Formal Verification of Neural PDE Surrogates via SMT Compilation
Researchers have developed a method to formally verify Fourier Neural Operators (FNOs) for PDE simulation using SMT solvers. By exploiting the piecewise-linear structure of FNOs after training, they compile the spectral convolution into linear real arithmetic in Z3. Two encodings are proposed: an exact encoding that is sound for proofs and counterexamples, and a faster frozen encoding that approximates the spectral path. On 10 small FNO surrogates for 1D advection-diffusion-reaction, the exact encoding produced 2 sound positivity proofs on linear models, 5 sound positivity counterexamples, and 10 sound mass-violation counterexamples. This work addresses the lack of formal guarantees in neural PDE surrogates, enabling verification of physical properties like positivity and mass conservation.
Key facts
- Fourier Neural Operators (FNOs) are used to accelerate PDE simulation.
- The spectral convolution in an FNO is a linear map after training weights and grid are fixed.
- The forward pass of an FNO is piecewise-linear and can be represented in Z3's linear real arithmetic.
- Two encodings are studied: exact encoding (dense matrix multiplication) and frozen encoding (constant spectral path).
- Experiments were conducted on 10 small FNO surrogates for 1D advection-diffusion-reaction.
- Models had 85 to 117 parameters and grids of size 8 to 32.
- Exact encoding gave 2 sound positivity proofs on linear (ReLU-free) models.
- Exact encoding gave 5 sound positivity counterexamples and 10 sound mass-violation counterexamples.
- The remaining 3 positivity proofs were not sound (likely due to ReLU nonlinearity).
- The work aims to provide formal guarantees for neural PDE surrogates.
Entities
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