Extreme value theory provides rigorous theory and statistical tools for extrapolation in machine learning, particularly in settings where traditional methods struggle due to data scarcity in the tails. A broad range of tasks benefit from these advances, including regression and classification beyond the training data, extreme quantile regression, supervised and unsupervised dimension reduction, generative artificial intelligence and anomaly detection. This review synthesizes recent developments in these fields at the intersection of statistical learning and extreme value theory, with a focus on principled methods based on asymptotically motivated representations of the tail of univariate and multivariate distributions. We consider different theoretical frameworks for both asymptotically dependent and independent data and discuss how they translate into efficient statistical methods for extrapolation to extreme regions. By addressing both theoretical and practical aspects, we offer a comprehensive overview of the state-of-the-art in this quickly evolving field, and identify promising directions for future research.

In contrast, we directly evaluate the matrix equations which arise from applying variable elimination on the Lagrange multiplier terms in the (differential) KKT system.

See Appendix Aforanoteontrain/testsplitfor Task 3. loss Testloss Energy Baseline HNNBaseline HNNBaseline HNN mass-spring170 20.38 .1 pendulum 42 10 25 5 pendulum 390 7 14 5 (6.3e4 3e4 39 5 pendulum.3

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