# DySLIM: Dynamics Stable Learning by Invariant Measure for Chaotic Systems

arXiv: https://arxiv.org/abs/2402.04467 · HTML: https://mlsanity.com/paper/2402.04467/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- Builds on 51 of its 138 references that are in the map.
- First posted on arXiv 6 Feb 2024.

## Builds on (most cited first, 30 of 51)

- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Demystifying MMD GANs](https://mlsanity.com/paper/1801.01401/) (2018)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Training Deep Nets with Sublinear Memory Cost](https://mlsanity.com/paper/1604.06174/) (2016)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [GraphCast: Learning skillful medium-range global weather forecasting](https://mlsanity.com/paper/2212.12794/) (2022)
- [Generative Moment Matching Networks](https://mlsanity.com/paper/1502.02761/) (2015)
- [MMD GAN: Towards Deeper Understanding of Moment Matching Network](https://mlsanity.com/paper/1705.08584/) (2017)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023)
- [Professor Forcing: A New Algorithm for Training Recurrent Networks](https://mlsanity.com/paper/1610.09038/) (2016)
- [Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers](https://mlsanity.com/paper/2007.00016/) (2020)
- [Forecasting Global Weather with Graph Neural Networks](https://mlsanity.com/paper/2202.07575/) (2022)
- [On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests](https://mlsanity.com/paper/1509.02237/) (2015)
- [Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks](https://mlsanity.com/paper/1802.07486/) (2018)
- [Training generative neural networks via Maximum Mean Discrepancy optimization](https://mlsanity.com/paper/1505.03906/) (2015)
- [Backpropagation Algorithms and Reservoir Computing in Recurrent Neural Networks for the Forecasting of Complex Spatiotemporal Dynamics](https://mlsanity.com/paper/1910.05266/) (2019)
- [Learning Deep Kernels for Non-Parametric Two-Sample Tests](https://mlsanity.com/paper/2002.09116/) (2020)
- [Optimal Transport Tools (OTT): A JAX Toolbox for all things Wasserstein](https://mlsanity.com/paper/2201.12324/) (2022)
- [Learned Coarse Models for Efficient Turbulence Simulation](https://mlsanity.com/paper/2112.15275/) (2021)

## Built on by (most cited first, 30 of 30)

- [Thermalizer: Stable autoregressive neural emulation of spatiotemporal chaos](https://mlsanity.com/paper/2503.18731/) (2025)
- [True Zero-Shot Inference of Dynamical Systems Preserving Long-Term Statistics](https://mlsanity.com/paper/2505.13192/) (2025)
- [Learning Interpretable Hierarchical Dynamical Systems Models from Time Series Data](https://mlsanity.com/paper/2410.04814/) (2024)
- [When are dynamical systems learned from time series data statistically accurate?](https://mlsanity.com/paper/2411.06311/) (2024)
- [Binned Spectral Power Loss for Improved Prediction of Chaotic Systems](https://mlsanity.com/paper/2502.00472/) (2025)
- [Generative emulation of chaotic dynamics with coherent prior](https://mlsanity.com/paper/2504.14264/) (2025)
- [Quantum-Informed Machine Learning for Predicting Spatiotemporal Chaos with Practical Quantum Advantage](https://mlsanity.com/paper/2507.19861/) (2025)
- [A probabilistic framework for learning non-intrusive corrections to long-time climate simulations from short-time training data](https://mlsanity.com/paper/2408.02688/) (2024)
- [Invariant Measures in Time-Delay Coordinates for Unique Dynamical System Identification](https://mlsanity.com/paper/2412.00589/) (2024)
- [Position: A Dynamical Systems Perspective is Needed to Advance Time Series Modeling](https://mlsanity.com/paper/2602.16864/) (2026)
- [Embed and Emulate: Contrastive representations for simulation-based inference](https://mlsanity.com/paper/2409.18402/) (2024)
- [Improved deep learning of chaotic dynamical systems with multistep penalty losses](https://mlsanity.com/paper/2410.05572/) (2024)
- [Adaptive Diffusion Posterior Sampling for Data and Model Fusion of Complex Nonlinear Dynamical Systems](https://mlsanity.com/paper/2603.12635/) (2026)
- [Calibrating Geophysical Predictions under Constrained Probabilistic Distributions](https://mlsanity.com/paper/2512.03081/) (2025)
- [Memory-Conditioned Flow-Matching for Stable Autoregressive PDE Rollouts](https://mlsanity.com/paper/2602.06689/) (2026)
- [Teacher Forcing as Generalized Bayes: Optimization Geometry Mismatch in Switching Surrogates for Chaotic Dynamics](https://mlsanity.com/paper/2604.25904/) (2026)
- [Divide And Conquer: Learning Chaotic Dynamical Systems With Multistep Penalty Neural Ordinary Differential Equations](https://mlsanity.com/paper/2407.00568/) (2024)
- [Invariant Measures for Data-Driven Dynamical System Identification: Analysis and Application](https://mlsanity.com/paper/2502.05204/) (2025)
- [ChaosNexus: A Foundation Model for ODE-based Chaotic System Forecasting with Hierarchical Multi-scale Awareness](https://mlsanity.com/paper/2509.21802/) (2025)
- [Towards Infinitely Long Neural Simulations: Self-Refining Neural Surrogate Models for Dynamical Systems](https://mlsanity.com/paper/2603.17750/) (2026)
- [The Dynamic-Probabilistic Consistency Gap in Chaotic Surrogate Modeling](https://mlsanity.com/paper/2605.31547/) (2026)
- [Ellipsoidal Time Series Forecasting](https://mlsanity.com/paper/2505.17370/) (2025)
- [PhyxMamba: Chaotic System Reconstruction from Short Context Observations with Generative State-Space Models](https://mlsanity.com/paper/2505.23863/) (2025)
- [Watch your neighbors: Training statistically accurate chaotic systems with local phase space information](https://mlsanity.com/paper/2605.14405/) (2026)
- [On Distributional Reinforcement Learning in Chaotic Dynamical Systems](https://mlsanity.com/paper/2605.30160/) (2026)
- [AugRelNet: Relation-Augmented Dynamics for Structure Discovery and Forecasting from Limited Data](https://mlsanity.com/paper/2606.11251/) (2026)
- [Quantum Statistical Memory Advantage Reveals Predictive Structure in Chaotic Invariant Measures](https://mlsanity.com/paper/2606.13422/) (2026)
- [Distillation of Foundation Models for Time-dependent PDEs](https://mlsanity.com/paper/2608.11937/) (2026)
- [Staying on the Attractor: Supervising Neural Surrogates of 3D Turbulence Where They Leave It](https://mlsanity.com/paper/2609.32864/) (2026)
- [Do Neural PDE Solvers Learn the Right Dynamics?](https://mlsanity.com/paper/2610.06952/) (2026)

## Abstract (arXiv)

Learning dynamics from dissipative chaotic systems is notoriously difficult due to their inherent instability, as formalized by their positive Lyapunov exponents, which exponentially amplify errors in the learned dynamics. However, many of these systems exhibit ergodicity and an attractor: a compact and highly complex manifold, to which trajectories converge in finite-time, that supports an invariant measure, i.e., a probability distribution that is invariant under the action of the dynamics, which dictates the long-term statistical behavior of the system. In this work, we leverage this structure to propose a new framework that targets learning the invariant measure as well as the dynamics, in contrast with typical methods that only target the misfit between trajectories, which often leads to divergence as the trajectories' length increases. We use our framework to propose a tractable and sample efficient objective that can be used with any existing learning objectives. Our Dynamics Stable Learning by Invariant Measure (DySLIM) objective enables model training that achieves better point-wise tracking and long-term statistical accuracy relative to other learning objectives. By targeting the distribution with a scalable regularization term, we hope that this approach can be extended to more complex systems exhibiting slowly-variant distributions, such as weather and climate models.