Diffusion models generate samples by denoising along the score of a perturbed target distribution. In practice, one trains a neural diffusion model, which is computationally expensive. Recent work suggests that score matching implicitly smooths the empirical score, and that this smoothing bias promotes generalization by capturing low-dimensional data geometry. We propose moment-matched score-smoothed overdamped Langevin dynamics (MM-SOLD), a training-free interacting particle sampler that enforces the target moments throughout the sampling trajectory. We prove that, in the large-particle limit, the empirical particle density converges to a deterministic limit whose one-particle stationary marginal is a Gibbs--Boltzmann density obtained by exponentially tilting a naive score-smoothed diffusion target. The mean and covariance of this distribution agree with the empirical moments of the training data. Experiments on 2D distributions and latent-space image generation show that MM-SOLD enables fast, robust, training-free sampling on CPUs, with sample fidelity and diversity competitive with neural diffusion baselines.

B.5 Langevin dynamics Under mild assumptions on r log p

We present simple differentially private estimators for the mean and covariance of multivariate sub-Gaussian data that are accurate at small sample sizes.

Ensuring the safety of neural networks under input uncertainty is a fundamental challenge in safety-critical applications. This paper builds on and expands Fazlyab's quadratic-constraint (QC) and semidefinite-programming (SDP) framework for neural network verification to a distributionally robust and tail-risk-aware setting by integrating worst-case Conditional Value-at-Risk (WC-CVaR) over a moment-based ambiguity set with fixed mean and covariance. The resulting conditions remain SDP-checkable and explicitly account for tail risk. This integration broadens input-uncertainty geometry-covering ellipsoids, polytopes, and hyperplanes-and extends applicability to safety-critical domains where tail-event severity matters. Applications to closed-loop reachability of control systems and classification are demonstrated through numerical experiments, illustrating how the risk level $\varepsilon$ trades conservatism for tolerance to tail events-while preserving the computational structure of prior QC/SDP methods for neural network verification and robustness analysis.

This paper introduces Uncertainty Propagation Network (UPN), a novel family of neural differential equations that naturally incorporate uncertainty quantification into continuous-time modeling. Unlike existing neural ODEs that predict only state trajectories, UPN simultaneously model both state evolution and its associated uncertainty by parameterizing coupled differential equations for mean and covariance dynamics. The architecture efficiently propagates uncertainty through nonlinear dynamics without discretization artifacts by solving coupled ODEs for state and covariance evolution while enabling state-dependent, learnable process noise. The continuous-depth formulation adapts its evaluation strategy to each input's complexity, provides principled uncertainty quantification, and handles irregularly-sampled observations naturally. Experimental results demonstrate UPN's effectiveness across multiple domains: continuous normalizing flows (CNFs) with uncertainty quantification, time-series forecasting with well-calibrated confidence intervals, and robust trajectory prediction in both stable and chaotic dynamical systems.

Legged robots possess a unique ability to traverse rough terrains and navigate cluttered environments, making them well-suited for complex, real-world unstructured scenarios. However, such robots have not yet achieved the same level as seen in natural systems. Recently, sampling-based predictive controllers have demonstrated particularly promising results. This paper investigates a sampling-based model predictive strategy combining model predictive path integral (MPPI) with cross-entropy (CE) and covariance matrix adaptation (CMA) methods to generate real-time whole-body motions for legged robots across multiple scenarios. The results show that combining the benefits of MPPI, CE and CMA, namely using model predictive optimized path integral (MPOPI), demonstrates greater sample efficiency, enabling robots to attain superior locomotion results using fewer samples when compared to typical MPPI algorithms. Extensive simulation experiments in multiple scenarios on a quadruped robot show that MPOPI can be used as an anytime control strategy, increasing locomotion capabilities at each iteration.