Sampling from heavy-tailed and multimodal distributions is challenging when neither the target density nor the proposal density can be evaluated, as encountered in α-stable Lévy-driven fractional Langevin algorithms. While the target distribution can be estimated from data using score-based or energy-based models, the α-stable proposal density and its score are generally unavailable, rendering classical density-based Metropolis-Hastings (MH) corrections impractical. Consequently, existing fractional Langevin methods operate in an unadjusted regime, often exhibiting substantial finite-time errors and poor empirical control over tail behavior. To address these limitations, we first introduce a general, fully score-based MH adjustment mechanism. Building upon this framework, we develop the Metropolis-Adjusted Fractional Langevin Algorithm (MAFLA). MAFLA employs designed proxies for fractional proposal score gradients under isotropic symmetric \alpha-stable noise and learns the acceptance function via Score Balance Matching. We empirically demonstrate the strong performance of MAFLA across a series of tasks, including combinatorial optimization problems, where the method significantly improves finite-time sampling accuracy over unadjusted fractional Langevin dynamics.

Generating extreme events poses two distinct obstacles, and generative models may easily fail at both. The difficulty is not confined to heavy tails, though those are the hardest case, and in the multivariate setting different directions may sit in different tail regimes, none of them known in advance. The first obstacle is in the generative objective itself, where every standard choice imposes regularity on the measures: KL and f-divergences require absolute continuity and can easily fail on singular or heavy-tailed targets, while Wasserstein distances require moments of both arguments and may not have well-defined variational derivatives. Lipschitz-regularized divergences, by contrast, are infimal convolutions of a divergence with a transport cost, i.e. a proximal regularization in the Wasserstein-1 geometry, and inherit the best of both, and more: the only assumption needed is on the source, which we design (a finite first moment), and, most importantly, none at all on the target, singular or heavy-tailed alike. The variational derivatives then exist, are finite and unique, and the Wasserstein gradient flow is well posed on targets other objectives cannot handle. Its dissipation inequality also yields a principled, data-driven stopping criterion in place of a preset number of steps. The second obstacle is that the transport itself cannot be regular for learning heavier tails than the source's. A Lipschitz map sends a light-tailed source to a light-tailed output, so a well-behaved flow never reaches the tail. CVaR, which measures tail risk, is blind to the bulk and aggregates the whole tail into a single scalar observable, so relatively few tail samples suffice to estimate it reliably, where densities and their gradients cannot be resolved at all; adding it to the generative objective - the two terms together forming a free energy - revives the transport in the sparsely sampled tail, through a bounded but non-Lipschitz velocity. The resulting correction is tail-agnostic and applies to the samples of any pre-trained model without access to its architecture; across synthetic and real target data it sharply improves both bulk and tail accuracy over pre-trained baselines and over recent tail-adapted GANs, diffusions and normalizing flows.

Extreme events in complex dynamical systems may arise from distinct mechanisms, but identifying their drivers and determining whether models represent the relevant pathways remain major challenges. This talk presents a unified information-propagation perspective on these two questions. First, assimilative causal inference combines causal diagnostics with data assimilation to identify the variables and dynamical pathways that contribute to an extreme event and to quantify their ranges of influence. Examples involving both noise-driven and nonlinearity-driven extremes illustrate how different mechanisms produce distinct causal signatures and precursor structures. Second, controlled state reconstruction is used to interrogate whether a dynamical model transmits observational information toward an extreme-event target through physically meaningful pathways. The resulting reconstruction fingerprints reveal not only whether additional observations improve the representation of an extreme, but also where information transfer is distorted or suppressed. Applications to ENSO extremes in CMIP models examine the coupled pathways linking sea-surface temperature, winds, thermocline variability, and ocean recharge. These methods aim to move beyond predicting extreme events toward understanding their mechanisms and assessing whether a model’s explanation of them can be trusted.

Stochastic optimal control (SOC) has application in many areas, including rare events, economics, and engineering. In this talk, we will present a highly parallel and derivative-free martingale neural network method, based on the probability theory of Varadhan’s martingale formulation of PDEs, to solve Hamilton-Jacobi-Bellman (HJB) equations arising from SOC problems in very high dimensions. The HJB equation is reformulated into a martingale problem such that loss functions will not require the computation of the gradient or Hessian matrix of the PDE solution and can be computed in parallel in both time and spatial domains. Moreover, the martingale conditions for the PDE are enforced using a Galerkin method realized with adversarial learning techniques, eliminating the need for direct computation of the conditional expectations associated with the martingale property. Also, a derivative-free implementation of the maximum principle for optimal controls is introduced. Numerical results demonstrate the effectiveness and efficiency of the proposed method, which is capable of solving HJB accurately and fast in dimensions as high as 10,000.

The quasipotential is an analog of the potential function for nongradient SDEs. It determines the exponential factors in transition rates and invariant measures and defines the most probable escape paths from the basins of attractors. Its computation in large regions surrounding the system’s attractors requires solving an optimal control problem on a grid. I will discuss two approaches for computing the quasipotential. The first one comprises Dijkstra-like ordered line-integral methods (OLIMs), including the latest version suitable for dimensions up to three and variable, anisotropic, and rank-deficient diffusion. The second one traces the unstable Lagrangian manifold of the associated Hamiltonian system. I will demonstrate the application of these approaches to problems coming from ecology, climate science, and engineering.

Long-time rare-event statistics can be estimated from ensembles of trajectories that are much shorter than the event timescale. Committors, mean first passage times, and related rare event statistics all solve Bellman equations defined in terms of stopped trajectories. Markov state models are Galerkin approximations of these equations. On finite-state spaces we can establish relative error bounds for these quantities that are nearly independent of a condition number or related quantity (return-time, spectral gap, etc), partially explaining the empirical success of these methods on rare event problems (where condition numbers are often extremely large). I'll also describe recent approaches to non-linear parametrization of these equations as well as their failure, in the case of state-of-the-art AI weather emulators, when training data does not sufficiently cover dynamically important regions of state space and show that combining parametrized approximation of rare event statistics with rare-event simulation is one promising route to generating the necessary training data.

We develop a data-driven switching-diffusion framework for modeling extreme and rare behavior in ensembles of nonlinear material-response trajectories. Using a progressive failure model that captures the salient features of hybrid composites, stress–strain paths are represented by latent discrete regimes coupled to continuous stochastic dynamics. State-dependent drift fields are learned nonparametrically in stress–strain space, while transitions between regimes are characterized through strain-dependent switching hazards. Particular attention is given to separating variability associated with regime switching, continuous fluctuations, and short-range nonwhite residual dependence. The resulting hybrid stochastic model provides a generative representation of the response ensemble and, through its associated generator, a route to computing threshold-crossing and extreme-event probabilities without relying exclusively on direct Monte Carlo sampling. The study illustrates how learning the local dynamics and switching mechanisms underlying an ensemble can support extrapolation toward rare events, while also exposing challenges associated with state identification, stochastic-model closure, and the accurate representation of tail behavior.

A classical approach for importance sampling (IS) of a rare event is to tilt the measure using its large deviation minimizer, thereby shifting the proposal mean towards the event set. This exponential tilt is log-efficient, yet we show that its coefficient of variation still diverges algebraically as the noise vanishes. The reason is geometric: the tilt ignores the curvature of the event boundary and the different scaling of fluctuations in the normal and tangential directions. Guided by sharp large-deviation asymptotics, we propose two remedies — a control variate with known mean built from the quadratic boundary approximation, and a non-Gaussian proposal that is exponential in the normal direction and curvature-corrected tangentially. The first keeps the coefficient of variation bounded, the second makes it vanish as the noise goes to zero. We analyze both samplers through their variance and through a Kullback–Leibler framework for the necessary sample size. Experiments on an SDE and a stochastic KdV equation down to extremely small probabilities confirm the rates, with constants independent of the discretization.