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

TL;DR

This paper introduces a new method for training surrogate models of chaotic systems that accurately captures both local Jacobian behavior and long-term statistical properties by analyzing phase space coverings.

Contribution

It bridges local Jacobian reproduction and statistical accuracy in surrogate models through a phase space covering approach and MMD-based training.

Findings

Improved Jacobian accuracy over existing methods.

Competitive statistical property reproduction.

Effective phase space covering analysis enhances model fidelity.

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.