Chaos arises in many complex dynamical systems, from weather to power grids, but is difficult to accurately model using data-driven emulators, including neural operator architectures. For chaotic systems, the inherent sensitivity to initial conditions makes exact long-term forecasts theoretically infeasible, meaning that traditional squared-error losses often fail when trained on noisy data. Recent work has focused on training emulators to match the statistical properties of chaotic attractors by introducing regularization based on handcrafted local features and summary statistics, as well as learned statistics extracted from a diverse dataset of trajectories. In this work, we propose a family of adversarial optimal transport objectives that jointly learn high-quality summary statistics and a physically consistent emulator. We theoretically analyze and experimentally validate a Sinkhorn divergence formulation (2-Wasserstein) and a WGAN-style dual formulation (1-Wasserstein). Our experiments across a variety of chaotic systems, including systems with high-dimensional chaotic attractors, show that emulators trained with our approach exhibit significantly improved long-term statistical fidelity.

We study the problem of reconstructing the latent geometry of a $d$-dimensional Riemannian manifold from a random geometric graph. While recent works have made significant progress in manifold recovery from random geometric graphs, and more generally from noisy distances, the precision of pairwise distance estimation has been fundamentally constrained by the volumetric barrier, namely the natural sample-spacing scale $n^{-1/d}$ coming from the fact that a generic point of the manifold typically lies at distance of order $n^{-1/d}$ from the nearest sampled point. In this paper, we introduce a novel approach, Orthogonal Ring Distance Estimation Routine (ORDER), which achieves a pointwise distance estimation precision of order $n^{-2/(d+5)}$ up to polylogarithmic factors in $n$ in polynomial time. This strictly beats the volumetric barrier for dimensions $d > 5$. As a consequence of obtaining pointwise precision better than $n^{-1/d}$, we prove that the Gromov--Wasserstein distance between the reconstructed metric measure space and the true latent manifold is of order $n^{-1/d}$. This matches the Wasserstein convergence rate of empirical measures, demonstrating that our reconstructed graph metric is asymptotically as good as having access to the full pairwise distance matrix of the sampled points. Our results are proven in a very general setting which includes general models of noisy pairwise distances, sparse random geometric graphs, and unknown connection probability functions.

Despite its success in practice, theoretical analyses of Adam are still limited.

Werecallthisresult from [4] (see Theorem 7 thereinformoredetails).

The action is to choose a set of arms, whereupon feedback for each arm in the chosen set is received.