This paper develops an Operator Learning framework for approximating the dynamic response of synchronous generators. The framework can be used to (i) build a neural network-based generator model that interacts with a power grid simulator or (ii) shadow the true generator's transient response. First, we develop a data-driven Deep Operator Network (DeepONet) to approximate the infinite-dimensional solution operator of the generators. Then, we design a numerical scheme based on DeepONet that simulates the generator's response over a given time horizon. The proposed scheme recursively employs the trained DeepONet to simulate the response for a given multi-dimensional input that describes the interaction between the generator and the power grid. In addition, we design a residual DeepONet numerical scheme that can incorporate information from existing mathematical models. We accompany this residual DeepONet scheme with an estimate for the prediction's cumulative error. Finally, we build a data aggregation (DAgger) strategy that allows fine-tuning of DeepONets using aggregated training data that the DeepONets will likely encounter during interactive simulations with other grid components. As a proof of concept, we demonstrate that the proposed frameworks can effectively approximate the transient model of a synchronous generator.

We explore the geometric aspects of the Furstenberg conjecture, proving that a non-atomic probability measure on the unit circle, invariant under both $p$- and $q$-actions for coprime integers $p,q>1$, must be the Lebesgue measure if it exhibits balanced geometry for one of these actions. Within rigidity theory, we show that balanced geometry is equivalent to the Lipschitz property. A consequence is that an orientation-preserving homeomorphism of the circle conjugating both $p$- and $q$-actions and preserving the Lebesgue measure must be the identity if one of these conjugations satisfies the Lipschitz property. Our approach does not rely solely on ergodicity, and we conclude by proposing conjectures and open problems that frame the Furstenberg conjecture through geometric and quasisymmetric perspectives.

We study mixing times of the one-sided $k$-transposition shuffle. We prove that this shuffle mixes relatively slowly, even for $k$ big. Using the recent ``lifting eigenvectors'' technique of Dieker and Saliola and applying the $\ell^2$ bound, we prove different mixing behaviors and explore the occurrence of cutoff depending on $k$.