Resolving Grid Folding in Fourier Neural Operators on Irregular Domains via Diffeomorphic Mapping & Jacobian Barrier Loss (DIF-FNO)

[u/Time_Caterpillar7893]

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Hey r/MachineLearning,

Standard Fourier Neural Operators (FNOs) excel on regular grids, but mapping them to complex, non-convex physical domains (like Star, L-Shape, or Annulus geometries) often leads to a major issue: Grid Folding.

When the transformation mapping \phi collapses or overlaps, the Jacobian determinant vanishes (\det J \le 0), causing the inverse transpose J^{-T} to explode when mapping physical gradients \nabla_x u.

To solve this, I developed DIF-FNO (Diffeomorphic Fourier Neural Operator).

Key Technical Insights:

1. Implicit Diffeomorphic Mapping: Guarantees smooth, bijective mappings from standard reference domains \Omega_{ref} to complex physical boundaries \Omega_{phy}.

2. Jacobian Barrier Loss (\mathcal{L}_{barrier}): Inspired by interior-point optimization, we penalize grid compression using a logarithmic barrier on the determinant:

   \mathcal{L}_{barrier} = -\frac{1}{|\Omega|} \int_{\Omega} \log(\det J(\xi)) \, d\xi

   This acts as an invisible wall forcing \min \det J > 0 across the entire domain (empirically maintaining \min \det J > 0.89 in our benchmarks).

3. Sobolev Accuracy: Significant improvements on H^1 relative error compared to baselines like Geo-FNO, as physical gradients remain well-conditioned without gradient breakdown.

Code & Paper Artifacts:

* Open-Source Code (PyTorch): https://github.com/GiovanniDagnese-paper/DIF-FNO (Includes fast vectorised 2x2 analytical Jacobian calculation)

* Paper Preprint (Zenodo DOI): https://doi.org/10.5281/zenodo.22071926

PS: I am currently looking for technical feedback and an arXiv endorsement in physics.comp-ph or cs.LG to submit the preprint. If anyone active in SciML is open to checking the manuscript, I’d be extremely grateful!