Chaotic systems make long-horizon forecasts difficult because small
perturbations in initial conditions cause trajectories to diverge at an
exponential rate. In this setting, neural operators trained to minimize squared
error losses, while capable of accurate short-term forecasts, often fail to
reproduce statistical or structural properties of the dynamics over longer time
horizons and can yield degenerate results. In this paper, we propose an
alternative framework designed to preserve invariant measures of chaotic
attractors that characterize the time-invariant statistical properties of the
dynamics. Specifically, in the multi-environment setting (where each sample
trajectory is governed by slightly different dynamics), we consider two novel
approaches to training with noisy data. First, we propose a loss based on the
optimal transport distance between the observed dynamics and the neural
operator outputs. This approach requires expert knowledge of the underlying
physics to determine what statistical features should be included in the
optimal transport loss. Second, we show that a contrastive learning framework,
which does not require any specialized prior knowledge, can preserve
statistical properties of the dynamics nearly as well as the optimal transport
approach. On a variety of chaotic systems, our method is shown empirically to
preserve invariant measures of chaotic attractors.

本文提出了一个针对混沌系统长期预测的框架，该框架旨在保留描述动力学的不变吸引子的不变统计特性。我们考虑两种方法来处理噪声数据的多环境设置中的训练，一种是基于观察到的动力学和神经操作器输出之间的最优输运距离的损失，另一种是不需要任何专业先前知识的对比学习框架。通过在各种混沌系统上进行实证验证，我们的方法在保持混沌吸引子的不变测度方面表现出很好的效果。

训练神经算子以保持混沌吸引子的不变量测度

Training neural operators to preserve invariant measures of chaotic  attractors

We discuss how to generate random unitary matrices from the classical compact
groups U(N), O(N) and USp(N) with probability distributions given by the
respective invariant measures. The algorithm is straightforward to implement
using standard linear algebra packages. This approach extends to the Dyson
circular ensembles too. This article is based on a lecture given by the author
at the summer school on Number Theory and Random Matrix Theory held at the
University of Rochester in June 2006. The exposition is addressed to a general
mathematical audience.

简述：讨论如何使用相应的不变测度概率分布从经典紧致群 U（N）、O（N）和 USp（N）中生成随机的幺正矩阵，该算法使用标准线性代数包实现，且可扩展到 Dyson 环集。