Accurately assessing financial risk requires capturing both individual asset volatility and the complex, asymmetric dependence structures that emerge during extreme market events. While modern diffusion-based models have advanced multivariate forecasting, they often suffer from a "normality bias" when trained end-to-end, sacrificing marginal calibration for joint coherence and consistently underestimating tail risk. To address this, we propose a Diffusion-Copula framework that explicitly decouples the learning of marginal distributions from their dependence structure. We employ deep Mixture Density Networks to capture heavy-tailed asset dynamics, followed by a Classification-Diffusion Copula to model the joint dependence. Applied to cryptocurrency markets, our approach demonstrates superior performance over state-of-the-art baselines in forecasting systemic extremes of both marginal and joint events. Crucially, we demonstrate that while baseline models classify simultaneous market crashes as statistically impossible "Black Swans" (high surprise), our framework identifies them as "Expected Crashes" (low surprise), successfully preserving the correlation structure necessary for robust risk management during contagion events.

Data assimilation (DA) integrates observational information with model predictions to improve state estimation in complex systems. While filtering provides the basis for online forecasts by using only past and present observations, it can exhibit delays and biases when the underlying dynamics evolve rapidly or undergo regime transitions. Smoothing, which additionally incorporates future observations, provides a natural pipeline for hindcasting and reanalysis that yields an uncertainty reduction beyond the filter. This paper introduces an ensemble Kalman-Bucy smoother (EnKBS) for continuous-time DA of nonlinear dynamical systems, where the smoother's conditional distributions are reconstructed using ensemble moments. The result is a derivative-free framework that does not require explicit computation of tangent-linear or adjoint models, which converges to the exact smoother solution at the infinite-ensemble limit for a wide class of complex systems. Incorporating standard regularization techniques for high-dimensional systems, such as covariance localization and inflation, the skill of the EnKBS is demonstrated in various important scientific problems. By integrating future observations, which reveal the underlying causal mechanisms for retrospective state updates, the EnKBS is used for Bayesian-based inference of causal relationships and their temporal influence range in a dyadic trigger-feedback model and the development of a causality-driven iterative learning algorithm that identifies the structure and recovers the hidden parameters of a nonlinear reduced-order model mimicking midlatitude atmospheric circulation. Notably, both tasks remain effective with an ensemble size of $O(10)$ under partial observations, suggesting that EnKBS can support the instantaneous discovery of high-dimensional complex systems over time.

Time series generation is a crucial aspect of data analysis, playing a pivotal role in learning the temporal patterns and their underlying dynamics across diverse fields. Conventional time series generation methods often struggle to capture extreme values adequately, diminishing their value in critical applications such as scenario planning and management for healthcare, finance, climate change adaptation, and beyond. In this paper, we introduce a conditional diffusion model called FIDE to address the challenge of preserving the distribution of extreme values in generative modeling for time series. FIDE employs a novel high-frequency inflation strategy in the frequency domain, preventing premature fade-out of the extreme value. It also extends traditional diffusion-based model, enabling the generation of samples conditioned on the block maxima, thereby enhancing the model's capacity to capture extreme events. Additionally, the FIDE framework incorporates the Generalized Extreme Value (GEV) distribution within its generative modeling framework, ensuring fidelity to both block maxima and overall data distribution.

This paper studies the use of a machine learning-based estimator as a control variate for mitigating the variance of Monte Carlo sampling. Specifically, we seek to uncover the key factors that influence the efficiency of control variates in reducing variance. We examine a prototype estimation problem that involves simulating the moments of a Sobolev function based on observations obtained from (random) quadrature nodes. Firstly, we establish an information-theoretic lower bound for the problem. We then study a specific quadrature rule that employs a nonparametric regression-adjusted control variate to reduce the variance of the Monte Carlo simulation. We demonstrate that this kind of quadrature rule can improve the Monte Carlo rate and achieve the minimax optimal rate under a sufficient smoothness assumption. Due to the Sobolev Embedding Theorem, the sufficient smoothness assumption eliminates the existence of rare and extreme events. Finally, we show that, in the presence of rare and extreme events, a truncated version of the Monte Carlo algorithm can achieve the minimax optimal rate while the control variate cannot improve the convergence rate.

Recent advancements in deep learning have led to the development of Foundation Models (FMs) for weather forecasting, yet their ability to predict extreme weather events remains limited. Existing approaches either focus on general weather conditions or specialize in specific-type extremes, neglecting the real-world atmospheric patterns of diversified extreme events. In this work, we identify two key characteristics of extreme events: (1) the spectral disparity against normal weather regimes, and (2) the hierarchical drivers and geographic blending of diverse extremes. Along this line, we propose UniExtreme, a universal extreme weather forecasting foundation model that integrates (1) an Adaptive Frequency Modulation (AFM) module that captures region-wise spectral differences between normal and extreme weather, through learnable Beta-distribution filters and multi-granularity spectral aggregation, and (2) an Event Prior Augmentation (EPA) module which incorporates region-specific extreme event priors to resolve hierarchical extreme diversity and composite extreme schema, via a dual-level memory fusion network. Extensive experiments demonstrate that UniExtreme outperforms state-of-the-art baselines in both extreme and general weather forecasting, showcasing superior adaptability across diverse extreme scenarios.

Extreme weather events are widely studied in fields such as agriculture, ecology, and meteorology. The spatio-temporal co-occurrence of extreme events can strengthen or weaken under changing climate conditions. In this paper, we propose a novel approach to model spatio-temporal extremes by integrating climate indices via a conditional variational autoencoder (cXVAE). A convolutional neural network (CNN) is embedded in the decoder to convolve climatological indices with the spatial dependence within the latent space, thereby allowing the decoder to be dependent on the climate variables. There are three main contributions here. First, we demonstrate through extensive simulations that the proposed conditional XVAE accurately emulates spatial fields and recovers spatially and temporally varying extremal dependence with very low computational cost post training. Second, we provide a simple, scalable approach to detecting condition-driven shifts and whether the dependence structure is invariant to the conditioning variable. Third, when dependence is found to be condition-sensitive, the conditional XVAE supports counterfactual experiments allowing intervention on the climate covariate and propagating the associated change through the learned decoder to quantify differences in joint tail risk, co-occurrence ranges, and return metrics. To demonstrate the practical utility and performance of the model in real-world scenarios, we apply our method to analyze the monthly maximum Fire Weather Index (FWI) over eastern Australia from 2014 to 2024 conditioned on the El Niño/Southern Oscillation (ENSO) index.

Understanding the plausible upper bounds of extreme weather events is essential for risk assessment in a warming climate. Existing methods, based on large ensembles of physics-based models, are often computationally expensive or lack the fidelity needed to simulate rare, high-impact extremes. Here, we present a novel framework that leverages a differentiable hybrid climate model, NeuralGCM, to optimize initial conditions and generate physically consistent worst-case heatwave trajectories. Applied to the 2021 Pacific Northwest heatwave, our method produces heatwave intensity up to 3.7 $^\circ$C above the most extreme member of a 75-member ensemble. These trajectories feature intensified atmospheric blocking and amplified Rossby wave patterns-hallmarks of severe heat events. Our results demonstrate that differentiable climate models can efficiently explore the upper tails of event likelihoods, providing a powerful new approach for constructing targeted storylines of extreme weather under climate change.

Extreme weather events pose escalating risks to global society, underscoring the urgent need to unravel their underlying physical mechanisms. Yet the prevailing expert-driven, labor-intensive diagnostic paradigm has created a critical analytical bottleneck, stalling scientific progress. While AI for Earth Science has achieved notable advances in prediction, the equally essential challenge of automated diagnostic reasoning remains largely unexplored. We present the Extreme Weather Expert (EWE), the first intelligent agent framework dedicated to this task. EWE emulates expert workflows through knowledge-guided planning, closed-loop reasoning, and a domain-tailored meteorological toolkit. It autonomously produces and interprets multimodal visualizations from raw meteorological data, enabling comprehensive diagnostic analyses. To catalyze progress, we introduce the first benchmark for this emerging field, comprising a curated dataset of 103 high-impact events and a novel step-wise evaluation metric. EWE marks a step toward automated scientific discovery and offers the potential to democratize expertise and intellectual resources, particularly for developing nations vulnerable to extreme weather.