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Chaotic systems make long-horizon forecasts difficult because smallperturbations in initial conditions cause trajectories to diverge at anexponential rate. In this setting, neural operators trained to minimize squarederror losses, while capable of accurate short-term forecasts, often fail toreproduce statistical or structural properties of the dynamics over longer timehorizons and can yield degenerate results. In this paper, we propose analternative framework designed to preserve invariant measures of chaoticattractors that characterize the time-invariant statistical properties of thedynamics. Specifically, in the multi-environment setting (where each sampletrajectory is governed by slightly different dynamics), we consider two novelapproaches to training with noisy data. First, we propose a loss based on theoptimal transport distance between the observed dynamics and the neuraloperator outputs. This approach requires expert knowledge of the underlyingphysics to determine what statistical features should be included in theoptimal transport loss. Second, we show that a contrastive learning framework,which does not require any specialized prior knowledge, can preservestatistical properties of the dynamics nearly as well as the optimal transportapproach. On a variety of chaotic systems, our method is shown empirically topreserve invariant measures of chaotic attractors.

Training neural operators to preserve invariant measures of chaotic attractors