Watch your neighbors: Training statistically accurate chaotic systems with local phase space information

Chaotic systems pose fundamental challenges for data-driven surrogate modeling, as small modeling errors lead to exponentially growing trajectory discrepancies. Since exact long-term prediction is unattainable, it is natural to ask what a good surrogate model for chaotic dynamics is. Prior work has largely focused either on reproducing the Jacobian of the underlying dynamics, which governs local expansion and contraction rates, or on reproducing the ground-truth dynamics' long-term statistical behavior. We propose a framework that bridges these two paradigms by training surrogate models with accurate Jacobians and long-term statistical properties. Our method constructs a local covering of a chaotic attractor in phase space and analyzes the expansion and contraction of these coverings under the dynamics. The surrogate model is trained by minimizing the maximum mean discrepancy between the pushforward distributions of the coverings under the surrogate and ground-truth dynamics. Experiments show that our method significantly improves Jacobian accuracy while remaining competitive with state-of-the-art statistically accurate dynamics learning methods, at a computational cost that is a constant factor independent of the system dimension. Our code is fully available at https://anonymous.4open.science/r/neighborwatch.