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.

翻译：混沌系统使得长期预测变得困难，因为初始条件的微小扰动会导致轨迹以指数速率发散。在这种情况下，经过训练以最小化平方误差损失的神经算子虽然能够进行准确的短期预测，但通常无法在更长的时间范围内再现动力学的统计或结构特性，并可能产生退化的结果。本文提出了一种替代框架，旨在保持混沌吸引子的不变测度，该测度刻画了动力学的时间不变统计特性。具体而言，在多环境设置（其中每个样本轨迹受略有不同的动力学支配）下，我们考虑了两种处理含噪数据的新方法。首先，我们提出了一种基于观测动力学与神经算子输出之间最优传输距离的损失函数。这种方法需要专家对底层物理学的深入了解，以确定应将哪些统计特征纳入最优传输损失中。其次，我们表明，一种不需要任何特定先验知识的对比学习框架，可以几乎与最优传输方法一样好地保持动力学的统计特性。在多种混沌系统上，我们的方法被实验证明能够保持混沌吸引子的不变测度。