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.

Mean-field Langevin dynamics (MFLD) minimizes an entropy-regularized nonlinear convex functional defined over the space of probability distributions. MFLD has gained attention due to its connection with noisy gradient descent for mean-field two-layer neural networks. Unlike standard Langevin dynamics, the nonlinearity of the objective functional induces particle interactions, necessitating multiple particles to approximate the dynamics in a finite-particle setting. Recent works (Chen et al., 2022; Suzuki et al., 2023b) have demonstrated the uniform-in-time propagation of chaos for MFLD, showing that the gap between the particle system and its mean-field limit uniformly shrinks over time as the number of particles increases. In this work, we improve the dependence on logarithmic Sobolev inequality (LSI) constants in their particle approximation errors, which can exponentially deteriorate with the regularization coefficient. Specifically, we establish an LSI-constant-free particle approximation error concerning the objective gap by leveraging the problem structure in risk minimization. As the application, we demonstrate improved convergence of MFLD, sampling guarantee for the mean-field stationary distribution, and uniform-in-time Wasserstein propagation of chaos in terms of particle complexity.

Recent works (Chen et al., 2022; Suzuki et al., 2023b) have demonstrated In this work, we improve the dependence on logarithmic Sobolev inequality (LSI) constants in their particle approximation errors which can exponentially deteriorate with the regularization coefficient. One may consider adding Gaussian noise to the gradient descent to make the method more stable.

Recurrent Neural Networks (RNNs) frequently exhibit complicated dynamics, and their sensitivity to the initialization process often renders them notoriously hardtotrain.

Then,fk is Li-Yorkechaoticwith constant (albeitsmall) probability. There suchthatk (whose minimumvalue ),fk is 3-periodic 1 .

"Gated Orthogonal Recurrent Units: On Learningto Forget".

Theirdifference,albeitobvious,shouldalsobeemphasized: GD isdeterministic, and the same constant initial condition will always lead to the same iterates.