Research map / Probabilistic and scientific modeling / Dynamical systems
DySLIM: Dynamics Stable Learning by Invariant Measure for Chaotic Systems
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Proposes DySLIM, a regularization term that matches the invariant measure (via MMD) of a learned chaotic-system surrogate to the true attractor measure, added to standard 1-step, curriculum, or pushforward losses.
Problem
Autoregressive surrogate models of chaotic dissipative systems become unstable over long rollouts: they blow up or converge to wrong attractors and yield inaccurate long-term statistics, even when short-term trajectory matching is good.
Approach
Treat learning as a constrained problem that preserves the system's invariant measure, relax it to a regularized loss, and estimate the measure mismatch with a sample-based MMD (rational quadratic kernel mixture) between true and rolled-out states, in both unconditional and conditional forms, added to any existing multi-step objective.
Key results
- On Kolmogorov flow (batch 128, LR 1e-4), DySLIM reduced Pfwd TCM from 3.68e-2 to 0.72e-2 and Wass1 from 4.64e-2 to 4.51e-2.
- On Kolmogorov flow, Curr objective MELR fell from 5.35e-2 to 1.64e-2 with DySLIM (lambda2=100).
- DySLIM gives improved short-term cosine similarity and longer decorrelation times on Lorenz 63 and KS, and keeps performance stable at large batch sizes and learning rates where baselines degrade.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add DySLIM regularization to 1-step (ablation) | 1-step model · Kolmogorov Flow | MELR: 2.77 -> 1.84 | Helped |
| Add DySLIM regularization to 1-step (ablation) | 1-step model · Kolmogorov Flow | MELRw: 0.44 -> 0.85 | Hurt |
| Add DySLIM regularization to 1-step (ablation) | 1-step model · Kolmogorov Flow | covRMSE: 7.93 -> 7.30 | Helped |
| Add DySLIM regularization to 1-step (ablation) | 1-step model · Kolmogorov Flow | Wass1: 16.2 -> 5.55 | Helped |
| Add DySLIM regularization to 1-step (ablation) | 1-step model · Kolmogorov Flow | TCM: 5.39 -> 2.45 | Helped |
| Add DySLIM regularization to curriculum training (ablation) | Curr model · Kolmogorov Flow | MELR: 5.35 -> 1.64 | Helped |
| Add DySLIM regularization to curriculum training (ablation) | Curr model · Kolmogorov Flow | MELRw: 0.95 -> 0.45 | Helped |
| Add DySLIM regularization to curriculum training (ablation) | Curr model · Kolmogorov Flow | covRMSE: 8.13 -> 6.95 | Helped |
| Add DySLIM regularization to curriculum training (ablation) | Curr model · Kolmogorov Flow | Wass1: 9.66 -> 4.76 | Helped |
| Add DySLIM regularization to curriculum training (ablation) | Curr model · Kolmogorov Flow | TCM: 3.50 -> 2.83 | Helped |
| Add DySLIM regularization to pushforward training (ablation) | Pfwd model · Kolmogorov Flow | MELR: 3.19 -> 2.46 | Helped |
| Add DySLIM regularization to pushforward training (ablation) | Pfwd model · Kolmogorov Flow | MELRw: 0.53 -> 0.53 | No clear effect |
Surprising findings
- Sinkhorn divergence, although competitive on small systems, performed worse than MMD as system size grew and batch size shrank.
- The 1-step objective was too unstable on the KS system even with regularization, so only Curr and Pfwd were evaluated there.
Limitations
- Assumes the learned model has an attractor and invariant measure, which is a key hypothesis that is not guaranteed.
- Unconditional regularization becomes uninformative as state dimension grows, and Sinkhorn divergence is less informative in high dimensions.
- Relies on MMD bandwidth tuning; squared-exponential kernel is sensitive to bandwidth.
Experimental setup
Datasets: Lorenz 63 simulated trajectories Kuramoto-Sivashinsky simulated trajectories Kolmogorov flow simulated trajectories
Benchmarks: Lorenz 63 Kuramoto-Sivashinsky Kolmogorov flow
Metrics: cosine similarity Sinkhorn divergence MELR MELRw covRMSE Wasserstein-1 time correlation metric
Baselines: 1-step objective Curriculum training Pushforward trick Sinkhorn-divergence regularization
Builds on: pushforward trick curriculum training maximum mean discrepancy Sinkhorn divergence
Closest papers by meaning
- Chaos Meets Attention: Transformers for Large-Scale Dynamical Prediction (ICML 2025 · similarity 0.88)
- Learning invariant representations of time-homogeneous stochastic dynamical systems (ICLR 2024 · similarity 0.87)
- The Dynamic-Probabilistic Consistency Gap in Chaotic Surrogate Modeling (NeurIPS 2026 · similarity 0.87)
- When are dynamical systems learned from time series data statistically accurate? (NeurIPS 2024 · similarity 0.87)
- Discovering Physics Laws of Dynamical Systems via Invariant Function Learning (ICML 2025 · similarity 0.86)
- Generalization at the Edge of Stability (NeurIPS 2026 · similarity 0.85)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.