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Neural Closure Certificates
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Proposes neural closure certificates: a data-driven method that trains a ReLU neural network as a closure certificate to formally verify unknown dynamical systems against omega-regular properties, with correctness under Lipschitz assumptions.
Problem
Verifying unknown (closed-form-free) controlled dynamical systems against omega-regular (liveness and safety) properties, where existing closure certificate methods need explicit models and SOS/SMT solvers.
Approach
Represent closure certificates as ReLU neural networks trained on grid samples with a loss built from a bilinear S-procedure reformulation of the implications; a Lipschitz-based margin argument (L*eps/2 <= eta) lifts sample-level zero loss to a formal guarantee over all states.
Key results
- Kuramoto oscillator: neural closure certificate found with delta=eta=0.01, estimated L=0.2798, satisfying condition (20); training took 5 hours.
- Two-room temperature model: delta=eta=0.2, L=0.6814, condition (20) satisfied; training around 4 hours.
- Inverted pendulum: delta=eta=0.2, L=1.002, condition (20) satisfied, certifying the omega-regular property.
Limitations
- Correctness requires a known Lipschitz constant (or an estimate) of the unknown transition function and the network.
- Conditions are only sufficient; the S-procedure reformulation may reject valid certificates.
- Sample complexity is high; compositional reasoning is left as future work.
Experimental setup
Datasets: simulated samples from known transition functions of three case-study systems
Benchmarks: Kuramoto oscillator two-room temperature model inverted pendulum
Metrics: loss zero on samples Lipschitz-based condition (20) satisfied training time
Builds on: closure certificates transition invariants neural barrier certificates
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