Fourier Neural Operator for Parametric Partial Differential Equations (Paper Explained)

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Numerical solvers for Partial Differential Equations are notoriously slow. They need to evolve their state by tiny steps in order to stay accurate, and they need to repeat this for each new problem. Neural Fourier Operators, the architecture proposed in this paper, can evolve a PDE in time by a single forward pass, and do so for an entire family of PDEs, as long as the training set covers them well. By performing crucial operations only in Fourier Space, this new architecture is also independent of the discretization or sampling of the underlying signal and has the potential to speed up many scientific applications.

OUTLINE: 0:00 - Intro & Overview 6:15 - Navier Stokes Problem Statement 11:00 - Formal Problem Definition 15:00 - Neural Operator 31:30 - Fourier Neural Operator 48:15 - Experimental Examples 50:35 - Code Walkthrough 1:01:00 - Summary & Conclusion

Paper: https://arxiv.org/abs/2010.08895 Blog: https://zongyi-li.github.io/blog/2020/fourier-pde/ Code: https://github.com/zongyi-li/fourier_neural_operator/blob/master/fourier_3d.py MIT Technology Review: https://www.technologyreview.com/2020/10/30/1011435/ai-fourier-neural-network-cracks-navier-stokes-and-partial-differential-equations/

Abstract: The classical development of neural networks has primarily focused on learning mappings between finite-dimensional Euclidean spaces. Recently, this has been generalized to neural operators that learn mappings between function spaces. For partial differential equations (PDEs), neural operators directly learn the mapping from any functional parametric dependence to the solution. Thus, they learn an entire family of PDEs, in contrast to classical methods which solve one instance of the equation. In this work, we formulate a new neural operator by parameterizing the integral kernel directly in Fourier space, allowing for an expressive and efficient architecture. We perform experiments on Burgers' equation, Darcy flow, and the Navier-Stokes equation (including the turbulent regime). Our Fourier neural operator shows state-of-the-art performance compared to existing neural network methodologies and it is up to three orders of magnitude faster compared to traditional PDE solvers.

Authors: Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, Anima Anandkumar

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