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Formal Synthesis of Barrier Certificates Using Fourier Kolmogorov-Arnold Network
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Replaces feed-forward networks with Fourier Kolmogorov-Arnold Networks as barrier certificate templates, verified by dReal SMT and a Lipschitz-based validity condition, implemented as the tool KAN4BC.
Problem
Synthesizing and formally verifying barrier certificates for safety of continuous and hybrid dynamical systems; FNN templates with fixed activation functions limit efficiency and scalability.
Approach
Uses Fourier KANs (learnable activations parameterized by sine/cosine Fourier coefficients) as the barrier certificate template, and combines an SMT (dReal) CEGIS verifier with a Lipschitz-constant validity condition enforced during training.
Key results
- KAN4BC synthesizes valid barrier certificates for all seven benchmarks within one hour, while FOSSIL2.0 fails on barr3 with all five activation functions.
- On barr3 KAN4BC synthesizes a certificate in 3 iterations and 66.34 s using a [2,15,1] Fourier KAN.
- KAN4BC generally needs fewer iterations and less total time than FOSSIL2.0 across most cases, though FOSSIL2.0 is faster on barr4 with tanh and barr7 with cosh.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace FNNs with Fourier KAN templates | Barrier-certificate synthesis · barr3 | synthesis success | Helped |
| Replace FNNs with Fourier KAN templates | Barrier-certificate synthesis · barr1, barr2, barr4, barr5, barr6, barr7 | total synthesis time | Mixed |
| Use Fourier KAN instead of FOSSIL2.0 with tanh activation | Barrier-certificate synthesis · barr1 | time (s): 7.96 vs 5.14 | Helped |
| Use Fourier KAN instead of FOSSIL2.0 with tanh activation | Barrier-certificate synthesis · barr4 | time (s): 124.85 vs 265.77 | Hurt |
| Use Fourier KAN instead of FOSSIL2.0 with cosh activation | Barrier-certificate synthesis · barr7 | time (s): 7.33 vs 300.30 | Hurt |
Surprising findings
- B-spline-based KAN failed to synthesize barrier certificates in the tested cases, while Fourier KAN succeeded.
- SMT verification was consistently faster than the Lipschitz method, contrary to the expectation that the Lipschitz method is the efficient alternative.
Limitations
- Lipschitz-based verification is slow and scales poorly with dataset size.
- Evaluated on only seven small benchmark systems (2 to 4 dimensions).
- FOSSIL2.0 is faster than KAN4BC on barr4 with tanh and barr7 with cosh.
Experimental setup
Datasets: barr1-barr7 benchmark dynamical systems (from FOSSIL2.0 and literature)
Benchmarks: barr1 barr2 barr3 barr4 barr5 barr6 barr7
Metrics: iterations to synthesize valid barrier synthesis time (s) success rate
Baselines: FOSSIL2.0 B-spline-based KAN
Builds on: Kolmogorov-Arnold Networks FOSSIL CEGIS dReal
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