Karthik's Blog Fourier Neural Operators Visually Explained

Fourier Neural Operators Visually Explained

How a Fourier neural operator turns an input into a prediction, step by step, with two small trained models running in your browser.

Karthik Srinivas VedulaStanford University

Oct 20, 2026 Learning Interactively Best viewed in a desktop browser

The flow the model predicts, with dye carried along

Notes

References

Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, A. Anandkumar. Fourier Neural Operator for Parametric Partial Differential Equations. ICLR 2021. arXiv:2010.08895.
N. Kovachki, Z. Li, B. Liu, K. Azizzadenesheli, K. Bhattacharya, A. Stuart, A. Anandkumar. Neural Operator: Learning Maps Between Function Spaces with Applications to PDEs. Journal of Machine Learning Research 24(89), 2023.
J. Kossaifi, N. Kovachki, Z. Li, D. Pitt, M. Liu-Schiaffini, R. J. George, B. Bonev, K. Azizzadenesheli, J. Berner, V. Duruisseaux, A. Anandkumar. A Library for Learning Neural Operators. arXiv:2412.10354, 2024. The two models on this page were trained with this library (neuraloperator 2.0.0).
NeuralOperator Team. Darcy Flow Dataset. Zenodo, 2021. doi:10.5281/zenodo.12784353.
Navier–Stokes dataset used for the fluid model: Zenodo record 12825163.
D. Hendrycks, K. Gimpel. Gaussian Error Linear Units (GELUs). arXiv:1606.08415, 2016.

Built with three.js (3D scenes), KaTeX (math) and fft.js by Fedor Indutny (the in-browser FFT). Written and built by Karthik Srinivas Vedula, Stanford University.

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