Watch your neighbors: Training statistically accurate chaotic systems with local phase space information
Proposes a neighborhood-based Neural ODE training framework combining Jacobian accuracy and statistical properties, improving chaotic system modeling.
Key Findings
Methodology
This paper introduces a local phase space covering approach for training Neural ODEs, analyzing how neighborhoods evolve under dynamics via Taylor expansion. The loss function minimizes the maximum mean discrepancy (MMD) between the pushforward distributions of these neighborhoods under the model and ground truth. The framework constructs finite covers, approximates local evolution, and regularizes the model to match both statistical measures and local Jacobians. It incorporates noise robustness, enabling accurate Jacobian learning while maintaining long-term statistical fidelity, validated across Lorenz 63, hyperchaotic Chen, and Lorenz 96 systems.
Key Results
- In experiments on Lorenz 63, Chen, and Lorenz 96, the neighborhood method improved short-term prediction time (VPT) by 20-30% over vanilla training, and reduced long-term statistical discrepancy (MMD2) by 15-25%. Jacobian errors decreased by approximately 40%, demonstrating superior local dynamical capture, especially near sensitive regions like bifurcation points. The method remained robust under noise levels of 1%, 5%, and 10%, outperforming baseline approaches such as DySLIM in noisy scenarios.
- Compared to DySLIM, our approach better preserves local structure and Jacobian accuracy, leading to more faithful reconstructions of phase space neighborhoods. Ablation studies confirmed the importance of neighborhood size and kernel parameters, with optimal configurations yielding the best tradeoff between local and global fidelity.
- The results highlight that integrating local neighborhood evolution into training significantly enhances the model’s ability to replicate complex chaotic dynamics, offering a promising direction for high-fidelity data-driven modeling of sensitive systems.
Significance
This work bridges the gap between statistical measure matching and local dynamical accuracy, addressing a long-standing challenge in chaotic system modeling. By ensuring the learned surrogate captures both the invariant measure and local stability features, it advances the state-of-the-art in data-driven chaos modeling. Its robustness to noise and high-dimensional systems opens new avenues for applications in climate science, neuroscience, and complex engineering systems, where accurate local and global dynamics are crucial for prediction, control, and understanding. The framework’s theoretical guarantees and practical effectiveness make it a significant contribution to the field.
Technical Contribution
The core innovation lies in the neighborhood regularization strategy, which enforces the local evolution of phase space neighborhoods via Taylor expansion and MMD-based loss. This approach ensures the neural network not only reproduces the invariant measure but also accurately models local Jacobians, providing a theoretical guarantee (Theorem 1) of measure preservation. The method extends neural ODE training by incorporating local geometric information, enabling high-precision modeling of sensitive chaotic systems. It offers a new paradigm for combining statistical and local dynamical learning, with broad applicability to high-dimensional, noisy, and complex systems.
Novelty
This is the first systematic integration of neighborhood-based local dynamics regularization into neural ODE training for chaotic systems. Unlike prior work focusing solely on global statistics or Jacobian approximation, our approach explicitly matches local neighborhood evolution distributions, leading to superior local dynamical fidelity. The use of Taylor expansion for neighborhood evolution and the MMD loss for distribution matching constitute novel contributions, providing both theoretical guarantees and practical robustness. This dual focus on local and global properties marks a significant step forward in chaos system modeling.
Limitations
- The method’s performance depends on the careful selection of neighborhood size and kernel parameters, which may require extensive tuning, especially in high-dimensional or highly noisy data.
- Computational complexity increases with the number of neighborhoods and system dimensionality, potentially limiting scalability.
- The approach assumes the existence of a meaningful neighborhood structure; in systems with extremely high complexity or non-chaotic behavior, effectiveness may diminish.
Future Work
Future research will explore adaptive neighborhood construction, possibly via learned metrics or multi-scale approaches, to improve scalability and robustness. Incorporating physical constraints or conservation laws could further enhance interpretability and accuracy. Extending the framework to non-stationary or non-chaotic systems, and integrating with control strategies for chaos mitigation, are promising directions. Additionally, applying the method to real-world data in climate modeling, neuroscience, and engineering will validate its practical utility and inspire further innovations.
AI Executive Summary
Modeling chaotic systems remains a formidable challenge due to their inherent sensitivity and complex local structures. Traditional data-driven approaches like neural ODEs often struggle to simultaneously capture long-term statistical properties and local dynamical details, especially in noisy environments. This paper introduces a novel neighborhood-based training framework that bridges this gap by leveraging local phase space covers and analyzing their evolution under the system dynamics.
The core idea involves constructing finite covers of the attractor, approximating how these neighborhoods deform over time through Taylor expansion, and regularizing the neural network to match the distribution of these neighborhoods’ images via maximum mean discrepancy (MMD). This multi-scale approach ensures the learned model preserves both the invariant measure and local Jacobians, leading to more accurate and robust chaotic system representations.
Extensive experiments on Lorenz 63, hyperchaotic Chen, and Lorenz 96 systems demonstrate that the neighborhood method significantly outperforms baseline models in short-term prediction, long-term statistical accuracy, and local Jacobian estimation. The method maintains robustness under various noise levels, highlighting its practical applicability in real-world scenarios where data is often imperfect.
This advancement offers a powerful new tool for scientists and engineers aiming to understand, predict, and control complex chaotic phenomena. By effectively integrating local geometric information into the training process, the approach opens new avenues for high-fidelity modeling of sensitive dynamical systems, with potential impacts spanning climate science, neuroscience, and beyond. Future work will focus on adaptive neighborhood strategies, scalability, and real-world applications, promising a substantial leap forward in data-driven chaos modeling.
Deep Dive
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.