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

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

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 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 $L$; (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.Comment: 5 pages, 4 figures, and Supplemental Materia

Homoclinic bifurcations in the quasi-geostrophic double-gyre circulation

The wind-driven double-gyre circulation in a rectangular basin goes through several dynamical regimes as the amount of lateral friction is decreased. This paper studies the transition to irregular flow in the double-gyre circulation by applying dynamical systems methodology to a quasi-geostrophic, equivalent-barotropic model with a 10-km resolution.The origin of the irregularities, in space and time, is the occurrence of homoclinic bifurcations that involve phase-space behavior far from stationary solutions. The connection between these homoclinic bifurcations and earlier transitions, which occur at larger lateral friction, is explained. The earlier transitions, such as pitchfork and asymmetric Hopf bifurcation, only involve the nonlinear saturation of linear instabilities, while the homoclinic bifurcations are associated with genuinely nonlinear behavior. The sequence of bifurcations—pitchfork, Hopf, and homoclinic—is independent of the lateral friction and may be described as the unfolding of a singularity that occurs in the frictionless, Hamiltonian limit of the governing equations.Two distinct chaotic regimes are identified: Lorenz chaos at relatively large lateral friction versus Shilnikov chaos at relatively small lateral friction. Both types of homoclinic bifurcations induce chaotic behavior of the recirculation gyres that is dominated by relaxation oscillations with a well-defined period.The relevance of these results to the mid-latitude oceans\u27 observed low-frequency variations is discussed. A previously documented 7-year peak in observed North-Atlantic variability is shown to exist across a hierarchy of models that share the gyre modes and homoclinic bifurcations discussed herein

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 review recent theoretical advances in studying the wind-driven circulation of the oceans. In doing so, we concentrate on the large-scale, wind-driven flow of the mid-latitude oceans, which is dominated by the presence of a larger, anticyclonic and a smaller, cyclonic gyre. The two gyres share the eastward extension of western boundary currents, such as the Gulf Stream or Kuroshio, and are induced by the shear in the winds that cross the respective ocean basins. The boundary currents and eastward jets carry substantial amounts of heat and momentum, and thus contribute in a crucial way to Earth's climate, and to changes therein. Changes in this double-gyre circulation occur from year to year and decade to decade. We study this low-frequency variability 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 low-frequency variability of the ocean circulation is but one of the causes of uncertainties in climate projections. The range of these uncertainties has barely decreased, or even increased, over the last three decades. Another major cause of such uncertainties could reside in the structural instability---in the classical, 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. The idea is to compare the climate simulations of distinct general circulation models (GCMs) used in climate projections, by applying stochastic-conjugacy methods and thus perform a stochastic classification of GCM families. This approach is particularly appropriate given recent interest in stochastic parametrization of subgrid-scale processes in GCMs. As a very first step in this direction, we study the behavior of the Arnol'd family of circle maps in the presence of noise. The maps' fine-grained resonant landscape is smoothed by the noise, thus permitting their coarse-grained classification

Regimes of low-frequency variability in a three-layer quasi-geostrophic ocean model

The temporal variability of the midlatitude double-gyre wind-driven ocean circulation is studied in a three-layer quasi-geostrophic model over a broad range in parameter space. Four different types of flow regimes are found, each characterized by a specific time-mean state and spatio-temporal variability. As the lateral friction is decreased, these regimes are encountered in the following order: the viscous antisymmetric regime, the asymmetric regime, the quasi-homoclinic regime and the inertial antisymmetric regime. The variability in the viscous and the inertial antisymmetric regimes (at high and low lateral friction, respectively) is mainly caused by Rossby basin modes. Low-frequency variability, i.e.on interannual to decadal time-scales, is present in the asymmetric and quasi-homoclinic regime and can be related to relaxation oscillations originating from low-frequency gyre modes. The focus of this paper is on the mechanisms of the transitions between the different regimes. The transition from the viscous antisymmetric regime to the asymmetric regime occurs through a symmetry-breaking pitchfork bifurcation. There are strong indications that the quasi-homoclinic regime is introduced through the existence of a homoclinic orbit. The transition to the inertial antisymmetric regime is due to the symmetrization of the time-mean state zonal velocity field through rectification effects

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.Comment: Invited survey paper for Special Issue on The Euler Equations: 250 Years On, in Physica D: Nonlinear phenomen

Predicting drowning from sea and weather forecasts: development and validation of a model on surf beaches of southwestern France

OBJECTIVE: To predict the coast-wide risk of drowning along the surf beaches of Gironde, southwestern France. METHODS: Data on rescues and drownings were collected from the Medical Emergency Center of Gironde (SAMU 33). Seasonality, holidays, weekends, weather and metocean conditions were considered potentially predictive. Logistic regression models were fitted with data from 2011 to 2013 and used to predict 2015-2017 events employing weather and ocean forecasts. RESULTS: Air temperature, wave parameters, seasonality and holidays were associated with drownings. Prospective validation was performed on 617 days, covering 232 events (rescues and drownings) reported on 104 different days. The area under the curve (AUC) of the daily risk prediction model (combined with 3-day forecasts) was 0.82 (95% CI 0.79 to 0.86). The AUC of the 3-hour step model was 0.85 (95% CI 0.81 to 0.88). CONCLUSIONS: Drowning events along the Gironde surf coast can be anticipated up to 3 days in advance. Preventative messages and rescue preparations could be increased as the forecast risk increased, especially during the off-peak season, when the number of available rescuers is low