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LILAD: Learning In-context Lyapunov-stable Adaptive Dynamics Models

Amit Jena, Na Li, Le Xie

TL;DR

LILAD addresses the challenge of learning dynamical models that are both adaptive to nonstationary environments and guaranteed to be stable. It jointly learns an adaptive dynamics model and a Lyapunov certificate through in-context learning, using a multi-task trajectory pool and a state-dependent attenuator gamma to enforce stability at test time. The framework employs adversarial training to couple dynamics and Lyapunov learning, and introduces an output-warping mechanism to ensure positive semi-definiteness of the Lyapunov function. Experiments across diverse autonomous systems and a high-dimensional PDE demonstrate that LILAD outperforms baselines in predictive accuracy while providing stability guarantees, highlighting its practical relevance for safety-critical, nonstationary settings.

Abstract

System identification in control theory aims to approximate dynamical systems from trajectory data. While neural networks have demonstrated strong predictive accuracy, they often fail to preserve critical physical properties such as stability and typically assume stationary dynamics, limiting their applicability under distribution shifts. Existing approaches generally address either stability or adaptability in isolation, lacking a unified framework that ensures both. We propose LILAD (Learning In-Context Lyapunov-stable Adaptive Dynamics), a novel framework for system identification that jointly guarantees adaptability and stability. LILAD simultaneously learns a dynamics model and a Lyapunov function through in-context learning (ICL), explicitly accounting for parametric uncertainty. Trained across a diverse set of tasks, LILAD produces a stability-aware, adaptive dynamics model alongside an adaptive Lyapunov certificate. At test time, both components adapt to a new system instance using a short trajectory prompt, which enables fast generalization. To rigorously ensure stability, LILAD also computes a state-dependent attenuator that enforces a sufficient decrease condition on the Lyapunov function for any state in the new system instance. This mechanism extends stability guarantees even under out-of-distribution and out-of-task scenarios. We evaluate LILAD on benchmark autonomous systems and demonstrate that it outperforms adaptive, robust, and non-adaptive baselines in predictive accuracy.

LILAD: Learning In-context Lyapunov-stable Adaptive Dynamics Models

TL;DR

LILAD addresses the challenge of learning dynamical models that are both adaptive to nonstationary environments and guaranteed to be stable. It jointly learns an adaptive dynamics model and a Lyapunov certificate through in-context learning, using a multi-task trajectory pool and a state-dependent attenuator gamma to enforce stability at test time. The framework employs adversarial training to couple dynamics and Lyapunov learning, and introduces an output-warping mechanism to ensure positive semi-definiteness of the Lyapunov function. Experiments across diverse autonomous systems and a high-dimensional PDE demonstrate that LILAD outperforms baselines in predictive accuracy while providing stability guarantees, highlighting its practical relevance for safety-critical, nonstationary settings.

Abstract

System identification in control theory aims to approximate dynamical systems from trajectory data. While neural networks have demonstrated strong predictive accuracy, they often fail to preserve critical physical properties such as stability and typically assume stationary dynamics, limiting their applicability under distribution shifts. Existing approaches generally address either stability or adaptability in isolation, lacking a unified framework that ensures both. We propose LILAD (Learning In-Context Lyapunov-stable Adaptive Dynamics), a novel framework for system identification that jointly guarantees adaptability and stability. LILAD simultaneously learns a dynamics model and a Lyapunov function through in-context learning (ICL), explicitly accounting for parametric uncertainty. Trained across a diverse set of tasks, LILAD produces a stability-aware, adaptive dynamics model alongside an adaptive Lyapunov certificate. At test time, both components adapt to a new system instance using a short trajectory prompt, which enables fast generalization. To rigorously ensure stability, LILAD also computes a state-dependent attenuator that enforces a sufficient decrease condition on the Lyapunov function for any state in the new system instance. This mechanism extends stability guarantees even under out-of-distribution and out-of-task scenarios. We evaluate LILAD on benchmark autonomous systems and demonstrate that it outperforms adaptive, robust, and non-adaptive baselines in predictive accuracy.

Paper Structure

This paper contains 26 sections, 1 theorem, 27 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Stability of Discrete-Time Dynamical Systems
  5. In-context Learning
  6. ICL of Adaptive and Stable Dynamics
  7. LILAD Framework Set-up
  8. Enforcing Semi-definiteness via Output Warping
  9. LILAD Training
  10. LILAD Evaluation and Stability Enforcement
  11. Experiments
  12. Baselines
  13. Simulations
  14. Scalability of LILAD to High-dimensional Systems
  15. Conclusion
  16. ...and 11 more sections

Key Result

Proposition 1

Let $G_{\theta^{*}}$ and $V_{\phi^{*}}$ be the optimal adaptive dynamics and Lyapunov models obtained from LILAD training, respectively. Let $\mathscr{C}_j^{M+1}$ be a test-time prompt-prefix containing $j$ state-next-state pairs corresponding to a new test-time system instance $x_{k+1} = f_{\tilde{

Figures (2)

  • Figure 1: Performance Comparison of LILAD. For the Simple Pendulum system with stochastic parameters $(g, l, b)$, the proposed LILAD method adapts more effectively to test-time instances than ICL. (a) ICL yields a sub-optimal approximation of the test-time vector field, causing predicted trajectories to deviate beyond the specified boundary. (b) In contrast, LILAD achieves a more accurate approximation, with guaranteed convergence of the predicted trajectories to the equilibrium. (c) Ground truth: the true vector field and corresponding reference trajectories of the test-time system.
  • Figure 2: LILAD scales effectively to a high-dimensional system derived from a discretized reaction-diffusion PDE. Its learned surrogate predicts trajectories that converge to the origin, matching the test-time ground truth. In contrast, ICL does not guarantee convergence to the origin. Each image shows the spatial temperature profile under a fixed test-time diffusion coefficient.

Theorems & Definitions (4)

  • Definition 1: Global exponential Stability (GES) grujic1974exponential
  • Definition 2: Lyapunov functions for discrete-time systems grujic1974exponential
  • Proposition 1
  • proof