Function-space certification for neural operators

Codebase for the paper Function-Space Certification for Neural Operators. Package version: 0.1.0.

System requirements

• Python 3.12 and uv.
• Dependencies and platform-specific versions are pinned in uv.lock; optional dependency groups are defined in pyproject.toml.
• Full paper experiments require an NVIDIA GPU.

Install

Use Python 3.12 and run commands from the repository root:

uv sync --frozen --group dev

Paper experiments

Each checkmark links to a standalone script with fixed experiment settings. Run a script as a module, for example:

uv run python -m scripts.poisson_1d_residual

PDE / Model	Solution error	Residual	Robustness	Max temp.
1D Poisson	✓	✓	✓	—
2D Poisson	✓	✓	✓	—
1D Helmholtz	✓	✓	✓	—
Burgers (space–time)	✓	✓	✓	—
KS (long rollout)	—	✓	—	—
SFNO (weather)	—	—	—	✓

For SFNO:

uv sync --frozen --group sfno
uv run --group sfno python -m scripts.sfno_temperature

Small examples and checks

uv run python -m examples.helmholtz_1d
uv run python -m examples.sfno

The Helmholtz demo generates its model and constant forcing input (a(x)=1) in code. It certifies the residual, solution error, and robustness to an input perturbation of size 0.001 on a 64-cell domain. Expected output is approximately:

Residual Linf upper: 1.0000318934200696
Solution Linf upper: 1.0000318934200696
Robustness cell-range width: 6.336254349409578e-08

See examples for other API demonstrations.

License

This project's code is available under the MIT License. Third-party code, model weights, and datasets are subject to their respective licenses and terms.

Layout

• lean_no/: certification library, model loading, integration, and PDE bounds.
• scripts/: the 14 checkmark experiments; _shared/ has reused helpers.
• models/: PDE and KS checkpoints.
• training/: PDE training and data generation.
• examples/ and tests/: small API demonstrations and correctness checks.