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Meta-Learning-Based Adaptive Stability Certificates for Dynamical Systems
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Integrates MAML meta-learning with neural Lyapunov functions so a meta-trained initialization adapts in a few gradient steps to test-time parametric shifts, yielding larger regions of attraction than non-adaptive baselines on control benchmarks.
Problem
Neural Lyapunov function stability certificates become conservative or invalid when system parameters shift over time, and retraining from scratch needs too much data and too many gradient steps for real-time use.
Approach
Meta-train a neural Lyapunov function initialization across sampled parameter instances with MAML, then adapt it with a few gradient steps on a small dataset from the test-time system; uses Lipschitz-based tightened Lyapunov conditions on a grid instead of an SMT solver for verification.
Key results
- On the inverted pendulum with stochastic (l, b), task-adapted meta-NLF attains ROA 28.04 vs 13.72 for T-NLF and 12.90 for QLF(TS).
- On the 5-microgrid system with all droop constants stochastic, task-adapted NLF reaches 44.35 ROA versus 17.10 for QLF(TS) and 0 for T-NLF; SOS-LF(TS) fails to produce ROAs.
- NLF(TS), which trains on the test-time system directly, generally yields the largest ROA (e.g., 58.93 on 5-MG), so meta-NLF stays below it.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Adapt the meta-NLF to the test-time system instead of using NLF (TS) trained directly on that system | Meta-NLF · Inverted Pendulum, stochastic l | ROA: 31.83 -> 15.04 | Hurt |
| Adapt the meta-NLF to the test-time system instead of using NLF (TS) trained directly on that system | Meta-NLF · Inverted Pendulum, stochastic l, b | ROA: 34.28 -> 28.04 | Hurt |
| Adapt the meta-NLF to the test-time system instead of using NLF (TS) trained directly on that system | Meta-NLF · Inverted Pendulum, stochastic l, m, g, b | ROA: 46.04 -> 42.67 | Hurt |
| Adapt the meta-NLF to the test-time system instead of using NLF (TS) trained directly on that system | Meta-NLF · 3-microgrid, stochastic dc1, dc2 | ROA: 71.94 -> 52.89 | Hurt |
| Adapt the meta-NLF to the test-time system instead of using NLF (TS) trained directly on that system | Meta-NLF · 3-microgrid, stochastic dc1, dc2, dc3 | ROA: 66.99 -> 60.41 | Hurt |
| Adapt the meta-NLF to the test-time system instead of using NLF (TS) trained directly on that system | Meta-NLF · 5-microgrid, stochastic dc1, dc2 | ROA: 42.83 -> 37.09 | Hurt |
| Adapt the meta-NLF to the test-time system instead of using NLF (TS) trained directly on that system | Meta-NLF · 5-microgrid, stochastic dc1, dc2, dc3, dc4, dc5 | ROA: 58.93 -> 44.35 | Hurt |
| Adapt the meta-NLF to the test-time system instead of using NLF (TS) trained directly on that system | Meta-NLF · Caltech Fan, stochastic m | ROA: 33.35 -> 25.16 | Hurt |
| Adapt the meta-NLF to the test-time system instead of using NLF (TS) trained directly on that system | Meta-NLF · Caltech Fan, stochastic m, r, d | ROA: 21.45 -> 13.72 | Hurt |
Surprising findings
- Naive transfer learning (T-NLF) often fails to produce any valid ROA (0 area) on several test systems, despite full nominal-system training.
- The meta-NLF itself does not satisfy Lyapunov conditions on test-time systems and requires adaptation before it can certify stability.
Limitations
- Requires a small dataset from the test-time system for a few gradient updates.
- Valid region D must be found by a computationally expensive shrinking procedure after meta-training.
- Verification relies on Lipschitz constants and a grid rather than formal SMT verification; the authors note meta-NLF alone is not a valid Lyapunov function on test-time systems.
Experimental setup
Datasets: simulated closed-loop dynamics (inverted pendulum, 3- and 5-microgrid power systems, Caltech ducted fan)
Benchmarks: Inverted Pendulum 3-microgrid power system 5-microgrid power system Caltech Ducted Fan
Metrics: region of attraction area
Baselines: SOS-LF(TS) QLF(TS) NLF(TS) T-NLF
Builds on: Model Agnostic Meta-learning (MAML) Neural Lyapunov Functions Lipschitz verification of Lyapunov conditions
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