Resolution-Invariant Fluid Dynamics Modeling: Fourier Neural Operators vs. Convolutional Networks

Modeling the dynamics of fluid flow from data has become an important problem in computational physics and data-driven PDE learning, especially as high-resolution simulations remain computationally expensive. Recent operator-learning approaches, such as Fourier Neural Operators (FNOs) \cite{li2020fno}, promise resolution-independent learning of PDE solution maps, but their practical advantages over traditional deep learning models are still not fully understood. In this work, we study the ability of FNOs to learn Navier–Stokes dynamics under changes in spatial resolution and directly compare their behavior to standard convolutional neural networks (CNNs) \cite{guo2016cnnfluid,fukami2019superres}. While CNNs learn local, grid-dependent filters, FNOs operate in Fourier space and are designed to approximate solution operators independent of discretization as Fourier transform is itself resolution invariant. We train both models on a single resolution of a high-fidelity Navier–Stokes dataset and evaluate their performance across coarser and finer grids without retraining. Our experiments show that FNOs retain stable accuracy across resolutions, whereas CNNs exhibit significant degradation once the test grid deviates from the training grid. We further evaluate both models on in-distribution and out-of-distribution initial conditions to probe whether FNOs truly capture the underlying fluid physics rather than memorizing a distribution of initial states. To our knowledge, no thorough comparative study between CNNs and FNOs has been conducted in this setting; this work provides a detailed examination of their relative strengths and limitations for data-driven fluid dynamics and climate modeling.