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

2026-05-14

AI Summary

Chaotic systems are highly sensitive to modeling errors, making it challenging to simultaneously achieve accuracy in local dynamics and long-term statistical fidelity. To address this, this work proposes a novel approach that constructs a local cover of the chaotic attractor in phase space and jointly optimizes both the Jacobian accuracy of the surrogate model and its long-term statistical behavior. The method uniquely integrates local pushforward distribution matching with Jacobian fidelity, thereby unifying local and global modeling paradigms. Training employs a loss function based on distribution matching, utilizing the Maximum Mean Discrepancy (MMD) metric. Experimental results demonstrate that the proposed method substantially improves Jacobian accuracy while achieving state-of-the-art performance in long-term statistical properties.

Abstract

Chaotic systems pose fundamental challenges for data-driven dynamics discovery, 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 training surrogate models that reproduce the ground-truth dynamics' long-term statistical behavior. In this work, we propose a new framework that aims to bridge these two paradigms by training surrogate dynamics 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. Our code is fully available at https://anonymous.4open.science/r/neighborwatch.

Problem

• chaotic systems
• dynamics discovery
• Jacobian accuracy
• statistical behavior
• surrogate modeling

Innovation

• chaotic systems
• Jacobian accuracy
• statistical dynamics
• maximum mean discrepancy
• phase space covering