Applying machine learning to improve simulations of a chaotic dynamical system using empirical error correction

Dynamical weather and climate prediction models underpin many studies of the Earth system and hold the promise of being able to make robust projections of future climate change based on physical laws. However, simulations from these models still show many differences compared with observations. Machine learning has been applied to solve certain prediction problems with great success, and recently it's been proposed that this could replace the role of physically-derived dynamical weather and climate models to give better quality simulations. Here, instead, a framework using machine learning together with physically-derived models is tested, in which it is learnt how to correct the errors of the latter from timestep to timestep. This maintains the physical understanding built into the models, whilst allowing performance improvements, and also requires much simpler algorithms and less training data. This is tested in the context of simulating the chaotic Lorenz '96 system, and it is shown that the approach yields models that are stable and that give both improved skill in initialised predictions and better long-term climate statistics. Improvements in long-term statistics are smaller than for single time-step tendencies, however, indicating that it would be valuable to develop methods that target improvements on longer time scales. Future strategies for the development of this approach and possible applications to making progress on important scientific problems are discussed.Comment: 26p, 7 figures To be published in Journal of Advances in Modeling Earth System

Thermodynamic analysis of snowball Earth hysteresis experiment: Efficiency, entropy production and irreversibility

We present an extensive thermodynamic analysis of a hysteresis experiment performed on a simplified yet Earth-like climate model. We slowly vary the solar constant by 20% around the present value and detect that for a large range of values of the solar constant the realization of snowball or of regular climate conditions depends on the history of the system. Using recent results on the global climate thermodynamics, we show that the two regimes feature radically different properties. The efficiency of the climate machine monotonically increases with decreasing solar constant in present climate conditions, whereas the opposite takes place in snowball conditions. Instead, entropy production is monotonically increasing with the solar constant in both branches of climate conditions, and its value is about four times larger in the warm branch than in the corresponding cold state. Finally, the degree of irreversibility of the system, measured as the fraction of excess entropy production due to irreversible heat transport processes, is much higher in the warm climate conditions, with an explosive growth in the upper range of the considered values of solar constants. Whereas in the cold climate regime a dominating role is played by changes in the meridional albedo contrast, in the warm climate regime changes in the intensity of latent heat fluxes are crucial for determining the observed properties. This substantiates the importance of addressing correctly the variations of the hydrological cycle in a changing climate. An interpretation of the climate transitions at the tipping points based upon macro-scale thermodynamic properties is also proposed. Our results support the adoption of a new generation of diagnostic tools based on the second law of thermodynamics for auditing climate models and outline a set of parametrizations to be used in conceptual and intermediate-complexity models or for the reconstruction of the past climate conditions. Copyright © 2010 Royal Meteorological Societ

Gauge Theory for Finite-Dimensional Dynamical Systems

Gauge theory is a well-established concept in quantum physics, electrodynamics, and cosmology. This theory has recently proliferated into new areas, such as mechanics and astrodynamics. In this paper, we discuss a few applications of gauge theory in finite-dimensional dynamical systems with implications to numerical integration of differential equations. We distinguish between rescriptive and descriptive gauge symmetry. Rescriptive gauge symmetry is, in essence, re-scaling of the independent variable, while descriptive gauge symmetry is a Yang-Mills-like transformation of the velocity vector field, adapted to finite-dimensional systems. We show that a simple gauge transformation of multiple harmonic oscillators driven by chaotic processes can render an apparently "disordered" flow into a regular dynamical process, and that there exists a remarkable connection between gauge transformations and reduction theory of ordinary differential equations. Throughout the discussion, we demonstrate the main ideas by considering examples from diverse engineering and scientific fields, including quantum mechanics, chemistry, rigid-body dynamics and information theory

Return-Map Cryptanalysis Revisited

As a powerful cryptanalysis tool, the method of return-map attacks can be used to extract secret messages masked by chaos in secure communication schemes. Recently, a simple defensive mechanism was presented to enhance the security of chaotic parameter modulation schemes against return-map attacks. Two techniques are combined in the proposed defensive mechanism: multistep parameter modulation and alternative driving of two different transmitter variables. This paper re-studies the security of this proposed defensive mechanism against return-map attacks, and points out that the security was much over-estimated in the original publication for both ciphertext-only attack and known/chosen-plaintext attacks. It is found that a deterministic relationship exists between the shape of the return map and the modulated parameter, and that such a relationship can be used to dramatically enhance return-map attacks thereby making them quite easy to break the defensive mechanism.Comment: 11 pages, 7 figure

Which is more appropriate: a multi-perspective comparison between systems dynamics and discrete event simulation

System Dynamics (SD) and Discrete Event Simulation (DES) are two established simulation tech-niques for simulating the dynamics of a system. Both have been widely used in modelling business de-cisions. This paper presents meta-comparison between the two approaches based on literature survey. Upon reviewing the existing literature it has been identified that existing comparisons could be classi-fied under three main perspectives: Systems perspective, Problems perspective and Methodology per-spective. The nature of system and nature of problem have been argued as primary factors for decid-ing modelling methodology. Therefore SD and DES comparisons have been classified on the basis of systems, problems and inherent aspects and capabilities of both modelling methods. It has been ar-gued that development of sound models need fit between system, problem and methodology. The suc-cess of model depends on it’s technical soundness as well as it’s successful implementation. In order to develop successful models this vision has been further extended to incorporate stakeholders, re-sources and time

Detecting event-related recurrences by symbolic analysis: Applications to human language processing

Quasistationarity is ubiquitous in complex dynamical systems. In brain dynamics there is ample evidence that event-related potentials reflect such quasistationary states. In order to detect them from time series, several segmentation techniques have been proposed. In this study we elaborate a recent approach for detecting quasistationary states as recurrence domains by means of recurrence analysis and subsequent symbolisation methods. As a result, recurrence domains are obtained as partition cells that can be further aligned and unified for different realisations. We address two pertinent problems of contemporary recurrence analysis and present possible solutions for them.Comment: 24 pages, 6 figures. Draft version to appear in Proc Royal Soc

A statistical mechanical approach for the computation of the climatic response to general forcings

The climate belongs to the class of non-equilibrium forced and dissipative systems, for which most results of quasi-equilibrium statistical mechanics, including the fluctuation-dissipation theorem, do not apply. In this paper we show for the first time how the Ruelle linear response theory, developed for studying rigorously the impact of perturbations on general observables of non-equilibrium statistical mechanical systems, can be applied with great success to analyze the climatic response to general forcings. The crucial value of the Ruelle theory lies in the fact that it allows to compute the response of the system in terms of expectation values of explicit and computable functions of the phase space averaged over the invariant measure of the unperturbed state. We choose as test bed a classical version of the Lorenz 96 model, which, in spite of its simplicity, has a well-recognized prototypical value as it is a spatially extended one-dimensional model and presents the basic ingredients, such as dissipation, advection and the presence of an external forcing, of the actual atmosphere. We recapitulate the main aspects of the general response theory and propose some new general results. We then analyze the frequency dependence of the response of both local and global observables to perturbations having localized as well as global spatial patterns. We derive analytically several properties of the corresponding susceptibilities, such as asymptotic behavior, validity of Kramers-Kronig relations, and sum rules, whose main ingredient is the causality principle. We show that all the coefficients of the leading asymptotic expansions as well as the integral constraints can be written as linear function of parameters that describe the unperturbed properties of the system, such as its average energy. Some newly obtained empirical closure equations for such parameters allow to define such properties as an explicit function of the unperturbed forcing parameter alone for a general class of chaotic Lorenz 96 models. We then verify the theoretical predictions from the outputs of the simulations up to a high degree of precision. The theory is used to explain differences in the response of local and global observables, to define the intensive properties of the system, which do not depend on the spatial resolution of the Lorenz 96 model, and to generalize the concept of climate sensitivity to all time scales. We also show how to reconstruct the linear Green function, which maps perturbations of general time patterns into changes in the expectation value of the considered observable for finite as well as infinite time. Finally, we propose a simple yet general methodology to study general Climate Change problems on virtually any time scale by resorting to only well selected simulations, and by taking full advantage of ensemble methods. The specific case of globally averaged surface temperature response to a general pattern of change of the CO2 concentration is discussed. We believe that the proposed approach may constitute a mathematically rigorous and practically very effective way to approach the problem of climate sensitivity, climate prediction, and climate change from a radically new perspective