Universal Approximation of Maximal Lyapunov Functions with Anchored Neural Networks
arXiv:2608.17290
2026
Dynamics
1 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper provides a constructive method for approximating maximal Lyapunov functions while preserving positive definiteness and strict decrease along system trajectories. Its transferable asset is an anchored, positivity-preserving neural parameterization together with a semiglobal certificate: if the function and its derivative are sufficiently close in C1 to a strict target on a compact invariant sublevel set, positivity and negative Lie derivative survive. This can be used to learn Lyapunov certificates for neural ODEs, learned dynamics, or neural controllers, and to reject parameter updates that violate a measurable stability margin. The strongest experiment should test the predicted margin threshold rather than only reporting task accuracy.
Ideas from this paper
✗ Failed on benchmark
2026
Replace an unconstrained scalar MLP certificate with an anchored positive-definite network whose value and gradient are fixed at the equilibrium. Train it so that its Lie derivative along a neural or physical vector field is strictly negative on a prescribed region of attraction. The construction makes stability robust to approximation error: a certificate remains valid whenever the value, gradient, and Lie-derivative errors stay below the target's strict-decrease margin.
Useful8/10
Difficulty5/10
Novelty8/10