Committor Functions for Climate Phenomena at the Predictability Margin: The example of El Ni\~no Southern Oscillation in the Jin and Timmerman model

Many phenomena in the climate system lie in the gray zone between weather and climate: they are not amenable to deterministic forecast, but they still depend on the initial condition. A natural example is medium-range forecasting, which is inherently probabilistic because it lies beyond the deterministic predictability time of the atmosphere, but for which statistically significant prediction can be made which depend on the current state of the system. Similarly, one may ask the probability of occurrence of an El Ni\~no event several months ahead of time. In this paper, we introduce a quantity which corresponds precisely to this type of prediction problem: the committor function is the probability that an event takes place within a given time window, as a function of the initial condition. We explain the main mathematical properties of this probabilistic concept, and compute it in the case of a low-dimensional stochastic model for El-Ni\~no, the Jin and Timmerman model. In this context, we show that the ability to predict the probability of occurrence of the event of interest may differ strongly depending on the initial state. The main result is the new distinction between intrinsic probabilistic predictability (when the committor function is smooth and probability can be computed which does not depend sensitively on the initial condition) and intrinsic probabilistic unpredictability (when the committor function depends sensitively on the initial condition). We also demonstrate that the Jin and Timmerman model might be the first example of a stochastic differential equation with weak noise for which transition between attractors do not follow the Arrhenius law, which is expected based on large deviation theory and generic hypothesis

Random exchange dynamics with bounds: H-theorem and negative temperature

Random exchange kinetic models are widely employed to describe the conservative dynamics of large interacting systems. Due to their simplicity and generality, they are quite popular in several fields, from statistical mechanics to biophysics and economics. Here we study a version where bounds on the individual shares of the globally conserved quantity are introduced. We analytically show that this dynamics allows stationary states with population inversion, described by Boltzmann statistics at negative absolute temperature. Their genuine equilibrium nature is verified by checking the detailed balance condition. An H-theorem is proven: the Boltzmann entropy monotonically increases during the dynamics. Finally, we provide analytical and numerical evidence that a large intruder in contact with the system thermalizes, suggesting a practical way to design a thermal bath at negative temperature. These results open new research perspectives, creating a bridge between negative temperature statistical descriptions and kinetic models with bounds.Comment: 5 pages, 2 figure

Statistical features of systems driven by non-Gaussian processes: theory & practice

Nowadays many tools, e.g. fluctuation relations, are available to characterize the statistical properties of non-equilibrium systems. However, most of these tools rely on the assumption that the driving noise is normally distributed. Here we consider a class of Markov processes described by Langevin equations driven by a mixture of Gaussian and Poissonian noises, focusing on their non-equilibrium properties. In particular, we prove that detailed balance does not hold even when correlation functions are symmetric under time reversal. In such cases, a breakdown of the time reversal symmetry can be highlighted by considering higher order correlation functions. Furthermore, the entropy production may be different from zero even for vanishing currents. We provide analytical expressions for the average entropy production rate in several cases. We also introduce a scale dependent estimate for entropy production, suitable for inference from experimental signals. The empirical entropy production allows us to discuss the role of spatial and temporal resolutions in characterizing non-equilibrium features. Finally, we revisit the Brownian gyrator introducing an additional Poissonian noise showing that it behaves as a two dimensional linear ratchet. It has also the property that when Onsager relations are satisfied its entropy production is positive although it is minimal. We conclude discussing estimates of entropy production for partially accessible systems, comparing our results with the lower bound provided by the thermodynamic uncertainty relations.Comment: 32 pages, 11 figures. 22 pages main text, 10 pages appendice

Out-of-Equilibrium Non-Gaussian Behavior in Driven Granular Gases

The characterization of the distance from equilibrium is a debated problem in particular in the treatment of experimental signals. If the signal is a 1-dimensional time-series, such a goal becomes challenging. A paradigmatic example is the angular diffusion of a rotator immersed in a vibro-fluidized granular gas. Here, we experimentally observe that the rotator's angular velocity exhibits significative differences with respect to an equilibrium process. Exploiting the presence of two relevant time-scales and non-Gaussian velocity increments, we quantify the breakdown of time-reversal asymmetry, which would vanish in the case of a 1d Gaussian process. We deduce a new model for the massive probe, with two linearly coupled variables, incorporating both Gaussian and Poissonian noise, the latter motivated by the rarefied collisions with the granular bath particles. Our model reproduces the experiment in a range of densities, from dilute to moderately dense, with a meaningful dependence of the parameters on the density.Comment: 5 pages, 4 figure

Prédiction des probabilités des extrêmes climatiques à partir des observations et de la dynamique

There is a large interest in predicting the occurrence of high impact climate events such as ENSO (El Niño Southern Oscillation) or rare events, for instance heat waves. Those are prediction problems at the predictability margin because the interesting time scale lies at the edge of the mixing time of the system. This thesis aims at introducing the relevant quantity for these prediction problems, the so-called committor function which is the probability for an event to occur in the future, as a function of the current state of the system. Computing the committor in a stochastic model for ENSO illustrates that the transition to strong El Niño regimes can have either intrinsic probabilistic predictability or unpredictability. The second goal is to illustrate how to compute and validate the committor function from observations, by discussing the analogue Markov chain which provides a way for learning effective dynamics from data. Starting from it, a new algorithm is developed, with the scope of computing the committor function more precisely than the other approaches, especially in case of lack of data. Moreover, it is shown, in the context of two stochastic systems, that coupling the learning of the com- mittor with a rare event algorithm improves the performance of the latter. Finally, this methodology is applied to a climate data-set, generated from a climate model, in order to study and predict the occurrence of extreme heat waves. After checking the consistency of the statistical quantities computed by the effective dynamics, a classifier based on the Markov chain is developed, with the capability of classifying heat waves better than other methods.Il existe un grand intérêt pour la prédiction de l'occurrence d'événements climatiques à fort impact tels que l'ENSO (El Niño Southern Oscillation) ou d'événements rares, par exemple les vagues de chaleur. Ce sont des problèmes de prédiction à la marge de prévisibilité car l'échelle de temps intéressante se situe à la limite du temps de mixing du système. Cette thèse vise à introduire la quantité pertinente pour ces problèmes de prédiction, la fonction dite committor qui est la probabilité qu'un événement se produise dans le futur, en fonction de l'état actuel du système. Le calcul du committor dans un modèle stochastique pour ENSO montre que la transition vers des régimes El Niño intenses peut avoir soit une prévisibilité probabiliste intrinsèque, soit une imprévisibilité. Le deuxième objectif est d'illustrer comment calculer et valider la fonction committor à partir d'observations, en discutant de la chaîne de Markov des analogues qui fournit un moyen d'apprendre une dynamique effective à partir de données. A partir de là, un nouvel algorithme est développé, avec pour objectif de calculer la fonction committor plus précisément que les autres approches, notamment en cas de manque de données. De plus, il est montré, dans le cadre de deux systèmes stochastiques, que coupler l'apprentissage du committor avec un algorithme d'événement rare améliore les performances de ce dernier. Enfin, cette méthodologie est appliquée à un ensemble de données climatiques, générées à partir d'un modèle climatique, afin d'étudier et de prédire l'occurrence de vagues de chaleur extrêmes. Après avoir vérifié la cohérence des quantités statistiques calculées par la dynamique effective, un classificateur basé sur la chaîne de Markov est développé, avec la capacité de mieux classer les vagues de chaleur que d'autres méthodes

Prédiction des probabilités des extrêmes climatiques à partir des observations et de la dynamique

There is a large interest in predicting the occurrence of high impact climate events such as ENSO (El Niño Southern Oscillation) or rare events, for instance heat waves. Those are prediction problems at the predictability margin because the interesting time scale lies at the edge of the mixing time of the system. This thesis aims at introducing the relevant quantity for these prediction problems, the so-called committor function which is the probability for an event to occur in the future, as a function of the current state of the system. Computing the committor in a stochastic model for ENSO illustrates that the transition to strong El Niño regimes can have either intrinsic probabilistic predictability or unpredictability. The second goal is to illustrate how to compute and validate the committor function from observations, by discussing the analogue Markov chain which provides a way for learning effective dynamics from data. Starting from it, a new algorithm is developed, with the scope of computing the committor function more precisely than the other approaches, especially in case of lack of data. Moreover, it is shown, in the context of two stochastic systems, that coupling the learning of the com- mittor with a rare event algorithm improves the performance of the latter. Finally, this methodology is applied to a climate data-set, generated from a climate model, in order to study and predict the occurrence of extreme heat waves. After checking the consistency of the statistical quantities computed by the effective dynamics, a classifier based on the Markov chain is developed, with the capability of classifying heat waves better than other methods.Il existe un grand intérêt pour la prédiction de l'occurrence d'événements climatiques à fort impact tels que l'ENSO (El Niño Southern Oscillation) ou d'événements rares, par exemple les vagues de chaleur. Ce sont des problèmes de prédiction à la marge de prévisibilité car l'échelle de temps intéressante se situe à la limite du temps de mixing du système. Cette thèse vise à introduire la quantité pertinente pour ces problèmes de prédiction, la fonction dite committor qui est la probabilité qu'un événement se produise dans le futur, en fonction de l'état actuel du système. Le calcul du committor dans un modèle stochastique pour ENSO montre que la transition vers des régimes El Niño intenses peut avoir soit une prévisibilité probabiliste intrinsèque, soit une imprévisibilité. Le deuxième objectif est d'illustrer comment calculer et valider la fonction committor à partir d'observations, en discutant de la chaîne de Markov des analogues qui fournit un moyen d'apprendre une dynamique effective à partir de données. A partir de là, un nouvel algorithme est développé, avec pour objectif de calculer la fonction committor plus précisément que les autres approches, notamment en cas de manque de données. De plus, il est montré, dans le cadre de deux systèmes stochastiques, que coupler l'apprentissage du committor avec un algorithme d'événement rare améliore les performances de ce dernier. Enfin, cette méthodologie est appliquée à un ensemble de données climatiques, générées à partir d'un modèle climatique, afin d'étudier et de prédire l'occurrence de vagues de chaleur extrêmes. Après avoir vérifié la cohérence des quantités statistiques calculées par la dynamique effective, un classificateur basé sur la chaîne de Markov est développé, avec la capacité de mieux classer les vagues de chaleur que d'autres méthodes

Prédiction des probabilités des extrêmes climatiques à partir des observations et de la dynamique

Il existe un grand intérêt pour la prédiction de l'occurrence d'événements climatiques à fort impact tels que l'ENSO (El Niño Southern Oscillation) ou d'événements rares, par exemple les vagues de chaleur. Ce sont des problèmes de prédiction à la marge de prévisibilité car l'échelle de temps intéressante se situe à la limite du temps de mixing du système. Cette thèse vise à introduire la quantité pertinente pour ces problèmes de prédiction, la fonction dite committor qui est la probabilité qu'un événement se produise dans le futur, en fonction de l'état actuel du système. Le calcul du committor dans un modèle stochastique pour ENSO montre que la transition vers des régimes El Niño intenses peut avoir soit une prévisibilité probabiliste intrinsèque, soit une imprévisibilité. Le deuxième objectif est d'illustrer comment calculer et valider la fonction committor à partir d'observations, en discutant de la chaîne de Markov des analogues qui fournit un moyen d'apprendre une dynamique effective à partir de données. A partir de là, un nouvel algorithme est développé, avec pour objectif de calculer la fonction committor plus précisément que les autres approches, notamment en cas de manque de données. De plus, il est montré, dans le cadre de deux systèmes stochastiques, que coupler l'apprentissage du committor avec un algorithme d'événement rare améliore les performances de ce dernier. Enfin, cette méthodologie est appliquée à un ensemble de données climatiques, générées à partir d'un modèle climatique, afin d'étudier et de prédire l'occurrence de vagues de chaleur extrêmes. Après avoir vérifié la cohérence des quantités statistiques calculées par la dynamique effective, un classificateur basé sur la chaîne de Markov est développé, avec la capacité de mieux classer les vagues de chaleur que d'autres méthodes.There is a large interest in predicting the occurrence of high impact climate events such as ENSO (El Niño Southern Oscillation) or rare events, for instance heat waves. Those are prediction problems at the predictability margin because the interesting time scale lies at the edge of the mixing time of the system. This thesis aims at introducing the relevant quantity for these prediction problems, the so-called committor function which is the probability for an event to occur in the future, as a function of the current state of the system. Computing the committor in a stochastic model for ENSO illustrates that the transition to strong El Niño regimes can have either intrinsic probabilistic predictability or unpredictability. The second goal is to illustrate how to compute and validate the committor function from observations, by discussing the analogue Markov chain which provides a way for learning effective dynamics from data. Starting from it, a new algorithm is developed, with the scope of computing the committor function more precisely than the other approaches, especially in case of lack of data. Moreover, it is shown, in the context of two stochastic systems, that coupling the learning of the com- mittor with a rare event algorithm improves the performance of the latter. Finally, this methodology is applied to a climate data-set, generated from a climate model, in order to study and predict the occurrence of extreme heat waves. After checking the consistency of the statistical quantities computed by the effective dynamics, a classifier based on the Markov chain is developed, with the capability of classifying heat waves better than other methods

Committor Functions for Climate Phenomena at the Predictability Margin: The example of El Niño Southern Oscillation in the Jin and Timmerman model

Many phenomena in the climate system lie in the gray zone between weather and climate: they are not amenable to deterministic forecast, but they still depend on the initial condition. A natural example is mediumrange forecasting, which is inherently probabilistic because it lies beyond the deterministic predictability time of the atmosphere, but for which statistically significant prediction can be made which depend on the current state of the system. Similarly, one may ask the probability of occurrence of an El Niño event several months ahead of time. In this paper, we introduce a quantity which corresponds precisely to this type of prediction problem: the committor function is the probability that an event takes place within a given time window, as a function of the initial condition. We explain the main mathematical properties of this probabilistic concept, and compute it in the case of a low-dimensional stochastic model for El-Niño, the Jin and Timmerman model. In this context, we show that the ability to predict the probability of occurrence of the event of interest may differ strongly depending on the initial state. The main result is the new distinction between intrinsic probabilistic predictability (when the committor function is smooth and probability can be computed which does not depend sensitively on the initial condition) and intrinsic probabilistic unpredictability (when the committor function depends sensitively on the initial condition). We also demonstrate that the Jin and Timmerman model might be the first example of a stochastic differential equation with weak noise for which transition between attractors do not follow the Arrhenius law, which is expected based on large deviation theory and generic hypothesis

Extreme heatwave sampling and prediction with analog Markov chain and comparisons with deep learning

We present a data-driven emulator, stochastic weather generator (SWG), suitable for estimating probabilities of prolonged heatwaves in France and Scandinavia. This emulator is based on the method of analogs of circulation to which we add temperature and soil moisture as predictor fields. We train the emulator on an intermediate complexity climate model run and show that it is capable of predicting conditional probabilities (forecasting) of heatwaves out of sample. Special attention is payed that this prediction is evaluated using proper score appropriate for rare events. To accelerate the computation of analogs dimensionality reduction techniques are applied and the performance is evaluated. The probabilistic prediction achieved with SWG is compared with the one achieved with Convolutional Neural Network (CNN). With the availability of hundreds of years of training data CNNs perform better at the task of probabilistic prediction. In addition, we show that the SWG emulator trained on 80 years of data is capable of estimating extreme return times of order of thousands of years for heatwaves longer than several days more precisely than the fit based on generalised extreme value distribution. Finally, the quality of its synthetic extreme teleconnection patterns obtained with stochastic weather generator is studied. We showcase two examples of such synthetic teleconnection patterns for heatwaves in France and Scandinavia that compare favorably to the very long climate model control run.Comment: 29 pages, 13 figures, presented at Climate Informatics 2023, UK Cambridg

Coupling rare event algorithms with data-based learned committor functions using the analogue Markov chain

Rare events play a crucial role in many physics, chemistry, and biology phenomena, when they change the structure of the system, for instance in the case of multistability, or when they have a huge impact. Rare event algorithms have been devised to simulate them efficiently, avoiding the computation of long periods of typical fluctuations. We consider here the family of splitting or cloning algorithms, which are versatile and specifically suited for far-from-equilibrium dynamics. To be efficient, these algorithms need to use a smart score function during the selection stage. Committor functions are the optimal score functions. In this work we propose a new approach, based on the analogue Markov chain, for a data-based learning of approximate committor functions. We demonstrate that such learned committor functions are extremely efficient score functions when used with the Adaptive Multilevel Splitting algorithm. We illustrate our approach for a gradient dynamics in a three-well potential, and for the Charney-DeVore model, which is a paradigmatic toy model of multistability for atmospheric dynamics. For these two dynamics, we show that having observed a few transitions is enough to have a very efficient data-based score function for the rare event algorithm. This new approach is promising for use for complex dynamics: the rare events can be simulated with a minimal prior knowledge and the results are much more precise than those obtained with a user-designed score function