Data derived from remote sensing or numerical simulations often have a regular gridded structure and are large in volume, making it challenging to find accurate spatial models that can fill in missing grid cells or simulate the process effectively, especially in the presence of spatial heterogeneity and heavy-tailed marginal distributions. To overcome this issue, we present a spatial autoregressive modeling framework, which maps observations at a location and its neighbors to independent random variables. This is a highly flexible modeling approach and well-suited for non-Gaussian fields, providing simpler interpretability. In particular, we consider the SAR model with Generalized Extreme Value distribution innovations to combine the observation at a central grid location with its neighbors, capturing extreme spatial behavior based on the heavy-tailed innovations. While these models are fast to simulate by exploiting the sparsity of the key matrices in the computations, the maximum likelihood estimation of the parameters is prohibitive due to the intractability of the likelihood, making optimization challenging. To overcome this, we train a convolutional neural network on a large training set that covers a useful parameter space, and then use the trained network for fast parameter estimation. Finally, we apply this model to analyze annual maximum precipitation data from ERA-Interim-driven Weather Research and Forecasting (WRF) simulations, allowing us to explore its spatial extreme behavior across North America.

Machine learning is vital in high-stakes domains, yet conventional validation methods rely on averaging metrics like mean squared error (MSE) or mean absolute error (MAE), which fail to quantify extreme errors. Worst-case prediction failures can have substantial consequences, but current frameworks lack statistical foundations for assessing their probability. In this work a new statistical framework, based on Extreme Value Theory (EVT), is presented that provides a rigorous approach to estimating worst-case failures. Applying EVT to synthetic and real-world datasets, this method is shown to enable robust estimation of catastrophic failure probabilities, overcoming the fundamental limitations of standard cross-validation. This work establishes EVT as a fundamental tool for assessing model reliability, ensuring safer AI deployment in new technologies where uncertainty quantification is central to decision-making or scientific analysis.

Issuing timely severe weather warnings helps mitigate potentially disastrous consequences. Recent advancements in Neural Weather Models (NWMs) offer a computationally inexpensive and fast approach for forecasting atmospheric environments on a 0.25{\deg} global grid. For thunderstorms, these environments can be empirically post-processed to predict wind gust distributions at specific locations. With the Pangu-Weather NWM, we apply a hierarchy of statistical and deep learning post-processing methods to forecast hourly wind gusts up to three days ahead. To ensure statistical robustness, we constrain our probabilistic forecasts using generalised extreme-value distributions across five regions in Switzerland. Using a convolutional neural network to post-process the predicted atmospheric environment's spatial patterns yields the best results, outperforming direct forecasting approaches across lead times and wind gust speeds. Our results confirm the added value of NWMs for extreme wind forecasting, especially for designing more responsive early-warning systems.

Data-driven software is increasingly being used as a critical component of automated decision-support systems. Since this class of software learns its logic from historical data, it can encode or amplify discriminatory practices. Previous research on algorithmic fairness has focused on improving average-case fairness. On the other hand, fairness at the extreme ends of the spectrum, which often signifies lasting and impactful shifts in societal attitudes, has received significantly less emphasis. Leveraging the statistics of extreme value theory (EVT), we propose a novel fairness criterion called extreme counterfactual discrimination (ECD). This criterion estimates the worst-case amounts of disadvantage in outcomes for individuals solely based on their memberships in a protected group. Utilizing tools from search-based software engineering and generative AI, we present a randomized algorithm that samples a statistically significant set of points from the tail of ML outcome distributions even if the input dataset lacks a sufficient number of relevant samples. We conducted several experiments on four ML models (deep neural networks, logistic regression, and random forests) over 10 socially relevant tasks from the literature on algorithmic fairness. First, we evaluate the generative AI methods and find that they generate sufficient samples to infer valid EVT distribution in 95% of cases. Remarkably, we found that the prevalent bias mitigators reduce the average-case discrimination but increase the worst-case discrimination significantly in 5% of cases. We also observed that even the tail-aware mitigation algorithm -- MiniMax-Fairness -- increased the worst-case discrimination in 30% of cases. We propose a novel ECD-based mitigator that improves fairness in the tail in 90% of cases with no degradation of the average-case discrimination.

This paper leverages the statistics of extreme values to predict the worst-case convergence times of machine learning algorithms. Timing is a critical non-functional property of ML systems, and providing the worst-case converge times is essential to guarantee the availability of ML and its services. However, timing properties such as worst-case convergence times (WCCT) are difficult to verify since (1) they are not encoded in the syntax or semantics of underlying programming languages of AI, (2) their evaluations depend on both algorithmic implementations and underlying systems, and (3) their measurements involve uncertainty and noise. Therefore, prevalent formal methods and statistical models fail to provide rich information on the amounts and likelihood of WCCT. Our key observation is that the timing information we seek represents the extreme tail of execution times. Therefore, extreme value theory (EVT), a statistical discipline that focuses on understanding and predicting the distribution of extreme values in the tail of outcomes, provides an ideal framework to model and analyze WCCT in the training and inference phases of ML paradigm. Building upon the mathematical tools from EVT, we propose a practical framework to predict the worst-case timing properties of ML. Over a set of linear ML training algorithms, we show that EVT achieves a better accuracy for predicting WCCTs than relevant statistical methods such as the Bayesian factor. On the set of larger machine learning training algorithms and deep neural network inference, we show the feasibility and usefulness of EVT models to accurately predict WCCTs, their expected return periods, and their likelihood.

We assume that a sufficiently large database is available, where a physical property of interest and a number of associated ruling primitive variables or observables are stored. We introduce and test two machine learning approaches to discover possible groups or combinations of primitive variables: The first approach is based on regression models whereas the second on classification models. The variable group (here referred to as the new effective good variable) can be considered as successfully found, when the physical property of interest is characterized by the following effective invariant behaviour: In the first method, invariance of the group implies invariance of the property up to a given accuracy; in the other method, upon partition of the physical property values into two or more classes, invariance of the group implies invariance of the class. For the sake of illustration, the two methods are successfully applied to two popular empirical correlations describing the convective heat transfer phenomenon and to the Newton's law of universal gravitation.

Understanding individual treatment effects in extreme regimes is important for characterizing risks associated with different interventions. This is hindered by the fact that extreme regime data may be hard to collect, as it is scarcely observed in practice. In addressing this issue, we propose a new framework for estimating the individual treatment effect in extreme regimes (ITE$_2$). Specifically, we quantify this effect by the changes in the tail decay rates of potential outcomes in the presence or absence of the treatment. Subsequently, we establish conditions under which ITE$_2$ may be calculated and develop algorithms for its computation. We demonstrate the efficacy of our proposed method on various synthetic and semi-synthetic datasets.

The heavy-tailed behavior of the generalized extreme-value distribution makes it a popular choice for modeling extreme events such as floods, droughts, heatwaves, wildfires, etc. However, estimating the distribution's parameters using conventional maximum likelihood methods can be computationally intensive, even for moderate-sized datasets. To overcome this limitation, we propose a computationally efficient, likelihood-free estimation method utilizing a neural network. Through an extensive simulation study, we demonstrate that the proposed neural network-based method provides Generalized Extreme Value (GEV) distribution parameter estimates with comparable accuracy to the conventional maximum likelihood method but with a significant computational speedup. To account for estimation uncertainty, we utilize parametric bootstrapping, which is inherent in the trained network. Finally, we apply this method to 1000-year annual maximum temperature data from the Community Climate System Model version 3 (CCSM3) across North America for three atmospheric concentrations: 289 ppm $\mathrm{CO}_2$ (pre-industrial), 700 ppm $\mathrm{CO}_2$ (future conditions), and 1400 ppm $\mathrm{CO}_2$, and compare the results with those obtained using the maximum likelihood approach.