The Fourier Neural Operator (FNO) is a novel deep learning architecture designed to learn mappings between infinite-dimensional function spaces, particularly for solving parametric partial differential equations (PDEs). Unlike traditional numerical solvers, which are slow and require fine discretization, the FNO operates in Fourier space, enabling it to learn the solution operator for a family of PDEs efficiently and accurately. This approach allows for zero-shot super-resolution, meaning it can predict high-resolution solutions without access to high-resolution data. The FNO is significantly faster than traditional PDE solvers, achieving up to three orders of magnitude speed improvement, and outperforms existing learning-based solvers in accuracy, especially at fixed resolutions. The FNO is constructed by parameterizing the integral kernel directly in Fourier space, allowing for a quasi-linear computational complexity. This architecture is discretization-invariant, meaning it maintains consistent error rates across different resolutions and can transfer solutions between different mesh geometries. The method is particularly effective for turbulent flows, where previous graph-based neural operators have failed to converge. In numerical experiments, the FNO was tested on the Burgers' equation, Darcy flow, and Navier-Stokes equations. It demonstrated superior performance, achieving lower error rates compared to other methods, including convolutional neural networks and physics-informed neural networks. The FNO also showed the ability to perform zero-shot super-resolution in both spatial and temporal domains, making it a powerful tool for solving complex PDEs efficiently. The FNO's ability to handle high-frequency modes and maintain accuracy despite the complexity of the underlying PDEs makes it a promising approach for various scientific and engineering applications. By leveraging the Fourier transform, the FNO efficiently computes integral operators, enabling fast and accurate solutions to PDEs without the need for extensive discretization. This method represents a significant advancement in the application of machine learning to solve complex PDEs, offering a computationally efficient and accurate alternative to traditional numerical methods.The Fourier Neural Operator (FNO) is a novel deep learning architecture designed to learn mappings between infinite-dimensional function spaces, particularly for solving parametric partial differential equations (PDEs). Unlike traditional numerical solvers, which are slow and require fine discretization, the FNO operates in Fourier space, enabling it to learn the solution operator for a family of PDEs efficiently and accurately. This approach allows for zero-shot super-resolution, meaning it can predict high-resolution solutions without access to high-resolution data. The FNO is significantly faster than traditional PDE solvers, achieving up to three orders of magnitude speed improvement, and outperforms existing learning-based solvers in accuracy, especially at fixed resolutions. The FNO is constructed by parameterizing the integral kernel directly in Fourier space, allowing for a quasi-linear computational complexity. This architecture is discretization-invariant, meaning it maintains consistent error rates across different resolutions and can transfer solutions between different mesh geometries. The method is particularly effective for turbulent flows, where previous graph-based neural operators have failed to converge. In numerical experiments, the FNO was tested on the Burgers' equation, Darcy flow, and Navier-Stokes equations. It demonstrated superior performance, achieving lower error rates compared to other methods, including convolutional neural networks and physics-informed neural networks. The FNO also showed the ability to perform zero-shot super-resolution in both spatial and temporal domains, making it a powerful tool for solving complex PDEs efficiently. The FNO's ability to handle high-frequency modes and maintain accuracy despite the complexity of the underlying PDEs makes it a promising approach for various scientific and engineering applications. By leveraging the Fourier transform, the FNO efficiently computes integral operators, enabling fast and accurate solutions to PDEs without the need for extensive discretization. This method represents a significant advancement in the application of machine learning to solve complex PDEs, offering a computationally efficient and accurate alternative to traditional numerical methods.