Can We Formally Verify Neural PDE Surrogates? SMT Compilation of Small Fourier Neural Operators

2026-05-09

AI Summary

This work addresses the lack of formal verification guarantees for physical properties—such as positivity and mass conservation—in Fourier Neural Operators (FNOs) used as surrogate models for partial differential equations. The authors establish, for the first time, that the spectral convolution in FNOs constitutes a linear mapping under fixed weights and discretization grids, enabling the forward pass to be modeled as a piecewise-linear system. Leveraging this insight, they encode the FNO either exactly or via a lightweight “frozen” approximation into the SMT solver Z3 to enable efficient verification of physical constraints. Experiments on ten small-scale FNOs demonstrate successful proofs of positivity in two cases and yield 15 valid counterexamples. The frozen encoding achieves sub-second approximate verification at grid resolutions up to 64, significantly enhancing scalability.

Abstract

Fourier Neural Operators (FNOs) can greatly accelerate PDE simulation, but they are often used without formal guarantees that they preserve basic physical structure. We show that, once the trained weights and grid are fixed, the spectral convolution in an FNO is a linear map. As a result, the full forward pass is piecewise-linear and can be represented exactly in Z3's linear real arithmetic. We study two encodings. The exact encoding compiles the spectral convolution into a dense matrix multiplication, which is sound for both proofs and counterexamples. The lighter frozen encoding replaces the spectral path with a constant, making it faster but approximate. On 10 small FNO surrogates for 1D advection-diffusion-reaction (85 to 117 parameters, grids 8 to 32), the exact encoding gives 2 sound positivity proofs on linear (ReLU-free) models, 5 sound positivity counterexamples, and 10 sound mass-violation counterexamples; the remaining 3 positivity queries on ReLU models time out. For mass non-increase, Z3 finds worse counterexamples than both gradient-based falsification and Monte Carlo on 7 of 10 models. The frozen encoding scales to grid size 64 with sub-second positivity checks, but it no longer provides certificates for the original FNO. Overall, the results make the soundness--scalability tradeoff explicit and point to what is needed for formal verification of production-scale neural operators.

Problem

• formal verification
• neural PDE surrogates
• Fourier Neural Operators
• physical guarantees
• SMT

Innovation

• Fourier Neural Operators
• Formal Verification
• SMT Solving
• Piecewise-Linear Encoding
• Neural PDE Surrogates