Generating samples from a complex (e.g., non-Gaussian, high-dimensional) probability distribution is a core computational challenge in diverse applications, ranging from computational statistics and machine learning to molecular simulation. A recurring setting is where the density ρ of the target distribution is specified up to a normalizing constant--for example, in Bayesian modeling, where ρ represents the posterior density. Here, evaluations of the score log ρ are often available as well, even for complex statistical models [Villa et al., 2021]. Alternatively, many new methods enable effective score estimation from data, without explicit density estimation; examples include score estimation from time series observations in chaotic dynamical systems [Chandramoorthy and Wang, 2022, Ni, 2020] and score-based modeling of image distributions [Song et al., 2020b,a]. In these settings, transport or "flow"-driven algorithms for generating samples have seen extensive success. The central idea is to construct a transport map from a simple, prescribed source distribution to the target distribution of interest. One class of transport approaches, e.g., as represented by variational inference with normalizing flows, involves constructing a parametric class of invertible maps and minimizing some statistical divergence between the pushforward (see Section 2) of the source by a member of this class and the target. A different, essentially nonparametric, class of transport approaches are based on particle systems, e.g., Stein variational gradient descent (SVGD)

Fast approximations of power flow results are beneficial in power system planning and live operation. In planning, millions of power flow calculations are necessary if multiple years, different control strategies or contingency policies are to be considered. In live operation, grid operators must assess if grid states comply with contingency requirements in a short time. In this paper, we compare regression and classification methods to either predict multi-variable results, e.g. bus voltage magnitudes and line loadings, or binary classifications of time steps to identify critical loading situations. We test the methods on three realistic power systems based on time series in 15 min and 5 min resolution of one year. We compare different machine learning models, such as multilayer perceptrons (MLPs), decision trees, k-nearest neighbours, gradient boosting, and evaluate the required training time and prediction times as well as the prediction errors. We additionally determine the amount of training data needed for each method and show results, including the approximation of untrained curtailment of generation. Regarding the compared methods, we identified the MLPs as most suitable for the task. The MLP-based models can predict critical situations with an accuracy of 97-98 % and a very low number of false negative predictions of 0.0-0.64 %.

We introduce a new algorithm for the numerical computation of Nash equilibria of competitive two-player games. Our method is a natural generalization of gradient descent to the two-player setting where the update is given by the Nash equilibrium of a regularized bilinear local approximation of the underlying game. It avoids oscillatory and divergent behaviors seen in alternating gradient descent. Using numerical experiments and rigorous analysis, we provide a detailed comparison to methods based on \emph{optimism} and \emph{consensus} and show that our method avoids making any unnecessary changes to the gradient dynamics while achieving exponential (local) convergence for (locally) convex-concave zero sum games. Convergence and stability properties of our method are robust to strong interactions between the players, without adapting the stepsize, which is not the case with previous methods.