Operator learning with Fourier neural operators : theoretical guarantees and applications

Abstract

This thesis investigates the use of Fourier Neural Operators (FNOs) to simulate steady fluid movement through porous materials, with a specific emphasis on the two-dimensional Darcy flow problem. The aim is to assess the effectiveness of FNOs in approximating solutions to partial differential equations (PDEs) purely through data, without needing explicit understanding of the underlying equations when making predictions. The study considers theoretical elements such as the universal approximation theorem and the discretization-invariance trait of FNOs, which bolster their ability to adapt to various grid sizes. To examine real-world performance, FNOs were trained on synthetic data produced by traditional numerical methods and evaluated using input fields with distinct structural characteristics, such as modified and transformed Gaussian random fields. The findings reveal that FNOs excel with smooth and orderly input fields, achieving low relative errors and little bias in predictions. However, when faced with irregular or discontinuous inputs, accuracy diminishes, showing sensitivity to the input's structure. This research underscores the capability and resilience of FNOs for solving PDEs and highlights their promise in computational physics and engineering, where efficient and precise modeling is crucial.