# Selection, Representation, and Execution in Sparse Fourier Neural Operators

arXiv: https://arxiv.org/abs/2608.30070 · HTML: https://mlsanity.com/paper/2608.30070/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 21 references that are in the map.
- First posted on arXiv 30 Aug 2026.

## Builds on (most cited first, 8 of 8)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Deep Compression: Compressing Deep Neural Networks with Pruning, Trained Quantization and Huffman Coding](https://mlsanity.com/paper/1510.00149/) (2015)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Mixtures of Neural Operators Reduce Active Complexity in Operator Learning](https://mlsanity.com/paper/2404.09101/) (2024)
- [Light-Weight Diffusion Multiplier and Uncertainty Quantification for Fourier Neural Operators](https://mlsanity.com/paper/2508.00643/) (2025)
- [When Spike Sparsity Does Not Translate to Deployed Cost: VS-WNO on Jetson Orin Nano](https://mlsanity.com/paper/2604.17040/) (2026)
- [Don't Fix the Basis -- Learn It: Spectral Representation with Adaptive Basis Learning for PDEs](https://mlsanity.com/paper/2605.10451/) (2026)
- [Operator Boosting Produces Pareto-Efficient PDE Surrogates](https://mlsanity.com/paper/2606.17460/) (2026)

## Abstract (arXiv)

Sparse representations are often expected to make models smaller and also reduce inference cost. For Fourier Neural Operators (FNOs), these objectives are not equivalent or do not always align: removing parts of the learned operator can leave the underlying transforms and dense computations unchanged, while changing the grid on which the model is evaluated can introduce overhead of its own. We therefore distinguish sparsity in the representation, in the stored parameters, in the theoretical operation count, and in measured runtime, and present an empirical study of several routes toward sparse FNOs that tests each transition between them separately. Coarsening the execution grid reduces the theoretical cost without reducing measured latency, and adding a correction term recovers accuracy at the cost of making the model slower. Even an 83% parameter reduction remains slower than the dense baseline under ordinary execution. These results motivate a stricter definition of useful sparsity: the deployed operator must preserve solution accuracy and map its reduced support to a genuinely cheaper execution path.