By name: neural-lyapunov-verification description: > Sound and complete verification of neural Lyapunov candidates for nonlinear control systems. Uses hyperplane partitioning of ReLU networks to verify stability guarantees. Use when: (1) verifying neural network controller stability, (2) Lyapunov function validation, (3) formal verification of learned control policies, (4) safety-critical neural control systems, (5) analyzing ReLU network dynamics for stability properties.
Neural Lyapunov Verification
Verify stability of neural network controllers using Lyapunov function analysis. Based on HyParLyVe (Hyperplane Partitioned Lyapunov Verifier) methodology.
Core Approach
- Partition ReLU networks into linear regions using hyperplane arrangements
- Check Lyapunov conditions (V(x) > 0, dV/dt < 0) in each linear region
- Sound and complete verification - no false positives or negatives
- Scalable to shallow ReLU networks (1-2 hidden layers)
Workflow
Step 1: Define the System
# System: dx/dt = f(x) + g(x) * u(x)
# where u(x) is the neural network controller
Step 2: Verify Lyapunov Candidate
For candidate V(x) (often a neural network):
- Check V(0) = 0 and V(x) > 0 for x != 0 (positive definite)
- Check dV/dt = grad(V) * f(x) < 0 for x != 0 (negative definite derivative)
Step 3: Hyperplane Partitioning
Partition input space by ReLU activation patterns:
- Each region = set of active/inactive neurons
- Within each region, network is affine: u(x) = Wx + b
- Verify Lyapunov conditions via linear programming in each region
Step 4: Region-of-Attraction Estimation
Find largest sublevel set {x : V(x) <= c} that satisfies all conditions.
Key References
- HyParLyVe: arXiv:2605.03992 - Hyperplane Partitioning for Neural Lyapunov Verification
- Related: neural-network-control-verification, formal-verification-ml
Limitations
- Works best for shallow ReLU networks (1-2 hidden layers)
- Computational cost grows exponentially with network width
- Requires known system dynamics f(x), g(x)