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

arXiv: https://arxiv.org/abs/2605.14405 · HTML: https://mlsanity.com/paper/2605.14405/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 56 references that are in the map.
- First posted on arXiv 14 May 2026.

## Builds on (most cited first, 16 of 16)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [AdaBelief Optimizer: Adapting Stepsizes by the Belief in Observed Gradients](https://mlsanity.com/paper/2010.07468/) (2020)
- [Stiff Neural Ordinary Differential Equations](https://mlsanity.com/paper/2103.15341/) (2021)
- [Generalized Teacher Forcing for Learning Chaotic Dynamics](https://mlsanity.com/paper/2306.04406/) (2023)
- [Training neural operators to preserve invariant measures of chaotic attractors](https://mlsanity.com/paper/2306.01187/) (2023)
- [DySLIM: Dynamics Stable Learning by Invariant Measure for Chaotic Systems](https://mlsanity.com/paper/2402.04467/) (2024)
- [A Review of Data-Driven Discovery for Dynamic Systems](https://mlsanity.com/paper/2210.10663/) (2022)
- [When are dynamical systems learned from time series data statistically accurate?](https://mlsanity.com/paper/2411.06311/) (2024)
- [Data-driven Estimation of the Power Flow Jacobian Matrix in High Dimensional Space](https://mlsanity.com/paper/1902.06211/) (2019)
- [Estimating the Jacobian matrix of an unknown multivariate function from sample values by means of a neural network](https://mlsanity.com/paper/2204.00523/) (2022)
- [Characterizing control between interacting subsystems with deep Jacobian estimation](https://mlsanity.com/paper/2507.01946/) (2025)

## Abstract (arXiv)

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.