Machine Learning

NORi combines a physics-based ocean boundary-layer mixing scheme with neural networks that learn the entrainment missing from a local diffusive closure. It trains against the evolution of temperature and salinity in large-eddy simulations while conserving tracers by construction. Tests examine unfamiliar forcing conditions and seasonal mixing at Ocean Weather Station Papa, where performance is comparable to established closures. An idealized double-gyre experiment remains numerically stable for 100 years despite training on two-day trajectories, although this does not establish accuracy in a realistic global ocean. Readers get a detailed example of designing, training, and evaluating a hybrid physics-and-ML parameterization that targets missing processes while retaining a simple physical foundation.

This paper introduces KGMM, a method for estimating the score function that describes how probability density changes across state space. It combines bisecting K-means clustering, a Gaussian mixture representation, and neural interpolation of the resulting score estimates. The construction addresses noise amplification that can make conventional Gaussian mixture scores unreliable when component covariances are small. Examples involving potential systems and chaotic Lorenz-type models show that the estimated scores can support stochastic models with accurate long-term distributions. Readers get a concrete score-estimation workflow and an understanding of when clustering can provide a useful foundation for generative reduced-order models.

This preprint builds reduced-order stochastic models for time series whose statistics vary periodically, as in the annual climate cycle. It uses score-based generative methods to reproduce probability distributions while also representing temporal dependence. A demonstration with the Planet Simulator climate model uses the 20 leading principal components of surface temperature. Validation examines marginal and joint distributions, autocorrelations, and spatial coherence, with synthetic centuries generated far faster than full simulations. Readers get a framework for emulating periodically forced systems and examples of the checks needed to judge statistical and temporal fidelity.

This paper asks how a small external forcing changes an entire probability distribution, including its variance, asymmetry, and tails. It combines the generalized fluctuation-dissipation theorem with score functions learned from unperturbed data. Clustering handles systems with a compact state-space representation, while a neural score model extends the approach to spatially distributed turbulence. Tests on stochastic models and two-dimensional Navier-Stokes flow show why Gaussian approximations can miss important changes in higher-order statistics. Readers get a practical framework for estimating distributional responses within the small-forcing regime, together with examples that connect response theory to generative modeling.