basins, via the bifurcation sequence that leads from steady states through periodic solutions and on to the chaotic, irregular flows documented in the observations. This sequence involves local, pitchfork and Hopf bifurcations, as well as global, homoclinic ones. The natural climate variability induced by the LFV of the ocean circulation is but one of the causes of uncertainties in climate projections. Another major cause of such uncertainties could reside in the structural instability in the topological sense, of the equations governing climate dynamics, including but not restricted to those of atmospheric and ocean dynamics. We propose a novel approach to understand, and possibly reduce, these uncertainties, based on the concepts and methods of random dynamical systems theory. As a very first step, we study the effect of noise on the topological classes of the Arnol'd family of circle maps, a paradigmatic model of frequency locking as occurring in the nonlinear interactions between the El Nino-Southern Oscillations (ENSO) and the seasonal cycle. It is shown that the maps' fine-grained resonant landscape is smoothed by the noise, thus permitting their coarse-grained classification. This result is consistent with stabilizing effects of stochastic parametrization obtained in modeling of ENSO phenomenon via some general circulation models.

math.DS↗

arXiv2008

Random changes of flow topology in two dimensional and geophysical turbulence

We study the two dimensional (2D) stochastic Navier Stokes (SNS) equations in the inertial limit of weak forcing and dissipation. The stationary measure is concentrated close to steady solutions of the 2D Euler equation. For such inertial flows, we prove that bifurcations in the flow topology occur either by changing the domain shape, the nonlinearity of the vorticity-stream function relation, or the energy. Associated to this, we observe in SNS bistable behavior with random changes from dipoles to unidirectional flows. The theoretical explanation being very general, we infer the existence of similar phenomena in experiments and in models of geophysical flows.

nlin.CD↗