Mickaël D. Chekroun

Detection and attribution studies have played a major role in shaping contemporary climate science and have provided key motivations supporting global climate policy negotiations. The goal of such studies is to associate observed climatic patterns of climate change with acting forcings - both anthropogenic and natural ones - with the goal of making statements on the acting drivers of climate change. The statistical inference is usually performed using regression methods referred to as optimal fingerprinting. We show here how a fairly general formulation of linear response theory relevant for nonequilibrium systems provides the physical and mathematical foundations behind the optimal fingerprinting approach for the climate change detection and attribution problem. Our angle allows one to clearly frame assumptions, strengths and potential pitfalls of the method.

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 Schrödinger operator having 32 metastable states. In addition, we address the so-called Gelfand superlinear problem having exponential nonlinearities, in dimension, 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.

We consider a three-dimensional slow-fast system with quadratic nonlinearity and additive noise. The associated deterministic system of this stochastic differential equation (SDE) exhibits a periodic orbit and a slow manifold. The deterministic slow manifold can be viewed as an approximate parameterization of the fast variable of the SDE in terms of the slow variables. In other words the fast variable of the slow-fast system is approximately "slaved" to the slow variables via the slow manifold. We exploit this fact to obtain a two dimensional reduced model for the original stochastic system, which results in the Hopf-normal form with additive noise. Both, the original as well as the reduced system admit ergodic invariant measures describing their respective long-time behaviour. We will show that for a suitable metric on a subset of the space of all probability measures on phase space, the discrepancy between the marginals along the radial component of both invariant measures can be upper bounded by a constant and a quantity describing the quality of the parameterization. An important technical tool we use to arrive at this result is Girsanov's theorem, which allows us to modify the SDEs in question in a way that preserves transition probabilities. This approach is then also applied to reduced systems obtained through stochastic parameterizing manifolds, which can be viewed as generalized notions of deterministic slow manifolds.

A central challenge in physics is to describe non-equilibrium systems driven by randomness, such as a randomly growing interface, or fluids subject to random fluctuations that account e.g. for local stresses and heat fluxes in the fluid which are not related to the velocity and temperature gradients. For deterministic systems with infinitely many degrees of freedom, normal form and center manifold theory have shown a prodigious efficiency to often completely characterize how the onset of linear instability translates into the emergence of nonlinear patterns, associated with genuine physical regimes. However, in presence of random fluctuations, the underlying reduction principle to the center manifold is seriously challenged due to large excursions caused by the noise, and the approach needs to be revisited.

In this study, we present an alternative framework to cope with these difficulties exploiting the approximation theory of stochastic invariant manifolds, on one hand, and energy estimates measuring the defect of parameterization of the high-modes, on the other. To operate for fluid problems subject to stochastic stirring forces, these error estimates are derived under assumptions regarding dissipation effects brought by the high-modes in order to suitably counterbalance the loss of regularity due to the nonlinear terms. As a result, the approach enables us to analyze, from reduced equations of the stochastic fluid problem, the occurrence in large probability of a stochastic analogue to the pitchfork bifurcation, as long as the noise’s intensity and the eigenvalue’s magnitude of the mildly unstable mode scale accordingly.

In the case of SPDEs forced by a multiplicative noise in the orthogonal subspace of e.g. its mildly unstable mode, our parameterization formulas show that the noise gets transmitted to this mode via non-Markovian coefficients, and that the reduced equation is only stochastically driven by the latter.  These coefficients depend explicitly on the noise path's history, and their memory content is self-consistently determined by the intensity of the random force and its interaction through the SPDE's nonlinear terms. Applications to a stochastic Rayleigh-B\'enard problem  are detailed, for which conditions for a stochastic pitchfork bifurcation (in large probability) to occur, are clarified.

 

 

 

One of the major sources of uncertainty in climate prediction results from the limitations in representing shallow cumulus (Cu) in models. Recently, a class of continental shallow convective Cu was shown to share distinct morphological properties and to emerge globally mostly over forests and vegetated areas, thus named greenCu. Using machine-learning supervised classification, we identify greenCu fields over three regions, from the tropics to mid- and higher-latitudes, and establish a novel satellite-based data set called greenCuDb, consisting of 1° × 1° sized, high-resolution MODIS images. Using greenCuDb in conjunction with ERA5 reanalysis data, we create greenCu composites for different regions and reveal that greenCu are driven by similar large-scale meteorological conditions, regardless of their geographical locations throughout the world's continents. These conditions include distinct profiles of temperature, humidity and large-scale vertical velocity. The boundary layer is anomalously warm and moderately humid, and is accompanied by a strong large-scale subsidence in the free troposphere.