Fourier Neural Operator

Also known as FNO

Fourier Neural Operators (FNOs) learn mappings from input functions, such as coefficients or initial conditions, to PDE solution fields, using Fourier transforms to mix information across space. Common extensions add physics-informed losses, multiscale or local components, geometry and symmetry handling, and temporal memory. They are used as fast surrogates for problems including fluid flow, wave propagation, porous media, and inverse design, and are evaluated on prediction accuracy, rollouts, and generalization across resolutions or parameter settings. A central challenge is that strong accuracy on familiar data does not ensure reliable behavior under high-frequency, boundary, geometry, or resolution shifts, while added physical structure and efficiency can trade off against one another.

Latest papers

71 papers14 papers in the last four weeks, up 180% on the four weeks before. 0.1% of all new papers.

FNO $^{\angle θ}$ : Extended Fourier neural operator for learning state and optimal control of distributed parameter systems
Apr 6, 2026Zhexian Li, Ketan SavlaFourier Neural Operator PDE Operator Learning
Sonny Astani Department of Civil and Environmental Engineering, University of Southern California, Los Angeles, CA 90089, USA. · Sonny Astani Department of Civil and Environmental Engineering, the Daniel Epstein Department of Industrial and Systems Engineering, and the Ming Hsieh Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA 90089, USA.
Hybrid Fourier Neural Operator for Surrogate Modeling of Laser Processing with a Quantum-Circuit Mixer
Apr 6, 2026Mateusz Papierz, Asel Sagingalieva, Alix Benoit +3Neural Surrogate Modeling Fourier Neural Operator
Terra Quantum AG, Kornhausstrasse 25, 9000 St. Gallen, Switzerland · Laboratory for Advanced Materials Processing, Feuerwerkerstrasse, Thun, 3603, BE, Switzerland
Performance of Neural and Polynomial Operator Surrogates
Apr 1, 2026Josephine Westermann, Benno Huber, Thomas O'Leary-Roseberry +1Surrogate Modeling PDE Surrogate Modeling
Heidelberg University · The Ohio State University
SIMR-NO: A Spectrally-Informed Multi-Resolution Neural Operator for Turbulent Flow Super-Resolution
Mar 30, 2026Muhammad Abid, Omer SanNeural Operators Computational Fluid Dynamics
Department of Mechanical and Aerospace Engineering, University of Tennessee, Knoxville, TN 37996, USA.
SLE-FNO: Single-Layer Extensions for Task-Agnostic Continual Learning in Fourier Neural Operators
Mar 20, 2026Mahmoud Elhadidy, Roshan M. D'Souza, Amirhossein ArzaniContinual Learning Fourier Neural Operator
Department of Mechanical Engineering, University of Utah, Salt Lake City, UT, USA · Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, UT, USA · Department of Mechanical Engineering, University of Wisconsin–Milwaukee, Milwaukee, WI, USA +1
Translation Invariance of Neural Operators for the FitzHugh-Nagumo Model
Mar 18, 2026Luca PellegriniFourier Neural Operator PDE Operator Learning
Department of Mathematics, University of Pavia, Pavia, Italy · Euler Institute, Faculty of Informatics, Universit`a della Svizzera italiana, Lugano, Switzerland
Windowed Fourier Propagator: A Frequency-Local Neural Operator for Wave Equations in Inhomogeneous Media
Mar 15, 2026Yiyang Cai, Zixuan Qiu, Yunlu Shu +4PDE Operator Learning Neural Operators
School of Mathematical Sciences, Fudan University, Shanghai, 200433, China · Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, 200438, China · Center for Applied Mathematics, Fudan University, Shanghai, 200438, China
Prediction of Steady-State Flow through Porous Media Using Machine Learning Models
Mar 6, 2026Jinhong Wang, Matei C. Ignuta-Ciuncanu, Ricardo F. Martinez-Botas +1Porous Media Flow PDE Surrogate Modeling
Department of Mechanical Engineering, Imperial College London, South Kensington Campus, London, SW7 2AZ, , United Kingdom
Tackling multiphysics problems via finite element-guided physics-informed operator learning
Mar 2, 2026Yusuke Yamazaki, Reza Najian Asl, Markus Apel +2Multiphysics Simulation Physics-Informed Neural Operators
Graduate School of Science and Technology, Keio University, Hiyoshi3-14-1,Kohoku-ku, Yokohama 223-8522, Japan · Chair of Structural Analysis, Technical University of Munich, Arcisstraße 21, 80333 Munich, Germany · ACCESS e.V., Intzestr. 5, 52072 Aachen, Germany +1
From Theory to Application: A Practical Introduction to Neural Operators in Scientific Computing
Mar 7, 2025Prashant K. JhaNeural Surrogate Modeling Fourier Neural Operator
Department of Mechanical Engineering, South Dakota School of Mines and Technology, Rapid City, SD 57701, USA.
FC-PINO: High Precision Physics-Informed Neural Operators via Fourier Continuation
Nov 29, 2022Adarsh Ganeshram, Valentin Duruisseaux, Haydn Maust +6Physics-Informed Neural Operators PDE Solving
Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, USA

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FNO $^{\angle θ}$ : Extended Fourier neural operator for learning state and optimal control of distributed parameter systems

Zhexian Li, Ketan Savla

Sonny Astani Department of Civil and Environmental Engineering, University of Southern California, Los Angeles, CA 90089, USA. · Sonny Astani Department of Civil and Environmental Engineering, the Daniel Epstein Department of Industrial and Systems Engineering, and the Ming Hsieh Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA 90089, USA.

Fourier Neural Operator PDE Operator Learning Optimal Control Linear Quadratic Regulator

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We propose an extended Fourier neural operator (FNO) architecture for learning state and linear quadratic additive optimal control of systems governed by partial differential equations. Using the Ehrenpreis-Palamodov fundamental principle, we show that any state and optimal control of linear PDEs with constant coefficients can be represented as an integral in the complex domain. The integrand of this representation involves the same exponential term as in the inverse Fourier transform, where the latter is used to represent the convolution operator in FNO layer. Motivated by this observation, we modify the FNO layer by extending the frequency variable in the inverse Fourier transform from the real to complex domain to capture the integral representation from the fundamental principle. We illustrate the performance of FNO in learning state and optimal control for the nonlinear Burgers' equation, showing order of magnitude improvements in training errors and more accurate predictions of non-periodic boundary values over FNO.