Can We Formally Verify Neural PDE Surrogates? SMT Compilation of Small Fourier Neural Operators
arXiv:2605.08938v2 Announce Type: replace-cross 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
The increasing deployment of AI in critical scientific and engineering applications necessitates formal verification methods to ensure reliability and safety.
This development addresses a fundamental challenge for deploying neural network surrogates in high-stakes environments, potentially broadening their adoption across industries requiring precision and predictability.
The ability to formally verify Fourier Neural Operators moves AI-driven PDE simulations from experimental to certifiable, increasing trust and enabling their use in practical, safety-critical systems.
- · Engineering simulation software providers
- · Certification bodies
- · AI ethicists and safety researchers
- · High-performance computing (HPC) sector
- · Organizations relying solely on empirical validation for AI models
- · Developers of uninterpretable black-box AI models for scientific applications
Formal verification tools for neural PDE surrogates will become standard in sectors like aerospace, energy, and defense.
This will accelerate the integration of AI into complex physical systems design and operational control, replacing traditional numerical solvers.
The established trust in AI-driven simulations could lead to a paradigm shift in scientific discovery, where verifiable AI models become primary research instruments.
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Read at arXiv cs.LG