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Understanding the interactions between the El Nino-Southern Oscillation (ENSO) and the Madden-Julian Oscillation (MJO) is essential to studying climate variabilities and predicting extreme weather events. Here, we develop a stochastic conceptual model for describing the coupled ENSO-MJO phenomenon. The model adopts a three-box representation of the interannual ocean component to characterize the ENSO diversity. For the intraseasonal atmospheric component, a low-order Fourier representation is used to describe the eastward propagation of the MJO. We incorporate decadal variability to account for modulations in the background state that influence the predominant types of El Nino events. In addition to dynamical coupling through wind forcing and latent heat, state-dependent noise is introduced to characterize the statistical interactions among these multiscale processes, improving the simulation of extreme events. The model successfully reproduces the observed non-Gaussian statistics of ENSO diversity and MJO spectra. It also captures the interactions between wind, MJO, and ENSO., Comment: 26 pages, 12 figures

Deep learning is widely used to predict complex dynamical systems in many scientific and engineering areas. However, the black-box nature of these deep learning models presents significant challenges for carrying out simultaneous data assimilation (DA), which is a crucial technique for state estimation, model identification, and reconstructing missing data. Integrating ensemble-based DA methods with nonlinear deep learning models is computationally expensive and may suffer from large sampling errors. To address these challenges, we introduce a deep learning framework designed to simultaneously provide accurate forecasts and efficient DA. It is named Conditional Gaussian Koopman Network (CGKN), which transforms general nonlinear systems into nonlinear neural differential equations with conditional Gaussian structures. CGKN aims to retain essential nonlinear components while applying systematic and minimal simplifications to facilitate the development of analytic formulae for nonlinear DA. This allows for seamless integration of DA performance into the deep learning training process, eliminating the need for empirical tuning as required in ensemble methods. CGKN compensates for structural simplifications by lifting the dimension of the system, which is motivated by Koopman theory. Nevertheless, CGKN exploits special nonlinear dynamics within the lifted space. This enables the model to capture extreme events and strong non-Gaussian features in joint and marginal distributions with appropriate uncertainty quantification. We demonstrate the effectiveness of CGKN for both prediction and DA on three strongly nonlinear and non-Gaussian turbulent systems: the projected stochastic Burgers--Sivashinsky equation, the Lorenz 96 system, and the El Ni\~no-Southern Oscillation. The results justify the robustness and computational efficiency of CGKN.

Understanding ENSO dynamics has tremendously improved over the past decades. However, one aspect still poorly understood or represented in conceptual models is the ENSO diversity in spatial pattern, peak intensity, and temporal evolution. In this paper, a physics-informed auto-learning framework is developed to derive ENSO stochastic conceptual models with varying degrees of freedom. The framework is computationally efficient and easy to apply. Once the state vector of the target model is set, causal inference is exploited to build the right-hand side of the equations based on a mathematical function library. Fundamentally different from standard nonlinear regression, the auto-learning framework provides a parsimonious model by retaining only terms that improve the dynamical consistency with observations. It can also identify crucial latent variables and provide physical explanations. Exploiting a realistic six-dimensional reference recharge oscillator-based ENSO model, a hierarchy of three- to six-dimensional models is derived using the auto-learning framework and is systematically validated by a unified set of validation criteria assessing the dynamical and statistical features of the ENSO diversity. It is shown that the minimum model characterizing ENSO diversity is four-dimensional, with three interannual variables describing the western Pacific thermocline depth, the eastern and central Pacific sea surface temperatures (SSTs), and one intraseasonal variable for westerly wind events. Without the intraseasonal variable, the resulting three-dimensional model underestimates extreme events and is too regular. The limited number of weak nonlinearities in the model are essential in reproducing the observed extreme El Ni\~nos and nonlinear relationship between the eastern and western Pacific SSTs.

Predicting the process of porosity-based ductile damage in polycrystalline metallic materials is an essential practical topic. Ductile damage and its precursors are represented by extreme values in stress and material state quantities, the spatial PDF of which are highly non-Gaussian with strong fat tails. Traditional deterministic forecasts using physical models often fail to capture the statistics of structural evolution during material deformation. This study proposes a data-driven statistical reduced-order modeling framework to provide a probabilistic forecast of the deformation process leading to porosity-based ductile damage, with uncertainty quantification. The framework starts with computing the time evolution of the leading moments of specific state variables from full-field polycrystal simulations. Then a sparse model identification algorithm based on causation entropy, including essential physical constraints, is used to discover the governing equations of these moments. An approximate solution of the time evolution of the PDF is obtained from the predicted moments exploiting the maximum entropy principle. Numerical experiments based on polycrystal realizations show that the model can characterize the time evolution of the non-Gaussian PDF of the von Mises stress and quantify the probability of extreme events. The learning process also reveals that the mean stress interacts with higher-order moments and extreme events in a strongly nonlinear and multiplicative fashion. In addition, the calibrated moment equations provide a reasonably accurate forecast when applied to the realizations outside the training data set, indicating the robustness of the model and the skill for extrapolation. Finally, an information-based measurement shows that the leading four moments are sufficient to characterize the crucial non-Gaussian features throughout the entire deformation history.

El Ni\~no-Southern Oscillation (ENSO) is the most predominant interannual variability in the tropics, significantly impacting global weather and climate. In this paper, a framework of low-order conceptual models for the ENSO is systematically derived from a spatially-extended stochastic dynamical system with full mathematical rigor. The spatially-extended stochastic dynamical system has a linear, deterministic, and stable dynamical core. It also exploits a simple stochastic process with multiplicative noise to parameterize the intraseasonal wind burst activities. A principal component analysis based on the eigenvalue decomposition method is applied to provide a low-order conceptual model that succeeds in characterizing the large-scale dynamical and non-Gaussian statistical features of the eastern Pacific El Ni\~no events. Despite the low dimensionality, the conceptual modeling framework contains outputs for all the atmosphere, ocean, and sea surface temperature components with detailed spatiotemporal patterns. This contrasts with many existing conceptual models focusing only on a small set of specified state variables. The stochastic versions of many state-of-the-art low-order models, such as the recharge-discharge and the delayed oscillators, become special cases within this framework. The rigorous derivation of such low-order models provides a unique way to connect models with different spatiotemporal complexities. The framework also facilitates understanding the instantaneous and memory effects of stochastic noise in contributing to the large-scale dynamics of the ENSO.

Discovering the underlying dynamics of complex systems from data is an important practical topic. Constrained optimization algorithms are widely utilized and lead to many successes. Yet, such purely data-driven methods may bring about incorrect physics in the presence of random noise and cannot easily handle the situation with incomplete data. In this paper, a new iterative learning algorithm for complex turbulent systems with partial observations is developed that alternates between identifying model structures, recovering unobserved variables, and estimating parameters. First, a causality-based learning approach is utilized for the sparse identification of model structures, which takes into account certain physics knowledge that is pre-learned from data. It has unique advantages in coping with indirect coupling between features and is robust to the stochastic noise. A practical algorithm is designed to facilitate the causal inference for high-dimensional systems. Next, a systematic nonlinear stochastic parameterization is built to characterize the time evolution of the unobserved variables. Closed analytic formula via an efficient nonlinear data assimilation is exploited to sample the trajectories of the unobserved variables, which are then treated as synthetic observations to advance a rapid parameter estimation. Furthermore, the localization of the state variable dependence and the physics constraints are incorporated into the learning procedure, which mitigate the curse of dimensionality and prevent the finite time blow-up issue. Numerical experiments show that the new algorithm succeeds in identifying the model structure and providing suitable stochastic parameterizations for many complex nonlinear systems with chaotic dynamics, spatiotemporal multiscale structures, intermittency, and extreme events.