Research map / Probabilistic and scientific modeling / Dynamical systems
Neural Continuous-Time Supermartingale Certificates
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Introduces a neural supermartingale certificate framework for continuous-time stochastic dynamical systems that gives formally verified lower bounds on reach-avoid-stay probabilities, using neural training plus IBP-based verification.
Problem
Formal probabilistic verification (reachability, avoidance, persistence) of nonlinear continuous-time stochastic dynamical systems with neural policies; existing neural supermartingale certificates assume discrete time.
Approach
A continuous-time proof rule for reach-avoid-stay supermartingale certificates based on the infinitesimal generator (Itô drift and diffusion terms), trained as a neural network from sampled drift/diffusion and verified soundly via interval bound propagation over AABB cells with analytic derivatives.
Key results
- Successfully trained RAS certificates for stochastic inverted pendulum and bivariate GBM with specifications eps=0.9 and delta=0.9.
- Verification time per phase averaged 14.9 s (pendulum) and 21.1 s (GBM) over five seeds.
- Inverted pendulum needed on average 3.2 verification phases and GBM 11.2.
Surprising findings
- Noise injection can stabilize a system: for GBM with policy sigma x, sigma > sqrt(2a) makes the system stabilize, contrary to the deterministic intuition.
Limitations
- No comparison to other methods since it is the first neural continuous-time stochastic certificate.
- Verification with IBP over cells is loose for generator bounds, requiring cell splitting up to a depth limit.
- Policy is pre-trained and fixed; controller synthesis alongside certificates is left to future work.
Experimental setup
Benchmarks: stochastic inverted pendulum bivariate geometric Brownian motion
Metrics: runtime per verification number of verifications certified reach-avoid and stay probabilities
Builds on: neural supermartingale certificates neural Lyapunov certificates interval bound propagation auto_LiRPA
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